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1
+ arXiv:2301.05710v1 [gr-qc] 13 Jan 2023
2
+ Perusing Buchbinder–Lyakhovich canonical formalism for Higher-Order
3
+ Theories of Gravity.
4
+ Dalia Saha†, Abhik Kumar Sanyal‡
5
+ January 18, 2023
6
+ † Dept. of Physics, University of Kalyani, West Bengal, India - 741235.
7
+ †,‡ Dept. of Physics, Jangipur College, Murshidabad, West Bengal, India - 742213.
8
+ Abstract
9
+ Ostrogradsky’s, Dirac’s and Horowitz’s techniques of higher order theories of gravity produce identical
10
+ phase-space structures. The problem is manifested in the case of Gauss-Bonnet-dilatonic coupled action in the
11
+ presence of higher-order term, in which case, classical correspondence can’t be established. Here, we explore yet
12
+ another technique developed by Buchbinder and his collaborators (BL) long back and show that it also suffers
13
+ from the same disease. However, expressing the action in terms of the three-space curvature, and removing “the
14
+ total derivative terms”, if Horowitz’s formalism or even Dirac’s constraint analysis is pursued, all pathologies
15
+ disappear. Here we show that the same is true for BL formalism, which appears to be the simplest of all the
16
+ techniques, to handle.
17
+ Keywords: Higher Order theory; Canonical Formulation.
18
+ 1
19
+ Introduction
20
+ Canonical formulation of higher-order theories was developed by Ostrogradsky almost two centuries back [1, 2].
21
+ However, it did not draw much attention, since other than toy mechanical models, practically no such physical
22
+ theories were persuaded at that time. Exactly a century elapsed, when it was applied to a physically motivated
23
+ problem, such as fourth order harmonic oscillator [3]. The real physical problem in this context appeared for
24
+ the first time, while a renormalized quantum theory of gravity was attempted to formulate [4]. Higher-derivative
25
+ theory of gravity is usually considered as a model of quantum gravity. The reason being, Einstein-Hilbert ac-
26
+ tion is supplemented by curvature squared terms (R2, RµνRµν ) to ensure renormalizability [4] and asymptotic
27
+ freedom [5–7]. Unfortunately, curvature-squared gravity theories have been found to suffer from the unresolved
28
+ problem of physical unitarity in perturbative analysis, which is usual for higher-derivative theories. However,
29
+ possibilities to overcome this difficulty were also discussed in some literatures [6, 8] and references therein. It is
30
+ also ascertained that curvature squared gravity would arise as a low-energy effective theory derived from super-
31
+ string theory in D = 10 dimensions [9–11]. Over the last couple of decades, higher order theories of gravity e.g.,
32
+ F(R), F(G), F(R, T ) etc, theories, (R, G, T being the Ricci scalar, the Gauss-Bonnet term, and the torsion
33
+ term respectively) have drawn much attention in search of alternatives to dark energy. Nonetheless, it is always
34
+ suggestive to test viability of such modified theories of gravity in different contexts. In the context of the very
35
+ early universe, a canonical formulation is required as a precursor, particularly to study quantum cosmology.
36
+ Since Ostrogradsky’s technique does not apply in the degenerate case of singular Lagrangian, for which the
37
+ Hessian determinant vanishes, Dirac’s constraint analysis [12] may be applied for the purpose. Nonetheless, a host
38
+ of theories have been formulated over decades to bypass the constraint analysis. One of these in this direction
39
+ was originally proposed by Boulware [13], and later reformulated by Horowitz’ [14], in particular in the context of
40
+ higher-order theory of gravity. Since the canonical formulation of higher order theories requires an extra degree
41
+ of freedom, in Horowitz’s formalism apart from the scale factor (‘a′ in the Robertson-Walker minisuperspace) an
42
+ auxiliary variable is introduced by taking derivative of the action (say A) with respect to the highest derivative of
43
+ 1Electronic address:
44
+ † daliasahamandal1983@gmail.com
45
+ ‡ sanyal ak@yahoo.com
46
+ 1
47
+
48
+ the field variable present (Q = ∂A
49
+ ∂¨a ). In the end, the auxiliary variable is replaced by the basic variable (extrinsic
50
+ curvature tensor) through a canonical transformation. The important finding in this regard is as follows: all the
51
+ three formalisms, viz, Ostrogradsky’s (once degeneracy has been removed), Dirac’s and Horowitz’s formalisms,
52
+ produce an identical phase-space structure [15]. Meanwhile, certain pathologies with Horowitz’ formalism have
53
+ been identified. For example, it was noticed that Horowitz’s formalism can even be applied in the case of linear
54
+ gravity theory (Einstein-Hilbert action) leading to wrong quantum dynamics [16–18], as well as some superfluous
55
+ total derivative terms are eliminated [18, 19], which neither may be obtained from the variational principle, nor
56
+ having any connection with Gibbons-Hawking–York term [20,21], nor any of its modified versions, associated with
57
+ higher-order gravity. Further, the coupling parameter, in the case of the “non-minimally coupled scalar tensor the-
58
+ ory of gravity associated with higher order term”, has not been found to play any particular role, since its derivative
59
+ does not appear in the Hamiltonian [22]. The same is true for the “Dilatonic coupled Gauss-Bonnet-theory in
60
+ the presence of higher order term”, where additionally, the classical correspondence with quantum counterpart,
61
+ could not be established [22]. In view of such an uncanny situation, yet another technique was developed, called
62
+ the “modified Horowitz’s formalism” (MHF), which was successfully applied to different modified higher-order
63
+ theories of gravity, to explore the evolution of the very early universe [15,17–19,22–32]. In the MHF, the action
64
+ is expressed in terms of the three-space curvature (instead of the scale factor), “the total derivative terms” are
65
+ removed by integrating the action by parts, and Horowitz’s formalism (the introduction of the auxiliary variable
66
+ etc.) was followed, thereafter.
67
+ To be very specific, let us consider the following isotropic and homogeneous Robertson–Walker (RW) metric:
68
+ ds2 = −N 2(t) dt2 + a2(t)
69
+
70
+ dr2
71
+ 1 − kr2 + r2(dθ2 + sin2θdφ2)
72
+
73
+ ,
74
+ (1)
75
+ for which the degeneracy in the Lagrangian disappears if the gauge (N ) is fixed a priori, in which case, Ostrograd-
76
+ sky’s technique applies as well. Once such degeneracy is removed, it is observed that Ostrogradsky’s technique
77
+ produces the same Hamiltonian, obtained following Horowitz’s as well as Dirac’s formalism [15]. Therefore, it
78
+ certainly follows that both the Ostrogradsky’s and Dirac’s formalism implicitly suffer from the same problem,
79
+ in disguise, as was noticed in Horowitz’s technique, as discussed above. Therefore in the MHF, instead of the
80
+ scale factor, the action is expressed in terms of the basic variable hij — the three space metric from the very
81
+ beginning—so that redundant total derivative terms do not appear [18, 19]. Thereafter, all the total derivative
82
+ terms are integrated out by parts, which become cancelled by the supplementary boundary (Gibbons–Hawking—
83
+ York and modified Gibbons–Hawking—York) terms. Subsequently, the auxiliary variable is introduced following
84
+ Horowitz’s proposal. In the end, the auxiliary variable is replaced by the other basic variable Kij — the extrinsic
85
+ curvature tensor. In this process, the unwanted problems that appeared following Horowitz’s formalism disappear,
86
+ while it produces a different Hamiltonian altogether. We mention that although both Hamiltonians (obtained
87
+ following the MHF and Ostrogradky’s, Dirac’s and Horowitz’s formalisms) are related through the canonical
88
+ transformation, they indeed produce different dynamics in the quantum domain. It is also important to mention
89
+ that it is not possible to carry over the classical canonical transformations to the quantum domain for higher-order
90
+ theories, due to the non-linearity. The MHF leads to an effective Hermitian Hamiltonian, a standard quantum
91
+ mechanical probabilistic interpretation, and a viable semiclassical treatment, which exhibit oscillation of the wave
92
+ function about the classical de-Sitter solution. As a result, the classical correspondence is established. In this
93
+ regard, the MHF may be considered as the most-viable technique to handle the higher-order theories. It has later
94
+ been established that, if the action is expressed in terms of the three-space metric (hij ) from the very beginning
95
+ and the total derivative terms are addressed, Dirac’s constraint analysis [12] also produces the Hamiltonian iden-
96
+ tical to that of the MHF [22,28–30].
97
+ Amongst other techniques, Hawking-Luttrell technique [33] has limited application, since conformal trans-
98
+ formation is not possible in general [19], Schmidt’s technique [34] is identical to the Horowitz’s formalism in
99
+ disguise [17]. However, there is yet another technique developed in the 80’s by Buchbinder and his collabora-
100
+ tors [35–39], which did not receive much attention. Querella [40] only noticed that although at a first glance, the
101
+ general formalism developed by Buchbinder and his collaborators (BL) appears to be satisfactory, nevertheless it
102
+ has pitfalls. BL formalism is our current concern. Here, we test this abstract theoretical settings of BL formalism
103
+ in simple minisuperspace model to explore the pitfall, if any. The underlying essence of this formalism is to bypass
104
+ Dirac’s constrained analysis, very much like Horowitz’s technique, but instead of introducing auxiliary variable,
105
+ here the program is initiated with the basic variables {hij, Kij}, the three-space curvature and the extrinsic
106
+ curvature tensors respectively from the very beginning. In our present attempt to explore the outcome of this
107
+ 2
108
+
109
+ technique, we discover that the formalism leads to identical phase space structure as was found in the case of
110
+ Ostrogradsky’s/Dirac’s/Horowitz’s formalism.
111
+ This paper is organized as follows. In the following section, we study scalar tensor theory of gravity (both the
112
+ minimal and non-minimal cases), and Gauss-Bonnet-Dilatonic coupled action being supplemented by the scalar
113
+ curvature squared (R2 ) term, following BL formalism. In Section ??, we explore the fact that once total derivative
114
+ terms are taken care of, the Hamiltonian does not differ from MHF. Section ?? discusses its physical application,
115
+ in connection with some earlier work. Section ??, concludes our work.
116
+ 2
117
+ BL Formalism in Three Different Higher Order Theories
118
+ In view of the very importance of higher-order curvature invariant terms required to construct a renormalizable
119
+ quantum theory of gravity when the curvature is extremely strong, a unique canonical formulation of the Einstein–
120
+ Hilbert action being supplemented by higher-order curvature invariant terms, is therefore necessary. Here, we shall
121
+ consider three different cases, minimally and non-minimally coupled scalar-tensor theory of gravity supplemented
122
+ by R2 term, and the scalar-tensor theory of gravity being supplemented by R2 and Gauss-Bonnet terms. In the
123
+ Robertson-Walker minisuperspace (1) under consideration, the Ricci scalar and the Gauss-Bonnet terms are
124
+ R =
125
+ 6
126
+ N 2
127
+
128
+ ¨a
129
+ a + ˙a2
130
+ a2 + N 2 k
131
+ a2 −
132
+ ˙N ˙a
133
+ Na
134
+
135
+ .
136
+ (2)
137
+ G = R2 − 4RµνRµν + RαβµνRαβµν =
138
+ 24
139
+ N 3a3
140
+
141
+ N¨a − ˙N ˙a
142
+ � � ˙a2
143
+ N 2 + k
144
+
145
+ .
146
+ (3)
147
+ respectively. For the sake of comparison with earlier results, we express actions in terms of the three space metric,
148
+ instead of the scale factor, as its importance has been mentioned already, and will be explicitly shown at the
149
+ beginning of Section ??. Since construction of higher-order theory to its canonical form requires an additional
150
+ degree of freedom, hence, in addition to the three-space metric hij , the extrinsic curvature tensor Kij is treated
151
+ as basic variable, as already stated. We therefore choose the basic variables hij = zδij = a2δij , so that Kij =
152
+
153
+ ˙hij
154
+ 2N = − a ˙a
155
+ N δij = −
156
+ ˙z
157
+ 2N δij . In terms of z = a2, the Ricci scalar and the Gauss-Bonnet terms take the following
158
+ forms,
159
+ R =
160
+ 6
161
+ N 2
162
+
163
+ ¨z
164
+ 2z + N 2 k
165
+ z − 1
166
+ 2
167
+ ˙N ˙z
168
+ Nz
169
+
170
+ ,
171
+ (4)
172
+ G = 12
173
+ N 2
174
+
175
+ ¨z
176
+ z − ˙z2
177
+ 2z2 −
178
+ ˙N ˙z
179
+ Nz
180
+ � �
181
+ ˙z2
182
+ 4N 2z2 + k
183
+ z
184
+
185
+ ,
186
+ (5)
187
+ respectively. It is noteworthy that since,
188
+ RµνRµν = 12
189
+ N 4
190
+
191
+ ¨a2
192
+ a2 + ˙a2¨a
193
+ a3 + ˙a4
194
+ a4 − 2
195
+ ˙N ˙a¨a
196
+ Na2 −
197
+ ˙N ˙a3
198
+ Na3 +
199
+ ˙N 2 ˙a2
200
+ N 2a2 + k N 2¨a
201
+ a3
202
+ + 2k N 2 ˙a2
203
+ a4
204
+ − k N ˙N ˙a
205
+ a3
206
+ + k2 N 4
207
+ a4
208
+
209
+ ,
210
+ (6)
211
+ therefore,
212
+ RµνRµν − 1
213
+ 3R2 = −
214
+ � 12
215
+ Na3
216
+ � d
217
+ dt
218
+ �1
219
+ 3
220
+ ˙a3
221
+ N 3 + k ˙a
222
+ N
223
+
224
+ ,
225
+ (7)
226
+ and as a result,
227
+ � �
228
+ RµνRµν − 1
229
+ 3R2
230
+ � √−gd4x = −12C
231
+ � � d
232
+ dt
233
+ �1
234
+ 3
235
+ ˙a3
236
+ N 3 + k ˙a
237
+ N
238
+ ��
239
+ dt
240
+ (8)
241
+ 3
242
+
243
+ is a total derivative term. Thus, RµνRµν term is redundant in RW metric, once R2 term is taken (the constant C
244
+ appears due to the integration of the three space). Hence, to scrutinize the BL formalism presented by Buchbinder
245
+ and his collaborators in RW minisuperspace model (1), we consider scalar-tensor theories of gravity and also
246
+ Gauss-Bonnet-Dilatonic coupled gravity theory, being associated with scalar curvature squared term R2.
247
+ 2.1
248
+ Minimal coupling:
249
+ Let us start with the following minimally coupled case,
250
+ A1 =
251
+ � √−g
252
+
253
+ αR + βR2 − 1
254
+ 2φ,µφ,ν − V (φ)
255
+
256
+ d4x + αΣR + βΣR2.
257
+ (9)
258
+ In the above, α =
259
+ 1
260
+ 16πG , β is a constant coupling parameter, αΣR = 2α
261
+
262
+ ∂V K
263
+
264
+ hd3x is the Gibbons-
265
+ Hawking–York boundary term [21] associated with Einstein–Hilbert sector of the above action, and βΣR2 =
266
+
267
+
268
+ ∂V RK
269
+
270
+ hd3x is its modified version corresponding to R2 term, while, K is the trace of the extrinsic curva-
271
+ ture tensor Kij . Note that, both the counter terms are required under the condition δR = 0, at the boundary.
272
+ Instead, if the condition δKij = 0 is chosen at the boundary, the counter terms are not required, as in the case
273
+ of Horowitz’s formalism [14], since both the boundary terms appearing under metric variation vanish. However,
274
+ in the case of Ostrogradsky’s technique [1] and Dirac constrained analysis [12], boundary terms are not taken
275
+ care of. This is true for BL formalism too, as we shall see shortly. Nevertheless, the modified Horowitz’s for-
276
+ malism [17–19, 23–25] fixes δhij = 0 = δR at the boundary, and hence requires supplementary boundary terms.
277
+ We have demonstrated earlier that proper attention to all the boundary terms is paid in modified Horowitz’s
278
+ formalism (MHF). As a result, it presents a different phase space structure of the Hamiltonian for a particular
279
+ action being supplemented by higher-order terms. Nonetheless, it is related to the others under a suitable set
280
+ of canonical transformation [15]. Although, as mentioned, such transformations cannot be carried over in the
281
+ quantum domain, due to non-linearity. So, it’s indeed required to check if BL formalism also produces the same.
282
+ The action (9) in the RW minisuperspace model (1) may be written in terms of the basic variable hij = zδij , as
283
+ A1 =
284
+ � �
285
+ 3α√z
286
+ � ¨z
287
+ N −
288
+ ˙N ˙z
289
+ N 2 + 2kN
290
+
291
+ + 9β
292
+ √z
293
+ � ¨z2
294
+ N 3 − 2 ˙N ˙z¨z
295
+ N 4
296
+ +
297
+ ˙N 2 ˙z2
298
+ N 5
299
+ − 4k ˙N ˙z
300
+ N 2
301
+ + 4k¨z
302
+ N + 4k2N
303
+
304
+ + z
305
+ 3
306
+ 2
307
+ � ˙φ2
308
+ 2N − V N
309
+ ��
310
+ dt + αΣR + βΣR2.
311
+ (10)
312
+ The (0
313
+ 0) component of the field equation in terms of the scale factor ‘a′ takes the following form
314
+
315
+ a2
316
+ � ˙a2
317
+ N 2 + k
318
+
319
+ + 36β
320
+ a2N 4
321
+
322
+ 2˙a...a − 2˙a2 ¨N
323
+ N − ¨a2 − 4˙a¨a
324
+ ˙N
325
+ N + 2˙a2 ¨a
326
+ a + 5˙a2 ˙N 2
327
+ N 2 − 2 ˙a3 ˙N
328
+ aN
329
+ − 3 ˙a4
330
+ a2 − 2kN 2 ˙a2
331
+ a2 + k2N 4
332
+ a2
333
+
334
+
335
+ � ˙φ2
336
+ 2N 2 + V
337
+
338
+ = 0,
339
+ (11)
340
+ which contains term upto third derivative. This is the energy constraint equation (E = 0), and when expressed
341
+ in terms of the phase space variables, becomes the Hamiltonian constraint equation, (due to diffeomorphic invari-
342
+ ance) of the theory under consideration. This we aim at, following the formalism presented by Buchbinder and
343
+ his collaborators (BL).
344
+ The action (9) has already been expressed in terms of the basic variable {hij}, instead of the scale factor.
345
+ Canonical formulation of higher order theories requires additional degree of freedom, and the only choice is the
346
+ extrinsic curvature tensor {Kij}.
347
+ In contrast to Horowitz’s formalism, where apart from {hij} an auxiliary
348
+ variable is introduced and at the end the Hamiltonian is expressed in terms of the basic variables {hij, Kij},
349
+ in BL formalism, these basic variables are associated from the very beginning. In Robertson-Walker metric, the
350
+ extrinsic curvature tensor is expressed as,
351
+ Kij = −
352
+ ˙hij
353
+ 2N = −2a˙a
354
+ 2N δij = − ˙z
355
+ 2N δij = −qij say.
356
+ (12)
357
+ 4
358
+
359
+ Since there is only one independent component, so instead of qij , the new generalized coordinate is chosen to be
360
+ its trace viz,
361
+ q = 3 ˙z
362
+ 2N ,
363
+ i.e. qij = q
364
+ 3δij.
365
+ (13)
366
+ To express the action in terms of velocities, we choose,
367
+ v ≡ ˙q, vφ ≡ ˙φ.
368
+ (14)
369
+ The scalar curvature (4) therefore takes the following form,
370
+ R = 2 ˙q
371
+ Nz + 6k
372
+ z ≡ Rq =
373
+ 2
374
+ Nz (v + 3Nk),
375
+ (15)
376
+ and action (10) can be expressed as,
377
+ A1q =
378
+ � �
379
+ 2α√z(v + 3kN) +
380
+
381
+ N√z (v + 3kN)2 + z
382
+ 3
383
+ 2
384
+
385
+ v2
386
+ φ
387
+ 2N − NV
388
+ ��
389
+ dt,
390
+ (16)
391
+ while the Lagrangian density is,
392
+ L1q = 2α√z(v + 3kN) +
393
+
394
+ N√z (v + 3kN)2 + z
395
+ 3
396
+ 2
397
+
398
+ v2
399
+ φ
400
+ 2N − NV
401
+
402
+ .
403
+ (17)
404
+ Note that the boundary terms remain intact in the action as well as in the point Lagrangian. Canonical momenta
405
+ are
406
+ pq = ∂Lq
407
+ ∂v = 2α√z +
408
+
409
+ N√z (v + 3kN), pN = ∂L1q
410
+ ∂vN
411
+ = 0, pz = ∂Lq
412
+ ∂vz
413
+ = 0 and pφ = ∂Lq
414
+ ∂vφ
415
+ = z
416
+ 3
417
+ 2 vφ
418
+ N
419
+ .
420
+ (18)
421
+ Clearly, there exists two primary constraints C ≡ pN ≈ 0, and D ≡ pz ≈ 0. Therefore, Dirac constraint analysis
422
+ appears to be essential. However, here is a wonderful twist in the BL formalism. For example, one can express
423
+ the modified Lagrangian density as,
424
+ L∗
425
+ 1 = L1q + pq ( ˙q − v) + pN
426
+
427
+ ˙N − vN
428
+
429
+ + pz
430
+
431
+ ˙z − 2Nq
432
+ 3
433
+
434
+ + pφ
435
+
436
+ ˙φ − vφ
437
+
438
+ ,
439
+ (19)
440
+ and equivalently, the Hamiltonian density as,
441
+ H∗
442
+ 1 = pq ˙q + pN ˙N + pφ ˙φ + pz ˙z − L∗
443
+ 1 = pqv + pNvN + pφvφ + pz
444
+ 2Nq
445
+ 3
446
+ − L1q.
447
+ (20)
448
+ As a consequence, one can immediately find that the primary constraint D ≡ pz ≈ 0 disappears. Further, since
449
+ N is a non-dynamical Lagrange multiplier, hence the constraint C vanishes strongly. Therefore, one arrives at,
450
+ H∗
451
+ 1 = pqv + CvN + pφvφ + pz
452
+ 2qN
453
+ 3
454
+ − L1q = CvN + pqv + pφvφ + pz
455
+ 2qN
456
+ 3
457
+ − L1q = CvN + NHm
458
+ BL,
459
+ (21)
460
+ where,
461
+ NHm
462
+ BL = pqv + pφvφ + 2N
463
+ 3 qpz − L1q
464
+ = pqv + pφvφ + 2N
465
+ 3 qpz − 2α√z(v + 3kN) −
466
+
467
+ N√z (v + 3kN)2 − z
468
+ 3
469
+ 2
470
+
471
+ v2
472
+ φ
473
+ 2N + NV
474
+
475
+ .
476
+ (22)
477
+ 5
478
+
479
+ In the above, m in the superscript stands for minimally coupled theory. Now upon substituting v and vφ from
480
+ the definition of momentum (18), we obtain,
481
+ NHm
482
+ BL = N
483
+ �2q
484
+ 3 pz +
485
+ √z
486
+ 16β p2
487
+ q −
488
+ �αz
489
+ 4β + 3k
490
+
491
+ pq + α2
492
+ 4β z
493
+ 3
494
+ 2 +
495
+ 1
496
+ 2z
497
+ 3
498
+ 2 p2
499
+ φ + V z
500
+ 3
501
+ 2
502
+
503
+ ,
504
+ (23)
505
+ so that the canonical Hamiltonian finally reads as,
506
+ Hm
507
+ BL = 2q
508
+ 3 pz +
509
+ √z
510
+ 16β p2
511
+ q −
512
+ �αz
513
+ 4β + 3k
514
+
515
+ pq + α2
516
+ 4β z
517
+ 3
518
+ 2 +
519
+ 1
520
+ 2z
521
+ 3
522
+ 2 p2
523
+ φ + V z
524
+ 3
525
+ 2 .
526
+ (24)
527
+ The action (10) may also be cast in the canonical form,
528
+ A1q =
529
+ � �
530
+ ˙zpz + ˙qpq + ˙φpφ − NHBL
531
+
532
+ dt d3x
533
+ =
534
+ � �
535
+ ˙hijπij + ˙KijΠij + ˙φpφ − NHBL
536
+
537
+ dt d3x,
538
+ (25)
539
+ where, πij and Πij are momenta canonically conjugate to hij and Kij respectively. For the sake of comparison,
540
+ let us make the following canonical transformation:
541
+ q → 3
542
+ 2x;
543
+ pq → 2
544
+ 3px,
545
+ (26)
546
+ to express the above Hamiltonian (24) as:
547
+ Hm
548
+ BL = xpz +
549
+ √z
550
+ 36β p2
551
+ x −
552
+ �αz
553
+ 6β + 2k
554
+
555
+ px + α2z
556
+ 3
557
+ 2
558
+
559
+ +
560
+ p2
561
+ φ
562
+ 2z
563
+ 3
564
+ 2 + V z
565
+ 3
566
+ 2 .
567
+ (27)
568
+ It is revealed that the above Hamiltonian (27) is exactly the one obtained earlier, following Ostrogradsky, Dirac
569
+ as well as Horowitz’s formalisms [15]. Note that, very much like the Ostrogradsky’s and Dirac’s formalisms here
570
+ also, once the formalism is initiated, i.e. (R is expressed in terms of {hij, Kij} (15) as well as the action (16)
571
+ and the point Lagrangian (17)), there remains no option to integrate the action by parts. As a result, even the
572
+ Gibbons-Hawking–York term [20,21] which is physically meaningful, being associated with the entropy of the black
573
+ hole, along with its higher-order counterpart, also remain obscure. On the contrary, following Modified Horowitz’s
574
+ Formalism (MHF), where boundary terms are taken care of, we earlier obtained [15]
575
+ Hm
576
+ MHF = xpz +
577
+ √z
578
+ 36β p2
579
+ x + 3αx2
580
+ 2√z − 18βkx2
581
+ z
582
+ 3
583
+ 2
584
+ − 36βk2
585
+ √z
586
+ − 6kα√z +
587
+ p2
588
+ φ
589
+ 2z
590
+ 3
591
+ 2 + V z
592
+ 3
593
+ 2 .
594
+ (28)
595
+ Although the two (27) and (28) exactly match under the following set of canonical transformations,
596
+ pz → pz − 18kβx
597
+ z
598
+ 3
599
+ 2
600
+ + 3αx
601
+ 2√z ,
602
+ z → z,
603
+ px → px + 36kβ
604
+ √z − 3α√z,
605
+ x → x,
606
+ pφ → pφ,
607
+ φ → φ.
608
+ and apparently there is no contradiction between the two, note the essential difference: linear term in the mo-
609
+ mentum (px ), which is very much present in (27), remains absent from the Hamiltonian (28). As a result, the
610
+ two Hamiltonians (27) and (28) induce completely different quantum dynamics, since in the quantum domain, as
611
+ mentioned, canonical transformation cannot be carried over due to non-linearity.
612
+ 6
613
+
614
+ 2.2
615
+ Non-minimally coupled case
616
+ We find that the two different Hamiltonians (27) and (28) render two different quantum descriptions of the
617
+ same classical model. Although, some of the essential features (Gibbons-Hawking-York term and its higher order
618
+ counterpart) are absent from the Hamiltonian (27), it is not clear, which one gives correct quantum description
619
+ of the theory. Further, there may exist a unitary transformation (we have not found it though) relating the two
620
+ Hamiltonian operators. Therefore, to inspect the situation more deeply, we consider the non-minimally coupled
621
+ case next, whose action
622
+ A2 =
623
+ � √−g d4x
624
+
625
+ f(φ)R + βR2 − 1
626
+ 2φ,µφ,ν − V (φ)
627
+
628
+ + f(φ)ΣR + BΣR2,
629
+ (29)
630
+ may be expressed in the RW metric (1) as
631
+ A2 =
632
+ � �
633
+ 3f(φ)√z
634
+ � ¨z
635
+ N −
636
+ ˙N ˙z
637
+ N 2 + 2kN
638
+
639
+ + 9β
640
+ √z
641
+ � ¨z2
642
+ N 3 − 2 ˙N ˙z¨z
643
+ N 4
644
+ +
645
+ ˙N 2 ˙z2
646
+ N 5
647
+ − 4k ˙N ˙z
648
+ N 2
649
+ + 4k¨z
650
+ N + 4k2N
651
+
652
+ + z
653
+ 3
654
+ 2
655
+ � ˙φ2
656
+ 2N − V N
657
+ ��
658
+ dt + f(φ)ΣR + BΣR2,
659
+ (30)
660
+ where, as already mentioned, the supplementary boundary terms are required when MHF is taken into account. In
661
+ the above, we consider an arbitrary functional coupling parameter f(φ). Pursuing the same procedure as above,
662
+ one finally arrives at the following Hamiltonian:
663
+ HnmBL = xpz +
664
+ √z
665
+ 36β p2
666
+ x −
667
+
668
+ f(φ) z
669
+ 6β + 2k
670
+
671
+ px + f 2(φ)z
672
+ 3
673
+ 2
674
+ 4β +
675
+ p2
676
+ φ
677
+ 2z
678
+ 3
679
+ 2 + V z
680
+ 3
681
+ 2 ,
682
+ (31)
683
+ which is again identical to the one found following Dirac formalism and may be found following Ostrogradsky’s
684
+ and Horowitz’s techniques as well [22]. In the superscript nm stands for non-minimal coupling. The action (30)
685
+ may also be cast in the canonical form as in (25). On the contrary, following MHF, one finds [22]
686
+ HnmMHF = xpz +
687
+ √z
688
+ 36β p2
689
+ x + 3f(φ)
690
+ � x2
691
+ 2√z − 2k√z
692
+
693
+ − 18kβ
694
+ √z
695
+ �x2
696
+ z + 2k
697
+
698
+ +
699
+ p2
700
+ φ
701
+ 2z
702
+ 3
703
+ 2
704
+ + 3xf ′(φ)pφ
705
+ z
706
+ + 9f ′(φ)2x2
707
+ 2√z
708
+ + V z
709
+ 3
710
+ 2 .
711
+ (32)
712
+ However, under the following set of canonical transformations,
713
+ pz → pz − 18βkx
714
+ z
715
+ 3
716
+ 2
717
+ + 3f(φ)x
718
+ 2√z ,
719
+ z → z,
720
+ px → px + 36β k
721
+ √z + 3f(φ)√z,
722
+ x → x,
723
+ pφ → pφ + 3f ′(φ)x√z,
724
+ φ → φ,
725
+ the two Hamiltonians (31) and (32) match again [22]. Nevertheless, here the difference is predominant and explicit.
726
+ Note that f ′(φ) term does not appear in (31), while it is coupled to pφ in (32). This coupled (f ′(φ)pφ ) term
727
+ requires operator ordering in the quantum domain, which is different for different form of f(φ). Hence even if the
728
+ two Hamiltonians are related through unitary transformation, such transformation would be different for different
729
+ form of f(φ).
730
+ 7
731
+
732
+ 2.3
733
+ Einstein-Gauss-Bonnet-Dilatonic action in the presence of higher order term
734
+ Although, it is clear that two different quantum descriptions follow from the same classical action using different
735
+ techniques, it is still abstruse to select the correct description. Therefore, we next consider Einstein-Gauss-Bonnet-
736
+ Dilatonic coupled action in the presence of higher order curvature invariant term. Gauss-Bonnet (GB) term arises
737
+ quite naturally as the leading order of the α′ expansion of heterotic superstring theory, where α′ is the inverse
738
+ string tension [41–46]. Several interesting features of GB term have been explored in the past and appear in
739
+ the literature [47–67]. However, Gauss–Bonnet term is topological invariant in 4-dimensions, and so to get its
740
+ contribution in the field equations, a dynamic dilatonic scalar coupling is required. It is worth mentioning that,
741
+ in string induced gravity near initial singularity, GB coupling with scalar field plays a very crucial role for the
742
+ occurrence of nonsingular cosmology [68,69]. The particular hallmark of GB term is the fact that, despite being
743
+ formed from a combination of higher order curvature invariant terms (G = R2 − 4RµνRµν + RαβµνRαβµν) (3),
744
+ it ends up only with second order field equations, avoiding Ostrogradsky’s instability, and equivalently, ghost
745
+ degrees of freedom. Nonetheless, such a wonderful feature ultimately leads to a serious pathology of ‘Branched
746
+ Hamiltonian’, which has no unique resolution till date [70–72].
747
+ Nevertheless, it has been revealed that, the
748
+ pathology may be bypassed upon supplementing the action with higher order curvature invariant term [24, 25].
749
+ We therefore consider the following action,
750
+ A3 =
751
+ � √−g d4x
752
+
753
+ αR + βR2 + γ(φ)G − 1
754
+ 2φ,µφ,ν − V (φ)
755
+
756
+ + αΣR + βΣR2 + γ(φ)ΣG.
757
+ (33)
758
+ In the above, Gauss-Bonnet term G , is coupled with γ(φ), while V (φ) is the dilatonic potential. Further, the
759
+ symbol K stands for K = K3 − 3KKijKij + 2KijKikKk
760
+ j , where, K is the trace of the extrinsic curvature
761
+ tensor Kij , and γ(φ)ΣG = 4γ(φ)
762
+
763
+ ∂V
764
+
765
+ 2GijKij + K
766
+ 3
767
+ � √
768
+ hd3x is the supplementary boundary term associated with
769
+ Gauss-Bonnet sector. The (0
770
+ 0) component of the Einstein’s field equation in terms of the scale factor here reads
771
+ as,
772
+
773
+ a2
774
+ � ˙a2
775
+ N 2 + k
776
+
777
+ + 36β
778
+ a2N 4
779
+
780
+ 2˙a...a − 2˙a2 ¨N
781
+ N − ¨a2 − 4˙a¨a
782
+ ˙N
783
+ N + 2˙a2 ¨a
784
+ a + 5˙a2 ˙N 2
785
+ N 2 − 2 ˙a3 ˙N
786
+ aN
787
+ − 3 ˙a4
788
+ a2 − 2kN 2 ˙a2
789
+ a2 + k2N 4
790
+ a2
791
+
792
+ + 24γ′ ˙a ˙φ
793
+ N 2a3
794
+ � ˙a2
795
+ N 2 + k
796
+
797
+
798
+ � ˙φ2
799
+ 2N 2 + V
800
+
801
+ = 0.
802
+ (34)
803
+ The action (33) in terms of the basic variable (hij = a2δij = zδij ) may be expressed as,
804
+ A3 =
805
+ � �
806
+ 3α√z
807
+
808
+ ¨z
809
+ N −
810
+ ˙N ˙z
811
+ N 2 + 2kN
812
+
813
+ + 9β
814
+ √z
815
+
816
+ ¨z2
817
+ N 3 − 2 ˙N ˙z¨z
818
+ N 4
819
+ +
820
+ ˙N 2 ˙z2
821
+ N 5
822
+ − 4k ˙N ˙z
823
+ N 2
824
+ + 4k¨z
825
+ N + 4k2N
826
+
827
+ + 3γ(φ)
828
+ N√z
829
+
830
+ ˙z2¨z
831
+ N 2z + 4k¨z −
832
+ ˙z4
833
+ 2N 2z2 −
834
+ ˙N ˙z3
835
+ N 3z − 2k ˙z2
836
+ z
837
+ − 4k ˙N ˙z
838
+ N
839
+
840
+ + z
841
+ 3
842
+ 2
843
+ � 1
844
+ 2N
845
+ ˙φ2 − V N
846
+ � �
847
+ dt
848
+ + αΣR + βΣR2 + γ(φ)ΣG,
849
+ (35)
850
+ where the additional supplementary boundary term γ(φ)ΣG = −γ(φ)
851
+ ˙z
852
+ N√z
853
+
854
+ ˙z2
855
+ N 2z + 12k
856
+
857
+ , is required in the case of
858
+ MHF. Inserting the other basic variable (Kij = − q
859
+ 3δij ) and considering ˙q = v (13), the action (35) finally may
860
+ be expressed as,
861
+ A3q =
862
+ � �
863
+ 2α√z(v + 3kN) +
864
+
865
+ N√z (v + 3kN)2 + 8γ(φ)
866
+ √z
867
+ �vq2
868
+ 9z − q4N
869
+ 27z2 + kv − kNq2
870
+ 3z
871
+
872
+ + z
873
+ 3
874
+ 2
875
+
876
+ v2
877
+ φ
878
+ 2N − NV
879
+ � �
880
+ dt + αΣR + βΣR2 + Λ(φ)ΣG.
881
+ (36)
882
+ Thus, the Lagrangian density takes the following form,
883
+ L3q = 2α√z(v + 3kN) +
884
+
885
+ N√z (v + 3kN)2 + 8γ(φ)
886
+ √z
887
+ �vq2
888
+ 9z − q4N
889
+ 27z2 + kv − kNq2
890
+ 3z
891
+
892
+ + z
893
+ 3
894
+ 2
895
+ � 1
896
+ 2N v2
897
+ φ − V N
898
+
899
+ ,
900
+ (37)
901
+ 8
902
+
903
+ where, boundary terms are not taken care of. The canonical momenta are
904
+ pq = ∂Lq
905
+ ∂v = 2α√z +
906
+
907
+ N√z (v + 3kN) + 8γ(φ)
908
+ √z
909
+ � q2
910
+ 9z + k
911
+
912
+ ,
913
+ pN = ∂L3q
914
+ ∂vN
915
+ = 0,
916
+ pφ = ∂Lq
917
+ ∂vφ
918
+ = z
919
+ 3
920
+ 2 vφ
921
+ N
922
+ ,
923
+ and,
924
+ pz = ∂L3q
925
+ ∂vz
926
+ = 0.
927
+ (38)
928
+ Clearly, there exists two primary constraints C ≡ pN ≈ 0, and D ≡ pz ≈ 0, which are usually handled by Dirac
929
+ constraint analysis. However, as mentioned, such analysis is not at all required in the BL formalism. For example,
930
+ one can express the modified Lagrangian density as,
931
+ L∗
932
+ 3 = L3q + pq ( ˙q − v) + pN
933
+
934
+ ˙N − vN
935
+
936
+ + pz
937
+
938
+ ˙z − 2Nq
939
+ 3
940
+
941
+ + pφ
942
+
943
+ ˙φ − vφ
944
+
945
+ ,
946
+ (39)
947
+ so that the corresponding Hamiltonian density takes the following form,
948
+ H∗
949
+ 3 = pq ˙q + pN ˙N + pφ ˙φ + pz ˙z − L∗
950
+ 3 = pqv + pNvN + pφvφ + 2Nq
951
+ 3
952
+ pz − L3q.
953
+ (40)
954
+ As a result, the primary constraint D ≡ pz ≈ 0 disappears and one obtains,
955
+ H∗
956
+ 3 = pqv + CvN + pφvφ + pz
957
+ 2qN
958
+ 3
959
+ − L3q = CvN +
960
+
961
+ pqv + pφvφ + 2qN
962
+ 3
963
+ pz − L3q
964
+
965
+ = CvN + NHGB
966
+ BL. (41)
967
+ In the superscript GB stands for Hamiltonian in connection with Einstein-Gauss-Bonnet-Dilatonic coupling.
968
+ Note that the constraint C ≡ pN strongly vanishes, since the lapse function N is simply a Lagrange multiplier.
969
+ Therefore,
970
+ NHGBBL = pqv + pφvφ + 2qN
971
+ 3
972
+ pz − L3q
973
+ = pqv + pφvφ + 2qN
974
+ 3
975
+ pz − 2α√z(v + 3kN) −
976
+
977
+ N√z (v + 3kN)2
978
+ − 8γ(φ)
979
+ √z
980
+ �vq2
981
+ 9z − q4N
982
+ 27z2 + kv − kNq2
983
+ 3z
984
+
985
+ − z
986
+ 3
987
+ 2
988
+ � 1
989
+ 2N v2
990
+ φ − V N
991
+
992
+ .
993
+ (42)
994
+ Now upon substituting v from the definition of momentum (38), one obtains,
995
+ NHGBBL =N
996
+ �2qpz
997
+ 3
998
+ +
999
+ √zp2
1000
+ q
1001
+ 16β − pq
1002
+ �αz
1003
+ 4β + 3k
1004
+
1005
+ + α2z
1006
+ 3
1007
+ 2
1008
+
1009
+ − pq
1010
+ � γq2
1011
+ 9βz + γk
1012
+ β
1013
+
1014
+ + 2αγ
1015
+ β
1016
+ � q2
1017
+ 9√z + k√z
1018
+
1019
+ + 4γq4
1020
+ 27z
1021
+ 3
1022
+ 2
1023
+ � γ
1024
+ 3β + 2
1025
+
1026
+ + 8γkq2
1027
+ 3z
1028
+ 3
1029
+ 2
1030
+ � γ
1031
+ 3β + 2
1032
+
1033
+ + 12γk2
1034
+ √z
1035
+ � γ
1036
+ 3β + 2
1037
+
1038
+ +
1039
+ p2
1040
+ φ
1041
+ 2z
1042
+ 3
1043
+ 2 + V z
1044
+ 3
1045
+ 2
1046
+
1047
+ .
1048
+ (43)
1049
+ The canonical Hamiltonian therefore finally reads as,
1050
+ HGB
1051
+ BL =2qpz
1052
+ 3
1053
+ +
1054
+ √zp2
1055
+ q
1056
+ 16β − pq
1057
+ �αz
1058
+ 4β + 3k
1059
+
1060
+ + α2z
1061
+ 3
1062
+ 2
1063
+
1064
+ − pq
1065
+ � γq2
1066
+ 9βz + γk
1067
+ β
1068
+
1069
+ + 2αγ
1070
+ β
1071
+ � q2
1072
+ 9√z + k√z
1073
+
1074
+ + 4γq4
1075
+ 27z
1076
+ 3
1077
+ 2
1078
+ � γ
1079
+ 3β + 2
1080
+
1081
+ + 8γkq2
1082
+ 3z
1083
+ 3
1084
+ 2
1085
+ � γ
1086
+ 3β + 2
1087
+
1088
+ + 12γk2
1089
+ √z
1090
+ � γ
1091
+ 3β + 2
1092
+
1093
+ +
1094
+ p2
1095
+ φ
1096
+ 2z
1097
+ 3
1098
+ 2 + V z
1099
+ 3
1100
+ 2 .
1101
+ (44)
1102
+ Again, for the sake of comparison, let us make the canonical transformation q → 3
1103
+ 2x; pq → 2
1104
+ 3px (26), to express
1105
+ the above Hamiltonian (44) in the following form,
1106
+ HGBBL = xpz +
1107
+ √zp2
1108
+ x
1109
+ 36β + α2z
1110
+ 3
1111
+ 2
1112
+
1113
+
1114
+ �αz
1115
+ 6β + γx2
1116
+ 6βz + 2kγ
1117
+ 3β + 2k
1118
+
1119
+ px +
1120
+ p2
1121
+ φ
1122
+ 2z
1123
+ 3
1124
+ 2 +
1125
+ � γ2
1126
+ 4βz
1127
+ 5
1128
+ 2 + 3γ
1129
+ 2z
1130
+ 5
1131
+ 2
1132
+
1133
+ x4
1134
+ +
1135
+ � αγ
1136
+ 2β√z + 12kγ
1137
+ z
1138
+ 3
1139
+ 2
1140
+ + 2kγ2
1141
+ βz
1142
+ 3
1143
+ 2
1144
+
1145
+ x2 + 2αkγ√z
1146
+ β
1147
+ + 24k2γ
1148
+ √z
1149
+ + 4k2γ2
1150
+ β√z + V z
1151
+ 3
1152
+ 2 ,
1153
+ (45)
1154
+ 9
1155
+
1156
+ and notice that, it is similar to the one already found, following Dirac formalism and may be found following
1157
+ Ostrogradsky’s and Horowitz’s techniques as well [22]. The action (35) may also be cast in the canonical form
1158
+ with respect to the basic variables as,
1159
+ A3q =
1160
+ � �
1161
+ ˙zpz + ˙qpq + ˙φvφ − NHBL
1162
+
1163
+ dt d3x =
1164
+ � �
1165
+ ˙hijπij + ˙KijΠij + ˙φvφ − NHMHF
1166
+
1167
+ dt d3x,
1168
+ (46)
1169
+ where, πij and Πij are momenta canonically conjugate to hij and Kij respectively. Hence, everything appears
1170
+ to be consistent. On the contrary, although following MHF, we found [22]
1171
+ HGB
1172
+ MHF = xpz +
1173
+ √zp2
1174
+ x
1175
+ 36β + 3α
1176
+ � x2
1177
+ 2√z − 2k√z
1178
+
1179
+ − 18kβ
1180
+ √z
1181
+ �x2
1182
+ z + 2k
1183
+
1184
+ +
1185
+ � x6
1186
+ 2z
1187
+ 9
1188
+ 2 + 12kx4
1189
+ z
1190
+ 7
1191
+ 2
1192
+ + 72k2x2
1193
+ z
1194
+ 5
1195
+ 2
1196
+
1197
+ γ′2
1198
+ +
1199
+ �x3
1200
+ z3 + 12kx
1201
+ z2
1202
+
1203
+ γ′pφ +
1204
+ p2
1205
+ φ
1206
+ 2Z
1207
+ 3
1208
+ 2 + V z
1209
+ 3
1210
+ 2 ,
1211
+ (47)
1212
+ nonetheless, under the following set of canonical transformations,
1213
+ pz → pz − 18βkx
1214
+ z
1215
+ 3
1216
+ 2
1217
+ + 3αx
1218
+ 2√z − 6kγ(φ)x
1219
+ z
1220
+ 3
1221
+ 2
1222
+ − 3γ(φ)x3
1223
+ 2z
1224
+ 5
1225
+ 2
1226
+ ,
1227
+ z → z,
1228
+ px → px + 36β k
1229
+ √z + 3α√z + 3γ(φ)x2
1230
+ z
1231
+ 3
1232
+ 2
1233
+ + 12kΛ
1234
+ √z ,
1235
+ x → x,
1236
+ pφ → pφ − γ′(φ)x3
1237
+ z
1238
+ 3
1239
+ 2
1240
+ − 12kγ′(φ)x
1241
+ √z
1242
+ ,
1243
+ φ → φ,
1244
+ (48)
1245
+ the two Hamiltonians (45) and (47) match again [22].
1246
+ Apparently therefore, there is absolutely no problem.
1247
+ Nevertheless note that, the Hamiltonian (47) contains a term (γ′(φ)pφ ), which is absent from (45). Now, during
1248
+ canonical quantization the presence of this term requires operator ordering, which is different for different form
1249
+ of γ(φ). As a result, even if the two may be related through unitary transformation, such transformation would
1250
+ be different for different form of γ(φ). Thus, there does not exist a unique unitary transformation. In a nutshell,
1251
+ we repeat that the two Hamiltonians (45) and (47) induce two different descriptions in the quantum domain, and
1252
+ apparently, there is no way to choose one to be the correct.
1253
+ 3
1254
+ The role of divergent terms:
1255
+ The very first important point to mention is, in all the formalisms the scale factor is treated as the basic variable,
1256
+ while we initiate our program treating three three-space curvature, instead. To explain the reason behind this
1257
+ choice, let us consider curvature squared action, A =
1258
+
1259
+ βR2d4x, as an example. Under variation, it gives a total
1260
+ derivative term σ = −4β
1261
+
1262
+ RK
1263
+
1264
+ h d3x, as mentioned earlier, where K is the trace of the extrinsic curvature
1265
+ tensor Kij . A counter term (−σ), known by the name modified Gibbons-Hawking–York term [20, 21], must be
1266
+ added to the action in case, instead of δ ˙q, δR is kept fixed at the boundary, as in MHF. In the RW (1) metric
1267
+ under consideration, the action reads as,
1268
+ A = 36β
1269
+ � �
1270
+ a¨a2 + 2˙a2¨a + 2k¨a + ˙a4
1271
+ a + 2k ˙a2
1272
+ a
1273
+ + k2
1274
+ a
1275
+
1276
+ dt
1277
+
1278
+ d3x.
1279
+ (49)
1280
+ Under integration by parts, we end up with,
1281
+ A = C
1282
+ � �
1283
+ a¨a2 + ˙a4
1284
+ a + 2k ˙a2
1285
+ a
1286
+ + k2
1287
+ a
1288
+
1289
+ dt + C
1290
+ �2
1291
+ 3 ˙a3 + 2k ˙a
1292
+
1293
+ .
1294
+ (50)
1295
+ where, C = 36β
1296
+
1297
+ d3x. Now following Horowitz’s program, we introduce an auxiliary variable Q = ∂A
1298
+ ∂¨a = 2Ca¨a,
1299
+ judiciously into the action in the following manner, such that it may be cast in canonical form,
1300
+ A =
1301
+ � �
1302
+ Q¨a − Q2
1303
+ 4Ca + C
1304
+ � ˙a4
1305
+ a + 2k ˙a2
1306
+ a
1307
+ + k2
1308
+ a
1309
+ ��
1310
+ dt + C
1311
+ �2
1312
+ 3 ˙a3 + 2k ˙a
1313
+
1314
+ .
1315
+ (51)
1316
+ 10
1317
+
1318
+ Integrating the action again by parts we find
1319
+ A =
1320
+
1321
+ − ˙Q˙a − Q2
1322
+ 4Ca + C
1323
+ � ˙a4
1324
+ a + 2k ˙a2
1325
+ a
1326
+ + k2
1327
+ a
1328
+ ��
1329
+ + C
1330
+ �Q˙a
1331
+ C + 2
1332
+ 3 ˙a3 + 2k ˙a
1333
+
1334
+ .
1335
+ (52)
1336
+ The action is canonical, since the Hessian determinant is non-zero. It is trivial to check that the above action
1337
+ gives correct field equations, but the left out total derivative term may be expressed as,
1338
+ σ′ = −4β
1339
+
1340
+ RK
1341
+
1342
+ h d3x + 16β
1343
+
1344
+ K
1345
+
1346
+ h
1347
+ � ˙a2
1348
+ a2
1349
+
1350
+ d3x,
1351
+ (53)
1352
+ and as a result σ ̸= σ′ . Thus some redundant total derivative terms are pulled out in the process, which has severe
1353
+ consequence in the quantum domain. On the contrary, if we start with, z = a2, the action reads as,
1354
+ A = C
1355
+ � � ¨z2
1356
+ 4√z + k¨z
1357
+ √z + k2
1358
+ √z
1359
+
1360
+ dt = C
1361
+ � � ¨z2
1362
+ 4√z +
1363
+ 2
1364
+ √z
1365
+
1366
+ + C k ˙z
1367
+ √z ,
1368
+ (54)
1369
+ where the last expression is found under integration by parts. Now following Horowitz’s program, we find the
1370
+ auxiliary variable as Q = ∂A
1371
+ ∂¨z = C
1372
+ ¨z
1373
+ 2√z , which is again judiciously introduced in the action as,
1374
+ A =
1375
+ � �
1376
+ Q¨z −
1377
+ √zQ2
1378
+ C
1379
+ + C
1380
+ � k¨z
1381
+ √z + k2
1382
+ √z
1383
+ ��
1384
+ dt + C k ˙z
1385
+ √z .
1386
+ (55)
1387
+ Finally, performing integration by parts again, one obtains,
1388
+ A =
1389
+ � �
1390
+ − ˙Q ˙z −
1391
+ √zQ2
1392
+ C
1393
+ + C
1394
+ � k¨z
1395
+ √z + k2
1396
+ √z
1397
+ ��
1398
+ dt + C
1399
+ �Q ˙z
1400
+ C + k ˙z
1401
+ √z
1402
+
1403
+ ,
1404
+ (56)
1405
+ The action is again canonical, the Euler-Lagrange equations here again lead to the appropriate field equations,
1406
+ while one can express the total derivative term as σ. In a nut-shell, although total derivative terms do not affect
1407
+ the classical field equations, for non-linear theories such as gravity, such terms tell upon the quantum dynam-
1408
+ ics. Therefore, to establish consistency in every respect, hij should be treated as the basic variable, instead of
1409
+ the scale factor. This is essentially the so-called MHF, which finally requires to replace the auxiliary variable
1410
+ by the the second basic variable, viz., the extrinsic curvature tensor Kij = −a˙a = − ˙z = x(say), in the Hamiltonian.
1411
+ Next, we observe that the phase-space structures obtained following BL formalism although are identical to
1412
+ the Ostrogradsky/Dirac/Horowitz’s formalism, they all differ from the MHF upto a canonical transformation. We
1413
+ quote from [22] the general argument in connection with the total derivative terms, which runs as; “it is just
1414
+ the change of the variables in the wave function and the phase transformation, plus the change of the integra-
1415
+ tion measure, and the transformation of the momenta respecting the change of the measure, and so a unitary
1416
+ transformation relates the two”. It’s possible (we have not found though) that each pair of quantum equations
1417
+ cast from {(27) and (28)}; {(31) and (32)}; {(45) and (47)}, are related by unitary transformation. However,
1418
+ it was also mentioned [22] that different forms of coupling parameter yield different quantum dynamics in the
1419
+ case of MHF, due to the presence of a coupling term (f ′(φ)pφ ) for non-minimal coupled case, and (γ′(φ)pφ ) for
1420
+ the Gauss-Bonnet-Dilaton coupled case, in the Hamiltonian. Thus, different unitary transformations (if exist)
1421
+ are required to relate the last two pairs. Such coupling as well as the derivative of coupling parameter remain
1422
+ absent in other formalisms. In a nutshell, unitary transformation relating each pair is not unique. Further, the
1423
+ semiclassical wave functions found for all the three cases studied here, exhibit different pre-factors and exponents
1424
+ for each pair [22]. This generates different probability amplitude and the evolution of the wave function while
1425
+ entering the classical domain.
1426
+ Finally, it is important to note that, if the coupling parameter f(φ) is treated as constant in Subsection ??,
1427
+ the Hamiltonian (32) merely reduces to (28), while the Hamiltonian (31) reduces to (27). Hence the question is:
1428
+ which of the two should be treated as the correct quantum description of the models under consideration? In
1429
+ this connection we mention that a serious problem arises with Ostrogradsky/Dirac/Horowitz as well as with BL
1430
+ 11
1431
+
1432
+ formalisms when considering Gauss-Bonnet-Dilaton induced action. To be specific, in Subsection ?? if γ(φ) is
1433
+ treated as a constant, then the contribution of Gauss-Bonnet term disappears from the Hamiltonian (47), and
1434
+ it reduces to (28).
1435
+ Indeed, it should since as mentioned, Gauss-Bonnet term is topologically invariant in 4-
1436
+ dimensions, and so without functional coupling, it does not contribute to the field equations and the Hamiltonian
1437
+ as well. On the contrary, a constant γ does not affect the form of the Hamiltonian (45), and it does not reduce
1438
+ to (27). This means, if we had started with a constant γ from the very beginning, all the terms appearing with
1439
+ γ in (45) would have been absent, and the end result would be (27). While, after constructing the Hamiltonian
1440
+ with arbitrary γ = γ(φ), if we set it equal to a constant, then its contribution remains present, and we obtain
1441
+ a different Hamiltonian, altogether. Clearly this is wrong. Hence, we realize that boundary terms indeed play
1442
+ a crucial role while constructing the phase-space structure of non-linear theories. In fact, if boundary terms are
1443
+ taken into account from the very beginning, treating hij as the basic variable, then Horowitz’s formalism reduces
1444
+ to the MHF, as already demonstrated. It was also noticed that if Dirac algorithm is applied after integrating the
1445
+ action by parts, then it also yields Hamiltonian identical to MHF [22]. It is therefore suggestive to test the same
1446
+ for BL formalism too. In this section we shall first integrate actions by parts to get rid of the total derivative
1447
+ terms and follow the BL formalism thereafter, to explore the outcome.
1448
+ 3.1
1449
+ Scalar-tensor theory: minimal coupling;
1450
+ Upon integrating the action (30) by parts, we obtain
1451
+ A1 =
1452
+ � 
1453
+ − 3α ˙z2
1454
+ 2N√z + 6αkN√z +
1455
+
1456
+ N√z
1457
+
1458
+
1459
+
1460
+
1461
+ ¨z
1462
+ N −
1463
+ ˙N ˙z
1464
+ N 2
1465
+ �2
1466
+ + 2k ˙z2
1467
+ z
1468
+ + 4k2N 2
1469
+
1470
+
1471
+  + z
1472
+ 3
1473
+ 2
1474
+ � ˙φ2
1475
+ 2N − NV
1476
+ �
1477
+  dt.
1478
+ (57)
1479
+ Replacing ˙z by 2N
1480
+ 3 q in view of (13), the above action may be cast as,
1481
+ A1q =
1482
+ � �
1483
+ −2
1484
+ 3αN q2
1485
+ √z + 6αkN√z +
1486
+
1487
+ N√z
1488
+ �4
1489
+ 9 ˙q2 + 8kN 2q2
1490
+ 9z
1491
+ + 4k2N 2
1492
+
1493
+ + z
1494
+ 3
1495
+ 2
1496
+ � ˙φ2
1497
+ 2N − NV
1498
+ ��
1499
+ dt.
1500
+ (58)
1501
+ Note that the action (58) cannot be expressed only in terms of velocities, due to the explicit presence of q unlike
1502
+ (16). However, similar situation arrived at, in the case of Gauss-Bonnet-Dilaton case, and so it doesn’t matter.
1503
+ The canonical momenta are the following:
1504
+ pq =
1505
+
1506
+ N√z ˙q;
1507
+ pφ = z
1508
+ 3
1509
+ 2
1510
+ N
1511
+ ˙φ;
1512
+ pz = 0 = pN.
1513
+ (59)
1514
+ Dirac constraint analysis appears to be inevitable, since the action is singular. However as mentioned, the lapse
1515
+ function N being the Lagrange multiplier, the constraint strongly vanishes, so that one can ignore it without
1516
+ loss of generality. Still, another primary constraint pz = 0 is apparent. Nonetheless, as already noticed, in BL
1517
+ formalism, Dirac analysis may be bypassed despite the presence of the constraint pz = 0 in the following manner.
1518
+ The Lagrangian density is:
1519
+ L1q = −2
1520
+ 3αN q2
1521
+ √z + 6αkN√z +
1522
+
1523
+ N√z
1524
+ �4
1525
+ 9 ˙q2 + 8kN 2q2
1526
+ 9z
1527
+ + 4k2N 2
1528
+
1529
+ + z
1530
+ 3
1531
+ 2
1532
+ � ˙φ2
1533
+ 2N − NV
1534
+
1535
+ ,
1536
+ (60)
1537
+ and hence the Hamiltonian reads as,
1538
+ NHm
1539
+ MBL = pq ˙q + pz ˙z + pφ ˙φ − L1q
1540
+ = N√z
1541
+ 16β p2
1542
+ q + 2
1543
+ 3Nqpz + N
1544
+ 2z
1545
+ 3
1546
+ 2 p2
1547
+ φ + 2αNq2
1548
+ 3√z
1549
+ − 6αkN√z − 8βkNq2
1550
+ z
1551
+ 3
1552
+ 2
1553
+ − 36βk2N
1554
+ √z
1555
+ + NV z
1556
+ 3
1557
+ 2 ,
1558
+ (61)
1559
+ where we have used (59) and replaced ˙z by 2N
1560
+ 3 q, in view of (13), and the suffix {MBL} now stands for ‘Modified
1561
+ Buchbinder-Lyakhovich’ formalism. Finally as before, for the sake of comparison, if we perform the canonical
1562
+ 12
1563
+
1564
+ transformation q → 3
1565
+ 2x,
1566
+ and
1567
+ pq → 2
1568
+ 3px, then the above Hamiltonian (61) may be expressed in the following
1569
+ form,
1570
+ Hm
1571
+ MBL = xpz +
1572
+ √z
1573
+ 36β p2
1574
+ x +
1575
+ p2
1576
+ φ
1577
+ 2z
1578
+ 3
1579
+ 2 + 3α
1580
+ 2√z (x2 − 4kz) − 18βk
1581
+ z
1582
+ 3
1583
+ 2
1584
+ (x2 + 2kz) + V z
1585
+ 3
1586
+ 2 ,
1587
+ (62)
1588
+ which is identical to Hm
1589
+ MHF presented in (28).
1590
+ 3.2
1591
+ Scalar-tensor theory: non-minimal coupling;
1592
+ Here again, upon integrating the action (30) by parts we obtain,
1593
+ A2 =
1594
+ � �
1595
+ − 3f ˙z2
1596
+ 2N√z − 3f ′ ˙φ ˙z√z
1597
+ N
1598
+ + 6fkN√z +
1599
+
1600
+ N√z
1601
+ �� ¨z
1602
+ N −
1603
+ ˙N ˙z
1604
+ N 2
1605
+ �2
1606
+ + 2k ˙z2
1607
+ z
1608
+ + 4k2N 2�
1609
+ + z
1610
+ 3
1611
+ 2
1612
+ � ˙φ2
1613
+ 2N − NV
1614
+ ��
1615
+ dt. (63)
1616
+ Now, replacing ˙z by 2N
1617
+ 3 q in view of (13), the above action (63) may be cast as,
1618
+ A2 =
1619
+ � �
1620
+ −2
1621
+ 3fN q2
1622
+ √z − 2f ′√zq ˙φ + 6fkN√z +
1623
+
1624
+ N√z
1625
+ �4
1626
+ 9 ˙q2 + 8kN 2q2
1627
+ 9z
1628
+ + 4k2N 2
1629
+
1630
+ + z
1631
+ 3
1632
+ 2
1633
+ � ˙φ2
1634
+ 2N − NV
1635
+ ��
1636
+ dt. (64)
1637
+ Canonical momenta may therefore be found as,
1638
+ pq =
1639
+
1640
+ N√z ˙q,
1641
+ pφ = −2f ′q√z + z
1642
+ 3
1643
+ 2
1644
+ N
1645
+ ˙φ,
1646
+ pN = 0 = pz.
1647
+ (65)
1648
+ As before, leaving out the constraint associate with the lapse function, and replacing ˙z = 2N
1649
+ 3 q in view of (13),
1650
+ the Hamiltonian may be cast as,
1651
+ NHnmMBL = pq ˙q + pz ˙z + pφ ˙φ − L
1652
+ = N
1653
+ � √z
1654
+ 16β p2
1655
+ q + 2
1656
+ 3qpz +
1657
+ p2
1658
+ φ
1659
+ 2z
1660
+ 3
1661
+ 2 + 2f ′
1662
+ z qpφ + 2fq2
1663
+ 3√z + 2f ′2q2
1664
+ √z
1665
+ − 6kf√z − 8βkq2
1666
+ z
1667
+ 3
1668
+ 2
1669
+ − 36βk2
1670
+ √z
1671
+ + V z
1672
+ 3
1673
+ 2
1674
+
1675
+ .
1676
+ (66)
1677
+ Finally, applying the canonical transformation relations q → 3
1678
+ 2x, and pq → 2
1679
+ 3px, we obtain
1680
+ HnmMBL = xpz +
1681
+ √z
1682
+ 36β p2
1683
+ x +
1684
+ p2
1685
+ φ
1686
+ 2z
1687
+ 3
1688
+ 2 + 3x
1689
+ z f ′pφ + 3f
1690
+ 2√z (x2 − 4kz) − 18βk
1691
+ z
1692
+ 3
1693
+ 2
1694
+ (x2 + 2kz) + 9x2f ′2
1695
+ 2√z
1696
+ + V z
1697
+ 3
1698
+ 2 .
1699
+ (67)
1700
+ Clearly, HnmMBL ∼= HnmMHF presented in (32).
1701
+ 3.3
1702
+ Einstein-Gauss-Bonnet-Dilatonic action
1703
+ Eventually, in order to construct the correct Hamiltonian in connection with the Einstein-Gauss-Bonnet-Dilatonic
1704
+ action (35), let us integrate it by parts to obtain,
1705
+ A3 =
1706
+ � �
1707
+ α
1708
+
1709
+
1710
+ 3 ˙z2
1711
+ 2N√z + 6kN√z
1712
+
1713
+ +
1714
+
1715
+ N√z
1716
+ �� ¨z
1717
+ N −
1718
+ ˙N ˙z
1719
+ N 2
1720
+ �2
1721
+ + 2k ˙z2
1722
+ z
1723
+ + 4k2N 2�
1724
+ − γ′(φ) ˙z ˙φ
1725
+ N√z
1726
+ � ˙z2
1727
+ N 2z + 12k
1728
+
1729
+ + z
1730
+ 3
1731
+ 2
1732
+ � ˙φ2
1733
+ 2N − NV
1734
+ ��
1735
+ dt.
1736
+ (68)
1737
+ 13
1738
+
1739
+ As before, replacing ˙z by 2N
1740
+ 3 q in view of (13), the above action may be cast as,
1741
+ A3q =
1742
+ � �
1743
+ − 2
1744
+ 3αN q2
1745
+ √z + 6αkN√z +
1746
+
1747
+ N√z
1748
+ �4
1749
+ 9 ˙q2 + 8kN 2q2
1750
+ 9z
1751
+ + 4k2N 2
1752
+
1753
+ − 2qγ′(φ) ˙φ
1754
+ 3√z
1755
+ �4q2
1756
+ 9z + 12k
1757
+
1758
+ + z
1759
+ 3
1760
+ 2
1761
+ � ˙φ2
1762
+ 2N − NV
1763
+ � �
1764
+ dt.
1765
+ (69)
1766
+ Canonical momenta are now found as,
1767
+ pq =
1768
+
1769
+ N√z ˙q,
1770
+ pφ = −2qγ′(φ)
1771
+ 3√z
1772
+ �4q2
1773
+ 9z + 12k
1774
+
1775
+ + z
1776
+ 3
1777
+ 2
1778
+ N
1779
+ ˙φ,
1780
+ pN = 0 = pz.
1781
+ (70)
1782
+ As always, leaving out the constraint associated with the lapse function, and replacing ˙z = 2N
1783
+ 3 q in view of (13),
1784
+ the Hamiltonian may be cast as,
1785
+ NHGB
1786
+ MBL =pq ˙q + pz ˙z + pφ ˙φ − L
1787
+ = N
1788
+ � √z
1789
+ 16β p2
1790
+ q + 2
1791
+ 3qpz +
1792
+ p2
1793
+ φ
1794
+ 2z
1795
+ 3
1796
+ 2 + 2αq2
1797
+ 3√z − 6kα√z − 8βkq2
1798
+ z
1799
+ 3
1800
+ 2
1801
+ − 36βk2
1802
+ √z
1803
+ + 2qγ′(φ)pφ
1804
+ 3z2
1805
+ �4q2
1806
+ 9z + 12k
1807
+
1808
+ + 2q2γ′2(φ)
1809
+ 9z
1810
+ 5
1811
+ 2
1812
+ �4q2
1813
+ 9z + 12k
1814
+ �2
1815
+ + V z
1816
+ 3
1817
+ 2
1818
+
1819
+ ,
1820
+ (71)
1821
+ Finally, the set of canonical transformations q → 3
1822
+ 2x, and pq → 2
1823
+ 3px, allows one to express the Hamiltonian (71)
1824
+ as,
1825
+ HGBMBL =xpz +
1826
+ √z
1827
+ 36β p2
1828
+ x +
1829
+ p2
1830
+ φ
1831
+ 2z
1832
+ 3
1833
+ 2 + 3α
1834
+ 2√z (x2 − 4kz) − 18βk
1835
+ z
1836
+ 3
1837
+ 2
1838
+ (x2 + 2kz)
1839
+ + xγ′(φ)
1840
+ z2
1841
+ �x2
1842
+ z + 12k
1843
+
1844
+ pφ + γ′2(φ)x2
1845
+ 2z
1846
+ 5
1847
+ 2
1848
+ �x2
1849
+ z + 12k
1850
+ �2
1851
+ + V z
1852
+ 3
1853
+ 2 .
1854
+ (72)
1855
+ As a result one finds, HGBMBL ∼= HGBMHF presented in (47). It is important to note that in the process of
1856
+ constructing the Hamiltonian starting from a divergent free action, the pathology discussed in regard of canonical
1857
+ formulation of Einstein-Gauss-Bonnet-Dilatonic action in the presence of higher-order term is also removed.
1858
+ 4
1859
+ Application
1860
+ It is mentioned in the introduction that canonical formulation is a precursor to canonical quantization. In the
1861
+ absence of a viable quantum theory of gravity, it is suggestive to canonically quantize the cosmological equa-
1862
+ tion and study quantum cosmology to extract some ethos of pre-Planck era. For example, one can explore the
1863
+ Euclidean wormhole solution. Nonetheless, ‘cosmological inflationary scenario’ has been developed since 1980,
1864
+ to solve horizon, flatness (fine tuning), structure formation and monopole problems, singlehandedly. Short-lived
1865
+ (10−36 −10−26)s. inflation, occurred just after Planck’s era and falls within the periphery of ‘quantum field theory
1866
+ in curved space-time’. To be more specific, ‘inflation is a quantum theory of perturbations on the top of the
1867
+ classical background’, so that the energy scale of the background remains much below Planck’s scale. Nonetheless
1868
+ in this context, Hartle [73] prescribed that, most of the important physics may still be extracted from the classical
1869
+ action provided, the semiclassical wave-function is strongly peaked. The reason being, in that case correlation
1870
+ between the geometrical and matter degrees of freedom is established, and hence the emergence of classical trajec-
1871
+ tories (i.e. the universe) is expected. Hence, quantization and an appropriate semiclassical approximation must
1872
+ be treated as a forerunner to study inflation.
1873
+ Canonical quantization and the semiclassical wave-function in connection with the Hamiltonian (67) for non-
1874
+ minimally coupled higher order theory had been presented in [26], which reduces to the minimally coupled case
1875
+ 14
1876
+
1877
+ when the coupling parameter becomes constant [19]. The Hamiltonian operator was found to be hermitian, stan-
1878
+ dard probabilistic interpretation holds, and the semiclassical wave-functions was found to be oscillatory about the
1879
+ classical inflationary solution. Inflation has been studied and the parameters are found with excellent agreement
1880
+ with the observational constraints [74,75]. Gravitational perturbation has also been studied.
1881
+ In [29] again, the quantum counterpart of the Hamiltonian (72) in connection with Einstein-Gauss-Bonnet-
1882
+ Dilatonic coupled action has been presented.
1883
+ Hermiticity of the Hamiltonian operator has been established,
1884
+ probabilistic interpretation is explored, and the semiclassical wave-function is found to be oscillatory about a
1885
+ classical inflationary solution. Finally, we have studied inflation and found that the inflationary parameters more-
1886
+ or-less satisfy observational constraints [74,75]. In a nut-shell, the results obtained in [29] are the following.
1887
+ iℏ∂Ψ
1888
+ ∂σ =
1889
+
1890
+ − ℏ2φ
1891
+ 54β0x
1892
+ � ∂2
1893
+ ∂x2 + n
1894
+ x
1895
+
1896
+ ∂x
1897
+
1898
+
1899
+ ℏ2
1900
+ 3xσ
1901
+ 4
1902
+ 3
1903
+ ∂2
1904
+ ∂φ2 + 2iℏα0
1905
+ σ
1906
+ � 1
1907
+ φ2
1908
+
1909
+ ∂φ − 1
1910
+ φ3
1911
+
1912
+ − 2iℏγ0x2
1913
+
1914
+ 7
1915
+ 3
1916
+
1917
+ 2φ ∂
1918
+ ∂φ + 1
1919
+
1920
+ + Ve
1921
+
1922
+ Ψ
1923
+ = �HeΨ,
1924
+ (73)
1925
+ where, the proper volume, σ = z
1926
+ 3
1927
+ 2 = a3 plays the role of internal time parameter, and n is the operator ordering
1928
+ index. In the above equation, �He is the effective hermitian Hamiltonian operator, while the the effective potential
1929
+ Ve is given by,
1930
+ Ve = 3α2
1931
+ 0x
1932
+ σ
1933
+ 2
1934
+ 3 φ4 − 4α0γ0x3
1935
+ σ2φ
1936
+ + 4γ2
1937
+ 0x5φ2
1938
+
1939
+ 10
1940
+ 3
1941
+ + α0x
1942
+ σ
1943
+ 2
1944
+ 3 φ
1945
+ + λ2σ
1946
+ 2
1947
+ 3 φ2
1948
+ 3x
1949
+ + 2σ
1950
+ 2
1951
+ 3 ΛM 2
1952
+ P
1953
+ x
1954
+ .
1955
+ (74)
1956
+ The effective Hamiltonian operator is found to be hermitian for n = −1, which selects the operator ordering
1957
+ parameter from physical consideration. Standard quantum mechanical probability interpretation also holds. Under
1958
+ a suitable (WKB) semiclassical approximation, the wave-function has been found to be,
1959
+ Ψ = Ψ0e
1960
+ i
1961
+
1962
+
1963
+ − 6α0λz2
1964
+ a0φ0 +16γ0a2
1965
+ 0φ2
1966
+ 0λ3√z
1967
+
1968
+ ,
1969
+ (75)
1970
+ which exhibits oscillatory behaviour about the classical inflationary solution a = a0eλt , where, α0, φ0, γ0 are
1971
+ constants. We have also presented several sets of inflationary parameters in [29], which depict that the spectral
1972
+ index of scalar perturbation and the scalar to tensor ratio lie within the range 0.967 ≤ ns ≤ 0.979 and 0.056 ≤
1973
+ r ≤ 0.089 respectively, showing reasonably good agreement with the recently released data [74,75]. The number
1974
+ of e-folding also remains within the acceptable range 46 < N < 73, which is sufficient to solve the horizon and
1975
+ flatness problems.
1976
+ 5
1977
+ Concluding remarks
1978
+ Although initiated two centuries back, canonical formulation of higher-order theory of gravity is particularly
1979
+ non-trivial. In fact, only after probing Dilatonic coupled Gauss-Bonnet action, it is learnt that divergent terms
1980
+ play a vital role to formulate correct quantum dynamics of non-linear gravity theory. The scheme is therefore
1981
+ first, to express the action in terms of the basic variable hij , otherwise if expressed in terms of the scale factor,
1982
+ as commonly done, some unwanted divergent terms are removed in the process of integration by parts, which are
1983
+ unaccredited by the variational principle. Next, unless divergent terms are taken care of, the Hamiltonian is found
1984
+ to be different, which is related through canonical transformation though, such transformation cannot be carried
1985
+ over in the quantum domain due to non-linearity. It is shown that in the case of Einstein-Gauss-Bonnet-Dilatonic
1986
+ coupled action in 4-dimension that, unless the action is divergent free, an erroneous Hamiltonian is constructed,
1987
+ since it does not reflect the topological invariance of the theory. This proves the importance of divergent terms in
1988
+ higher order theories. In this respect the difference of BL formalism with MHF is apparent. In fact BL formalism
1989
+ produces identical Hamiltonian as obtained earlier following Ostrogradsky’s, Dirac’s or Horowitz’s formalisms.
1990
+ However, MHF is essentially the Horowitz formalism, after expressing the action in terms of the three space
1991
+ curvature and taking care of the total derivative terms under integration by parts. It was shown that following the
1992
+ same route if Dirac’s algorithm is applied, the Hamiltonian becomes identical to the one found following MHF,
1993
+ 15
1994
+
1995
+ and one obtains unique quantum description. Here, we reveal that the same is true with BL formalism. In fact,
1996
+ BL formalism not only bypasses constraint analysis, as in the case of Horowitz’s formalism, it also does not require
1997
+ auxiliary variable to cast the action in canonical form, which is a bit intricate. In a straightforward manner, it
1998
+ establishes diffeomorphic invariance, and therefore is the easiest technique to handle higher-order theories.
1999
+ References
2000
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2001
+ Acad. St. Petersbourg Ser. VI, 1850, 4, 385-517.
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1
+ Exploring the Use of WebAssembly in HPC
2
+ Mohak Chadha, Nils Krueger, Jophin John,
3
+ Anshul Jindal, Michael Gerndt
4
+ {firstname.lastname}@tum.de
5
+ Chair of Computer Architecture and Parallel Systems,
6
+ Technische Universität München, Germany
7
+ Shajulin Benedict
8
+ shajulin@iiitkottayam.ac.in
9
+ Department of Computer Science and Engg., Indian
10
+ Institute of Information Technology Kottayam, Kerala
11
+ Abstract
12
+ Containerization approaches based on namespaces offered
13
+ by the Linux kernel have seen an increasing popularity in the
14
+ HPC community both as a means to isolate applications and
15
+ as a format to package and distribute them. However, their
16
+ adoption and usage in HPC systems faces several challenges.
17
+ These include difficulties in unprivileged running and build-
18
+ ing of scientific application container images directly on HPC
19
+ resources, increasing heterogeneity of HPC architectures, and
20
+ access to specialized networking libraries available only on
21
+ HPC systems. These challenges of container-based HPC appli-
22
+ cation development closely align with the several advantages
23
+ that a new universal intermediate binary format called We-
24
+ bAssembly (Wasm) has to offer. These include a lightweight
25
+ userspace isolation mechanism and portability across oper-
26
+ ating systems and processor architectures. In this paper, we
27
+ explore the usage of Wasm as a distribution format for MPI-
28
+ based HPC applications. To this end, we present MPIWasm, a
29
+ novel Wasm embedder for MPI-based HPC applications that
30
+ enables high-performance execution of Wasm code, has low-
31
+ overhead for MPI calls, and supports high-performance net-
32
+ working interconnects present on HPC systems. We evaluate
33
+ the performance and overhead of MPIWasm on a production
34
+ HPC system and AWS Graviton2 nodes using standardized
35
+ HPC benchmarks. Results from our experiments demonstrate
36
+ that MPIWasm delivers competitive native application per-
37
+ formance across all scenarios. Moreover, we observe that
38
+ Wasm binaries are 139.5x smaller on average as compared
39
+ to the statically-linked binaries for the different standardized
40
+ benchmarks.
41
+ CCS Concepts: • Software and its engineering → Process
42
+ management.
43
+ Keywords: WebAssembly, Wasmer, Wasm, MPI, HPC
44
+ Permission to make digital or hard copies of all or part of this work for
45
+ personal or classroom use is granted without fee provided that copies are not
46
+ made or distributed for profit or commercial advantage and that copies bear
47
+ this notice and the full citation on the first page. Copyrights for components
48
+ of this work owned by others than ACM must be honored. Abstracting with
49
+ credit is permitted. To copy otherwise, or republish, to post on servers or to
50
+ redistribute to lists, requires prior specific permission and/or a fee. Request
51
+ permissions from permissions@acm.org.
52
+ PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
53
+ © 2023 Association for Computing Machinery.
54
+ ACM ISBN 979-8-4007-0015-6/23/02...$15.00
55
+ https://doi.org/10.1145/nnnnnnn.nnnnnnn
56
+ ACM Reference Format:
57
+ Mohak Chadha, Nils Krueger, Jophin John, Anshul Jindal, Michael
58
+ Gerndt and Shajulin Benedict. 2023. Exploring the Use of We-
59
+ bAssembly in HPC. In The 28th ACM SIGPLAN Annual Sympo-
60
+ sium on Principles and Practice of Parallel Programming (PPoPP
61
+ ’23), February 25-March 1, 2023, Montreal, QC, Canada. ACM,
62
+ New York, NY, USA, 16 pages. https://doi.org/10.1145/nnnnnnn.
63
+ nnnnnnn
64
+ 1
65
+ Introduction
66
+ Linux containers, due to their portability and high availability,
67
+ have become the de-facto standard for developing, testing,
68
+ and deploying a wide range of applications from enterprise
69
+ to web services in cloud environments [29]. This is because
70
+ containers enable users to package their application along
71
+ with its custom software dependencies as a single unit into
72
+ easy-to-deploy images. Motivated by their popularity in the
73
+ cloud, containers have also seen a growing interest in the HPC
74
+ community [30, 74, 90]. For HPC systems, containers provide
75
+ flexibility to users and allow them to define custom software
76
+ stacks, i.e., user-defined software stack (UDSS) for their large-
77
+ scale scientific applications. Moreover, they enable easy, reli-
78
+ able, and verifiable environments that can be reproduced in
79
+ the future. To this end, several HPC-focused containerization
80
+ solutions, such as Charliecloud [72], Shifter [49], Singular-
81
+ ity [58], Podman [48], and Sarus [33] have been introduced.
82
+ In contrast to previous approaches, this paper investigates us-
83
+ ing a new novel technology called WebAssembly (Wasm) [52],
84
+ dubbed as an alternative to Linux containers [81], for packag-
85
+ ing and distributing HPC applications.
86
+ Despite their increasing popularity, the adoption and us-
87
+ age of containers in HPC systems is still significantly lim-
88
+ ited [36]. This can be attributed to the several challenges
89
+ commonly faced by users in running and building container
90
+ images for their applications on HPC systems. For execut-
91
+ ing containers, most containerization solutions require root
92
+ privileges which are not possible for normal HPC users due
93
+ to shared filesystems and their UNIX permissions in HPC.
94
+ While HPC-focused containerization solutions such as Sin-
95
+ gularity [58] and Podman [48] support rootless-containers
96
+ through fakeroot [61], their current implementations do
97
+ not support distributed filesystems such as GPFS commonly
98
+ found on HPC systems [70, 80]. Moreover, as argued by [71],
99
+ building Open Container Initiative (OCI) [69] compliant con-
100
+ tainer images on HPC resources by unprivileged (normal)
101
+ arXiv:2301.03982v1 [cs.DC] 10 Jan 2023
102
+
103
+ PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
104
+ Mohak Chadha et al.
105
+ users where the applications will eventually run is signifi-
106
+ cantly hard and requires support from the supercomputing
107
+ center. This is because most container building solutions such
108
+ as Docker [43] also require root privileges. As a result, most
109
+ users use their own local systems for building/developing their
110
+ application container images and then transfer the built image
111
+ to a login/front-end node of an HPC system for execution.
112
+ However, this scenario leads to several problems in container-
113
+ based HPC application development. First, HPC nodes are
114
+ becoming more heterogeneous [11] with different processor
115
+ architectures such as x86_64 or aarch64 and have special-
116
+ ized accelerators such as GPUs. As application performance
117
+ is critical in HPC, compiling an application using the specific
118
+ microarchitectural features of a particular processor is signifi-
119
+ cantly important. While building container images for mul-
120
+ tiple platforms either by cross-compiling HPC applications
121
+ or by emulation with QEMU is possible with plugins such as
122
+ build-x [44], it is not widely supported by HPC application
123
+ build procedures and requires the presence of specific Linux
124
+ kernel features (binfmt_misc [62]). Moreover, testing and
125
+ developing HPC applications offers insights only on the target
126
+ system. In addition, most container images can range from
127
+ several MiBs to several GiBs. As a result, frequent network
128
+ transfers from the local to the HPC system can be cumber-
129
+ some. Second, building HPC applications requires access to
130
+ specialized networking libraries and licenses to compilers
131
+ that are not available on the local user systems. Finally, while
132
+ different containerization solutions have almost no impact on
133
+ the performance of the containerized application [72, 75, 84],
134
+ building high-performant HPC application container images
135
+ is non-trivial, involves a steep learning curve, and requires
136
+ knowledge about specific MPI library versions (e.g., Open-
137
+ MPI [10] 4.0) and high performance network interconnect
138
+ hardware (e.g., Intel OmniPath [4]) and libraries (e.g., Intel
139
+ Performance Scaled Messaging [5]) present on the target sys-
140
+ tem. These challenges of container-based HPC application
141
+ development closely align with the several advantages and
142
+ core problems that Wasm [52] aims to solve.
143
+ Wasm is a low-level, statically typed universal binary in-
144
+ struction format for memory-safe, sandboxed execution in a
145
+ virtual machine. It offers portability across modern proces-
146
+ sor architectures and operating systems, fast execution, and
147
+ a low-level memory model [52]. Although originally meant
148
+ for execution in Web browsers, due to its simplicity and gen-
149
+ erality, Wasm has seen widespread adoption and usage in
150
+ non-Web domains such as serverless computing [78], edge
151
+ computing [47, 53], and Internet of Things [51]. It does not
152
+ require garbage collection and is designed to be a universal
153
+ compilation target with mature support for programming lan-
154
+ guages with an LLVM [59] front-end such as C, C++, C#,
155
+ and Rust [32, 35, 40, 45, 77].
156
+ Figure 1 demonstrates a general workflow for using Wasm
157
+ in HPC. Developers can compile their HPC applications to
158
+ Wasm once on their local systems ahead-of-time (AoT) and
159
+ HPC Application
160
+ WebAssembly
161
+ x86_64
162
+ aarch64
163
+ WebAssembly embedder
164
+ Compile
165
+ Figure 1. An HPC application can be compiled to WebAssem-
166
+ bly and distributed to multiple platforms where it can be
167
+ executed efficiently by a supporting WebAssembly embedder.
168
+ distribute it across multiple platforms instead of distribut-
169
+ ing source code or building application containers. Typically,
170
+ Wasm binaries have a smaller size as compared to native
171
+ x86_64 binaries [52, 56, 91]. Following this, the resulting
172
+ binary can be executed on any platform using a standalone
173
+ Wasm embedder [52]. The Wasm embedder serves two major
174
+ purposes. First, it provides an isolated execution environ-
175
+ ment for running a Wasm binary on a platform. In contrast to
176
+ container-based approaches that utilize different Linux names-
177
+ paces [71] for isolation and security, Wasm provides light-
178
+ weight isolation at the application level based on software
179
+ fault isolation (SFI) [85] and control flow integrity (§2.2).
180
+ Second, it is responsible for compiling Wasm binaries to
181
+ native machine code, either by using Just-in-Time (JIT) en-
182
+ gines at the time of execution, or AoT by using the same
183
+ JIT engines or AoT compilers. Note that, Wasm binaries
184
+ can be executed by normal users and are completely unpriv-
185
+ ileged. Several open-source standalone embedders such as
186
+ Wasmer [87], Wasmtime [37], and Wasm3 [79] are currently
187
+ available. However, none of them support the execution of
188
+ HPC applications.
189
+ As the first step towards bringing Wasm to the HPC ecosys-
190
+ tem, we only focus on MPI-based [6] HPC applications in
191
+ this paper. We chose MPI due to its understanding and in-
192
+ fluence in the HPC community [34]. Towards this, our key
193
+ contributions are:
194
+ • We implement and present MPIWasm, a novel Wasm
195
+ embedder for MPI-based HPC applications based on
196
+ Wasmer [87]. MPIWasm enables high performance exe-
197
+ cution of Wasm code, has low-overhead for MPI calls
198
+ through zero-copy memory operations, and supports
199
+ high-performance networking interconnects such as
200
+ Intel OmniPath [4].
201
+ • We demonstrate with extensive experiments the low-
202
+ overhead and performance of MPIWasm using standard-
203
+ ized HPC benchmarks on a production HPC system and
204
+ AWS Graviton2 [1] nodes based on the x86_64 and the
205
+ aarch64 architectures respectively.
206
+ • We elaborate on the different possible future directions
207
+ for using Wasm in the HPC ecosystem.
208
+
209
+ </>WA口口Exploring the Use of WebAssembly in HPC
210
+ PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
211
+ The rest of this paper is structured as follows. §2 provides a
212
+ detailed overview on Wasm. In §3, we describe our embedder
213
+ MPIWasm in detail. Our experimental results are presented
214
+ in §4. In §5, we describe the different possible directions
215
+ for using Wasm in the HPC ecosystem. §6 describes some
216
+ previous approaches related to our work. In §7, we conclude
217
+ the paper and present an outlook.
218
+ 2
219
+ A primer on WebAssembly
220
+ 2.1
221
+ WebAssembly Overview
222
+ WebAssembly (Wasm) was introduced in 2015 as an alter-
223
+ native to JavaScript for web-browser based applications. It
224
+ superseded asm.js [67], a previous attempt by Mozilla which
225
+ focused on a subset of Javascript code that can be optimized
226
+ AoT.
227
+ When an application is compiled to Wasm, the resulting
228
+ binary is called a module. Wasm modules contain function
229
+ definitions, declarations of global variables, tables, and a lin-
230
+ ear memory address space. All of the application code in
231
+ Wasm is organized in functions. The conceptual machine in
232
+ Wasm is stack-based and does not contain registers, there-
233
+ fore all instructions pop their operands from the stack of
234
+ the machine. However, since application control flow is an
235
+ explicit part of the module and Wasm operations are typed,
236
+ it is possible to statically predict the layout of the stack at
237
+ any point in the program which allows compilers to trans-
238
+ late the stack semantics to a register-based instruction set.
239
+ Similar to other higher-level programming languages, Wasm
240
+ allows the definition of global variables that are not scoped
241
+ to a specific function or block. Tables in Wasm modules are
242
+ used for storing references to functions [52]. The Wasm ISA
243
+ currently supports only four data types for variables: (i) i32,
244
+ 32-bit integers, (ii) i64, 64-bit integers, (iii) f32, 32-bit IEEE
245
+ 754 floating point numbers, and (iv) f64 64-bit IEEE 754
246
+ floating point numbers. For constructing, complex types a
247
+ combination of these basic types is commonly used.
248
+ Wasm provides the capability for data and code to be shared
249
+ between the module and its embedder using the import/export
250
+ system. All of the function definitions that can occur in a
251
+ Wasm module can be imported from the embedder instead of
252
+ being defined within it. Similarly, function definitions that are
253
+ present in the module can be exported so that the embedder
254
+ can utilize them (§2.3).
255
+ 2.2
256
+ WebAssembly Security and Sandboxing Model
257
+ Wasm utilizes software fault isolation techniques (SFI) [85]
258
+ to sandbox the executing Wasm module. By default, a Wasm
259
+ module cannot interact with the host system or perform I/O
260
+ operations of any kind. Any system interaction that is to be
261
+ initiated by the Wasm module’s code must be done through
262
+ the functions imported from the embedder (§2.1). As a result,
263
+ the embedder can act both as a translation layer and as an
264
+ arbiter to enforce isolation requirements. As a translator, it
265
+ 1
266
+ (type (;1;) (func (param i32) (result i32)))
267
+ 2
268
+ ...
269
+ 3
270
+ (type (;5;) (func (param i32 i32) (result i32)))
271
+ 4
272
+ ...
273
+ 5
274
+ (type (;14;) (func (param i32 i32 i32 i32 i32 i32)
275
+ 6
276
+ (result i32)))
277
+ 7
278
+ (type (;15;) (func (param i32 i32 i32 i32) (result i32 )))
279
+ 8
280
+ ...
281
+ 9
282
+ (import "wasi_snapshot_preview1" "path_open"
283
+ 10
284
+ (func $__wasi_path_open (type 22)))
285
+ 11
286
+ (import "wasi_snapshot_preview1" "fd_close"
287
+ 12
288
+ (func $__wasi_fd_close (type 1)))
289
+ 13
290
+ (import "wasi_snapshot_preview1" "fd_seek"
291
+ 14
292
+ (func $__wasi_fd_seek (type 23)))
293
+ 15
294
+ (import "wasi_snapshot_preview1" "fd_read"
295
+ 16
296
+ (func $__wasi_fd_read (type 15)))
297
+ 17
298
+ (import "wasi_snapshot_preview1" "proc_exit"
299
+ 18
300
+ (func $__wasi_proc_exit (type 0)))
301
+ 19
302
+ ...
303
+ 20
304
+ (export "_start" (func $_start ))
305
+ 21
306
+ (export "memory" (memory 0))
307
+ Listing 1. Example representation of a compiled C++ appli-
308
+ cation’s Wasm module using the WASI-SDK in WebAssembly
309
+ text format (WAT) [68]. Ellipses signify sections that are
310
+ omitted for brevity.
311
+ is possible for the embedder to provide a common interface
312
+ to the Wasm module even though the underlying system may
313
+ have different native interfaces, while as an arbiter it is possi-
314
+ ble for the embedder to restrict access of the Wasm module
315
+ to system resources based on an application-level security
316
+ policy. For instance, it is possible for the embedder to allow
317
+ file I/O only to files that reside in a specific directory to iso-
318
+ late the Wasm module from the rest of the filesystem. While
319
+ in principle similar to kernel-level system call filtering tech-
320
+ niques such as Seccomp-BPF [83] on Linux, performing such
321
+ filtering on the application level allows to define semantically
322
+ more meaningful policies.
323
+ In Wasm, all memory access is confined to a module’s lin-
324
+ ear memory which is separate from the code space. Currently,
325
+ the Wasm specification [52] supports 32-bit addresses to in-
326
+ dex the memory that a module has access to. While this limits
327
+ a single module’s memory to 4GiB, it also enables hardware
328
+ accelerated bound checks of memory accesses at runtime [42].
329
+ If an embedder is a process with a 64-bit memory address
330
+ space, it can safely execute an untrusted Wasm module in its
331
+ memory space without requiring additional isolation by re-
332
+ serving a continuous range of virtual memory for the module
333
+ to use. Not all pages in this range need to be mapped to phys-
334
+ ical memory, it is sufficient to only map the required number
335
+ of pages to fit the amount of memory used by the module at
336
+ a given point in time. This ensures that a Wasm module can
337
+ only operate in its own execution environment and cannot
338
+ corrupt the memory of the embedder, since any out-of-bounds
339
+ memory access will result in a page fault which can then be
340
+ handled by it. Moreover, since the memory instructions in
341
+ Wasm’s specification [52] work with offsets, it is not possible
342
+ to read and write to arbitrary memory locations in Wasm.
343
+ In the assembly produced by C programs, where a func-
344
+ tion call is expressed as a jump instruction to the address of
345
+ the function’s first instruction, a typical exploit is to change
346
+ this address to take control of the program’s control flow.
347
+
348
+ PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
349
+ Mohak Chadha et al.
350
+ 1
351
+ typedef int MPI_Comm;
352
+ 2
353
+ typedef int MPI_Datatype;
354
+ 3
355
+ ...
356
+ 4
357
+ int MPI_Init(int* argc , char *** argv);
358
+ 5
359
+ int MPI_Finalize(void);
360
+ 6
361
+ int MPI_Send(
362
+ 7
363
+ const void* buf , int count , MPI_Datatype datatype ,
364
+ 8
365
+ int dest , int tag , MPI_Comm comm
366
+ 9
367
+ );
368
+ 10
369
+ int MPI_Recv(
370
+ 11
371
+ void* buf , int count , MPI_Datatype datatype ,
372
+ 12
373
+ int source , int tag , MPI_Comm comm , MPI_Status* status
374
+ 13
375
+ );
376
+ Listing 2. Excerpt of the custom MPIWasm mpi.h header
377
+ file.
378
+ However, such exploits are not possible with Wasm since it
379
+ features control flow integrity by enforcing structured pro-
380
+ gram control flow. This is because of two reasons. First, in
381
+ Wasm, a function is represented as an index in a table (§2.1)
382
+ which adds an additional level of indirection to express the
383
+ function address. Second, the Wasm specification prevents
384
+ constructing arbitrary memory addresses [42] and the sepa-
385
+ ration of the embedder and the module’s memory prevents
386
+ overwriting function instructions.
387
+ 2.3
388
+ WebAssembly System Interface
389
+ Since Wasm was originally designed for web browsers, a sys-
390
+ tem interface that targets POSIX environments and enables
391
+ execution of Wasm modules on them was not part of the orig-
392
+ inal specification [52]. To overcome this, the WebAssembly
393
+ System Interface (WASI) specification [89] was designed.
394
+ WASI specifies the interface an embedder needs to implement
395
+ to execute most POSIX applications. Embedders that imple-
396
+ ment the WASI specification will be able to run any generic
397
+ application compiled with the WASI-SDK [28]. The WASI-SDK
398
+ includes the clang compiler and its own C library based on
399
+ musl libc that call WASI systemcalls imported from the em-
400
+ bedder instead of relying on Linux systemcalls [22]. Note that,
401
+ due to the ubiquity of glibc [26] on Linux systems, some ap-
402
+ plications have come to depend on glibc-specific functions or
403
+ behavior. Such applications will require modifications before
404
+ they can be compiled to a WASI-compliant Wasm module.
405
+ Listing 1 shows a compiled Wasm module of a C++ appli-
406
+ cation using the WASI-SDK in the WebAssembly text format
407
+ (WAT). WAT is a human readable format that enables de-
408
+ velopers to examine the source code of a Wasm module. It
409
+ can be observed that the module contains several functions
410
+ with integers as parameter and return types (Lines 1-7) (§2.1),
411
+ imports WASI functions (Lines 9-18), and exports its _start
412
+ (main function) and memory (Lines 20-21). Exporting these
413
+ two definitions allows the embedder that executes this module
414
+ to call its entrypoint function and to read from and write to
415
+ the module’s linear memory. While the import statements on
416
+ Lines 9-16 enable the Wasm module to open and read from
417
+ a file, the function proc_exit is used by the embedder to
418
+ handle the termination of the application, e.g., by deallocat-
419
+ ing the memory reserved for the module. For the module to
420
+ 1
421
+ (import "env" "MPI_Init" (
422
+ 2
423
+ func $MPI_Init (param i32 i32) (result i32)
424
+ 3
425
+ ))
426
+ 4
427
+ (import "env" "MPI_Finalize" (func $MPI_Finalize (result i32 )))
428
+ 5
429
+ (import "env" "MPI_Send" (
430
+ 6
431
+ func $MPI_Send (param i32 i32 i32 i32 i32 i32) (result i32)
432
+ 7
433
+ ))
434
+ 8
435
+ (import "env" "MPI_Recv" (
436
+ 9
437
+ func $MPI_Recv (param i32 i32 i32 i32 i32 i32 i32)
438
+ 10
439
+ (result i32)
440
+ 11
441
+ ))
442
+ Listing 3. WAT representation of module imports that corre-
443
+ spond to the functions shown in Listing 2.
444
+ execute, the imported functions need to be implemented by
445
+ the embedder.
446
+ 3
447
+ MPIWasm
448
+ In this section, we describe MPIWasm, our embedder for
449
+ executing Wasm modules that utilize functions from the MPI
450
+ standard in detail.
451
+ 3.1
452
+ Overview
453
+ The purpose of MPIWasm is to support the execution of MPI
454
+ applications compiled to Wasm on HPC systems. To facili-
455
+ tate its adoption and suitability in HPC environments, it (i)
456
+ supports high-performance execution of MPI-based HPC ap-
457
+ plications compiled to Wasm (§3.3), (ii) has low-overhead
458
+ for MPI calls through zero-copy memory operations (§3.6),
459
+ and (iii) supports high-performance interconnects such as
460
+ Infiniband [64] and Intel OmniPath [4]. These network inter-
461
+ connects are utilized by MPI libraries on HPC systems for
462
+ high-performance inter-rank communication. To enable the
463
+ immediate support for network interconnects present on mod-
464
+ ern HPC systems, MPIWasm links against the MPI library on
465
+ the target HPC system at runtime and provides a translation
466
+ layer between the Wasm module and the host1 MPI library.
467
+ As a result, the developer doesn’t need to be aware about
468
+ the particular networking libraries or network interconnects
469
+ present on the target HPC system. Depending on the partic-
470
+ ular host MPI library such as OpenMPI [10] or MPICH [8],
471
+ MPIWasm needs to be built separately. Both of these libraries
472
+ are currently supported by MPIWasm.
473
+ Our embedder currently supports the execution of MPI
474
+ applications written in C/C++ and conforming to the MPI-2.2
475
+ standard [65]. Integrating the support for MPI-3.1 [66] is of
476
+ our interest for the future but is out of scope for this work.
477
+ We chose to focus on C/C++ applications due to the stability
478
+ and maturity of the Wasm backend in the LLVM/Clang [59]
479
+ project since llvm-8. As the base for MPIWasm, we use the
480
+ open-source Wasm embedder called Wasmer [87]. Wasmer
481
+ supports the execution of Wasm modules on three major plat-
482
+ forms, i.e., Linux, Windows, and macOS, and supports both
483
+ x86_64 and aarch64 instruction set architectures. Moreover,
484
+ it implements the WASI specification (§2.3) and provides
485
+ 1We use the term target and host interchangeably for the system on which
486
+ the Wasm module is executing.
487
+
488
+ Exploring the Use of WebAssembly in HPC
489
+ PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
490
+ ergonomic mechanisms to define additional functions that
491
+ are provided to the module. This dynamic extension of the
492
+ embedder’s functionality enables the addition of MPI func-
493
+ tions to the functionality it provides to the Wasm module. For
494
+ implementing MPIWasm, we use the Rust programming lan-
495
+ guage. This is because of two reasons. First, it provides high
496
+ performance comparable to C/C++ with memory-safety [82].
497
+ Second, it has extensive support and documentation for em-
498
+ bedding Wasmer and using it as a library.
499
+ 3.2
500
+ Compiling C/C++ MPI applications to Wasm
501
+ Most MPI applications expect POSIX functionality to be
502
+ available in their execution environment, for instance the abil-
503
+ ity to read from and write to file descriptors. WASI (§2.3)
504
+ defines the WebAssembly exports that enable Wasm mod-
505
+ ules that target it to call most of the functions defined in the
506
+ C standard libraries shipped on POSIX systems. Towards
507
+ this, the WASI-SDK [28] combines the clang compiler and
508
+ the wasi-libc C library to enable the compilation of C/C++
509
+ applications that only make use of POSIX functions and no
510
+ additional libraries to Wasm. The compilation of C/C++ MPI
511
+ applications is not supported by the stock WASI-SDK. To this
512
+ end, we implement a custom mpi.h MPI header file and add
513
+ it to the WASI-SDK. The header file includes the definitions
514
+ for the different MPI types such as MPI_Op, MPI_Comm, and
515
+ MPI_Datatype and the definition for the MPI_Status struc-
516
+ ture. Moreover, it defines the signatures for the MPI functions
517
+ according to the MPI-2.2 [65] standard. An excerpt from the
518
+ header file is shown in Listing 2. It is a reduced version of
519
+ a traditional header file found with MPI libraries with most
520
+ types defined as integers (§3.6). By combining our header file
521
+ with the WASI-SDK, a C/C++ MPI application conforming to
522
+ the MPI-2.2 standard can be compiled to Wasm. Moreover,
523
+ to facilitate the ease-of-use and enable adoption, we imple-
524
+ ment a custom python-based tool that simplifies the entire
525
+ compilation process for MPI applications. Listing 3 shows
526
+ the different MPI-specific imports present in a Wasm module
527
+ corresponding to the functions shown in Listing 2. MPIWasm
528
+ provides definitions for these imports to enable the execution
529
+ of MPI-based HPC applications. In addition, it supports the
530
+ WASI specification which enables the POSIX functionality
531
+ for MPI applications.
532
+ 3.3
533
+ Executing Wasm Code with High Performance
534
+ There exist several strategies for executing Wasm modules.
535
+ These include using an interpreter [79], Ahead-of-Time (AoT)
536
+ compilation [37], or Just-in-Time (JIT) compilation [37].
537
+ However, for HPC systems the most useful approach is trans-
538
+ lating the Wasm instructions (Wasm ISA) to the native instruc-
539
+ tion set of the host machine before the application is executed,
540
+ i.e., AoT. Towards this, MPIwasm builds on the code genera-
541
+ tion infrastructure provided by Wasmer [87]. Wasmer currently
542
+ supports three compiler backends, i.e., Singlepass [86],
543
+ Cranelift [3], and LLVM [59]. The SinglePass compiler
544
+ Table 1. Comparing compile duration and performance for
545
+ the different compiler backends supported by Wasmer [87]
546
+ for the HPCG [73] Wasm module. The Wasm module was
547
+ generated using our WASI-SDK (§3.3). The Wasm module is
548
+ executed using MPIWasm on an x86_64 system.
549
+ Compiler
550
+ Compile Duration (ms)
551
+ Single-Core Performance (GFLOP/s)
552
+ Singlepass [86]
553
+ 52
554
+ 0.3769
555
+ Cranelift [3]
556
+ 150
557
+ 1.3240
558
+ LLVM [59]
559
+ 2811
560
+ 1.5426
561
+ is designed to emit machine code in linear time and does
562
+ not perform many code optimizations. The Cranelift com-
563
+ piler is completely based on Rust and is similar to LLVM.
564
+ With Cranelift, the WASM instructions are first translated
565
+ to the intermediate representation (IR) of Cranelift, i.e.,
566
+ (Cranelift-IR) which are then translated to the native in-
567
+ struction set of the host machine by taking microarchitecture-
568
+ specific optimizations into account. On the other hand, with
569
+ LLVM the Wasm ISA is first translated to LLVM-IR followed
570
+ by the generation of native machine code. Cranelift-IR is
571
+ similar to LLVM-IR but at a lower level of abstraction which
572
+ hinders mid-level code optimizations2. At the end of the com-
573
+ pilation process, all three compilers produce a shared ob-
574
+ ject, which can be loaded with a fast dlopen call using the
575
+ libloading library [27].
576
+ Table 1 shows a comparison of the compile-time and run-
577
+ time performance of the three different compilers supported
578
+ by Wasmer for the HPCG benchmark. While LLVM is the slow-
579
+ est to compile the Wasm module, it also results in the fastest
580
+ runtime performance for the HPCG application. As a result,
581
+ we chose LLVM as the compiler backend in MPIWasm. To
582
+ offset the longer compilation times required by LLVM as com-
583
+ pared to the other two compilers, we implement a caching
584
+ mechanism for the generated machine code. Our caching
585
+ mechanism builds on the FileSystemCache [46] provided
586
+ by Wasmer. In our implementation, we generate a hash for
587
+ each Wasm module using the Blake-3 hash function [2].
588
+ Moreover, we store the generated shared object from LLVM
589
+ as the generated hash in the local filesystem. As a result, any
590
+ changes to the Wasm module lead to the generation of a new
591
+ hash which triggers the recompilation of the module. To this
592
+ end, repeated execution of the same application on a system
593
+ with MPIWasm will not lead to recompilation overhead for
594
+ execution.
595
+ 3.4
596
+ Filesystem Isolation with MPIWasm
597
+ Since in Wasm all system interactions by the application have
598
+ to be performed by calling functions implemented by the
599
+ embedder (§2.2,§2.3), it enables the embedder to place addi-
600
+ tional restrictions on their use and to employ checks on the
601
+ arguments supplied to them. In Wasmer, all exported functions
602
+ 2A more detailed discussion between Cranelift-IR and LLVM-IR can be
603
+ found here [25].
604
+
605
+ PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
606
+ Mohak Chadha et al.
607
+ Figure 2. Memory address space of MPIWasm with an instan-
608
+ tiated Wasm module. All memory access instructions to the
609
+ Wasm module’s linear address space are given offsets relative
610
+ to the base address.
611
+ that handle file I/O perform their own permission handling
612
+ that is separate from the one employed by the OS. This in-
613
+ process indirection of filesystem accesses allows Wasmer to
614
+ present a virtual directory tree to the Wasm module that only
615
+ contains directories that the module is allowed to access. In
616
+ addition, access rights to individual directories can be more
617
+ restrictive than the permissions granted to the user that is exe-
618
+ cuting the embedder. For instance, a user can have read and
619
+ write access to their home directory and all of its subdirecto-
620
+ ries, but grant read-only access to one specific subdirectory
621
+ to a Wasm module executed by the embedder. MPIWasm ex-
622
+ poses this isolation functionality with its -d flag that grants
623
+ read-write access to the given directory to the Wasm module.
624
+ Note that the full absolute path to the exposed directories
625
+ is not presented to the Wasm module. In the virtual direc-
626
+ tory tree presented to it, all of the subdirectories it has been
627
+ given access to are direct children of the root directory. This
628
+ approach to mapping directory paths avoids exposure of in-
629
+ formation contained in the full path to the directories, such as
630
+ a username in the case of a home directory.
631
+ 3.5
632
+ Translating from Wasm to Host Memory Address
633
+ A major part of the Wasm security model is the separation
634
+ of the host and the module’s linear memory address space
635
+ (§2.2). Since it is the responsibility of the Wasm embedder to
636
+ uphold capability restrictions, protecting it’s data structures
637
+ from unintended or malicious access by the modules’ code is
638
+ significantly important. However, this separation presents a
639
+ challenge for supporting MPI applications, because the MPI
640
+ API is based on the library being able to read and write di-
641
+ rectly to the memory of the application. The executing Wasm-
642
+ based MPI application can only provide memory addresses
643
+ in its own linear memory address space, while the target MPI
644
+ library requires addresses in the host memory address space.
645
+ For executing Wasm modules, MPIWasm reserves a part
646
+ of its own address space for use by the Wasm module. As a
647
+ result, every byte contained in this range can be addressed
648
+ either with a memory address in the module’s memory space
649
+ or with a memory address in the embedder’s (host’s) memory
650
+ space. Moreover, while instantiating the module’s linear mem-
651
+ ory, MPIWasm records its base address. Following this, it is
652
+ possible to convert an address from the linear address space of
653
+ the Wasm module to the embedder’s address space and vice-
654
+ versa by treating the address in the linear address space as
655
+ an offset relative to the module’s base address. This is shown
656
+ in Figure 2. In particular, MPIWasm directly converts 32-bit
657
+ Wasm pointers that refer to the module’s linear address space
658
+ to 64-bit pointers that refer to the embedder’s address space
659
+ and vice-versa. To this end, MPIWasm directly utilizes the
660
+ MPI library present on the host system without copying any
661
+ data from the module’s address space to a different location,
662
+ i.e., it supports zero-copy memory operations.
663
+ Our mechanism for memory address translation does not
664
+ violate memory-safety because: (i) a malicious Wasm module
665
+ cannot violate control flow integrity (§2.2) and (ii) since the
666
+ size of the linear memory is always known, MPIWasm can
667
+ perform runtime bound checks for all memory accesses. As a
668
+ result, a module cannot access the memory of the embedder
669
+ or the memory of the underlying operating system unless
670
+ explicitly given access to it.
671
+ 3.6
672
+ Translating MPI Datatypes
673
+ MPI is implemented as a library with the most common being
674
+ OpenMPI [10], MPICH [8], and MVAPICH [9]. Hence, it
675
+ does not guarantee an Application Binary Interface (ABI)
676
+ and interoperability between libraries. This means that chang-
677
+ ing the MPI implementation requires recompilation of the
678
+ entire application code. One of the reasons for ABI incom-
679
+ patibility is that the MPI standard does not specify explicit
680
+ types for its datatypes such as MPI_Op and their implemen-
681
+ tation is completely up to the MPI library. However, since
682
+ Wasm modules are designed to be portable not just between
683
+ the different MPI libraries but also between different CPU
684
+ architectures, it becomes necessary to add an abstraction be-
685
+ tween the datatypes used by the host’s MPI library and the
686
+ datatypes exposed to the Wasm module by MPIWasm. An
687
+ abstraction is possible since most MPI datatypes such as
688
+ MPI_Comm, MPI_Datatype and MPI_Op are opaque to the ap-
689
+ plication and only used as arguments to MPI functions. MPI-
690
+ Wasm defines most MPI datatypes as 32-bit integers from
691
+ the perspective of the Wasm module (Listing 2) and transpar-
692
+ ently translates these datatypes to the host equivalents (§3.7).
693
+ We use integers as datatypes since MPIWasm internally uses
694
+ IDs to identify data structures that it creates on behalf of the
695
+ module in order to communicate with the host MPI library.
696
+ 3.7
697
+ Implementing MPI Functions in MPIWasm
698
+ Wasm imports are referred to by namespace and name of
699
+ the definition to import. By default, any symbols that are
700
+ not defined while compiling C/C++ applications to Wasm
701
+ will be resolved by making them imports of the module in
702
+ the env namespace. This is also demonstrated in Listing 3
703
+ with the function imports related to the MPI standard. MPI-
704
+ Wasm provides definitions for all these functions with the
705
+ same name as the original MPI function and exports them
706
+ in the env namespace. For implementing these functions, we
707
+
708
+ 0x0
709
+ OxFFFF_FFFFExploring the Use of WebAssembly in HPC
710
+ PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
711
+ combine the memory address and MPI datatype translations
712
+ as described in §3.5 and §3.6 respectively. Towards this, we
713
+ maintain a structure called Env that stores the global state
714
+ required by these translations. This structure includes infor-
715
+ mation about the memory allocated to the Wasm module, it’s
716
+ base pointer (§3.5) and information about the different used
717
+ datatypes such as MPI_Comm by the module. For directly utiliz-
718
+ ing the host MPI library, we use the project rsmpi [7] in MPI-
719
+ Wasm. rsmpi provides MPI bindings for Rust and supports
720
+ OpenMPI [10] and MPICH [8]. It utilizes the rust-bindgen
721
+ project to generate foreign function interfaces tailored to spe-
722
+ cific MPI libraries. Each MPI function in MPIWasm directly
723
+ calls the equivalent function in rsmpi with the appropriate
724
+ arguments.
725
+ While for most functions in the MPI-2.2 standard MPI-
726
+ Wasm directly defers the execution to the host MPI library,
727
+ the implementation of the MPI functions MPI_Alloc_mem and
728
+ MPI_Free_mem is done differently. With these functions, it is
729
+ possible to allocate memory for use with other MPI functions.
730
+ When a Wasm module calls MPI_Alloc_mem, it expects a 32-
731
+ bit memory address in the module’s address space, while call-
732
+ ing the MPI_Alloc_mem function of the host MPI library re-
733
+ turns a 64-bit memory address in the embedder’s memory ad-
734
+ dress space which is not inside the chunk of memory reserved
735
+ for the Wasm module. To overcome this, MPIWasm only
736
+ supports MPI_Alloc_mem and MPI_Free_mem if the Wasm
737
+ module defines and exports the functions malloc and free.
738
+ When MPI_Alloc_mem is called, MPIWasm simply invokes
739
+ the exported malloc and receives a suitable 32-bit module
740
+ memory address. This address can then be used as the return
741
+ value for MPI_Alloc_mem. We implement MPI_Free_mem in
742
+ a similar way.
743
+ 3.8
744
+ Limitations
745
+ The Wasm specification currently assumes little-endian byte
746
+ order for multi-byte values [52] in execution environments.
747
+ By giving direct access to the Wasm module’s memory to the
748
+ host MPI library, we assume that the byte order of values in
749
+ the module address space and embedder address space is the
750
+ same. As a result, MPIWasm does not support big-endian CPU
751
+ architectures. This is not a disadvantage since most processor
752
+ architectures in HPC systems are little-endian. Moreover, due
753
+ to the current linear 32-bit memory space for a Wasm module,
754
+ HPC applications compiled to Wasm cannot have more than
755
+ 4GiB of memory. The support for 64-bit memory addresses
756
+ is an important milestone for the Wasm specification and is
757
+ highlighted in the Wasm Memory64 proposal [20], but is out
758
+ of scope for this work.
759
+ 4
760
+ Experimental Results
761
+ In this section, we present performance results for our embed-
762
+ der MPIWasm across different processor architectures. For
763
+ 1
764
+ mpirun -np <number -of-processes > ./ mpiWasm mpi -app.wasm <args >
765
+ Listing 4. Executing MPI applications compiled to Wasm
766
+ with MPIWasm.
767
+ Table 2. Comparing the size of native dynamically-linked,
768
+ statically-linked, and Wasm binaries for the different MPI
769
+ applications. The native applications are compiled for the
770
+ x86_64 architecture.
771
+ Application
772
+ Native Size Dynamic (KiB)
773
+ Native Size Static (MiB)
774
+ Wasm Size (KiB)
775
+ Intel MPI Benchmarks [55].
776
+ 1087
777
+ 27
778
+ 893
779
+ HPCG [73].
780
+ 164
781
+ 26
782
+ 722
783
+ IOR [60].
784
+ 364
785
+ 16
786
+ 315.32
787
+ IS [31].
788
+ 36
789
+ 15
790
+ 57.88
791
+ DT [31].
792
+ 40
793
+ 15
794
+ 49.51
795
+ all our experiments, we follow best practices while reporting
796
+ results [54].
797
+ 4.1
798
+ System Description
799
+ For analyzing the performance of our implemented Wasm
800
+ embedder, we use two systems. First, a production HPC clus-
801
+ ter located at our institute, i.e., SuperMUC-NG. Second, an
802
+ AWS EC2 virtual machine (VM) instance with the Gravi-
803
+ ton2 processor [1]. Our HPC cluster contains eight islands
804
+ comprising a total of 6480 compute nodes based on the Intel
805
+ Skylake-SP architecture. Each compute node has two sockets,
806
+ comprising two Intel Xeon Platinum 8174 processors, with
807
+ 24 cores each and a total of 96GiB of main memory. The
808
+ nominal operating core frequency for each core is 3.10 GHz.
809
+ Hyper-Threading and Turbo Boost are disabled on the system.
810
+ The internal interconnect on our system is a fast Intel Omni-
811
+ Path [4] network with a bandwidth of 100 Gbit/s. Moreover,
812
+ our cluster provides a general parallel filesystem based on
813
+ the Lenovo DSS-G for IBM Spectrum Scale [19] with an
814
+ aggregate bandwidth of 200 GiB/s. For our experiments, we
815
+ use up to 128 nodes of the HPC system, i.e., 6144 cores. On
816
+ the other hand, the AWS Graviton2 processor based on the
817
+ 64-bit ARMv8-A Neoverse-N1 [24] architecture consists of 32
818
+ cores each with a nominal frequency of 2.50 GHz and a total
819
+ main memory of 64GiB. We limit our experiments to one
820
+ node for the Graviton2 processor.
821
+ 4.2
822
+ HPC Benchmarks
823
+ For our experiments with MPIWasm, we use the Intel MPI
824
+ Benchmarks [55], two benchmarks from the the NASA Ad-
825
+ vanced Supercomputing (NAS) Parallel Benchmark (NPB)
826
+ suite [31], the IOR benchmark [60], and the High Perfor-
827
+ mance Compute Gradient (HPCG) benchmark [73].
828
+ The Intel MPI benchmarks perform a set of MPI perfor-
829
+ mance measurements for point-to-point and global communi-
830
+ cation operations for a range of message sizes. We use them
831
+ since they characterize the performance of a cluster and are
832
+ an indication of the efficiency of the used MPI implementa-
833
+ tion. The NPB suite includes a set of benchmarks that aim
834
+ to evaluate the overall performance of HPC clusters. Due to
835
+ the support for compiling Fortran to Wasm being in the early
836
+ stages (§5), only the Integer Sort (IS) and Data Transfer (DT)
837
+
838
+ PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
839
+ Mohak Chadha et al.
840
+ 20
841
+ 22
842
+ 24
843
+ 26
844
+ 28
845
+ 210
846
+ 1
847
+ 1.5
848
+ Bytes
849
+ Iteration Time (usec)
850
+ PingPong (time) ≤ 1024 Bytes
851
+ Native
852
+ WASM
853
+ 212 214 216 218 220 222
854
+ 101
855
+ 102
856
+ Bytes
857
+ Iteration Time (usec)
858
+ PingPong (time) > 1024 Bytes
859
+ Native
860
+ WASM
861
+ (a) PingPong.
862
+ 20
863
+ 22
864
+ 24
865
+ 26
866
+ 28
867
+ 210
868
+ 2
869
+ 4
870
+ Bytes
871
+ Iteration Time (usec)
872
+ Sendrecv 6144 Ranks (time) ≤ 1024 Bytes
873
+ Native
874
+ WASM
875
+ 212 214 216 218 220 222
876
+ 101
877
+ 102
878
+ 103
879
+ 104
880
+ Bytes
881
+ Iteration Time (usec)
882
+ Sendrecv 6144 Ranks (time) > 1024 Bytes
883
+ Native
884
+ WASM
885
+ (b) SendRecv.
886
+ 20
887
+ 22
888
+ 24
889
+ 26
890
+ 28
891
+ 210
892
+ 0
893
+ 20
894
+ 40
895
+ Bytes
896
+ Iteration Time (usec)
897
+ Bcast 6144 Ranks (time) ≤ 1024 Bytes
898
+ Native
899
+ WASM
900
+ 212 214 216 218 220 222
901
+ 100
902
+ 102
903
+ 104
904
+ Bytes
905
+ Iteration Time (usec)
906
+ Bcast 6144 Ranks (time) > 1024 Bytes
907
+ Native
908
+ WASM
909
+ (c) Broadcast.
910
+ 22
911
+ 24
912
+ 26
913
+ 28
914
+ 210
915
+ 40
916
+ 60
917
+ Bytes
918
+ Iteration Time (usec)
919
+ Allreduce 6144 Ranks (time) ≤ 1024 Bytes
920
+ Native
921
+ WASM
922
+ 212 214 216 218 220 222
923
+ 102
924
+ 104
925
+ Bytes
926
+ Iteration Time (usec)
927
+ Allreduce 6144 Ranks (time) > 1024 Bytes
928
+ Native
929
+ WASM
930
+ (d) AllReduce.
931
+ 20
932
+ 22
933
+ 24
934
+ 26
935
+ 28
936
+ 210
937
+ 0
938
+ 1
939
+ 2
940
+ ·104
941
+ Bytes
942
+ Iteration Time (usec)
943
+ Allgather 6144 Ranks (time) ≤ 1024 Bytes
944
+ Native
945
+ WASM
946
+ 212
947
+ 214
948
+ 216 217
949
+ 105
950
+ 106
951
+ Bytes
952
+ Iteration Time (usec)
953
+ Allgather 6144 Ranks (time) > 1024 Bytes
954
+ Native
955
+ WASM
956
+ (e) AllGather.
957
+ 20
958
+ 22
959
+ 24
960
+ 26
961
+ 28
962
+ 210
963
+ 0
964
+ 1
965
+ 2
966
+ ·105
967
+ Bytes
968
+ Iteration Time (usec)
969
+ Alltoall 6144 Ranks (time) ≤ 1024 Bytes
970
+ Native
971
+ WASM
972
+ 212
973
+ 214
974
+ 216
975
+ 106
976
+ Bytes
977
+ Iteration Time (usec)
978
+ Alltoall 6144 Ranks (time) > 1024 Bytes
979
+ Native
980
+ WASM
981
+ (f) Alltoall.
982
+ 22
983
+ 24
984
+ 26
985
+ 28
986
+ 210
987
+ 0
988
+ 10
989
+ 20
990
+ 30
991
+ Bytes
992
+ Iteration Time (usec)
993
+ Reduce 768 Ranks (time) ≤ 1024 Bytes
994
+ Native
995
+ WASM
996
+ 212 214 216 218 220 222
997
+ 100
998
+ 102
999
+ 104
1000
+ Bytes
1001
+ Iteration Time (usec)
1002
+ Reduce 768 Ranks (time) > 1024 Bytes
1003
+ Native
1004
+ WASM
1005
+ 22
1006
+ 24
1007
+ 26
1008
+ 28
1009
+ 210
1010
+ 0
1011
+ 20
1012
+ 40
1013
+ 60
1014
+ Bytes
1015
+ Iteration Time (usec)
1016
+ Reduce 6144 Ranks (time) ≤ 1024 Bytes
1017
+ Native
1018
+ WASM
1019
+ 212 214 216 218 220 222
1020
+ 100
1021
+ 102
1022
+ 104
1023
+ Bytes
1024
+ Iteration Time (usec)
1025
+ Reduce 6144 Ranks (time) > 1024 Bytes
1026
+ Native
1027
+ WASM
1028
+ (g) Reduce.
1029
+ 20
1030
+ 22
1031
+ 24
1032
+ 26
1033
+ 28
1034
+ 210
1035
+ 0
1036
+ 50
1037
+ 100
1038
+ 150
1039
+ Bytes
1040
+ Iteration Time (usec)
1041
+ Gather 768 Ranks (time) ≤ 1024 Bytes
1042
+ Native
1043
+ WASM
1044
+ 212
1045
+ 214
1046
+ 216217218219220
1047
+ 100
1048
+ 102
1049
+ 104
1050
+ Bytes
1051
+ Iteration Time (usec)
1052
+ Gather 768 Ranks (time) > 1024 Bytes
1053
+ Native
1054
+ WASM
1055
+ 20
1056
+ 22
1057
+ 24
1058
+ 26
1059
+ 28
1060
+ 210
1061
+ 0
1062
+ 200
1063
+ 400
1064
+ 600
1065
+ Bytes
1066
+ Iteration Time (usec)
1067
+ Gather 6144 Ranks (time) ≤ 1024 Bytes
1068
+ Native
1069
+ WASM
1070
+ 212
1071
+ 214
1072
+ 216 217
1073
+ 100
1074
+ 102
1075
+ 104
1076
+ Bytes
1077
+ Iteration Time (usec)
1078
+ Gather 6144 Ranks (time) > 1024 Bytes
1079
+ Native
1080
+ WASM
1081
+ (h) Gather.
1082
+ 20
1083
+ 22
1084
+ 24
1085
+ 26
1086
+ 28
1087
+ 210
1088
+ 0
1089
+ 100
1090
+ 200
1091
+ 300
1092
+ Bytes
1093
+ Iteration Time (usec)
1094
+ Scatter 768 Ranks (time) ≤ 1024 Bytes
1095
+ Native
1096
+ WASM
1097
+ 212
1098
+ 214
1099
+ 216217218219220
1100
+ 101
1101
+ 103
1102
+ 105
1103
+ Bytes
1104
+ Iteration Time (usec)
1105
+ Scatter 768 Ranks (time) > 1024 Bytes
1106
+ Native
1107
+ WASM
1108
+ 20
1109
+ 22
1110
+ 24
1111
+ 26
1112
+ 28
1113
+ 210
1114
+ 0
1115
+ 1,000
1116
+ 2,000
1117
+ Bytes
1118
+ Iteration Time (usec)
1119
+ Scatter 6144 Ranks (time) ≤ 1024 Bytes
1120
+ Native
1121
+ WASM
1122
+ 212
1123
+ 214
1124
+ 216 217
1125
+ 101
1126
+ 103
1127
+ 105
1128
+ Bytes
1129
+ Iteration Time (usec)
1130
+ Scatter 6144 Ranks (time) > 1024 Bytes
1131
+ Native
1132
+ WASM
1133
+ (i) Scatter.
1134
+ Figure 3. Performance comparison of the Intel MPI benchmarks for MPIWasm and their native execution on our HPC system.
1135
+ 20
1136
+ 22
1137
+ 24
1138
+ 26
1139
+ 28
1140
+ 210
1141
+ 0.4
1142
+ 0.6
1143
+ 0.8
1144
+ 1
1145
+ Bytes
1146
+ Iteration Time (usec)
1147
+ PingPong (time) ≤ 1024 Bytes
1148
+ Native
1149
+ WASM
1150
+ 212 214 216 218 220 222
1151
+ 100
1152
+ 101
1153
+ 102
1154
+ Bytes
1155
+ Iteration Time (usec)
1156
+ PingPong (time) > 1024 Bytes
1157
+ Native
1158
+ WASM
1159
+ (a) PingPong.
1160
+ 20
1161
+ 22
1162
+ 24
1163
+ 26
1164
+ 28
1165
+ 210
1166
+ 0.5
1167
+ 1
1168
+ 1.5
1169
+ Bytes
1170
+ Iteration Time (usec)
1171
+ Sendrecv 32 Ranks (time) ≤ 1024 Bytes
1172
+ Native
1173
+ WASM
1174
+ 212 214 216 218 220 222
1175
+ 100
1176
+ 101
1177
+ 102
1178
+ 103
1179
+ Bytes
1180
+ Iteration Time (usec)
1181
+ Sendrecv 32 Ranks (time) > 1024 Bytes
1182
+ Native
1183
+ WASM
1184
+ (b) SendRecv.
1185
+ 22
1186
+ 24
1187
+ 26
1188
+ 28
1189
+ 210
1190
+ 2
1191
+ 4
1192
+ 6
1193
+ 8
1194
+ Bytes
1195
+ Iteration Time (usec)
1196
+ Allreduce 32 Ranks (time) ≤ 1024 Bytes
1197
+ Native
1198
+ WASM
1199
+ 212 214 216 218 220 222
1200
+ 101
1201
+ 102
1202
+ 103
1203
+ 104
1204
+ Bytes
1205
+ Iteration Time (usec)
1206
+ Allreduce 32 Ranks (time) > 1024 Bytes
1207
+ Native
1208
+ WASM
1209
+ (c) AllReduce.
1210
+ 20
1211
+ 22
1212
+ 24
1213
+ 26
1214
+ 28
1215
+ 210
1216
+ 0
1217
+ 10
1218
+ 20
1219
+ Bytes
1220
+ Iteration Time (usec)
1221
+ Allgather 32 Ranks (time) ≤ 1024 Bytes
1222
+ Native
1223
+ WASM
1224
+ 212 214 216 218 220 222
1225
+ 102
1226
+ 103
1227
+ 104
1228
+ 105
1229
+ Bytes
1230
+ Iteration Time (usec)
1231
+ Allgather 32 Ranks (time) > 1024 Bytes
1232
+ Native
1233
+ WASM
1234
+ (d) AllGather.
1235
+ 20
1236
+ 22
1237
+ 24
1238
+ 26
1239
+ 28
1240
+ 210
1241
+ 20
1242
+ 40
1243
+ Bytes
1244
+ Iteration Time (usec)
1245
+ Alltoall 32 Ranks (time) ≤ 1024 Bytes
1246
+ Native
1247
+ WASM
1248
+ 212 214 216 218 220 222
1249
+ 102
1250
+ 104
1251
+ Bytes
1252
+ Iteration Time (usec)
1253
+ Alltoall 32 Ranks (time) > 1024 Bytes
1254
+ Native
1255
+ WASM
1256
+ (e) Alltoall.
1257
+ 12 4
1258
+ 8
1259
+ 16
1260
+ 32
1261
+ 0
1262
+ 10
1263
+ 20
1264
+ Ranks
1265
+ GFLOP/s
1266
+ HPCG GFLOPS
1267
+ Native
1268
+ WASM
1269
+ 12 4
1270
+ 8
1271
+ 16
1272
+ 32
1273
+ 0
1274
+ 50
1275
+ 100
1276
+ 150
1277
+ Ranks
1278
+ GB/s
1279
+ HPCG Bandwidth
1280
+ Native
1281
+ WASM
1282
+ (f) HPCG
1283
+ Figure 4. Performance comparison of selected Intel MPI benchmarks and HPCG for MPIWasm against their native execution on
1284
+ the AWS Graviton2 Processor.
1285
+ benchmarks from this suite were used since they are written
1286
+ in pure C. The IS benchmark performs bucketed parallel sort-
1287
+ ing of integers across all participating processes, while the
1288
+ DT benchmark tests the communication and the performance
1289
+ of 64-bit floating point operations of a HPC cluster by send-
1290
+ ing data through a topology of nodes. We use the topologies
1291
+ Black-Hole (bh), White-Hole (wh), and Shuffle (sh) for
1292
+ the DT benchmark. For our experiments, we use the classes
1293
+ C and B for the IS and DT benchmarks respectively. The IOR
1294
+ Benchmark measures the filesystem I/O performance avail-
1295
+ able to MPI processes. It supports multiple backends that
1296
+ utilize different APIs to perform system I/O. For our experi-
1297
+ ments with MPIWasm, we use the POSIX API backend since
1298
+ the POSIX filesystem APIs are included in the WASI specifi-
1299
+ cation (§2.3, §3.2). The HPCG benchmark aims to evaluate
1300
+ the real-world performance of HPC systems by solving a sys-
1301
+ tem of linear equations with the conjugate gradient method.
1302
+ For our experiments, we use the default available problem
1303
+
1304
+ Exploring the Use of WebAssembly in HPC
1305
+ PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
1306
+ size for HPCG. Note that, in our experiments we use the ver-
1307
+ sions 2019 Update 6 and 3.3.1 for the Intel MPI and NAS
1308
+ parallel benchmarks respectively.
1309
+ 4.3
1310
+ Experiment Setup
1311
+ For all our experiments, we execute the benchmarks in a pure-
1312
+ MPI configuration without shared memory parallelization
1313
+ with OpenMP, as it is currently not supported by MPIWasm.
1314
+ We use OpenMPI-4.0 as the MPI library since it is available
1315
+ on our HPC system and can be easily installed on the AWS
1316
+ Graviton2 nodes. For compiling the native applications on
1317
+ our HPC system (§4.1), we use the clang-11 compiler, while
1318
+ for the AWS Graviton2 node, we use the gcc7.1 compiler.
1319
+ In both cases, the applications were compiled with the -O3
1320
+ optimization flag. For compiling the different benchmarks
1321
+ to Wasm, we use the clang-11 compiler along with our cus-
1322
+ tomized WASI-SDK with -O3 -msimd128 flags for both test
1323
+ systems (§3.2). The -msimd128 flag enables the generation
1324
+ of SIMD instructions in Wasm. We compile the applications
1325
+ to Wasm only once on our local systems and execute them
1326
+ directly with MPIWasm on the different test systems. We
1327
+ build MPIWasm on our local system for the different plat-
1328
+ forms, i.e., x86_64 and aarch64 with OpenMPI to generate
1329
+ bindings for rsmpi (§3.7). Following this, we directly exe-
1330
+ cute the applications compiled to Wasm on the test systems
1331
+ as shown in Listing 4. Each MPI rank corresponds to one
1332
+ instance of the embedder with it’s own Wasm module. The
1333
+ native applications were executed directly using mpirun.
1334
+ 4.4
1335
+ Comparing Wasm Binary Size
1336
+ Table 2 shows the comparison between the absolute binary
1337
+ sizes for the different applications. The static versions of the
1338
+ binaries are generated by supplying the -static flag to the
1339
+ clang-11 compiler (§4.3) and linking the different applica-
1340
+ tions with the static versions of the required libraries such
1341
+ as libmpi.a, libopen-rte.a, and libz.a. To this end, we
1342
+ made necessary changes to the Make [50] and CMake [39]
1343
+ files used by the different applications (§4.2). While the stack-
1344
+ based instruction set and compact binary format give Wasm
1345
+ the potential to produce smaller binaries for the same appli-
1346
+ cations as compared to the native dynamically-linked bina-
1347
+ ries [52], three out of five applications that we used had a
1348
+ bigger binary size when compiled to WebAssembly in com-
1349
+ parison to the equivalent dynamically-linked native binary.
1350
+ While Wasm can benefit from a smaller representation on
1351
+ a function-by-function basis, in practice dynamically-linked
1352
+ native binaries can offset that advantage by being able to rely
1353
+ on commonly used libraries to be present on the system. For
1354
+ instance, a native binary can dynamically link against glibc,
1355
+ while a Wasm binary must statically include functions from
1356
+ wasi-libc (§2.3). However, in contrast to containers, Wasm
1357
+ binaries are significantly smaller making them more feasible
1358
+ for application distribution in HPC environments. In addi-
1359
+ tion, Wasm binaries are 139.5x smaller on average than the
1360
+ statically-linked binaries of the different applications. This is
1361
+ because the linker, i.e., lld copies all library routines from
1362
+ the different libraries used by an application into the binary
1363
+ during static linking.
1364
+ 4.5
1365
+ Benchmarking MPIWasm
1366
+ Figure 3 and Figure 4 show the iteration times for the different
1367
+ Intel MPI benchmarks for their native execution as compared
1368
+ to their execution with MPIWasm on our HPC system and
1369
+ the AWS Graviton2 processor respectively. For execution
1370
+ with MPIWasm, the iteration times don’t include the time
1371
+ required for compiling the Wasm modules to native machine
1372
+ code (§3.3). To avoid repetition, we omit some results for the
1373
+ Graviton2 processor. Error bars in the graphs represent mini-
1374
+ mum and maximum values for iteration timings as reported
1375
+ by the Intel MPI Benchmarks, while points in the graphs
1376
+ represent the average timings as reported by the benchmarks.
1377
+ For the PingPong benchmark using MPIWasm leads to a
1378
+ geometric mean (GM) average slowdown of 0.05x for the
1379
+ x86_64 system and a GM average speedup of 1.01x for the
1380
+ aarch64 system across all message sizes (Figures 3a, 4a). We
1381
+ calculate this value by dividing the metric t_avg_us reported
1382
+ by the benchmarks for their native execution by the value
1383
+ reported for execution with MPIWasm, followed by a GM of
1384
+ the obtained values. For computing slowdown, we subtract
1385
+ one from the obtained GM value. We observe a maximum
1386
+ bandwidth of 12.80 GiB/s and 10.98 GiB/s for the native exe-
1387
+ cution of the PingPong benchmark on the HPC and Graviton2
1388
+ processor respectively. On the other hand, with MPIWasm,
1389
+ we observe a maximum bandwidth of 13.44 GiB/s and 10.61
1390
+ GiB/s on the two systems. For the SendRecv benchmark,
1391
+ we observe a GM average slowdown of 0.06x and 0.07x
1392
+ with MPIWasm across all message sizes on the x86_64 and
1393
+ aarch64 systems respectively (Figures 3b, 4b). For the native
1394
+ version of the benchmark, we observe a maximum bandwidth
1395
+ of 7.24 GiB/s and 11.01 GiB/s on the two systems, while
1396
+ with MPIWasm, we observe a maximum bandwidth of 7.50
1397
+ GiB/s and 10.83 GiB/s. For the collective communication
1398
+ Broadcast routine, we observe an average GM slowdown of
1399
+ 0.13x with MPIWasm across all message sizes for 128 nodes
1400
+ as shown in Figure 3c. We observe an average GM slow-
1401
+ down of 0.06x and 0.10x with MPIWasm across all message
1402
+ sizes for the collective communication AllReduce routine
1403
+ as shown in Figures 3d and 4c. For AllGather with MPI-
1404
+ Wasm, we observe an average GM slowdown of 0.06x and
1405
+ 0.09x across all message sizes for the HPC system and Gravi-
1406
+ ton2 processor respectively (Figure 3e, 4d). Similarly, for the
1407
+ Alltoall collective communication routine, we observe an
1408
+ average GM slowdown of 0.10x for the two systems across all
1409
+ message sizes with MPIWasm as shown in Figures 3f and 4e.
1410
+ For 16 nodes of our HPC system, we observe an average GM
1411
+ slowdown of 0.12x, 0.14x, and 0.05x across message sizes
1412
+ for the routines Reduce, Gather, and Scatter as shown in
1413
+ Figures 3g, 3h, and 3i. On the other hand, for 128 nodes,
1414
+
1415
+ PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
1416
+ Mohak Chadha et al.
1417
+ 64128
1418
+ 256
1419
+ 512
1420
+ 1,024
1421
+ 0.4
1422
+ 0.6
1423
+ 0.8
1424
+ 1
1425
+ 1.2
1426
+ ·104
1427
+ Ranks
1428
+ Mop/s
1429
+ IS Total Operations/s
1430
+ Native
1431
+ WASM
1432
+ bh
1433
+ wh
1434
+ sh
1435
+ 0
1436
+ 500
1437
+ 1,000
1438
+ 1,500
1439
+ Topology
1440
+ MB/s
1441
+ DT Total Throughput
1442
+ Native
1443
+ WASM w/o SIMD
1444
+ WASM w SIMD
1445
+ (a) NPB.
1446
+ 1
1447
+ 4
1448
+ 8
1449
+ 12
1450
+ 16
1451
+ 2.4
1452
+ 2.6
1453
+ 2.8
1454
+ 3
1455
+ 3.2
1456
+ ·104
1457
+ Block Size (MiB)
1458
+ MiB/s
1459
+ IOR Bandwidth (Read)
1460
+ Native
1461
+ WASM
1462
+ 1
1463
+ 4
1464
+ 8
1465
+ 12
1466
+ 16
1467
+ 3
1468
+ 4
1469
+ ·104
1470
+ Block Size (MiB)
1471
+ MiB/s
1472
+ IOR Bandwidth (Write)
1473
+ Native
1474
+ WASM
1475
+ (b) IOR.
1476
+ 48 16
1477
+ 48
1478
+ 96
1479
+ 144
1480
+ 0
1481
+ 20
1482
+ 40
1483
+ 60
1484
+ 80
1485
+ Ranks
1486
+ GFLOP/s
1487
+ HPCG GFLOPS
1488
+ Native
1489
+ WASM
1490
+ 48 16
1491
+ 48
1492
+ 96
1493
+ 144
1494
+ 0
1495
+ 200
1496
+ 400
1497
+ 600
1498
+ Ranks
1499
+ GB/s
1500
+ HPCG Bandwidth
1501
+ Native
1502
+ WASM
1503
+ 192 768 1,536
1504
+ 3,072
1505
+ 6,144
1506
+ 0
1507
+ 2,000
1508
+ 4,000
1509
+ Ranks
1510
+ GFLOP/s
1511
+ HPCG GFLOPS
1512
+ Native
1513
+ WASM
1514
+ 192 768 1,536
1515
+ 3,072
1516
+ 6,144
1517
+ 0
1518
+ 1
1519
+ 2
1520
+ 3
1521
+ ·104
1522
+ Ranks
1523
+ GB/s
1524
+ HPCG Bandwidth
1525
+ Native
1526
+ WASM
1527
+ (c) HPCG.
1528
+ Figure 5. Performance comparison of standardized HPC benchmarks for MPIWasm against their native execution on our HPC
1529
+ system.
1530
+ we observe an average GM slowdown of 0.05x, 0.10x, and
1531
+ 0.08x for the three routines. The results for testing MPIWasm
1532
+ with the Intel MPI Benchmarks compiled to Wasm demon-
1533
+ strate that neither the mechanism for calling host functions
1534
+ in Wasmer [87] nor the translation layer implemented in MPI-
1535
+ Wasm induce significant overhead for MPI communication
1536
+ (§3.5,§3.6,§3.7). We expand on the translation overhead in
1537
+ MPIWasm in §4.6. Overall, our results indicate that MPI-
1538
+ Wasm delivers close to native performance for the different
1539
+ MPI routines on both x86_64 and aarch64 architectures.
1540
+ The performance of MPIWasm on our HPC system for
1541
+ the IS and DT benchmarks is shown in Figure 5a. For the IS
1542
+ benchmark with MPIWasm, we observe 8260 average mega
1543
+ operations per second across all processes as compared to
1544
+ 8546 average mega operations per second for the native exe-
1545
+ cution. For the DT benchmark with different topologies (§4.2),
1546
+ execution with MPIWasm leads to decreased throughput as
1547
+ compared to the native execution. The DT benchmark per-
1548
+ forms a significant number of pairwise comparison opera-
1549
+ tions which benefit greatly from vectorization with SIMD
1550
+ instructions. To demonstrate the effect of SIMD for the DT
1551
+ benchmark, we compile it to Wasm by disabling and enabling
1552
+ the generation of SIMD instructions. The Wasm version of the
1553
+ DT benchmark with SIMD leads to 1.36x better throughput
1554
+ as compared to the Wasm version without SIMD (Figure 5a).
1555
+ The difference in performance as compared to the native ver-
1556
+ sion of the DT benchmark can be attributed to the support
1557
+ for only 128-bit SIMD instructions in the Wasm specifica-
1558
+ tion [52] as compared to 512-bit SIMD instructions present
1559
+ in modern Intel processors [38, 57, 76] (§3.3). Support for
1560
+ higher-width SIMD in Wasm is an important milestone in its
1561
+ road-map but out of scope for this work (§5).
1562
+ Figure 5b shows total aggregated read and write band-
1563
+ width available to all MPI processes for the IOR benchmark.
1564
+ Points in the graph represent the average bandwidth reported
1565
+ by the benchmark, while error bars in the graph represent
1566
+ the maximum and minimum bandwidth observed over all
1567
+ iterations of the benchmark with the same block size. With
1568
+ four nodes of our HPC system, the upper bound for achiev-
1569
+ able bandwidth in our setup (§4.1) with IOR is 400 GBit/s
1570
+ (≈ 47684 MiB/s). With MPIWasm, we observe similar read
1571
+ (29411 MiB/s) and write (40206 MiB/s) bandwidth averaged
1572
+ across all block sizes as compared to the native execution
1573
+ of the benchmark. Testing the filesystem I/O performance
1574
+ of the MPIWasm demonstrates that the userspace permission
1575
+ handling and virtual directory tree implemented by Wasmer
1576
+ to provide filesystem isolation (§3.4) has no significant im-
1577
+ pact on the achievable bandwidth when performing I/O with
1578
+ the POSIX filesystem API. For the HPCG benchmark, we ob-
1579
+ serve similar performance when executed with MPIWasm as
1580
+ compared to it’s native execution on the HPC system and
1581
+ the Graviton2 processor up to 192 MPI processes (Figures 5c
1582
+ and 4f). On increasing the number of processes, the native exe-
1583
+ cution of the HPCG benchmark outperforms the execution with
1584
+ MPIWasm as shown in Figure 5c. For 6144 MPI processes, we
1585
+ observe a 14% reduction in GFLOP/s on execution with MPI-
1586
+ Wasm. This behavior can be attributed to the significantly fre-
1587
+ quent amount of communication required by the HPCG bench-
1588
+ mark [63]. HPCG repeatedly uses the Allreduce routine to
1589
+ reduce a single variable of size double over all MPI processes
1590
+ to finalize vector-vector dot operations. With increasing num-
1591
+ ber of processes, the number of times the Allreduce routine
1592
+ is called also increases. For instance, executing HPCG with
1593
+ 768 processes results in four times more calls to Allreduce
1594
+ as compared to the execution with 192 processes. As a re-
1595
+ sult, the repeated datatype translations in MPIWasm increase
1596
+ the cost for invoking the collective communication routine
1597
+ leading to performance degradation (§4.6).
1598
+ 4.6
1599
+ Analyzing Datatype Translation Overhead
1600
+ To measure the datatype translation overhead in MPIWasm,
1601
+ we implement a custom PingPong application that sends/re-
1602
+ ceives messages of varying sizes between two processes and
1603
+ iterates over the different MPI datatypes, i.e., BYTE, CHAR,
1604
+ INT, FLOAT, DOUBLE, and LONG. Following this, we instru-
1605
+ ment the Send routine in MPIWasm to determine the latency
1606
+ for translating the different datatypes. Finally, we execute the
1607
+ application on our HPC system. Figure 6 shows the trans-
1608
+ lation overhead for different datatypes and message sizes
1609
+ in MPIWasm. We observe an average overhead of 85.44ns,
1610
+
1611
+ Exploring the Use of WebAssembly in HPC
1612
+ PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
1613
+ 8
1614
+ 64
1615
+ 256
1616
+ 1024
1617
+ 32768
1618
+ 262144 1048576 2097152 4194304
1619
+ Bytes
1620
+ 0
1621
+ 50
1622
+ 100
1623
+ 150
1624
+ 200
1625
+ 250
1626
+ 300
1627
+ Translation Time (ns)
1628
+ MPI_BYTE
1629
+ MPI_CHAR
1630
+ MPI_INT
1631
+ MPI_FLOAT
1632
+ MPI_DOUBLE
1633
+ MPI_LONG
1634
+ Figure 6. Comparing the datatype translation overhead in
1635
+ MPIWasm.
1636
+ 84.72ns, 99.78ns, 96.32ns, 103.35ns, and 104.79ns across all
1637
+ message sizes for the MPI datatypes BYTE, CHAR, INT, FLOAT,
1638
+ DOUBLE, and LONG respectively. We observe an increase in the
1639
+ translation overhead for message sizes greater than 256KB.
1640
+ This can be attributed to an increased latency for acquiring
1641
+ read locks from the Env structure that maintains the global
1642
+ state for translations in MPIWasm (§3.7).
1643
+ 5
1644
+ Into the Future: Wasm and HPC
1645
+ In this section, we highlight and discuss the different exten-
1646
+ sions proposed by the Wasm community to the current Wasm
1647
+ specification [52] that can be implemented in an embedder
1648
+ for HPC applications to enhance performance and portability.
1649
+ Controlled Threading for Wasm modules. The Wasm
1650
+ Threads proposal [16] lays the foundation for utilizing Wasm
1651
+ for multithreaded algorithms. It enables Wasm modules to de-
1652
+ fine shared memories, informing the embedder that the mod-
1653
+ ule expects the memory to be accessed by multiple threads.
1654
+ To enable safe multithreaded access to shared memory, the
1655
+ proposal also defines atomic Wasm instructions that can be
1656
+ used to implement locks and atomic data structures in the
1657
+ functions of the module. To enable HPC applications to make
1658
+ use of the functionality added by the threads proposal, an API
1659
+ that allows Wasm modules to create additional threads on its
1660
+ own needs to be added to the embedder. Implementing the
1661
+ POSIX threads [15] and OpenMP [41] APIs in the embed-
1662
+ der would enable compatibility with the threading code in
1663
+ existing HPC applications.
1664
+ Wasm Extended SIMD. The current Wasm SIMD pro-
1665
+ posal [52] only specifies 128-bit SIMD instructions, while
1666
+ modern CPUs support higher-width-SIMD, for instance the
1667
+ AVX-512 instruction set extensions for x86_64, which speci-
1668
+ fies 512-bit SIMD instructions. Towards this, the Wasm Flex-
1669
+ ible Vector proposal [13] aims to provide support for SIMD
1670
+ instructions that are wider than 128-bit. Moreover, the Wasm
1671
+ relaxed SIMD instructions [21] aim to make it possible to uti-
1672
+ lize hardware SIMD instructions that are not well defined, i.e.,
1673
+ they differ in rounding behavior from the Wasm specification.
1674
+ Implementing these proposals in the embedder would allow
1675
+ compiled Wasm modules for HPC applications that contain
1676
+ vectorizable code to make better use of SIMD instructions
1677
+ available in modern CPU architectures.
1678
+ 20
1679
+ 22
1680
+ 24
1681
+ 26
1682
+ 28
1683
+ 210
1684
+ 2
1685
+ 4
1686
+ 6
1687
+ Bytes
1688
+ Iteration Time (usec)
1689
+ PingPong (time) ≤ 1024 Bytes
1690
+ MPIWasm
1691
+ Faasm
1692
+ 212 214 216 218 220 222
1693
+ 100
1694
+ 101
1695
+ 102
1696
+ 103
1697
+ Bytes
1698
+ Iteration Time (usec)
1699
+ PingPong (time) > 1024 Bytes
1700
+ MPIWasm
1701
+ Faasm
1702
+ Figure 7. Comparing the performance of MPIWasm and
1703
+ Faasm [78].
1704
+ Wasm PGAS. Partitioned Global Address Space (PGAS)
1705
+ is a programming model for parallel distributed memory ap-
1706
+ plications that introduces a memory address space that spans
1707
+ the local memory of multiple processes. With a memory ad-
1708
+ dress from this global address space a process can read from
1709
+ and write to the memory of other processes. Since Wasm
1710
+ already specifies the concept of defining and importing mem-
1711
+ ories, the embedder could be extended to provide non-local
1712
+ memory to the Wasm module. To support this use-case, the
1713
+ Wasm Multi-Memory proposal [14] needs to be implemented,
1714
+ which allows a Wasm module to define or import more than
1715
+ one memory.
1716
+ Dynamic Linking of Wasm Modules. While there is ex-
1717
+ isting work on establishing an ABI for dynamic linking be-
1718
+ tween Wasm modules [12], it has not been standardized yet.
1719
+ Supporting dynamic linking would significantly decrease the
1720
+ size of Wasm binaries for more complex applications as they
1721
+ would no longer need to statically link parts of wasi-libc.
1722
+ For HPC, it would enable commonly used libraries such as
1723
+ BLAS to be provided by MPIWasm. Combining dynamic link-
1724
+ ing with efforts to provide repositories for Wasm modules
1725
+ such as WAPM [88] could enable automatic dependency man-
1726
+ agement for Wasm applications.
1727
+ Compiling Fortran applications to Wasm Currently, the
1728
+ support for compiling Fortran-based applications to Wasm is
1729
+ very nascent with only one known attempt based on Dragon-
1730
+ egg [17]. However, the implementation of the Memory64
1731
+ proposal [20] should enable the usage of existing Fortran
1732
+ LLVM compilers such as F18 for easily compiling Fortran-
1733
+ based applications to Wasm [18].
1734
+ Wasm and Accelerators The module execution hints pro-
1735
+ posal [23] highlights the changes required in the Wasm speci-
1736
+ fication to enable the support for executing Wasm modules
1737
+ on hardware accelerators such as GPUs. Implementing the
1738
+ proposal in the embedder would enable compatibility with
1739
+ existing GPU-based HPC applications.
1740
+ 6
1741
+ Related Work
1742
+ Solutions for Packaging and distributing HPC applica-
1743
+ tions. Recently several HPC-focused tools such as Char-
1744
+ liecloud [72] and Singularity [58] have been introduced for
1745
+ distributing HPC applications through containerization. In
1746
+ contrast, we utilize the universal binary instruction format
1747
+ Wasm to package and distribute HPC applications. Moreover,
1748
+
1749
+ PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
1750
+ Mohak Chadha et al.
1751
+ while containerization requires building HPC application con-
1752
+ tainers for different platforms, HPC applications can be com-
1753
+ piled once to Wasm and executed on any platform using a
1754
+ supporting Wasm embedder.
1755
+ MPI and WebAssembly. To the best of our knowledge,
1756
+ Faasm [78] is the only compute platform that enables the exe-
1757
+ cution of MPI applications compiled to Wasm. It is based on
1758
+ a gRPC-based distributed messaging library called Faabric
1759
+ and contains a Wasm runtime, a workload scheduler, and
1760
+ a distributed state store. In order to run an application on
1761
+ Faasm, it needs to be compiled to Wasm and uploaded to the
1762
+ shared function storage. Following this, the application can
1763
+ be invoked using events such as HTTP requests. For support-
1764
+ ing MPI applications, Faasm implements only a subset of
1765
+ the MPI-1 specification on top of its messaging library and
1766
+ its own workload scheduler. Moreover, it also does not sup-
1767
+ port user-defined communicators required by the Intel MPI
1768
+ benchmarks (§4.1). In contrast, we take an inverted approach,
1769
+ where MPIWasm builds on top of existing native MPI libraries
1770
+ and provides a way for Wasm modules to call functions from
1771
+ them efficiently. Figure 7 shows the performance comparison
1772
+ between MPIWasm and Faasm for the PingPong benchmark
1773
+ (§4.2). With MPIWasm, we achieve a GM average speedup
1774
+ of 4.28x across all message sizes as compared to Faasm.
1775
+ 7
1776
+ Conclusion and Future Work
1777
+ In this paper, we took the first step towards bringing We-
1778
+ bAssembly to the HPC ecosystem and presented MPIWasm, a
1779
+ Wasm embedder that enables high performance execution of
1780
+ MPI applications compiled to Wasm across different proces-
1781
+ sor architectures. In the future, we plan to extend MPIWasm
1782
+ to support acceralators such as GPUs found on HPC sys-
1783
+ tems [23].
1784
+ 8
1785
+ Acknowledgement
1786
+ We thank our shepherd Milind Chabbi for his help in prepar-
1787
+ ing the final version of this paper. Furthermore, we thank
1788
+ the anonymous reviewers for their insightful comments and
1789
+ valuable feedback that significantly improved the quality of
1790
+ this paper. This work was supported by the funding of the
1791
+ German Federal Ministry of Education and Research (BMBF)
1792
+ in the scope of the Software Campus program.
1793
+ References
1794
+ [1] [n.d.]. AWS Graviton 2 Processors.
1795
+ https://aws.amazon.com/ec2/
1796
+ graviton/
1797
+ [2] [n.d.]. Blake-3 Hash function.
1798
+ https://github.com/BLAKE3-team/
1799
+ BLAKE3
1800
+ [3] [n.d.]. Cranelift Compiler.
1801
+ https://github.com/bytecodealliance/
1802
+ wasmtime/tree/main/cranelift
1803
+ [4] [n.d.]. Intel Omni-Path Fabric. https://www.intel.com/content/www/
1804
+ us/en/high-performance-computing-fabrics/omni-path-fabric-
1805
+ software-components.html
1806
+ [5] [n.d.]. Intel Performance Scaled Messaging 2. https://github.com/
1807
+ cornelisnetworks/opa-psm2/blob/master/README
1808
+ [6] [n.d.]. MPI: A Message-Passing Interface Standard. https://www.mpi-
1809
+ forum.org/docs/mpi-3.1/mpi31-report.pdf
1810
+ [7] [n.d.]. MPI bindings for Rust. https://github.com/rsmpi/rsmpi
1811
+ [8] [n.d.]. MPICH: High-Performance portable MPI. https://www.mpich.
1812
+ org/
1813
+ [9] [n.d.]. MVAPICH:MPI over InfiniBand, Omni-Path, Ethernet/iWARP,
1814
+ and RoCE. http://mvapich.cse.ohio-state.edu/
1815
+ [10] [n.d.]. OpenMPI: Open-Source High-Performance computing. https:
1816
+ //www.open-mpi.org/
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+ A
2125
+ Artifact Appendix
2126
+ A.1
2127
+ Description
2128
+ MPIWasm is an embedder for MPI-based HPC applications
2129
+ based on Wasmer [87]. It enables the high performance execu-
2130
+ tion of these applications compiled to WebAssembly (Wasm)
2131
+ and serves two purposes:
2132
+ 1. Delivering close to native application performance, i.e.,
2133
+ when applications are executed directly on a host ma-
2134
+ chine without using Wasm.
2135
+ 2. Enabling the distribution of MPI-based HPC applica-
2136
+ tions as Wasm binaries.
2137
+ Our artifact contains the source code for MPIWasm, toolchain
2138
+ for compiling C/C++ based MPI applications to Wasm, the
2139
+ Wasm binaries for the different standardized HPC bench-
2140
+ marks used in this paper, pre-built versions of our embedder
2141
+ for different operating systems, and scripts for parsing experi-
2142
+ ment data and generating plots. The artifact is available at:
2143
+ https://doi.org/10.5281/zenodo.7468121
2144
+ or
2145
+ https://github.com/kky-fury/MPIWasm
2146
+ A.2
2147
+ Getting Started
2148
+ For testing our Wasm embedder for executing MPI appli-
2149
+ cations compiled to WebAssembly, we provide a pre-built
2150
+ docker image for the linux/amd64 platform with all the re-
2151
+ quired dependencies.
2152
+ 1
2153
+ sudo
2154
+ docker
2155
+ run − i t
2156
+ kkyfury / ppoppae : v2
2157
+ / bin / bash
2158
+ 2
2159
+ #Executing the HPCG benchmark compiled to Wasm
2160
+ 3
2161
+ mpirun −−allow −run −as − r o o t −np 4
2162
+ . / t a r g e t / r e l e a s e / embedder
2163
+ \
2164
+ 4
2165
+ examples / xhpcg . wasm
2166
+ 5
2167
+ #Executing the IntelMPI benchmarks compiled to Wasm
2168
+ 6
2169
+ mpirun −−allow −run −as − r o o t −np 4
2170
+ . / t a r g e t / r e l e a s e / embedder
2171
+ \
2172
+ 7
2173
+ examples / imb . wasm
2174
+ Listing 5. Getting started with MPIWasm.
2175
+ Towards this, a user can follow the steps described in Listing 5.
2176
+ Following this, MPIWasm should successfully execute the
2177
+ HPCG and IntelMPI benchmarks. We provide sample output
2178
+ for the two benchmarks in the provided artifact. The user can
2179
+ increase/decrease the number of processes (-np) for executing
2180
+ the benchmarks. However, depending on the system where
2181
+
2182
+ Exploring the Use of WebAssembly in HPC
2183
+ PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
2184
+ the container is executing, the user might need to provide the
2185
+ -oversubscribe flag to mpirun.
2186
+ A.3
2187
+ Running Experiments with MPIWasm
2188
+ This section describes how to run experiments with our em-
2189
+ bedder to obtain plots similar to the ones in this paper.
2190
+ A.3.1
2191
+ Running small-scale experiments. To run small-scale
2192
+ experiments inside the docker container, we provide an end-
2193
+ to-end script. This script:
2194
+ 1. Executes the HPCG, IS, and IntelMPI benchmarks for
2195
+ their native execution and when they are executed using
2196
+ MPIWasm.
2197
+ 2. Parses the obtained results and generates the relevant
2198
+ plots.
2199
+ 1
2200
+ sudo
2201
+ docker
2202
+ run − i t
2203
+ kkyfury / ppoppae : v2
2204
+ / bin / bash
2205
+ 2
2206
+ cd
2207
+ run_experiments
2208
+ 3
2209
+ . / runme . sh
2210
+ Listing 6. Running small-scale experiments with MPIWasm.
2211
+ For running the experiments, the user can follow the steps de-
2212
+ scribed in Listing 6. The script can take around 10-15 minutes
2213
+ to finish execution. After completion, you can see the gener-
2214
+ ated data in the run_experiments/experiment_data folder. The
2215
+ generated plots can be found in the run_experiments/Plots
2216
+ folder. We provide sample plots for the different benchmarks
2217
+ in our artifact. However, on executing benchmarks inside
2218
+ the container, the performance difference between the native
2219
+ execution of the application and MPIWasm can be around
2220
+ 8-12%.
2221
+ A.3.2
2222
+ Running large-scale experiments on an HPC sys-
2223
+ tem. For running large-scale experiments with our embedder,
2224
+ a user needs to do the following:
2225
+ 1. Build a version of the embedder for your HPC sys-
2226
+ tem depending on the particular architecture, operating
2227
+ system, and the MPI library on the system. MPIWasm
2228
+ currently supports the OpenMPI library with limited sup-
2229
+ port for MPICH and MVAPICH. We provide examples for
2230
+ building MPIWasm for different operating systems and
2231
+ architectures in our artifact.
2232
+ 2. Execute the MPI applications using the built embedder
2233
+ on the HPC system. This can be done via submitting
2234
+ jobs to a RJMS software on an HPC system such as
2235
+ SLURM [92]. We provide sample job scripts for our HPC
2236
+ system, i.e., SuperMUC-NG that uses SLURM in our
2237
+ artifact.
2238
+ 3. After executing the applications, the user can utilize
2239
+ the different parsers provided in our artifact to parse
2240
+ the benchmark data. Following this, the results can
2241
+ be visualized using the plotting helper provided in the
2242
+ artifact.
2243
+ A.4
2244
+ Compiling C/C++ applications to Wasm
2245
+ We have setup a docker container with the required depen-
2246
+ dencies for compiling different MPI applications conformant
2247
+ to the MPI-2.2 standard to Wasm. The artifact also includes
2248
+ HPCG, IntelMPI, and IS benchmarks as examples.
2249
+ 1
2250
+ sudo
2251
+ docker
2252
+ run − i t
2253
+ kkyfury / w a s i t o o l c h a i n : v1
2254
+ / bin / bash
2255
+ 2
2256
+ #Compiling HPCG
2257
+ 3
2258
+ cd
2259
+ / work / example / hpcg −benchmark
2260
+ 4
2261
+ . / wasi −cmake . sh
2262
+ 5
2263
+ cd cmake− build −wasi
2264
+ 6
2265
+ make
2266
+ Listing 7. Compiling applications to Wasm.
2267
+ Listing 7 describes the steps a user can follow for compiling
2268
+ the HPCG benchmark to Wasm. For steps to compile the other
2269
+ benchmarks to Wasm, please look at the base Readme.md
2270
+ file provided with the artifact. All the different applications
2271
+ compiled to Wasm that we used in this paper are also present
2272
+ in the artifact.
2273
+ A.5
2274
+ Using MPIWasm
2275
+ For detailed usage instructions, please look at the base Readme.md
2276
+ file provided with the artifact.
2277
+ A.6
2278
+ Modifying MPIWasm
2279
+ For modifying our embedder, we recommend using our pro-
2280
+ vided docker-compose file in the artifact. This docker-compose
2281
+ file mounts the volume with the embedder’s source code in-
2282
+ side the container. As a result, any changes to it’s source code
2283
+ will be reflected inside it. For our embedder, we currently
2284
+ support the following operating systems:
2285
+ 1. CentOS-8.2
2286
+ 2. Opensuse-15-1
2287
+ 3. Ubuntu-20-04
2288
+ 4. MacOS-monterey
2289
+ 1
2290
+ cd wasi −mpi− r s
2291
+ 2
2292
+ docker
2293
+ compose run
2294
+ centos −8−2
2295
+ 3
2296
+ cargo
2297
+ b u i l d −− r e l e a s e
2298
+ 4
2299
+ #After the build process , you can see the built embedder in
2300
+ 5
2301
+ #
2302
+ the /target/release/ folder.
2303
+ Listing 8. Building MPIWasm.
2304
+ Listing 8 describes the steps for building MPIWasm for
2305
+ CentOS-8.2 after modifications. The instructions for building
2306
+ the embedder for other operating systems are provided in the
2307
+ base Readme.md file inside the artifact. After the build process,
2308
+ the embedder can be copied to the user’s local filesystem
2309
+ using the docker-cp command as shown in Listing 9.
2310
+ 1
2311
+ docker
2312
+ cp < c o n t a i n e r −id > : / s / t a r g e t / r e l e a s e / embedder
2313
+ \
2314
+ 2
2315
+ < d e s t i n a t i o n −path −user − f i l e s y s t e m >
2316
+ Listing 9. Copying MPIWasm.
2317
+ We provide provide the base image dockerfiles for the dif-
2318
+ ferent supported operating systems inside the artifact. These
2319
+ example dockerfiles can be easily extended to support other
2320
+ different linux distributions.
2321
+
2322
+ PPoPP ’23, February 25-March 1, 2023, Montreal, QC, Canada
2323
+ Mohak Chadha et al.
2324
+ A.7
2325
+ Support for aarch64
2326
+ Our embedder also supports execution on linux/arm64 plat-
2327
+ forms. We provide pre-built versions of our embedder for
2328
+ arm64 for the different supported operating systems in the
2329
+ artifact.
2330
+ A.7.1
2331
+ Building images for aarch64. If the user is building
2332
+ the docker image on an x86_64 system then docker-buildx
2333
+ is required. Note that, in this case, building the image might
2334
+ take around 12 hours.
2335
+ 1
2336
+ sudo
2337
+ docker
2338
+ buildx
2339
+ c r e a t e
2340
+ −−name mybuilder −−use −− b o o t s t r a p
2341
+ 2
2342
+ cd wasi −mpi− r s / . g i t l a b / c i / images /
2343
+ 3
2344
+ sudo
2345
+ docker
2346
+ buildx
2347
+ b u i l d −−push −f
2348
+ ubuntu −20 −04. D o c k e r f i l e
2349
+ \
2350
+ 4
2351
+ −− p l at f or m
2352
+ l i n u x / arm64
2353
+ \
2354
+ 5
2355
+ − t
2356
+ kkyfury / ubuntumodifiedbase : v1
2357
+ .
2358
+ 6
2359
+ cd
2360
+ . . / . . / . . /
2361
+ 7
2362
+ sudo
2363
+ docker
2364
+ buildx
2365
+ b u i l d −−push −f
2366
+ D o c k e r f i l e
2367
+ \
2368
+ 8
2369
+ −− p l at f or m
2370
+ l i n u x / arm64
2371
+ \
2372
+ 9
2373
+ − t
2374
+ kkyfury / embedderarm : v1
2375
+ .
2376
+ Listing 10. Building MPIWasm for aarch64 on x86_64.
2377
+ Listing 10 describes the steps required for building MPI-
2378
+ Wasm for arm systems with docker-buildx. The user should
2379
+ change the docker image tags according to their docker reg-
2380
+ istry account, i.e., replace kkyfury with your registry user-
2381
+ name. On the other hand, if the user is using an aardch64
2382
+ system then please follow the instructions described in §A.6.
2383
+
69E2T4oBgHgl3EQfkweU/content/tmp_files/load_file.txt ADDED
The diff for this file is too large to render. See raw diff
 
69E4T4oBgHgl3EQfCAuy/content/tmp_files/2301.04857v1.pdf.txt ADDED
@@ -0,0 +1,1309 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ Neural Spline Search for Quantile Probabilistic Modeling
2
+ Ruoxi Sun1*, Chun-Liang Li1*, Sercan Ö. Arık1, Michael W. Dusenberry2, Chen-Yu Lee1, Tomas
3
+ Pfister 1
4
+ 1Google Cloud AI 2Google Research, Brain Team
5
+ {ruoxis, chunliang, soarik, dusenberrymw, chenyulee, tpfister}@google.com
6
+ Abstract
7
+ Accurate estimation of output quantiles is crucial in many use
8
+ cases, where it is desired to model the range of possibility.
9
+ Modeling target distribution at arbitrary quantile levels and at
10
+ arbitrary input attribute levels are important to offer a compre-
11
+ hensive picture of the data, and requires the quantile function
12
+ to be expressive enough. The quantile function describing the
13
+ target distribution using quantile levels is critical for quantile
14
+ regression. Althought various parametric forms for the distri-
15
+ butions (that the quantile function specifies) can be adopted,
16
+ an everlasting problem is selecting the most appropriate one
17
+ that can properly approximate the data distributions. In this
18
+ paper, we propose a non-parametric and data-driven approach,
19
+ Neural Spline Search (NSS), to represent the observed data
20
+ distribution without parametric assumptions. NSS is flexible
21
+ and expressive for modeling data distributions by transform-
22
+ ing the inputs with a series of monotonic spline regressions
23
+ guided by symbolic operators. We demonstrate that NSS out-
24
+ performs previous methods on synthetic, real-world regression
25
+ and time-series forecasting tasks.
26
+ Introduction
27
+ For many machine learning applications, modeling the pre-
28
+ diction intervals (e.g. estimating the ranges all individual
29
+ predictions observation fall), beyond point estimates, is cru-
30
+ cial (Salinas et al. 2020; Wen et al. 2017; Tagasovska and
31
+ Lopez-Paz 2019; Gasthaus et al. 2019; Pearce et al. 2018).
32
+ The prediction intervals can help with decision making for
33
+ retail sales optimization (Simchi-Levi et al. 2008), medi-
34
+ cal diagnoses (Begoli, Bhattacharya, and Kusnezov 2019;
35
+ Mhaskar, Pereverzyev, and van der Walt 2017; Jiang et al.
36
+ 2012), information safety (Smith, Dinev, and Xu 2011), fi-
37
+ nancial investment management (Engle 1982), robotics and
38
+ control (Buckman et al. 2018), autonomous transformation
39
+ (Xu et al. 2014) and many others.
40
+ To estimate prediction intervals, we would need to estimate
41
+ different levels of quantiles for the target distribution using
42
+ quantile regression (Koenker and Regression 2005; Wald-
43
+ mann 2018). A real-world challenge is to select the paramet-
44
+ ric forms of target distributions, which is specified by the
45
+ quantile function (also known as the inverse CDF function),
46
+ *These authors contributed equally.
47
+ Copyright © 2023, Association for the Advancement of Artificial
48
+ Intelligence (www.aaai.org). All rights reserved.
49
+ 2.0
50
+ 1.5
51
+ 1.0
52
+ 0.5
53
+ 0.0
54
+ 0.5
55
+ 1.0
56
+ 1.5
57
+ 2.0
58
+ X
59
+ 2
60
+ 1
61
+ 0
62
+ 1
63
+ 2
64
+ Y
65
+ X=X0
66
+ P(Y|X)
67
+ True
68
+ Quantile 25%
69
+ Quantile 75%
70
+ Figure 1: Modeling multiple quantiles at different
71
+ condition-levels with a universal quantile function. The
72
+ goal is to model target data distribution y at any arbitrary
73
+ quantile level and attribute level X, using one versatile quan-
74
+ tile function. Gray dots are observed data points, while green
75
+ and blue lines indicate 25% and 75% quantile levels. The
76
+ data distribution y varies at different levels of X, say variance
77
+ of y increases when X is away from zero. Red dots are data
78
+ points at X = X0, p(Y |X0)).
79
+ to properly align with observed data distribution. Different
80
+ choices for the target distribution (Gaussian, Poisson, Neg-
81
+ ative Binomial, Student-t etc.) may yield different quantile
82
+ predictions, and misalignment of the assumption with the real
83
+ distribution may hinder the performance of the model. There-
84
+ fore, such heuristic or empirical hand-picking based paramet-
85
+ ric assumptions for the distribution can be sub-optimal. An
86
+ approach based on learning from the data in an automated
87
+ way, would be highly desirable, from both foundational and
88
+ practical perspectives.
89
+ For learnable parametric modeling, one challenge is how to
90
+ model all quantiles for all input attributes level in a com-
91
+ putationally efficient way. First, modeling an any arbitrary
92
+ quantile, as opposed to a couple of pre-defined quantile levels,
93
+ offers a more comprehensive view on the target distribution,
94
+ and provides convenience to use the quantile model (e.g. no
95
+ need to re-train the model when quantiles at testing are dif-
96
+ ferent from the ones at training). Second, real-world data can
97
+ have complex distributions beyond what simple assumptions
98
+ can model. Fig. 1 shows different input attribute X levels
99
+ arXiv:2301.04857v1 [cs.AI] 12 Jan 2023
100
+
101
+ 0
102
+ 1
103
+ 2
104
+ 3
105
+ 4
106
+ 5
107
+ Y|X
108
+ 0.000
109
+ 0.005
110
+ 0.010
111
+ 0.015
112
+ 0.020
113
+ Probability Density
114
+ PDF(Y|x)
115
+ 0
116
+ 1
117
+ 2
118
+ 3
119
+ 4
120
+ 5
121
+ Y|X
122
+ 0.0
123
+ 0.2
124
+ 0.4
125
+ 0.6
126
+ 0.8
127
+ 1.0
128
+ Probability Density
129
+ CDF(Y|x)
130
+ Figure 2: An example target distribution with a complex
131
+ shape, in PDF and CDF space. Black lines are observed tar-
132
+ get distributions, in the form of mixture of the other three dis-
133
+ tributions shown with color. Fitting the black line accurately
134
+ would be extremely difficult for most of the commonly-used
135
+ single parametric splines, motivating for the use of learnable
136
+ spline family composed of multiple splines.
137
+ have different dependency dynamics with target y level (i.e.
138
+ the variance of y increases when X apart from 0). Fig. 2
139
+ shows that the observed distribution cannot trivially fit well
140
+ with one single distribution. Therefore, in order to model all
141
+ quantiles at all X, we need a quantile function with a com-
142
+ plexity that does not increase significantly with number of
143
+ input attributes and the number of quantiles. This necessitates
144
+ a versatile and highly-expressive quantile function.
145
+ There has been many efforts on improving various aspects
146
+ of quantile regression. Gasthaus et al. (2019) proposes linear
147
+ spline interpolation between knots in the inverse CDF space
148
+ to model the target distribution in time-series forecasting
149
+ setup. This is proposed to avoid the assumption on paramet-
150
+ ric form of the target distribution. Park et al. (2022) and Moon
151
+ et al. (2021) focus on learning a valid quantile function with-
152
+ out quantile crossing (e.g. quantiles violate monotonically
153
+ increasing property), via special design of the neural network
154
+ architecture or first-order inequality constraint optimization.
155
+ Despite being distribution agnostic, these approaches for de-
156
+ scribing the target distribution (specified by quantile function)
157
+ are restricted to one function family (e.g. linear spline), which
158
+ may limit the expressiveness to represent the target distribu-
159
+ tion. In this paper, with the goal of designing an expressive
160
+ quantile function for various quantiles and input levels, we
161
+ propose a data-driven approach Neural Spline Search (NSS),
162
+ which transforms the inputs with a series of monotonic spline
163
+ regressions guided by symbolic operators. The contributions
164
+ of our paper can be summarized as:
165
+ 1. We propose an efficient search space and mechanism to
166
+ find an expressive quantile function to model the data
167
+ distribution, avoiding specifying a parametric form of the
168
+ observed distribution as prior.
169
+ 2. We propose a novel approach to generate an expressive
170
+ quantile function using a combination of different distri-
171
+ butions and operators guided by symbolic operators.
172
+ 3. The proposed method can be incorporated into other tasks
173
+ (including but not limited to time series forecasting) as
174
+ their quantile function.
175
+ 4. We demonstrate significant accuracy improvements across
176
+ numerous regression or time series forecasting tasks. For
177
+ example, on UCI benchmarks, we show 3.5%-7.0% im-
178
+ provement compared to next best methods.
179
+ Related Work
180
+ Quantile regression is used to estimate the target distribu-
181
+ tion at different quantile levels. The α-quantile estimator
182
+ is the solution when minimizing quantile loss at level α
183
+ (Koenker and Bassett Jr 1978). Another quantile regression
184
+ related loss is continuous ranked probability score (CRPS)
185
+ (Gneiting and Raftery 2007), which is the averaging over all
186
+ quantile levels, instead of one single quantile.
187
+ Neural network quantile forecasting. To model sequential
188
+ dependency of time series, several forecasting models pro-
189
+ pose a hidden state-emission framework ((Salinas et al. 2020;
190
+ Wen et al. 2017; Gasthaus et al. 2019; de Bézenac et al. 2020;
191
+ Wang et al. 2019)), where the dynamics of hidden states
192
+ are modeled by auto-regressive recurrent neural works (e.g.
193
+ LSTM), which takes previous hidden states and current ob-
194
+ servations as input and outputs current observation. Different
195
+ from modeling the likelihood with parametric distributions
196
+ (e.g. Gaussian (Salinas et al. 2020)), emission models for
197
+ quantile estimation is to learn the parameters of quantile
198
+ function. The overall framework is optimized by employing
199
+ a quantile (Wen et al. 2017) or CRPS (Gasthaus et al. 2019)
200
+ loss.
201
+ Symbolic regression has shown great success in many fields,
202
+ including program synthesis (Parisotto et al. 2016), mathe-
203
+ matical expressions extraction (Cranmer et al. 2020), physics-
204
+ based learning (Li et al. 2019; Petersen et al. 2019). As the
205
+ search space is enormous and scaled exponentially with the
206
+ length of operators, symbolic regression rule operators are
207
+ usually set to be a small number and are learned by Monte
208
+ Carlo Tree Search guided evolutionary strategies (Li et al.
209
+ 2019) or reinforcement learning (Petersen et al. 2019).
210
+ Methods
211
+ Learning quantile function in quantile regression
212
+ Let the input data attributes X and the target variable y are
213
+ jointly distributed as p(X, y). The conditional cumulative
214
+ distribution function (CDF) is F(Y = y|X) = P(Y ≤
215
+ y|X). The quantile function, which is also called the inverse
216
+ CDF function, takes quantile level as inputs and returns a
217
+ threshold value Y below which random draws from the given
218
+ CDF would fall quantile percent of the time. Specifically, the
219
+ α-th quantile function of y|X = x is denoted as:
220
+ q(α, x) = F −1
221
+ y|X=x(α) = inf{y : F(y|X = x) ≥ α}
222
+ (1)
223
+ Here we can think the quantile function is to perform a
224
+ transformation on a uniform-distributed random variable
225
+ α ∼ U(0, 1) to the target distribution p(y|X). Quantile
226
+ function is able to fully specify a distribution. So specifying
227
+ the quantile function is describing the target distribution
228
+ p(y|X).
229
+ Quantile regression estimates different conditional quantile
230
+ levels of the target variable given a certain level of input
231
+
232
+ P(y|X)
233
+ Inverse CDF
234
+ alpha
235
+ y
236
+ P(y|X)
237
+ Inverse CDF
238
+ alpha
239
+ y
240
+ P(y|X)
241
+ Inverse CDF
242
+ alpha
243
+ y
244
+ P(y|X)
245
+ Inverse CDF
246
+ alpha
247
+ y
248
+ Spline Basis
249
+ P(y|X)
250
+ Inverse CDF
251
+ alpha
252
+ y
253
+ S
254
+ C
255
+ +
256
+ P(y|X)
257
+ Inverse CDF
258
+ alpha
259
+ y
260
+ Spline Basis
261
+ P(y|X)
262
+ Inverse CDF
263
+ alpha
264
+ y
265
+ ....
266
+ P(y|X)
267
+ Inverse CDF
268
+ alpha
269
+ y
270
+ P(y|X)
271
+ Inverse CDF
272
+ alpha
273
+ y
274
+ ...
275
+ ....
276
+ ....
277
+ P(y|X)
278
+ Inverse CDF
279
+ alpha
280
+ y
281
+ P(y|X)
282
+ Inverse CDF
283
+ alpha
284
+ y
285
+ ....
286
+ ....
287
+ NSS-sum
288
+ Initial distribution
289
+ target distribution
290
+ Spline Basis
291
+ Spline Basis
292
+ NSS-chain
293
+ P(y|X)
294
+ Inverse CDF
295
+ alpha
296
+ y
297
+ Operators
298
+ +
299
+ Sum
300
+ S
301
+ Scale
302
+ C
303
+ Chaining
304
+ ....
305
+ ....
306
+ Figure 3: Overview of Neural Spline Search (NSS). Modeling the target data distribution can be done by learning the quantile
307
+ function (e.g. inverse CDF), which maps a [0, 1]-variable (quantile) to a target value y. Unlike parametric methods which specify
308
+ a distribution family and learn the parameters, NSS can generate the target distribution through a set of transformations on the
309
+ inverse CDF space (quantile space), where the transformation is guided by a series of operators. Here, the bottom gray box shows
310
+ possible operators (denoted as circles), including but not limited to summation (“+”), scale (“S”), and chaining (“C”). The basis
311
+ splines are shown with color-shaded squares. The initial distribution is a uniform distribution, as shown in the leftmost panel
312
+ (blue shaded), and the target distribution is the rightmost distribution (purple shaded). There is no obvious parametric distribution
313
+ to achieve this transformation. Therefore, NSS is used to search for the suitable transformation through simple operators. In
314
+ the first row of the middle panel, we show operators for NSS-sum, where the initial uniform distribution is transformed by
315
+ the red- and the yellow-shaded splines (e.g. c-spline) through sum (“+”) and scale (“S”) operators. The second row shows the
316
+ chaining transformation of the initial distribution, where the orange and cyan splines are used to transform the initial spline. The
317
+ parameters of the splines are learned by a neural network. In general, the operators and transformations in NSS are not limited to
318
+ two splines (we represent them as the gray splines next to the yellow and cyan shaded splines).
319
+ attributes, as opposed to regression, which estimates the con-
320
+ ditional mean of the target variable. In quantile regression,
321
+ a particular quantile level α of the conditional distribution
322
+ of y given X = x, q(α, x) is estimated by minimizing the
323
+ pinball loss ρ (or quantile loss), as the the quantile function
324
+ q is shown to be the minimizer of the expected pinball loss
325
+ (Koenker and Bassett Jr 1978):
326
+ ρα(y, q) = (y − q)(α − 1(y < q)),
327
+ (2)
328
+ q(α, x) = arg min
329
+ q
330
+ Ey[ρα(y, q)].
331
+ (3)
332
+ where 1 is the indicator function. One shortcoming of pinball
333
+ loss is only measuring the loss at a single quantile level,
334
+ which hinders the estimated q for a global picture of the
335
+ distribution (i.e. other α levels). On contrast, the continuous
336
+ ranked probability score (CRPS) considers all quantile levels
337
+ by integrating the pinball loss over α = [0, 1] (Matheson and
338
+ Winkler 1976; Gneiting and Raftery 2007).
339
+ CRPS(y, q) =
340
+ � 1
341
+ 0
342
+ 2ρα(y, q)dα
343
+ (4)
344
+ As a proper scoring rule (Gneiting and Raftery 2007), CRPS
345
+ is minimized when the quantile function is q = F. That is,
346
+ F −1
347
+ y
348
+ = arg min
349
+ q
350
+ Ey[CRPS(y, q)].
351
+ (5)
352
+ Please refer (Koenker and Regression 2005) for detailed
353
+ proof.
354
+ Improving the expressiveness of quantile function
355
+ Fig. 2 demonstrate the need of an expressive quantile func-
356
+ tion for modeling target distribution. Inspired from neural
357
+ architecture search (NAS) (Elsken, Metzen, and Hutter 2019),
358
+ we propose an approach to search for the suitable combina-
359
+ tion of distributions. The search is over different operations
360
+ and basis distributions. We first introduce parametrization of
361
+ quantile function, and the two non-parametric spline-based
362
+ distributions.
363
+ Parameterizing quantile functions
364
+ We propose to param-
365
+ eterize the quantile function qθ(α, x) using a deep neural
366
+ network with parameters θ. The quantile function is aimed
367
+ to be accurate for any quantile levels α and input attributes
368
+ level X = x. X is high dimensional in real data, not as the
369
+ one dimensional in the toy examples in Fig. 1 and Fig. 2.
370
+ C-spline distribution
371
+ The c-spline (yα = qcsplie
372
+ θ
373
+ (α, x))
374
+ describes the CDF (Fig. 2, Right Panel) of a probability
375
+ distribution Fy|X by setting K anchor points (denoted as
376
+ knots) on the CDF curve and performing linear interpolation
377
+ to fill in the gap between the knots. Specifically, the knots
378
+ split CDF curve into bins and c-spline learns the width wi
379
+ and height hi of bins by neural networks NN that depend on
380
+ the input attributes level X = x.
381
+ {wi, hi}K = NNθ(x)
382
+ yα = r({wi, hi}K, α)
383
+ ∀α ∈ [0 : 1]
384
+
385
+ where hi and wi are non-negative delta values imposed by
386
+ non-negative activation (i.e. Relu or Sigmoid), and the loca-
387
+ tion of each bin (e.g. Y|X) is Li = �i
388
+ k=0 wk and quantile
389
+ level αi = �i
390
+ k=0 hk. The accumulation sum design is to en-
391
+ sure that quantile function is monotically increasing and there
392
+ is no quantile crossing. r is a function to convert knots to
393
+ output of quantile function: for quantile level αi that is on the
394
+ knots, we can directly read from li , for quantile levels that
395
+ are off the knots, quantile values can be computed through
396
+ linear algebra operations on the two nearby knots r(α) =
397
+
398
+ li + (α−αi)(lj−li)
399
+ αj−αi
400
+ ,
401
+ if αi ≤ α ≤ αj
402
+ 0 ≤ i, j ≤ K
403
+ lk,
404
+ if hk = α
405
+ P-spline distribution
406
+ The difference between p-spline
407
+ from c-spline is having anchor knots in PDF space, instead
408
+ of CDF space. Similarly with C-spline, P-spline also per-
409
+ form linear interpolation over knots, and the quantile level is
410
+ achieved by integration over pdf via polynomial operations.
411
+ Neural Spline Search (NSS)
412
+ We describe our proposed method, Neural Spline Search
413
+ (NSS), which is overviewed in Fig. 3. Similar to symbolic
414
+ regression (Parisotto et al. 2016; Li et al. 2019), NSS effec-
415
+ tively searches in the space of discrete symbolic operators
416
+ and distribution space for a candidate that can better fit the
417
+ target data distribution. Specifically, let T(O, S, k) denote the
418
+ space of all transformations, via operators O on all distribu-
419
+ tion S with a maximum sequence length k. NSS aims to find
420
+ the function f(x) selecting operators and distributions in the
421
+ space T such that {f(x) ∈ T(O, S, k) : ℓ(f(x), xtrain) ≤ δ
422
+ }, where ℓ denotes loss function CRPS, xtrain is training
423
+ data and δ is the acceptance threshold. Given the large search
424
+ space composed of combinations of numerous splines and
425
+ operators, we restrict to use spline-based distribution as the
426
+ basis distribution, and limit the operator search space to sum-
427
+ mation and chaining operations upon the transformation basis
428
+ spline regressions. Note that this work can be easily extend
429
+ to other operations and distributions, which we leave to fu-
430
+ ture work. We describe the following NSS transformations as
431
+ they are observed to work well consistently across different
432
+ datasets: NSS with summation (NSS-sum) and NSS with
433
+ chaining (NSS-chain). Algorithm 1 and Fig. 4(b)
434
+ NSS-sum
435
+ NSS-sum performs transformations using the scale and sum-
436
+ mation operators. We represent this scenario with two splines:
437
+ Spline 1: c-spline and Spline 2: p-spline, and two operators:
438
+ scale O1 : O(a) = λa and summation O2 : O(a, b) : a + b;
439
+ therefore, the overall transformation is (Spline 1-Operator 1) -
440
+ (Spline 2-Operator 2), which yields: f = c-spline + λ p-spline.
441
+ Essentially, NSS-sum performs weighted sum of different
442
+ splines. The motivation behind is that c-spline with fewer
443
+ parameters can be more robust against overfitting, whereas
444
+ p-spline increases the expressiveness of the splines.
445
+ NSS-chain
446
+ Another proposed NSS design is NSS-chain. We focus on
447
+ the chaining operator due to its expressiveness. This design
448
+ is inspired by the success of normalizing flow (Rezende and
449
+ Mohamed 2015), where a sequence of bijector transforms is
450
+ utilized to transform distributions. Different from normaliz-
451
+ ing flow which has practical applicability challenges, NSS-
452
+ chain only requires the forward pass of the transformation,
453
+ not the inverse as normalizing flow does. This significantly
454
+ reduces the computational complexity and broadens the fea-
455
+ sibility of transformations. As mentioned, quantile function
456
+ takes input attributes level (X) to predict the target value (y)
457
+ at quantile level (α).
458
+ y = qθ(X, α),
459
+ (6)
460
+ where X ∈ Rm and α ∈ [0, 1]. We present two designs to
461
+ chain different transformations (see Fig. 4 (a)). We note that
462
+ chaining of transformation is not limited to the two designs.
463
+ Algorithm 1: Neural Spline Search
464
+ Operators = {+, ×, Scale, Chain, ...}
465
+ Splines = {c-spline, p-spline, Gaussian, Cauchy ...}
466
+ Data: Quantile level α ∈ [0, 1], N data points
467
+ {X ∈ Rd, y ∈ R1}N, d ≥ 1, with chain depth k.
468
+ Transform indicates the transformation using the
469
+ input spline Sθ and operator O.
470
+ Result: p(y|X) and F −1
471
+ y|X(α)
472
+ k ← 1;
473
+ while k ≤ K do
474
+ Select O = {Oi}no ∈ Operators ;
475
+ Select S = {Sj}ns ∈ Splines ;
476
+ θ ← MLP(X) ;
477
+ ypred ← Transform(Sθ, O, α);
478
+ if α NSS-chain then
479
+ Normalize ypred to [0, 1] as y′
480
+ pred ;
481
+ α ← y′
482
+ pred;
483
+ else
484
+ X ← Y
485
+ ▷ if X-NSS-chain ;
486
+ end
487
+ k ← k + 1;
488
+ end
489
+ • α-chaining
490
+ The α-chaining is when we consider the condition level
491
+ (X) unchanged during the chain of transformation, and
492
+ the output of each transformation is a scaled version of
493
+ quantile level for the next transformation. In particular,
494
+ after each transformation, we normalize the output y to
495
+ be in the range [0, 1], and then the normalized output is
496
+ re-input as the new α to the next transformation. This is
497
+ repeated until the maximum depth is reached. This design
498
+ is more similar with normalizing ��ow methods.
499
+ y = qθK(X, ...fn(qθ2(X, fn(qθ1(X, α)))))
500
+ (7)
501
+ θk for k=1,2,..K are parameters for different splines in
502
+ K-length chain. fn is the normalization function.
503
+ • X-chaining
504
+ X-chaining is when we consider quantile level α level is
505
+ unchanged during chaining, as each transformation learns
506
+
507
+ alpha
508
+ P(y|X)
509
+ y_alpha
510
+ Inverse CDF
511
+ alpha
512
+ y
513
+ MLP
514
+ X
515
+ Spline's parameters
516
+ alpha
517
+ P(y|X)
518
+ y_alpha
519
+ Inverse CDF
520
+ alpha
521
+ y
522
+ MLP
523
+ X
524
+ Spline's parameters
525
+ x-chaining
526
+ alpha
527
+ P(y|X)
528
+ y_alpha
529
+ Inverse CDF
530
+ alpha
531
+ y
532
+ MLP
533
+ X
534
+ Spline's parameters
535
+ alpha-chaining
536
+ NNS Chain
537
+ Figure 4: (a) Illustration of NSS-chain methods. The dia-
538
+ gram demonstrates chaining for NSS-chain. Left: α-chaining.
539
+ The output y of the spline, after re-scaling to [0, 1], is re-
540
+ inputted to the quantile spline at quantile level α. Right: X-
541
+ chaining. The output y is instead re-inputted to the quantile
542
+ spline as X. Both rely on input attributes X.
543
+ a suitable condition level (or feature) for next iteration.
544
+ Similarly with α-chaining in the iterative manner, except
545
+ that the output y of each transformation is projected to
546
+ generate X for the next iteration of Eq. 6.
547
+ y = qθK(...qθ2(qθ1(X, α), α), α)
548
+ (8)
549
+ The advantage of this approach, compared tp α-chaining,
550
+ is that we keep quantile levels α unchanged, and re-
551
+ normalizing output is not needed.
552
+ Remarks on NSS: . (1) why a simple spline-based algo-
553
+ rithm, e.g. C-spline, is not enough? Although in theory
554
+ spline-based algorithms can represent any arbitrary distribu-
555
+ tions with sufficiently high number of knots K, in practice,
556
+ we find a large K often lead to unstable training, as also
557
+ studied in (Park et al. 2022). In contrast, we find the combina-
558
+ tion (combined or chained) over a relatively restricted splines
559
+ are more robust in capturing the overall of the target distribu-
560
+ tion (2) Include both spline-based distribution and classic
561
+ parametric distribution In addition to spline-based distribu-
562
+ tion, we also encourage incorporating parametric distribution
563
+ (e.g. Gaussian) as basis distribution for NSS, especially when
564
+ prior knowledge (say Gaussian noise) is available. Because, it
565
+ is challenging for spline based methods to reconstruct Gaus-
566
+ sian distribution even with infinite number of knots; and , the
567
+ benefits of combining the two are the parametric distribution
568
+ offers advantage of classic statistics and robust to noise, and
569
+ the non-parametric spline offers flexibility.
570
+ Training
571
+ Once we select the operators and splines, the parameters of
572
+ the splines are trained in an end-to-end way by optimizing
573
+ CRPS (Eq. 4). Specifically, during training, we fit parameters
574
+ by optimizing over with the empirical mean of CRPS over N
575
+ data points:
576
+ θ∗ = arg min
577
+ θ
578
+ 1/N
579
+ N
580
+
581
+ i=1
582
+ Ey[CRPS(y, qθ(Xi, α))].
583
+ (9)
584
+ Algorithm 2 overviews the training of NSS for spline parame-
585
+ ter selection. Because of the form of the transformations, the
586
+ analytical solution of CRPS integration is intractable. Thus,
587
+ we use a Monte Carlo estimation for the CRPS loss. In par-
588
+ ticular, we sample m number of α values from the range of
589
+ [0, 1] and average them for the corresponding pinball loss.
590
+ Algorithm 2: Training with CRPS
591
+ Data: N data points {Xi ∈ Rd, yi ∈ R1}N
592
+ i=1, m
593
+ quantile levels, T transformation, which
594
+ takes selected splines Sselect and selected
595
+ operators Oselect from NSS. lr is learning
596
+ rate.
597
+ Result: Neural network weights θ
598
+ e ← 1;
599
+ while e ≤ Nepoch do
600
+ f = Transform(Sselect, Oselect) ℓ ← 0 ;
601
+ for α in [0, 1
602
+ m, 2
603
+ m, ..1] do
604
+ ypred
605
+ α
606
+ = fθ(X, α) ;
607
+ ℓ ← ℓ + pinball_loss (ypred
608
+ α
609
+ , y, α)
610
+ end
611
+ CRPS = ℓ/m ;
612
+ θ ← θ − lr · ∇θ CRPS ;
613
+ e ← e + 1;
614
+ end
615
+ Experiments
616
+ Comparison methods
617
+ QD (Pearce et al. 2018) generates prediction intervals (PIs)
618
+ for estimating uncertainty for regression tasks with the as-
619
+ sumption that high-quality PIs should be as narrow as possi-
620
+ ble. Deep Quantile Aggregation (Kim et al. 2021) proposes
621
+ weighted ensembling strategies where aggregation weights
622
+ vary over both individual models and feature values plus
623
+ (pairs of) quantile levels. The monotonization layer in the
624
+ network is applied to avoid crossing of quantile estimates.
625
+ RQspline (Durkan et al. 2019) proposes a fully-differentiable
626
+ module based on monotonic rational-quadratic splines, which
627
+ enhances the flexibility of coupling and autoregressive trans-
628
+ forms while retaining analytic invertibility. Global-Coarse
629
+ (Ratcliff 1979) provides an analysis of distribution statis-
630
+ tics of group reaction time distributions. MLE (NB) and
631
+ Mix. MLE are Negative Binomial and mixture likelihood
632
+ based methods (Awasthi et al. 2021). C-spline is proposed in
633
+ (Gasthaus et al. 2019), where C-spline is used as the quantile
634
+ function in time-series forecasting.
635
+ Metrics
636
+ For point predictions, we focus on the following metrics:
637
+ Mean absolute error (MAE): 1
638
+ n
639
+ �n
640
+ t=1 |Tt − Pt| where Tt and
641
+ Pt are true and predicted value; Mean Absolute Percentage
642
+ Error (MAPE): 1
643
+ n
644
+ �n
645
+ t=1 | Tt−Pt
646
+ Tt
647
+ |. Weighted Average Percent-
648
+ age Error (WAPE):
649
+ �n
650
+ t=1 |Tt−Pt|
651
+ �n
652
+ t=1 |Tt|
653
+ ; and Root Mean Square
654
+ Error (RMSE):
655
+ � �N
656
+ t (Tt−Pt)2
657
+ n
658
+ . For quantile predictions, we
659
+ use the Pinball Loss (Eq. 2), with 50%-th, Q50; 90%-th, Q90;
660
+ and 10%-th Q10 quantiles.
661
+
662
+ Methods
663
+ Boston
664
+ Concrete
665
+ kin8nm
666
+ Power
667
+ Protein
668
+ Wine
669
+ Gaussian
670
+ 0.0754
671
+ 0.0564
672
+ 0.048
673
+ 0.0449
674
+ 0.2116
675
+ 0.0978
676
+ QD
677
+ 0.5003
678
+ 0.4150
679
+ 0.3945
680
+ 0.3688
681
+ 0.6689
682
+ 0.4456
683
+ RQspline
684
+ 0.0917
685
+ 0.0622
686
+ 0.0479
687
+ 0.0485
688
+ 0.2153
689
+ 0.0912
690
+ p-sline
691
+ 0.0778
692
+ 0.0570
693
+ 0.0444
694
+ 0.0453
695
+
696
+ 0.0966
697
+ c-spline
698
+ 0.0806
699
+ 0.0543
700
+ 0.0430
701
+ 0.0447
702
+ 0.2002
703
+ 0.0947
704
+ NSS-X-chain
705
+ 0.0787
706
+ 0.0588
707
+ 0.0430
708
+ 0.0448
709
+ 0.2052
710
+ 0.0962
711
+ NSS-α-chain
712
+ 0.0846
713
+ 0.0568
714
+ 0.0417
715
+ 0.0448
716
+ 0.2067
717
+ 0.0976
718
+ NSS-sum
719
+ 0.0709
720
+ 0.0512
721
+ 0.0414
722
+ 0.0442
723
+ 0.1949
724
+ 0.0957
725
+ Gain percentage
726
+ 12.0%
727
+ 17.7%
728
+ 3.7%
729
+ 1.1%
730
+ 2.6%
731
+ -
732
+ Table 1: Mean Absolute Error (MAE) on UCI benchmarks. Test performance of the proposed method (NSS) and existing
733
+ methods on UCI benchmarks. We use the 50th quantile estimator as our estimates. The dash indicates unavailability. The shaded
734
+ area is the proposed methods. Bold is the top one. Lower is better. Gaussian: Gaussian kernel; QD is quantity-driven methods
735
+ proposed in (Pearce et al. 2018); RQ spline proposed in (Durkan et al. 2019); c-spline proposed in (Gasthaus et al. 2019). Boston,
736
+ Concrete, Power is short for Boston Housing, Concrete Strength, Power Plant. Gain percentage is computed as (best nss - best
737
+ baseline)/best baseline.
738
+ Training
739
+ For simplicity, the proposed NSS methods use depth-
740
+ 2
741
+ splines,
742
+ which
743
+ contain
744
+ {(c-spline,
745
+ p-spline),
746
+ (c-
747
+ spline,
748
+ p-spline),
749
+ (c-spline,
750
+ c-spline),
751
+ (p-spline,
752
+ p-
753
+ spline)}. NSS-sum is tuned with λ in the range of
754
+ [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9].
755
+ NSS-chain
756
+ nor-
757
+ malizing of y in α chaining can be achieved by applying
758
+ sigmoid layer or scaling by max value. As splines are
759
+ monotonically-increasing functions, the spline value y with
760
+ α = 0 is the minimum value of y and α = 1 yields the
761
+ maximum value of y. Scale is yscale =
762
+ y−ymin
763
+ ymax−ymin . We use
764
+ a batch size=128 and a learning rate of 0.005 for 100 epochs.
765
+ Results
766
+ To demonstrate the effectiveness of proposed methods, we
767
+ conduct experiments on synthetic, real-world tabular regres-
768
+ sion, and time series forecasting datasets.
769
+ Synthetic data
770
+ Dataset. We generate 2000 data points (X
771
+ ∈ R1 and
772
+ y ∈ R1), where X is in the range of [−2, 2] and y has Gaus-
773
+ sian distribution y ∼ N(0.3 sin(3x), 0.2x2), where sin is the
774
+ sinusodial function. We construct the validation and test sets
775
+ to come from the same distribution. Unlike real-world data,
776
+ the synthetic data would have known quantile levels, that can
777
+ be used for evaluating the accuracy of quantile estimates. We
778
+ make the task more challenging by setting a data-dependent
779
+ variance for the Gaussian noise to evaluate the ability of learn-
780
+ ing condition-specific quantile values. Fig. 5 shows that the
781
+ proposed NSS-chain and NSS-sum can capture the true under-
782
+ lying quantiles, whereas QD (Pearce et al. 2018) struggles on
783
+ the varying variance locations (e.g. around x = 0). The upper
784
+ and lower black lines are the predicted 2.5%-th and 97.5%-th
785
+ quantiles for the observed data (e.g. red dots), shown along
786
+ with the ground truth quantiles (e.g. shaded red area). The
787
+ results indicate that more expressive NSS transformations
788
+ are superior in more challenging scenarios, where true data
789
+ points are distributed differently (e.g., distributions depend
790
+ on the value of the inputs"). Fig. 6 shows the calibration plot
791
+ of the predicted vs. true distributions at different quantile
792
+ 2.0
793
+ 1.5
794
+ 1.0
795
+ 0.5
796
+ 0.0
797
+ 0.5
798
+ 1.0
799
+ 1.5
800
+ 2.0
801
+ x
802
+ 2
803
+ 1
804
+ 0
805
+ 1
806
+ 2
807
+ y
808
+ QD
809
+ 2.0
810
+ 1.5
811
+ 1.0
812
+ 0.5
813
+ 0.0
814
+ 0.5
815
+ 1.0
816
+ 1.5
817
+ 2.0
818
+ x
819
+ 2
820
+ 1
821
+ 0
822
+ 1
823
+ 2
824
+ y
825
+ NSS-SUM.
826
+ 2.0
827
+ 1.5
828
+ 1.0
829
+ 0.5
830
+ 0.0
831
+ 0.5
832
+ 1.0
833
+ 1.5
834
+ 2.0
835
+ x
836
+ 2
837
+ 1
838
+ 0
839
+ 1
840
+ 2
841
+ y
842
+ NSS-CHAIN.
843
+ Figure 5: NSS on Synthetic data. We compare the per-
844
+ formance of proposed NSS against existing methods QD
845
+ (Pearce et al. 2018). The red dots are observed data points,
846
+ shaded red area is the ground truth 2.5% and 97.5% quantile
847
+ levels, and the dark black lines are the predicted 2.5% and
848
+ 97.5% quantile levels.
849
+ 0.0
850
+ 0.2
851
+ 0.4
852
+ 0.6
853
+ 0.8
854
+ 1.0
855
+ True percentile
856
+ 0.0
857
+ 0.2
858
+ 0.4
859
+ 0.6
860
+ 0.8
861
+ 1.0
862
+ Predicted percentile
863
+ QD
864
+ NSS-SUM
865
+ NSS-CHAIN
866
+ 2
867
+ 0
868
+ 2
869
+ X
870
+ y
871
+ X=0.5
872
+ 0.0
873
+ 0.2
874
+ 0.4
875
+ 0.6
876
+ 0.8
877
+ 1.0
878
+ True percentile
879
+ Calibration plot
880
+ QD
881
+ NSS-SUM
882
+ NSS-CHAIN
883
+ 2
884
+ 0
885
+ 2
886
+ X
887
+ y
888
+ X=1.0
889
+ 0.0
890
+ 0.2
891
+ 0.4
892
+ 0.6
893
+ 0.8
894
+ 1.0
895
+ True percentile
896
+ QD
897
+ NSS-SUM
898
+ NSS-CHAIN
899
+ 2
900
+ 0
901
+ 2
902
+ X
903
+ y
904
+ X=1.5
905
+ Figure 6: Calibration plots. Predicted vs. ground truth per-
906
+ centiles at condition levels: X=0.5, 1.0 and 1.5. The perfect
907
+ calibration would correspond to the diagonal (red dotted)
908
+ line.
909
+ levels. Here, we show the true percentile p as the fraction of
910
+ data in the dataset such that the p percentile of the predictive
911
+ distribution is larger than the ground truth data. The perfect
912
+ prediction would be the diagonal line. Fig. 6 indicates that
913
+ the proposed methods NSS-sum and NSS-chain can capture
914
+ the proposed true distribution at various levels by close to the
915
+ red line, whereas QD does not fit as well.
916
+ Real-world tabular regression
917
+ We use UCI benchmarks (Asuncion and Newman 2007) that
918
+ contain tabular data from diverse domains (e.g. real estate
919
+ and physics). Following (Salem, Langseth, and Ramampiaro
920
+ 2020), the datasets are normalized with z-score standardiza-
921
+
922
+ Methods
923
+ Boston
924
+ Concrete
925
+ kin8nm
926
+ Power
927
+ Protein
928
+ Wine
929
+ Gaussian
930
+ 0.0276
931
+ 0.0203
932
+ 0.0171
933
+ 0.0158
934
+ 0.0725
935
+ 0.0357
936
+ Global-Coarse∗
937
+ 0.0745
938
+ 0.0596
939
+ 0.0681
940
+ 0.0473
941
+ 0.1321
942
+
943
+ Deep Quantile Aggregation∗
944
+ 0.0754
945
+ 0.0541
946
+ 0.0684
947
+ 0.0441
948
+ 0.1253
949
+
950
+ QD
951
+ 0.1212
952
+ 0.1076
953
+ 0.1004
954
+ 0.0972
955
+ 0.1547
956
+ 0.1164
957
+ RQspline
958
+ 0.0458
959
+ 0.0418
960
+ 0.0203
961
+ 0.0189
962
+ 0.0863
963
+ 0.0424
964
+ p-sline
965
+ 0.0308
966
+ 0.0211
967
+ 0.016
968
+ 0.0160
969
+
970
+ 0.0358
971
+ c-spline
972
+ 0.0312
973
+ 0.0198
974
+ 0.0157
975
+ 0.0159
976
+ 0.0688
977
+ 0.0351
978
+ NSS-X-chain
979
+ 0.0311
980
+ 0.0216
981
+ 0.0165
982
+ 0.0162
983
+ 0.0707
984
+ 0.0358
985
+ NSS-α-chain
986
+ 0.0322
987
+ 0.0208
988
+ 0.0151
989
+ 0.0159
990
+ 0.0726
991
+ 0.0363
992
+ NSS-sum
993
+ 0.0265
994
+ 0.0191
995
+ 0.0152
996
+ 0.0157
997
+ 0.0674
998
+ 0.0357
999
+ Gain percentage
1000
+ 4.0%
1001
+ 3.5%
1002
+ 3.8%
1003
+ 0.6%
1004
+ 7.0%
1005
+ -
1006
+ Table 2: Average pinball loss on UCI benchmarks. The test pinball loss (the lower, the better) is over 99 quantile levels,
1007
+ α = {0.01, 0.02, ...0.99}. The compared methods are Global-Coarse proposed in (Ratcliff 1979); QD (Pearce et al. 2018); Deep
1008
+ Quantile Aggregation (DQA) (Kim et al. 2021); RQspline (Durkan et al. 2019); ∗ indicates entries are from (Kim et al. 2021)
1009
+ (under the same experiment setup).
1010
+ Methods
1011
+ MAPE
1012
+ WAPE
1013
+ RMSE
1014
+ Q50
1015
+ Q90
1016
+ Q10
1017
+ MLE (NB)
1018
+ 0.44434
1019
+ 0.27240
1020
+ 7.70958
1021
+ 0.27240
1022
+ 0.10907
1023
+ 0.15275
1024
+ Mix MLE
1025
+ 0.44839
1026
+ 0.26838
1027
+ 7.22556
1028
+ 0.26838
1029
+ 0.10293
1030
+ 0.14508
1031
+ c-spline
1032
+ 0.44672
1033
+ 0.26635
1034
+ 7.06332
1035
+ 0.26635
1036
+ 0.10238
1037
+ 0.14241
1038
+ p-spline
1039
+ 0.44912
1040
+ 0.26834
1041
+ 7.14643
1042
+ 0.26834
1043
+ 0.10343
1044
+ 0.14333
1045
+ NSS-sum
1046
+ 0.44501
1047
+ 0.26545
1048
+ 6.96697
1049
+ 0.26545
1050
+ 0.10238
1051
+ 0.14266
1052
+ NSS-chain
1053
+ 0.44883
1054
+ 0.26420
1055
+ 6.91726
1056
+ 0.26420
1057
+ 0.10243
1058
+ 0.14149
1059
+ Table 3: Performance comparisons for time series forecasting on M5. Different evaluation metrics are included in this table
1060
+ for M5. Detailed descriptions of the metrics are in Sec . Qk indicates the pinball loss of k-th quantile. e.g. Q50 is the pinball loss
1061
+ of 50th quantile. Lower is better.
1062
+ tion.
1063
+ We evaluate the accuracy for both point predictions and quan-
1064
+ tiles. As the point predictions, we use the 50th quantile es-
1065
+ timator as our estimates. Table 1 shows that the proposed
1066
+ NSS methods outperform the other existing methods on most
1067
+ datasets in mean absolute error (MAE). In mean square error
1068
+ (MSE), the results are provided in Appendix Table 4. We
1069
+ observe that the NSS-sum performs better than NSS-chain.
1070
+ For quantile metrics, we use the pinball loss (Eq. 2) over
1071
+ 100 quantile levels α = {0.01, 0.02, ...0.99} in Table 2. The
1072
+ results indicate that NSS consistently outperforms other al-
1073
+ ternatives across different UCI benchmarks. In pinball loss,
1074
+ NSS-sum performs better than NSS-chain. We attribute the
1075
+ superiority of NSS-sum for regression to make balance be-
1076
+ tween different transformation, which is helpful in explaining
1077
+ the variance in the data.
1078
+ Retail demand forecasting
1079
+ For time series forecasting, we focus on the M5 dataset,
1080
+ which contains time-varying sales data for retail goods, along
1081
+ with other relevant covariates like price, promotions, day
1082
+ of the week, special events etc. It represents an important
1083
+ real-world scenario, where the accurate estimation of the
1084
+ output distribution is crucial, as retailers use them to optimize
1085
+ prices or promotions.
1086
+ The time series forecasting experiments are conducted by
1087
+ performing one-step ahead prediction, yielding predictions
1088
+ in an autoregressive way. Table 3 shows the results of
1089
+ our method compared to other alternatives. We observe
1090
+ consistent outperformance of NSS in various forecasting
1091
+ evaluation metrics. Different from regression tasks, we
1092
+ observe that NSS-chain is better than NSS-sum, indicating
1093
+ its benefit in capturing time-dependent relationship.
1094
+ Remarks on NSS-sum vs NSS-chain. The results show that
1095
+ NSS-sum is superior on regression, while NSS-chain has
1096
+ advantages on time series forecasting. The observations may
1097
+ indicate NSS-sum is suitable for more constrained tasks (e.g.
1098
+ regression, one time step time series-forecasting), where be-
1099
+ ing moderately expressive would suffice. NSS-sum is also
1100
+ more robust and easier to train. On the other hand, NSS-chain
1101
+ may be more expressive, which is beneficial to fit tasks re-
1102
+ quires more complex distributions at different time steps of
1103
+ the time series, but for individual step NSS-chain is not as
1104
+ accurate as NSS-sum in fitting the distribution.
1105
+ Conclusion
1106
+ We propose a novel approach for modeling uncertainty. The
1107
+ proposed Neural Spline Search (NSS) method employs a se-
1108
+ ries of monotonic spline regression transformations, guided
1109
+ by symbolic operators. We demonstrate the effectiveness of
1110
+ NSS for superior modeling of output distributions, on both
1111
+ synthetic and real-world datasets. We leave the extensions to
1112
+ different operators and splines, including parametric distribu-
1113
+ tion transformations to future work.
1114
+
1115
+ .
1116
+ References
1117
+ Asuncion, A.; and Newman, D. 2007. UCI machine learning
1118
+ repository.
1119
+ Awasthi, P.; Das, A.; Sen, R.; and Suresh, A. T. 2021. On the
1120
+ benefits of maximum likelihood estimation for Regression
1121
+ and Forecasting. arXiv preprint arXiv:2106.10370.
1122
+ Begoli, E.; Bhattacharya, T.; and Kusnezov, D. 2019. The
1123
+ need for uncertainty quantification in machine-assisted med-
1124
+ ical decision making. Nature Machine Intelligence, 1(1):
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+ 20–23.
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+ Buckman, J.; Hafner, D.; Tucker, G.; Brevdo, E.; and
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+ Lee, H. 2018.
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+ with stochastic ensemble value expansion. arXiv preprint
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+ arXiv:1807.01675.
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+ Cranmer, M.; Sanchez-Gonzalez, A.; Battaglia, P.; Xu, R.;
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+ Cranmer, K.; Spergel, D.; and Ho, S. 2020. Discovering
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+ bilistic forecasting with spline quantile function RNNs. In
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+ AISTATS.
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+ Gneiting, T.; and Raftery, A. E. 2007. Strictly proper scoring
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+ rules, prediction, and estimation. Journal of the American
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+ Jiang, X.; Osl, M.; Kim, J.; and Ohno-Machado, L. 2012.
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+ arXiv:2103.00083.
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+ Koenker, R.; and Bassett Jr, G. 1978. Regression quantiles.
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+ Econometrica: journal of the Econometric Society, 33–50.
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+ Koenker, R.; and Regression, Q. 2005. Econometric Society
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+ Monographs. Quantile regression.
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+ Li, L.; Fan, M.; Singh, R.; and Riley, P. 2019. Neural-guided
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+ arXiv
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+ preprint arXiv:1901.07714.
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+ Matheson, J. E.; and Winkler, R. L. 1976. Scoring rules for
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+ continuous probability distributions. Management science,
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+ Mhaskar, H. N.; Pereverzyev, S. V.; and van der Walt, M. D.
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+ prediction. Frontiers in Applied Mathematics and Statistics,
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+ 3: 14.
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+ Moon, S. J.; Jeon, J.-J.; Lee, J. S. H.; and Kim, Y. 2021.
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+ Computational and Graphical Statistics, 30(4): 1238–1248.
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+ NASA. 2015. Pluto: The ’Other’ Red Planet. https://www.
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+ nasa.gov/nh/pluto-the-other-red-planet. Accessed: 2018-12-
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+ Parisotto, E.; Mohamed, A.-r.; Singh, R.; Li, L.; Zhou, D.;
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+ and Kohli, P. 2016. Neuro-symbolic program synthesis. arXiv
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+ preprint arXiv:1611.01855.
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+ Park, Y.; Maddix, D.; Aubet, F.-X.; Kan, K.; Gasthaus, J.;
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+ and Wang, Y. 2022. Learning quantile functions without
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+ quantile crossing for distribution-free time series forecasting.
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+ In International Conference on Artificial Intelligence and
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+ Statistics, 8127–8150. PMLR.
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+ Pearce, T.; Brintrup, A.; Zaki, M.; and Neely, A. 2018. High-
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+ quality prediction intervals for deep learning: A distribution-
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+ free, ensembled approach. In ICML.
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+ Petersen, B. K.; Larma, M. L.; Mundhenk, T. N.; Santiago,
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+ C. P.; Kim, S. K.; and Kim, J. T. 2019. Deep symbolic regres-
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+ sion: Recovering mathematical expressions from data via risk-
1198
+ seeking policy gradients. arXiv preprint arXiv:1912.04871.
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+ Ratcliff, R. 1979. Group reaction time distributions and
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+ an analysis of distribution statistics. Psychological bulletin,
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+ 86(3): 446.
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+ Rezende, D.; and Mohamed, S. 2015. Variational inference
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+ with normalizing flows. In ICML.
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+ Salem, T. S.; Langseth, H.; and Ramampiaro, H. 2020. Pre-
1205
+ diction intervals: Split normal mixture from quality-driven
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+ deep ensembles. In Conference on Uncertainty in Artificial
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+ Intelligence.
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+ Salinas, D.; Flunkert, V.; Gasthaus, J.; and Januschowski, T.
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+ 2020. DeepAR: Probabilistic forecasting with autoregressive
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+ recurrent networks. International Journal of Forecasting,
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+ 36(3): 1181–1191.
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+ Simchi-Levi, D.; Kaminsky, P.; Simchi-Levi, E.; and Shankar,
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+ R. 2008. Designing and managing the supply chain: concepts,
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+ strategies and case studies. Tata McGraw-Hill Education.
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+ Smith, H. J.; Dinev, T.; and Xu, H. 2011. Information privacy
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+ research: an interdisciplinary review. MIS quarterly, 989–
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+ 1015.
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+ Tagasovska, N.; and Lopez-Paz, D. 2019. Single-model un-
1219
+ certainties for deep learning. Advances in Neural Information
1220
+ Processing Systems, 32.
1221
+ Waldmann, E. 2018. Quantile regression: a short story on
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+ how and why. Statistical Modelling, 18(3-4): 203–218.
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+ Wang, Y.; Smola, A.; Maddix, D.; Gasthaus, J.; Foster, D.;
1224
+ and Januschowski, T. 2019. Deep factors for forecasting. In
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+ International conference on machine learning, 6607–6617.
1226
+ PMLR.
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+
1228
+ Wen, R.; Torkkola, K.; Narayanaswamy, B.; and Madeka, D.
1229
+ 2017. A multi-horizon quantile recurrent forecaster. arXiv
1230
+ preprint arXiv:1711.11053.
1231
+ Xu, W.; Pan, J.; Wei, J.; and Dolan, J. M. 2014. Motion
1232
+ planning under uncertainty for on-road autonomous driving.
1233
+ In ICRA.
1234
+
1235
+ Appendix
1236
+ Mean square error for UCI dataset
1237
+
1238
+ Methods
1239
+ Bost House
1240
+ Concr Stren
1241
+ kin8nm
1242
+ Power plant
1243
+ Protein
1244
+ Wine
1245
+ Gaussian
1246
+ 0.0105
1247
+ 0.0054
1248
+ 0.0042
1249
+ 0.0032
1250
+ 0.0648
1251
+ 0.0164
1252
+ (Salem, Langseth, and Ramampiaro 2020)∗
1253
+ 0.1120
1254
+ 0.0560
1255
+ 0.0600
1256
+ 0.0420
1257
+ 0.3100
1258
+ 0.5970
1259
+ QD
1260
+ 0.2705
1261
+ 0.1839
1262
+ 0.1613
1263
+ 0.1393
1264
+ 0.5277
1265
+ 0.2164
1266
+ RQspline
1267
+ 0.0255
1268
+ 0.0070
1269
+ 0.0040
1270
+ 0.0037
1271
+ 0.0809
1272
+ 0.0195
1273
+ p-sline
1274
+ 0.0136
1275
+ 0.0058
1276
+ 0.0032
1277
+ 0.0032
1278
+
1279
+ 0.0162
1280
+ c-spline
1281
+ 0.0162
1282
+ 0.0050
1283
+ 0.0031
1284
+ 0.0032
1285
+ 0.0757
1286
+ 0.0159
1287
+ NSS-X-chain
1288
+ 0.0128
1289
+ 0.0056
1290
+ 0.0031
1291
+ 0.0032
1292
+ 0.0751
1293
+ 0.0164
1294
+ NSS-α-chain
1295
+ 0.0184
1296
+ 0.0058
1297
+ 0.0029
1298
+ 0.0032
1299
+ 0.0760
1300
+ 0.0169
1301
+ NSS-sum
1302
+ 0.0112
1303
+ 0.0046
1304
+ 0.0029
1305
+ 0.0032
1306
+ 0.0711
1307
+ 0.0160
1308
+ Table 4: Mean Square Error of UCI datasets
1309
+
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1
+ Identification of lung nodules CT scan using YOLOv5 based on
2
+ convolution neural network
3
+ Haytham Al Ewaidat
4
+ ID 1,*, Youness El Brag
5
+ ID 2
6
+ 1Jordan University of Science and Technology, Faculty of Applied Medical Sciences, Department of Allied Medical
7
+ Sciences-Radiologic Technology, Irbid, Jordan, 22110
8
+ 2Abdelmalek Essaˆadi University of Science and Technology, Faculty of Multi-Disciplinary Larache, Department of
9
+ Computer Sciences, ksar el kebir , Morocco, 92150
10
+ Correspondence author: Dr Haytham Al Ewaidat, Department of Allied Medical Sciences-Radiologic Technology,
11
+ Faculty of Applied Medical Sciences, Jordan University of Science and Technology. PO Box 3030, Irbid 22110,
12
+ Jordan Tel: (+962)27201000-26939; Fax: (+962)27201087; E-mail: haewaidat@just.edu.jo
13
+ arXiv:2301.02166v1 [eess.IV] 31 Dec 2022
14
+
15
+ Abstract
16
+ Purpose: The lung nodules localization in CT scan images is the most difficult task due to the complexity of the
17
+ arbitrariness of shape, size, and texture of lung nodules. This is a challenge to be faced when coming to developing
18
+ different solutions to improve detection systems. the deep learning approach showed promising results by using
19
+ convolutional neural network (CNN), especially for image recognition and it’s one of the most used algorithm in
20
+ computer vision.
21
+ Approach: we use (CNN) building blocks based on YOLOv5 (you only look once) to learn the features representations
22
+ for nodule detection labels, in this paper, we introduce a method for detecting lung cancer localization. Chest X-rays
23
+ and low-dose computed tomography are also possible screening methods, When it comes to recognizing nodules
24
+ in radiography, computer-aided diagnostic (CAD) system based on (CNN) have demonstrated their worth. One-
25
+ stage detector YOLOv5 trained on 280 annotated CT SCAN from a public dataset LIDC-IDRI based on segmented
26
+ pulmonary nodules.
27
+ Results: we analyze the predictions performance of the lung nodule locations, and demarcates the relevant CT scan
28
+ regions. In lung nodule localization the accuracy is measured as mean average precision (mAP). the mAP takes into
29
+ account how well the bounding boxes are fitting the labels as well as how accurate the predicted classes for those
30
+ bounding boxes, the accuracy we got 92.27% .
31
+ Conclusion: this study was to identify the nodule that were developing in the lungs of the participants. It was difficult
32
+ to find information on lung nodules in medical literature,
33
+ Keywords: computer-aided diagnostic, deep learning, Convolutional Neural Networks ,Lung Nodule.
34
+ *Address all correspondence to Haytham Al Ewaidat , haewaidat@just.edu.jo
35
+ 1
36
+ introduction
37
+ As far as noninvasive therapy and clinical assessment are concerned, medical image analysis offers
38
+ a tremendous advantage. X-rays, CTs, MRIs, and ultrasounds are utilized to make precise diag-
39
+ noses based on the obtained restorative images. By using attractive fields, CT can capture pictures
40
+ on film in medical imaging. One-of-a-kind lung cancer is responsible for 1.61 million fatalities per
41
+ year. Most of the cases of lung cancer in Indonesia are observed in the MIoT centers. If the tumor
42
+ is identified early, the survival percentage is better then. It’s not an easy task to find lung cancer
43
+ in its early stages. Approximately 80% of cancer patients are diagnosed at the core or accelerated
44
+ phase of the disease. Lung cancer is the second most common cancer among men and the tenth
45
+ most common among women worldwide. After breast and colorectal cancer, lung cancer is the
46
+ thirdly most common cancer among women. Features extraction in image processing is one of the
47
+ simplest and most efficient dimensionality reduction approaches. The non-invasive nature of CT
48
+ imaging is one of its most notable characteristics. It’s surprising to see angles increasing when
49
+ compared to other imaging modalities.
50
+ Computed tomography imaging is the best technique for examining lung disorders. CT scans, on
51
+ the other hand, have a high probability of false-positive results and are associated with cancer-
52
+ causing radiation exposure. When compared to standard-dose CT, low-dose CT utilizes a lot less
53
+ radiation contact power. The findings reveal that the detection sensitivity of low-dose and standard-
54
+ dose CT images is not significantly different. A well know database the National Lung Screening
55
+ Trial database shows that cancer-related fatalities were considerably decreased in the group that
56
+ was subjected to low-dose CT scans rather than chest radiography. The sensitivity of lung nodule
57
+ 1
58
+
59
+ identification may be improved by the use of more detailed anatomical information, and better
60
+ image registration methods. As a result, the datasets have grown enormously. Up to 500 seg-
61
+ ments/slice may be generated from a single scan, depending on how thick the slice is. A single
62
+ slice is examined by a competent radiologist in 2–3.5 minutes. A radiologist’s workload keeps
63
+ rising while screening a Ct for the presence of a suspicious nodule. The detection sensitivity of
64
+ nodules is influenced by a variety of factors, including the size, location, form, nearby structures,
65
+ edges, and density, in addition to the CT slice section thickness.
66
+ Only 68 percent of lung cancer nodules are properly identified when only one radiologist doctor
67
+ views the scan, and up to 82% of the time when two radiologists check the scan, according to
68
+ the study results. Early diagnosis of malignant lung nodules by radiologists is a tough, time-
69
+ consuming, and laborious process in and of itself. The radiologist needs a lot of time to carefully
70
+ screen a large number of images, but this method is prone to mistake when looking for microscopic
71
+ nodules.
72
+ An aid for radiologists is required in this case to speed up readings, catch any missing nodules,
73
+ and enable improved localization. A primary goal of computer-aided detection systems was to
74
+ minimize radiologists’ labor and boost the detection rate of nodules. Newer CAD systems, on
75
+ the other hand, can distinguish between benign, and malignant nodules, which is helpful in the
76
+ screening process. CAD systems frequently beat professional radiologists in nodule identification
77
+ and localization tasks because of recent breakthroughs in deep learning models, particularly in
78
+ image processing. CAD systems, on the other hand, have an FP rate of 1–8.2 per scan and a
79
+ detection range of 38–100%, according to different studies. As a result of their likeness to one
80
+ other, benign, and malignant nodule remain a difficult challenge to solve.
81
+ During the screening process, a variety of mistakes might occur. For example, if a scan fails to
82
+ capture or recognize a specific region of the lesion or fails to distinguish between benign, and
83
+ malignant lesions in a patient’s body, the patient may be at risk of misdiagnosis. Most people
84
+ die as a result of misdiagnoses and delays in treatment because of these mistakes. In radiology,
85
+ over 4% of reports include diagnostic mistakes on a daily, and about 30% of aberrant radiological
86
+ diagnoses are ignored. Early-stage lung nodules may be detected and classified more accurately
87
+ using different methodologies such as deep learning.
88
+ Lung nodule identification using deep learning with a specific methodology is presented in this
89
+ research. lung CT images, physiological symptoms, and clinical indicators, the suggested ap-
90
+ proach reduces false-positive findings and eventually prevents invasive procedures. YOLOv5 is
91
+ used which has convolutional networks were built to identify and classify nodules. For nodule
92
+ identification. Nodule identification and classification using the publicly accessible data set LIDC-
93
+ IDRI surpasses state-of-the-art deep learning techniques. Using a variety of techniques, we were
94
+ able to reduce the number of false positives in the learning algorithm.
95
+ Lung nodule computer-aided detection (CAD) systems were originally developed in the late 1980s,
96
+ but the processing resources required for sophisticated image analysis methods at the time made
97
+ these efforts unattractive. For image analysis, and decision support systems based on computers,
98
+ the graphics processing unit and convolutional neural networks revolutionized their performance.
99
+ Some of the most important lung nodule identification and classification approaches have been
100
+ suggested by researchers in deep learning based medical images analysis models. For lung nodule
101
+ 2
102
+
103
+ classification, Yutong Xie et al1 . proposed a method that utilizes Texture, Shape, and Deep Model
104
+ learned Data at the choice level.
105
+ Nodule heterogeneity may be shown with the use of this algorithm’s GLCM-based surface de-
106
+ scriptor, Fourier-shape descriptor, and a DCNN. Based on CNNs Chougrad et al.2 studied the
107
+ classification of breast cancer using a CAD framework. Transfer learning, on the other hand, takes
108
+ just a small number of medical images to train a system. With the use of the transfer learning
109
+ approach, the CNNs were taught to their fullest potential. In terms of accuracy, CNN came out on
110
+ top with a score of 98.94 percent. Using the wavelet transform and principal component analysis,
111
+ Heba Mohsen et al3 developed a DNN classifier for brain tumor classification. A technique of reg-
112
+ ularized linear discriminant analysis was developed in 2015 by Sharma et al,4 and it used a regular-
113
+ ization parameter to perform a standard cross-validation methodology. An appropriate collection
114
+ of characteristics is needed to evaluate medical data for illness prediction. Several evolutionary
115
+ algorithms have been used to find the best possible traits. Gravitational search and Elephant Herd
116
+ optimizations have recently been used to choose the best features.5 Another deep learning-based
117
+ model created by Kuruvilla, and Gunavathi, K. in 2014, an ANN-based cancer classification for
118
+ CT scans. Development of the statistical model used to classify the data was completed. Compared
119
+ to feed-forward networks feed-forward backpropagation networks are more accurate, according to
120
+ research. Classifier accuracy may be improved even more by using the skewness feature.6
121
+ Lung cancer detection categorization is becoming more and more popular due to the rapid advance-
122
+ ment of pattern recognition and image processing methods. Textural evaluation of thin-section CT
123
+ images has been used in the literature to help distinguish various obstructive lung disorders. At-
124
+ tenuation distribution statistics, acquisition-length parameter, and co-occurrence descriptor are all
125
+ included in 13-dimensional vectors of local textures information developed by Chabat et al.7 A
126
+ Bayesian classifier is used for feature segmentation. These five scalar metrics, max, entropy, en-
127
+ ergy, contrast, and homogeneity were recovered per each co-occurrence matrix to minimize the
128
+ feature vector’s dimensionality. The textural characteristics of Solitary Pulmonary Nodules dis-
129
+ covered by CT have been described and assessed by Yanjie Zhu et al.8 It took 300 generations for
130
+ 67 characteristics to be retrieved, however, only 25 features were picked. SVM-based classifiers
131
+ are used for classification. For Interstitial Lung Disease, Sang Cheol Park and colleagues9 used a
132
+ genetic algorithm to identify the best picture attributes (ILD). Hiram et al10 used the frequency do-
133
+ main, and SVM with RBF to classify lung nodule classifications. Solitary pulmonary nodules may
134
+ be automatically detected using an algorithm provided by Hong et al.11 True nodules are identified
135
+ and labeled on original images using an SVM classifier. The LIDC-IDRI images database was
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+ used by Antonio et al,12 to classify lung nodules. Ecological taxonomic diversity and taxonomic
137
+ distinctness indexes are used for classification using SVM.13 Results show a 98.11% accuracy rate.
138
+ The mesh grid region growth approach was used in CT to select and analyze just the pixels that
139
+ were most likely to be relevant to the diagnosis. The ILD status of all unselected pixels was deter-
140
+ mined to be negative. To recognize lung cancer cells, Zhi-Hua et al14 presented Neural Ensemble-
141
+ based Detection (NED), which makes use of an artificial neural network ensemble. Using this
142
+ technology, it is possible to accurately identify cancer cells. An algorithm developed by Hui Chen
143
+ et al,15 uses a Neural Networks Ensemble to construct the categorization of a lung nodule on a thin
144
+ section CT image (NNE). A model suggested by Aggarwal, Furquan, and Kalra16 is characterized
145
+ by normal lung architecture by which segmentation is done using the best possible thresholds. Ge-
146
+ 3
147
+
148
+ ometric, statistical, and grey level properties are used to extract features. Classification is done
149
+ using LDA. The accuracy is 84%, the sensitivity is 97%, and the specificity is 53%. An inference-
150
+ based approach to identify lung cancer nodules has been developed by Roy, Sirohi, and Patle.17 To
151
+ improve contrast, this technique employs grey transformations using an active contour model, the
152
+ image is segmented. Training the classifier is done by extracting features such as area, mean, major
153
+ axis, and minor axis length. Overall, the system’s accuracy rate is 94.12%. This approach has a
154
+ disadvantage in that it does not distinguish between benign cancers and those that are malignant.
155
+ Authors have used wavelet feature descriptors to classify lung nodules.18 One and two-level de-
156
+ compositions of wavelet transformations are used in this example. A total of 19 characteristics are
157
+ derived from each wavelet sub-band. SVM is used to distinguish between CT images that include
158
+ malignant nodules and those that do not.
159
+ 2
160
+ Material and Methods
161
+ In this section, we introduce our methods for Lung Nodules localization We use a one-stage-
162
+ method based on YOLOv5 detection , the methodology has been split into the following Subsec-
163
+ tions to explain the whole process of our method .
164
+ 2.1
165
+ Dataset
166
+ For this research, the dataset has been collected from LIDC-IDRI. In LIDC-IDRI image collection,
167
+ thoracic CT scans with marked-up annotated lesions are included. For the development, training,
168
+ and assessment of computer-assisted diagnostic (CAD) approaches for the detection and diagnosis
169
+ of lung cancer is a worldwide web-accessible resource One example of a public-private partnership
170
+ founded on consensus-based decision-making is this collaboration between the National Cancer
171
+ Institute, the Foundation for the National Institute of Health, and Food and Drug Administration
172
+ (FDA), which was spearheaded by NCI and supported by the FDA. This data collection, which
173
+ includes 239 Ct images for training and 41 images for validation. is a subset of the original dataset.
174
+ Some of the samples are given below in the following Fig.1.
175
+ Fig. 1: Samples from dataset LIDC-IDRI Lung Cancer
176
+ 2.2
177
+ Pre-Processing Data
178
+ Real-world data tends to be fragmentary, noisy, and inconclusive. This may lead to low-quality data
179
+ collection, which in turn can lead to low-quality models. Data Preprocessing offers procedures that
180
+ 4
181
+
182
+ LIDC-IDRI-0001
183
+ LightSpeed Plus
184
+ 1-January-2000
185
+ ST:2.50SL
186
+ ST
187
+ LittleEndianExplicit
188
+ Images:1/1
189
+ 400mA120.00kV
190
+ Series:
191
+ 3000566
192
+ WL:
193
+ -600WW:1600LIDC-IDRI-0003
194
+ LightSpeed16
195
+ 1-January-2008
196
+ ST:2.50S
197
+ T
198
+ LittleEndianExplicit
199
+ Images:1/1
200
+ 300mA120.00kV
201
+ Series:
202
+ 3000611
203
+ WL:
204
+ -600WW:1600may properly organize the data for better comprehension in the deep learning process to solve these
205
+ challenges. Data Preprocessing steps that have been used in this research study are given in the
206
+ following Fig.2.
207
+ Fig. 2: Preprocessing steps for images
208
+ 2.3
209
+ Model architecture
210
+ As discussed in the introduction in this research YOLOv5 model is used for feature extraction and
211
+ detection of lung nodules in CT scans. Let have a brief discussion about Yolov5 and its architecture.
212
+ 2.3.1
213
+ YOLOv5 for lung nodules localization
214
+ the whole structure of Yolov4 Optimal speed and accuracy of object detection19 is shown in Fig.3
215
+ and YOLOv5 illustration representation shown in Fig.4. the YOLO family of models consists of
216
+ three main components to every single-stage object detector, and YOLOv5 has its own three main
217
+ modules
218
+ Fig. 3: Overview of YOLOv5 building blocks model architecture
219
+ (1) Backbone Figure 3:it’s mostly used to extract the elements of the most significant feature
220
+ from the images that have been provided. Cross Stage Partial Networks(CSP) is the back-
221
+ bone of YOLOv5’s feature extraction, which uses them to extract an image’s most informa-
222
+ tive details
223
+ 5
224
+
225
+ input image
226
+ output
227
+ image
228
+ Gray
229
+ Noise
230
+ Edge
231
+ Filter
232
+ scale
233
+ Removal
234
+ DetectionBackbone: CSPDarknet
235
+ Neck: PANet
236
+ Head: Yolo Layer
237
+ BottleNeckCSP
238
+ Concat
239
+ BottleNeckCSP
240
+ Convlx1
241
+ input image
242
+ UpSample
243
+ Conv3×3 S2
244
+ Conv1×1
245
+ Concat
246
+ BottleNeckCSP
247
+ Final
248
+ BottleNeckCSP
249
+ Concat
250
+ BottleNeckCSP
251
+ Output
252
+ Convl×1
253
+ UpSample
254
+ Conv3x3 S2
255
+ Convl×1
256
+ Concat
257
+ SPP
258
+ BottleNeckCSP
259
+ BottleNeckCSP
260
+ Convl×1
261
+ CSP
262
+ Cross Stage Partial Netword
263
+ Conv
264
+ Convolutional Layer
265
+ SPP
266
+ Spatial pyramid pooling
267
+ Concat
268
+ Concatenate Function(2) Neck Figure 3: it used to create feature pyramids. Feature pyramids aid models in generaliz-
269
+ ing successfully when it comes to object scaling. It aids in the identification of the same item
270
+ at various scales and dimensions. Feature pyramids are quite valuable and can help models
271
+ perform effectively on data that has never been examined. It’s not only FPN, BiFPN, and
272
+ PANet that are used in feature pyramid models
273
+ (3) Head Figure 3: it has layers that generate predictions from anchor boxes on features and
274
+ generated final output vectors with probabilities, object classes scores, and bounding boxes.,
275
+ YOLOv5 uses the following choices for training20
276
+ Fig. 4: Model detection can be considered a regression problem. The image is divided into S * S grids in
277
+ which bounding boxes are predicted for each grid cell, along with their confidence value
278
+ 2.3.2
279
+ Training Model
280
+ During the training and validation process, a total of 270 CT Scan images are used of which 239
281
+ CT Scans are used for training and 41 are used for validations. For training, the Google Colab is
282
+ used which is an online platform for training models. Which provides 16GB GPU free for training.
283
+ The batch size was kept to 16 and the number of epochs was kept to 100. Splitting of data can be
284
+ seen in Fig.5 .
285
+ 6
286
+
287
+ Head
288
+ Bounding Boxes + confidence Score
289
+ Backbone
290
+ Neck
291
+ images
292
+ Extraction of
293
+ Elaboration in
294
+ informative
295
+ Featuer
296
+ Labels
297
+ features
298
+ Pyramids
299
+ S x S Grid on input
300
+ image
301
+ Bounding Boxes + confidence Score
302
+ Localization of Lung Nodule
303
+ Class Probability
304
+ MapFig. 5: Dataset Splitting Diagram CT Scan images
305
+ 3
306
+ Results
307
+ the model had initial leverage to train faster and predict the location of lung nodules and demarcates
308
+ the relevant CT scan regions. before diving into the analysis of the results is necessary to explain
309
+ the statistical machine learning knowledge behind those results, the explanations have been split
310
+ into the following Subsections to explain the whole analysis of the method we use.
311
+ 3.1
312
+ Evaluation Metrics
313
+ In this section, we describe Charts of evaluation metrics that got from our experiment. It is known
314
+ to us that, in the computer Aide system, the main part is detecting the object inside the image.
315
+ Common metrics for measuring the performance of classification algorithms such as YOLOv5
316
+ that are based on CNN include, Recall, precision, F-score, mAP, PR curve, F1 curve , IOU,21
317
+ overlapping error, and boundary-based evaluation, the evaluation metrics we used is the mean
318
+ Average Precision (mAP),22 the precision, and F1-Curve. We will briefly explain them in the
319
+ following part. According to the theory of the statistical machine learning , precision is a two-
320
+ category statistical indicator whose formula is .
321
+ Precision : measures how accurate is our predictions was. the percentage of our predictions are
322
+ correct as shown in Fig.8,and following equation1.
323
+ Precision =
324
+ TP
325
+ TP + FP
326
+ (1)
327
+ Recall: measures how much of the true bbox were correctly predicted as shown in the following
328
+ equation.2.
329
+ Recall =
330
+ TP
331
+ TP + FN
332
+ (2)
333
+ 7
334
+
335
+ Total Data
336
+ Distrubtion CT Scan
337
+ 270Samples
338
+ Traning
339
+ Validation
340
+ images 239 Samples
341
+ images 41 Samplesmoreover, it is necessary to know TP, FP, and FN in the localization Nodules task.
342
+ (1) True positive (TP): IoU>[formula] (in this work, [formula] takes 0.2) the number of Local-
343
+ ization frames (the same Ground Truth is only calculated once23)
344
+ (2) False positive (FP): the number of check boxes for IoU<=[formula] or the number of re-
345
+ dundant check boxes that detect the same Ground Truth
346
+ (3) False negative (FN): the number of Ground Truths not detected
347
+ the IoU is a measures of the degree of overlap between two boundaries. We use that to measure
348
+ how much our predicted frame overlaps with the ground truth (the actual ground frame) ,the IOU
349
+ is shown with Fig.7 as follows, and the formula is as following Fig.6:
350
+ Fig. 6: Graphical representation of the Intersection over Union (IoU=0.2) calculation on a narrow-band
351
+ imaging. The light blue rectangle represents the ground truth bounding box, while the red rectangle repre-
352
+ sents the model prediction. The IoU is calculated by dividing the overlap area by the total area of union
353
+ after getting familiar with these definitions of statistical learning formulas, we introduce the mAP
354
+ (mean Average Precision). The mAP compares the ground-truth bounding box to the detected box
355
+ and returns a score. The higher the score, the more accurate the model is in its detections.
356
+ F1-score is defined as the harmonic average of precision and recall as shown in figure 10a:
357
+ F1 Score = 2 ∗ Precision ∗ Recall
358
+ Precision + Recall
359
+ (3)
360
+ 8
361
+
362
+ LIDC-IDRI-0003
363
+ LightSpeed16
364
+ 1-January-20e6
365
+ Ground truth
366
+ intersection
367
+ 0
368
+ area of overlap
369
+ Prediction
370
+ Iou =
371
+ area of union
372
+ Ground truth
373
+ Ground truth
374
+ Prediction
375
+ ST:2.58
376
+ Prediction
377
+ ittleEndianExplicit
378
+ ges: 1/1
379
+ 300mA120.00kl
380
+ WL:-600hW:1606(a) the Predicted location of a
381
+ Single Nodule
382
+ (b) the Predicted location of
383
+ tow Nodules in region
384
+ Fig. 7: Example of output Results
385
+ 3.2
386
+ Experiment’s Setting
387
+ the set Hyper-parameters of our fine tuning model are shown in Table 1, Our experiment uses
388
+ Pytorch framework deep learning on GPU Tesla K80 by Google open Platform Colab-research .
389
+ Parameters
390
+ Value
391
+ Batch size
392
+ 16
393
+ Image size
394
+ 416
395
+ Epoch
396
+ 145
397
+ Learning rate
398
+ 0.01
399
+ Optimizer
400
+ SGD
401
+ Table 1: Parameters and their value.
402
+ 3.3
403
+ Experiment’s Result and Analysis
404
+ To check the model’s predictions, and generalizations a few evaluation parameters must be tracked
405
+ during training and validation. There are several criteria to keep in mind while evaluating a box
406
+ loss, Precision, and recall values. The variable box benefits from objectivity and categorization.
407
+ Fig.9 shows all of the graphs that were used for this work. And Figure 10a shows the F1 indicator
408
+ training process for a single category that we want to be detected. The F1 score tends to be 0 with
409
+ increasing confidence . Training and validation box losses are reduced Fig.9, suggesting that the
410
+ model is sound good. the mAP is the abbreviation of median accuracy performances. The high
411
+ number indicates that this parameter is correct 92.27% as shown blow in Fig.8.
412
+ 9
413
+
414
+ THORAXW/OCONTRAST
415
+ LightSpeed1e
416
+ 1-January-2000 9:01:09
417
+ Nodules 0.76
418
+ ST:2.502
419
+ ST
420
+ LittleEndianExplicit
421
+ Images:1/1
422
+ 265mA120.00kV
423
+ Series:
424
+ WL:
425
+ -600WW:1600LIDC-IDRI-0011
426
+ LightSpeed16
427
+ 1-January-200e
428
+ Nodules 0.33
429
+ Nodules 0.78
430
+ ST:2.50SL:
431
+ ST
432
+ LittleEndianExplicit
433
+ Images:1/1
434
+ 265mA120.00kV
435
+ Series:
436
+ 3000559
437
+ WL:
438
+ -600wW:1600(a) mean Average Precision Evaluation
439
+ (b) Precision Evaluation
440
+ Fig. 8: the important Evaluation Metrics
441
+ (a) the training confidence of object pres-
442
+ ence loss
443
+ (b) the validation confidence of object pres-
444
+ ence loss
445
+ (c) the training bounding box regression
446
+ loss
447
+ (d) the validation bounding box regression
448
+ loss
449
+ Fig. 9: Results of feature extraction training and validation
450
+ Precision is needed to determine how accurate the model forecasts are 92.82% following the figure
451
+ 8. Only excellent results may be achieved by using the recall method. the model performance
452
+ showed a good benefit of using Hyper-parameter tuning to make better Learning from data samples
453
+ and generalize good knowledge from distribution can be seen the following Fig.9 . Due to the
454
+ importance of both precision and recall, there is a precision-recall curve the shows the tradeoff
455
+ between the precision and recall values for different thresholds. This curve helps to select the best
456
+ threshold to maximize both metrics, tin the following Fig.10b
457
+ 10
458
+
459
+ metrics/mAP 0.5
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+ 0.8
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+ 0.6
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+ 0.4
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+ 0.2
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+ Epoch
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+ 0
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+ 0
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+ 20
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+ 40
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+ 60
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+ 80
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+ 100
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+ 120
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+ 140metrics/precision
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+ 0.8
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+ 0.6
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+ 0.4
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+ 0.2
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+ Epoch
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+ 0
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+ 0
481
+ 20
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+ 40
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+ 60
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+ 80
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+ 100
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+ 120
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+ 140train/obj_loss
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+ 0.012
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+ 0.01
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+ 0.008
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+ 0.006
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+ 0.004
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+ 0.002
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+ Epoch
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+ 0
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+ 0
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+ 20
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+ 40
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+ 60
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+ 80
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+ 100
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+ 120
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+ 140val/obj_loss
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+ 0.012
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+ 0.01
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+ 0.008
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+ 0.006
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+ 0.004
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+ 0.002
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+ Epoch
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+ 0
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+ 20
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+ 40
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+ 60
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+ 80
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+ 100
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+ 120
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+ 0
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+ 140train/box_loss
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+ 0.12
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+ 0.1
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+ 0.08
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+ 0.06
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+ 0.04
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+ 0.02
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+ Epoch
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+ 0
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+ 0
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+ 20
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+ 40
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+ 60
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+ 80
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+ 100
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+ 120
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+ 140val/box_loss
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+ 0.1
537
+ 0.08
538
+ 0.06
539
+ 0.04
540
+ 0.02
541
+ Epoch
542
+ 0
543
+ 0
544
+ 20
545
+ 40
546
+ 60
547
+ 80
548
+ 100
549
+ 120
550
+ 140(a) F1 indicator training process for single
551
+ category
552
+ (b) Precision — Recall Curve of the valida-
553
+ tion data
554
+ Fig. 10: the important Evaluation Metrics
555
+ 4
556
+ Discussion Ana Conclusion
557
+ This research examined at how an AI model can help readers detect viewable lung cancer in Ct
558
+ images. Residents identified more viewable lung cancer when AI was being used as a second
559
+ reader.In this research, the dataset has been collected from LIDC-IDRI. In LIDC-IDRI image col-
560
+ lection, thoracic CT scans with marked-up annotated lesions are included. Yolov5 model is used
561
+ for feature extraction and detection of lung nodules in CT scans.During the training and validation
562
+ process, a total of 270 CT Scan images are used of which 239 CT Scans are used for training and
563
+ 41 are used for validations. In this study, the model’s performance was assessed using accuracy,
564
+ precision, and recall. The accuracy metric indicates how well the model recognised both positive
565
+ and negative instances. The precision metric measures how well the model predicts both negative
566
+ and positive cases. The model’s high accuracy, precision, and recall imply that it has a small error
567
+ possibility. Our findings imply that the AI technique assists low experienced individuals in terms
568
+ of recall while benefiting more-experienced audience in terms of precision. Previous research has
569
+ revealed that inexperienced readers are more likely to overlook lung malignancies, particularly le-
570
+ sions with a limited visibility score. In this research LIDC-IDRI dataset is used which have lung
571
+ nodules in it. The purpose of this study was to identify the nodule that were developing in the
572
+ lungs of the participants. It was difficult to find information on lung nodules in medical literature.
573
+ Research in the medical field often use deep learning. Deep learning will be utilised to develop
574
+ an algorithm with the support of previous medical imaging research, according to the findings of
575
+ a literature review. Using over 270 CT images , we were able to classify and identify nodules
576
+ using a deep learning algorithm. Using medical images analysis based on deep neural networks,
577
+ this study found that as much as 92.27% of cancer could be detected. Nodules on radiographs are
578
+ easier to see with its help. Using this technology in the future will help treat illnesses including
579
+ brain tumours and breast cancer.
580
+ 5
581
+ Disclosures
582
+ The authors declare that they have no conflict of interest
583
+ 6
584
+ Acknowledgments
585
+ We would like to thank our respectful research assistant Moath Alawaqla, for his distinguished role
586
+ of data collection.
587
+ 11
588
+
589
+ 1.0
590
+ Nodules
591
+ all classes 0.91 at 0.437
592
+ 0.8
593
+ 0.6
594
+ 0.4
595
+ 0.2
596
+ 0.0 +
597
+ 0.0
598
+ 0.2
599
+ 0.4
600
+ 0.6
601
+ 0.8
602
+ 1.0
603
+ Confidence1.0
604
+ Nodules 0.923
605
+ all classes 0.923 mAP@0.5
606
+ 0.8
607
+ 0.6
608
+ Precision
609
+ 0.4
610
+ 0.2
611
+ 0.0
612
+ 0.0
613
+ 0.2
614
+ 0.4
615
+ 0.6
616
+ 0.8
617
+ 1.0
618
+ Recall7
619
+ Funding
620
+ This work supported by Jordan University of Science and Technology, Irbid-Jordan,
621
+ References
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+ 17 T. S. Roy, N. Sirohi, and A. Patle, “Classification of lung image and nodule detection us-
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+ 19 A. Bochkovskiy, C.-Y. Wang, and H.-Y. M. Liao, “Yolov4: Optimal speed and accuracy of
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+ 20 M. Kasper-Eulaers, N. Hahn, S. Berger, et al., “Detecting heavy goods vehicles in rest areas
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+ in winter conditions using yolov5,” Algorithms 14(4), 114 (2021).
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+ 21 D. Zhou, J. Fang, X. Song, et al., “Iou loss for 2d/3d object detection,” in 2019 International
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
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ Parametric “Non-nested” Discriminants
2
+ for Multiplicities of Univariate Polynomials
3
+ Hoon Hong
4
+ Department of Mathematics, North Carolina State University
5
+ Box 8205, Raleigh, NC 27695, USA
6
+ hong@ncsu.edu
7
+ Jing Yang∗
8
+ SMS–HCIC–School of Mathematics and Physics,
9
+ Center for Applied Mathematics of Guangxi,
10
+ Guangxi Minzu University, Nanning 530006, China
11
+ yangjing0930@gmail.com
12
+ Abstract
13
+ We consider the problem of complex root classification, i.e., finding the conditions on the coefficients
14
+ of a univariate polynomial for all possible multiplicity structures on its complex roots. It is well known
15
+ that such conditions can be written as conjunctions of several polynomial equations and one inequation
16
+ in the coefficients. Those polynomials in the coefficients are called discriminants for multiplicities. It
17
+ is well known that discriminants can be obtained by using repeated parametric gcd’s. The resulting
18
+ discriminants are usually nested determinants, that is, determinants of matrices whose entries are deter-
19
+ minants, and so son. In this paper, we give a new type of discriminants which are not based on repeated
20
+ gcd’s. The new discriminants are simpler in that they are non-nested determinants and have smaller
21
+ maximum degrees.
22
+ 1
23
+ Introduction
24
+ In this paper, we consider the problem of complex root classification, i.e., finding the conditions on the
25
+ coefficients of a polynomial over the complex field C for every potential multiplicity structure its complex
26
+ roots may have. For example, consider a quintic polynomial F = a5x5 +a4x4 +a3x3 +a2x2 +a1x+a0 where
27
+ ai’s take values over C. We would like to find conditions C0, C1, . . . , C6 on a = (a0, . . . , a5) such that
28
+ multiplicity structure of F =
29
+
30
+
31
+
32
+
33
+
34
+
35
+
36
+
37
+
38
+
39
+
40
+
41
+
42
+
43
+
44
+
45
+
46
+
47
+
48
+ (1, 1, 1, 1, 1)
49
+ if
50
+ C0 (a) holds
51
+ (2, 1, 1, 1)
52
+ if
53
+ C1 (a) holds
54
+ (2, 2, 1)
55
+ if
56
+ C2 (a) holds
57
+ (3, 1, 1)
58
+ if
59
+ C3 (a) holds
60
+ (3, 2)
61
+ if
62
+ C4 (a) holds
63
+ (4, 1)
64
+ if
65
+ C5 (a) holds
66
+ (5)
67
+ if
68
+ C6 (a) holds
69
+ In general, the problem is stated as follows:
70
+ Problem: For every µ = (µ1, . . . , µm) such that µ1 ≥ . . . ≥ µm > 0 and µ1 +· · ·+µm = n, find a condition
71
+ on the coefficients of a polynomial Fover C of degree n such that the multiplicity structure of F is µ.
72
+ ∗Corresponding author.
73
+ 1
74
+ arXiv:2301.00315v1 [cs.SC] 1 Jan 2023
75
+
76
+ The problem is important because many tasks in mathematics, science and engineering can be reduced to
77
+ the problem. Due to its importance, the problem and several related problems have been already carefully
78
+ studied [4, 6, 7, 8, 9, 11].
79
+ The problem can be viewed as a generalization of a well known problem of finding a condition on
80
+ coefficients such that the polynomial has a given number of distinct roots.
81
+ This subproblem has been
82
+ extensively studied. For instance, the subdiscriminant theory provides a complete solution to the subproblem:
83
+ a univariate polynomial of degree n has m distinct roots if and only if its 0-th, . . ., (n − m − 1)-th psd’s
84
+ (i.e., principal subdiscriminant coefficient) vanish and the (n−m)-th psd does not. For details, see standard
85
+ textbooks on computational algebra (e.g., [1]).
86
+ In [11], Yang, Hou and Zeng gave an algorithm to generate conditions for discriminating different mul-
87
+ tiplicity structures of a univariate polynomial (referred as YHZ’s condition hereinafter) by making use of
88
+ repeated gcd computation for parametric polynomials [2, 3, 10]. It is based on a similar idea adopted by
89
+ Gonzalez-Vega et al. [4] for solving the real root classification and quantifier elimination problems by using
90
+ Sturm-Habicht sequences. The conditions produced by these methods are conjunctions of several polynomial
91
+ equations and one inequation on the coefficients. Those polynomials in he coefficients are called discrimi-
92
+ nants for multiplicities. The maximum degree of the discriminants grow exponentially in the degree of F.
93
+ Furthermore, each discriminant is a “nested” determinant, that is, it is a determinant of a matrix whose
94
+ entries are again determinants and so on.
95
+ In [6], the authors developed a new type of multiplicity discriminants to distinguish different multiplicities
96
+ when the number of distinct roots is fixed. The main idea is to convert the multiplicity condition expressed
97
+ as a permanent inequation in roots into a sum of determinants in coefficients. In order to generate conditions
98
+ for all the possible multiplicity structures of a univariate polynomial, one may first use subdiscriminants in
99
+ classical resultant theory to decide the number of distinct complex roots and then add one more inequation
100
+ to discriminate different multiplicity structures with the same number of distinct roots. In the new condition,
101
+ the maximum degree of the discriminants grow linearly in the degree of F, which makes the size of discrim-
102
+ inants significantly smaller. However, the form of resulting discriminants is a sum of many determinants,
103
+ which makes the further analysis (reasoning) difficult.
104
+ The main contribution in this paper is to provide a new type of discriminants, which are non-nested
105
+ determinants and whose max degrees are smaller than those in the previous methods.
106
+ The method is
107
+ based on a significantly different theory and techniques from the previous methods (which are essentially
108
+ based on repeated parametric gcd or subdiscriminant theory). The new condition is given by a newly devised
109
+ multiplicity discriminant in coefficients for every potential multiplicity vector of a given degree, which can be
110
+ viewed as a generalization of subdiscriminant theory to higher order derivatives. To build up the connection
111
+ between the new discriminants and multiple roots, we first convert it into the ratio of two determinants
112
+ in terms of generic roots (without considering the multiplicities). Then by making use of the connection
113
+ between divided difference with multiple nodes and the derivatives of higher orders at the nodes, we integrate
114
+ the multiplicity information into the expression and convert it into an expression in terms of multiple roots.
115
+ After careful manipulation, it is shown that the new discriminant can capture the multiplicity information.
116
+ The paper is structured as follows. In Section 2, we first present the problem to be solved in a formal
117
+ way. In Section 3, we give a precise statement of the main result of the paper (Theorem 9). Then a proof
118
+ of Theorem 9 is provided in Section 4. The proof is long thus we divide the proof into three subsections
119
+ which are interesting on their own.
120
+ In Section 5, we compare the form and size of polynomials in the
121
+ multiplicity-discriminant condition in Theorem 9 and those given by previous works.
122
+ 2
123
+ Problem
124
+ Definition 1 (Multiplicity vector). Let F ∈ C [x] with m distinct complex roots, say r1, . . . , rm, with mul-
125
+ tiplicities µ1, . . . , µm respectively. Without losing generality, we assume that µ1 ≥ · · · ≥ µm > 0. Then the
126
+ multiplicity vector of F, written as mult (F), is defined by
127
+ mult (F) = (µ1, . . . , µm)
128
+ 2
129
+
130
+ Example 2. Let F = x5 − 5x4 + 7x3 + x2 − 8x + 4. Then mult (F) = (2, 2, 1), since it can be verified that
131
+ F = (x − 1)2 (x + 1)1 (x − 2)2. Note that the multiplicity vector is a partition of 5, which is the degree of F.
132
+ Definition 3 (Potential multiplicity vectors). Let n be a positive integer. Let M(n) stand for the set of all
133
+ the potential multiplicity vectors of polynomials of degree n, equivalently, the set of all partitions of n, that
134
+ is,
135
+ M(n) = {(µ1, . . . , µm) : µ1 + · · · + µm = n, µ1 ≥ · · · ≥ µm > 0}
136
+ Example 4. M (5) = { (1, 1, 1, 1, 1) ,
137
+ (2, 1, 1, 1) , (2, 2, 1) , (3, 1, 1) , (3, 2) , (4, 1) , (5) }.
138
+ Problem 5 (Parametric multiplicity problem). The parametric multiplicity problem is stated as:
139
+ In : n, a positive integer standing for the polynomial of degree n with parametric coefficients a, that is,
140
+ F =
141
+ n
142
+
143
+ i=0
144
+ aixi where an ̸= 0
145
+ Out: For each µ ∈ M(n), find a condition Cµ on a such that mult (F) = µ.
146
+ 3
147
+ Main Result
148
+ Definition 6 (Determinant polynomial). Consider a vector of univariate polynomials
149
+ P =
150
+
151
+ ��
152
+ P0
153
+ ...
154
+ Pk
155
+
156
+ �� ∈ C[x]k+1
157
+ where deg Pi ≤ k and Pi = �
158
+ 0≤j≤k aijxj. The coefficient matrix of P, written as C (P) , is defined by
159
+ C (P) = coef (P) =
160
+
161
+ ��
162
+ coef (P0)
163
+ ...
164
+ coef (Pk)
165
+
166
+ �� =
167
+
168
+ ��
169
+ a0k
170
+ · · ·
171
+ a00
172
+ ...
173
+ ...
174
+ akk
175
+ · · ·
176
+ ak0
177
+
178
+ ��
179
+ The determinant polynomial of P, written as dp (P) , is defined by
180
+ dp (P) = |C (P) |
181
+ Definition 7 (Multiplicity Discriminant). Let F = �n
182
+ i=0 aixi where an ̸= 0. Let γ = (γ1, . . . , γs) ∈ M (n).
183
+ The the γ-discriminant of F, written as D (γ) , is defined by
184
+ D (γ) = 1
185
+ an
186
+ dp
187
+
188
+ ��������������������
189
+ F (0)xγ0−1
190
+ ...
191
+ F (0)x0
192
+ F (1)xγ1−1
193
+ ...
194
+ F (1)x0
195
+ ...
196
+ F (s)xγs−1
197
+ ...
198
+ F (s)x0
199
+
200
+ ��������������������
201
+ where γ0 is the smallest so that the above matrix is square. It is straightforward to show that γ0 = γ1 − 1.
202
+ 3
203
+
204
+ Example 8. Let n = 5 and F = �n
205
+ i=0 aixi and an ̸= 0. Then
206
+ D (5)
207
+ =
208
+ dp
209
+
210
+ �������������
211
+ F (0)x3
212
+ F (0)x2
213
+ F (0)x1
214
+ F (0)x0
215
+ F (1)x4
216
+ F (1)x3
217
+ F (1)x2
218
+ F (1)x1
219
+ F (1)x0
220
+
221
+ �������������
222
+ =
223
+ 1
224
+ a5
225
+ ������������������
226
+ a5
227
+ a4
228
+ a3
229
+ a2
230
+ a1
231
+ a0
232
+ a5
233
+ a4
234
+ a3
235
+ a2
236
+ a1
237
+ a0
238
+ a5
239
+ a4
240
+ a3
241
+ a2
242
+ a1
243
+ a0
244
+ a5
245
+ a4
246
+ a3
247
+ a2
248
+ a1
249
+ a0
250
+ 5a5
251
+ 4a4
252
+ 3a3
253
+ 2a2
254
+ 1a1
255
+ 5a5
256
+ 4a4
257
+ 3a3
258
+ 2a2
259
+ 1a1
260
+ 5a5
261
+ 4a4
262
+ 3a3
263
+ 2a2
264
+ 1a1
265
+ 5a5
266
+ 4a4
267
+ 3a3
268
+ 2a2
269
+ 1a1
270
+ 5a5
271
+ 4a4
272
+ 3a3
273
+ 2a2
274
+ 1a1
275
+ ������������������
276
+ D (4, 1)
277
+ =
278
+ dp
279
+
280
+ �����������
281
+ F (0)x2
282
+ F (0)x1
283
+ F (0)x0
284
+ F (1)x3
285
+ F (1)x2
286
+ F (1)x1
287
+ F (1)x0
288
+ F (2)x0
289
+
290
+ �����������
291
+ =
292
+ 1
293
+ a5
294
+ ����������������
295
+ a5
296
+ a4
297
+ a3
298
+ a2
299
+ a1
300
+ a0
301
+ a5
302
+ a4
303
+ a3
304
+ a2
305
+ a1
306
+ a0
307
+ a5
308
+ a4
309
+ a3
310
+ a2
311
+ a1
312
+ a0
313
+ 5a5
314
+ 4a4
315
+ 3a3
316
+ 2a2
317
+ 1a1
318
+ 5a5
319
+ 4a4
320
+ 3a3
321
+ 2a2
322
+ 1a1
323
+ 5a5
324
+ 4a4
325
+ 3a3
326
+ 2a2
327
+ 1a1
328
+ 5a5
329
+ 4a4
330
+ 3a3
331
+ 2a2
332
+ 1a1
333
+ 5 · 4a5
334
+ 4 · 3a4
335
+ 3 · 2a3
336
+ 2 · 1a2
337
+ ����������������
338
+ D (3, 2)
339
+ =
340
+ dp
341
+
342
+ ���������
343
+ F (0)x1
344
+ F (0)x0
345
+ F (1)x2
346
+ F (1)x1
347
+ F (1)x0
348
+ F (2)x1
349
+ F (2)x0
350
+
351
+ ���������
352
+ =
353
+ 1
354
+ a5
355
+ ��������������
356
+ a5
357
+ a4
358
+ a3
359
+ a2
360
+ a1
361
+ a0
362
+ a5
363
+ a4
364
+ a3
365
+ a2
366
+ a1
367
+ a0
368
+ 5a5
369
+ 4a4
370
+ 3a3
371
+ 2a2
372
+ 1a1
373
+ 5a5
374
+ 4a4
375
+ 3a3
376
+ 2a2
377
+ 1a1
378
+ 5a5
379
+ 4a4
380
+ 3a3
381
+ 2a2
382
+ 1a1
383
+ 5 · 4a5
384
+ 4 · 3a4
385
+ 3 · 2a3
386
+ 2 · 1a2
387
+ 5 · 4a5
388
+ 4 · 3a4
389
+ 3 · 2a3
390
+ 2 · 1a2
391
+ ��������������
392
+ D (3, 1, 1)
393
+ =
394
+ dp
395
+
396
+ ���������
397
+ F (0)x1
398
+ F (0)x0
399
+ F (1)x2
400
+ F (1)x1
401
+ F (1)x0
402
+ F (2)x0
403
+ F (3)x0
404
+
405
+ ���������
406
+ =
407
+ 1
408
+ a5
409
+ ��������������
410
+ a5
411
+ a4
412
+ a3
413
+ a2
414
+ a1
415
+ a0
416
+ a5
417
+ a4
418
+ a3
419
+ a2
420
+ a1
421
+ a0
422
+ 5a5
423
+ 4a4
424
+ 3a3
425
+ 2a2
426
+ 1a1
427
+ 5a5
428
+ 4a4
429
+ 3a3
430
+ 2a2
431
+ 1a1
432
+ 5a5
433
+ 4a4
434
+ 3a3
435
+ 2a2
436
+ 1a1
437
+ 5 · 4a5
438
+ 4 · 3a4
439
+ 3 · 2a3
440
+ 2 · 1a2
441
+ 5 · 4 · 3a5
442
+ 4 · 3 · 2a4
443
+ 3 · 2 · 1a3
444
+ ��������������
445
+ D (2, 2, 1)
446
+ =
447
+ dp
448
+
449
+ �������
450
+ F (0)x0
451
+ F (1)x1
452
+ F (1)x0
453
+ F (2)x1
454
+ F (2)x0
455
+ F (3)x0
456
+
457
+ �������
458
+ =
459
+ 1
460
+ a5
461
+ ������������
462
+ a5
463
+ a4
464
+ a3
465
+ a2
466
+ a1
467
+ a0
468
+ 5a5
469
+ 4a4
470
+ 3a3
471
+ 2a2
472
+ 1a1
473
+ 5a5
474
+ 4a4
475
+ 3a3
476
+ 2a2
477
+ 1a1
478
+ 5 · 4a5
479
+ 4 · 3a4
480
+ 3 · 2a3
481
+ 2 · 1a2
482
+ 5 · 4a5
483
+ 4 · 3a4
484
+ 3 · 2a3
485
+ 2 · 1a2
486
+ 5 · 4 · 3a5
487
+ 4 · 3 · 2a4
488
+ 3 · 2 · 1a3
489
+ ������������
490
+ D (2, 1, 1, 1)
491
+ =
492
+ dp
493
+
494
+ �������
495
+ F (0)x0
496
+ F (1)x1
497
+ F (1)x0
498
+ F (2)x0
499
+ F (3)x0
500
+ F (4)x0
501
+
502
+ �������
503
+ =
504
+ 1
505
+ a5
506
+ ������������
507
+ a5
508
+ a4
509
+ a3
510
+ a2
511
+ a1
512
+ a0
513
+ 5a5
514
+ 4a4
515
+ 3a3
516
+ 2a2
517
+ 1a1
518
+ 5a5
519
+ 4a4
520
+ 3a3
521
+ 2a2
522
+ 1a1
523
+ 5 · 4a5
524
+ 4 · 3a4
525
+ 3 · 2a3
526
+ 2 · 1a2
527
+ 5 · 4 · 3a5
528
+ 4 · 3 · 2a4
529
+ 3 · 2 · 1a3
530
+ 5 · 4 · 3 · 2a5
531
+ 4 · 3 · 2 · 1a4
532
+ ������������
533
+ D (1, 1, 1, 1, 1)
534
+ =
535
+ dp
536
+
537
+ �����
538
+ F (1)x0
539
+ F (2)x0
540
+ F (3)x0
541
+ F (4)x0
542
+ F (5)x0
543
+
544
+ �����
545
+ =
546
+ 1
547
+ a5
548
+ ����������
549
+ 5a5
550
+ 4a4
551
+ 3a3
552
+ 2a2
553
+ 1a1
554
+ 5 · 4a5
555
+ 4 · 3a4
556
+ 3 · 2a3
557
+ 2 · 1a2
558
+ 5 · 4 · 3a5
559
+ 4 · 3 · 2a4
560
+ 3 · 2 · 1a3
561
+ 5 · 4 · 3 · 2a5
562
+ 4 · 3 · 2 · 1a4
563
+ 5 · 4 · 3 · 2 · 1a5
564
+ ����������
565
+ 4
566
+
567
+ Note that the last one D (1, 1, 1, 1, 1) = 5544332211a4
568
+ 5. Since a5 ̸= 0, we see that D (1, 1, 1, 1, 1) ̸= 0.
569
+ Theorem 9 (Main Result). Let F = �n
570
+ i=0 aixi
571
+ where an ̸= 0. Let M(n) =
572
+
573
+ µ0, µ1, . . . , µp
574
+
575
+ where the
576
+ entries are ordered in the lexicographically increasing order, that is, µ0 ≺lex µ1 ≺lex · · · ≺lex µp. Then we
577
+ have the following conditions for the multiplicity vectors.
578
+ mult(F) =
579
+
580
+
581
+
582
+
583
+
584
+
585
+
586
+
587
+
588
+ µ0
589
+ if
590
+ D
591
+
592
+ µp
593
+
594
+ ̸= 0
595
+ µ1
596
+ else if
597
+ D
598
+
599
+ µp−1
600
+
601
+ ̸= 0
602
+ ...
603
+ ...
604
+ ...
605
+ ̸= 0
606
+ µp
607
+ else if
608
+ D (µ0)
609
+ ̸= 0
610
+ Equivalently,
611
+ mult(F) = µi
612
+ ⇐⇒
613
+ D
614
+
615
+ µp
616
+
617
+ = · · · = D
618
+
619
+ µp−i−1
620
+
621
+ = 0 ∧ D
622
+
623
+ µp−i
624
+
625
+ ̸= 0
626
+ Example 10. We have the following condition for each multiplicity vector for degree 5.
627
+ mult(F) =
628
+
629
+
630
+
631
+
632
+
633
+
634
+
635
+
636
+
637
+
638
+
639
+
640
+
641
+
642
+
643
+
644
+
645
+
646
+
647
+ (1, 1, 1, 1, 1)
648
+ if
649
+ D (5)
650
+ ̸= 0
651
+ (2, 1, 1, 1)
652
+ else if
653
+ D (4, 1)
654
+ ̸= 0
655
+ (2, 2, 1)
656
+ else if
657
+ D (3, 2)
658
+ ̸= 0
659
+ (3, 1, 1)
660
+ else if
661
+ D (3, 1, 1)
662
+ ̸= 0
663
+ (3, 2)
664
+ else if
665
+ D (2, 2, 1)
666
+ ̸= 0
667
+ (4, 1)
668
+ else if
669
+ D (2, 1, 1, 1)
670
+ ̸= 0
671
+ (5)
672
+ else if
673
+ D (1, 1, 1, 1, 1)
674
+ ̸= 0
675
+ Equivalently, for instance,
676
+ mult(F) = (2, 2, 1)
677
+ ⇐⇒
678
+ D (5) = D (4, 1) = 0 ∧ D (3, 2) ̸= 0
679
+ Remark 11.
680
+ 1. Note that µ0 = (1, . . . , 1) and
681
+ D (µ0) = 1
682
+ an
683
+ ���������
684
+ nan
685
+ · · ·
686
+ 1a1
687
+ n (n − 1) an
688
+ · · ·
689
+ 2 · 1a2
690
+ ...
691
+ ...
692
+ n (n − 1) · · · 1an
693
+ ���������
694
+ =
695
+ n
696
+
697
+ i=1
698
+ ii · an−1
699
+ n
700
+ ̸= 0
701
+ Hence the last condition is always satisfied and there is no need to check the condition.
702
+ 2. Note that µi and µp−i are conjugates of each other.
703
+ 4
704
+ Proof of the Main Theorem
705
+ Here is a high level view of the proof.
706
+ We start with converting D (µ) into the equivalent symmetric
707
+ polynomials in generic roots (though displayed as a ratio of two determinants) which is easier to embed the
708
+ multiplicity information. Then by making use of the connection between divided difference with multiple
709
+ nodes and the derivatives of higher orders at the nodes, we convert the expression in generic roots to that
710
+ in distinct roots with multiplicity information integrated. The theorem will be proved by eliminating the
711
+ entries in the determinantal expression obtained from the second stage which may vanish under the given
712
+ multiplicity structure.
713
+ 5
714
+
715
+ 4.1
716
+ Multiplicity discriminant in terms of roots
717
+ We first understand what the multiplicity discriminants look like in terms of roots. .
718
+ Notation 12. V (α1, . . . , αn) :=
719
+ �������
720
+ αn−1
721
+ 1
722
+ · · ·
723
+ αn−1
724
+ n
725
+ ...
726
+ ...
727
+ α0
728
+ 1
729
+ · · ·
730
+ α0
731
+ n
732
+ �������
733
+ Lemma 13 (Multiplicity discriminant in generic roots). Let F = an(x−α1) · · · (x−αn) and γ = (γ1, . . . , γs) ∈
734
+ M(n). Then
735
+ D(γ) =
736
+ aγ1−2
737
+ n
738
+ ·
739
+ ������������������
740
+ F (1)(α1)αγ1−1
741
+ 1
742
+ · · ·
743
+ F (1)(αn)αγ1−1
744
+ n
745
+ ...
746
+ ...
747
+ F (1)(α1)α0
748
+ 1
749
+ · · ·
750
+ F (1)(αn)α0
751
+ n
752
+ ...
753
+ ...
754
+ F (s)(α1)αγs−1
755
+ 1
756
+ · · ·
757
+ F (s)(αn)αγs−1
758
+ n
759
+ ...
760
+ ...
761
+ F (s)(α1)α0
762
+ 1
763
+ · · ·
764
+ F (s)(αn)α0
765
+ n
766
+ ������������������
767
+ V (α1, . . . , αn)
768
+ (1)
769
+ Proof.
770
+ 1. Since γ1 ≥ · · · ≥ γs and γ0 = γ1 − 1, we have
771
+ deg(F (0)xn−2) > · · · > deg(F (0)xγ1−1) > max(F (0)xγ0−1, F (1)xγ1−1, . . . , F (s)xγs−1)
772
+ Thus
773
+ D(γ) = 1
774
+ an
775
+ dp
776
+
777
+ ��������������������
778
+ F (0)xγ1−2
779
+ ...
780
+ F (0)x0
781
+ F (1)xγ1−1
782
+ ...
783
+ F (1)x0
784
+ ...
785
+ F (s)xγs−1
786
+ ...
787
+ F (s)x0
788
+
789
+ ��������������������
790
+ = 1
791
+ an
792
+ · aγ1−n
793
+ n
794
+ dp
795
+
796
+ ���������������������������
797
+ F (0)xn−2
798
+ ...
799
+ F (0)xγ1−1
800
+ F (0)xγ1−2
801
+ ...
802
+ F (0)x0
803
+ F (1)xγ1−1
804
+ ...
805
+ F (1)x0
806
+ ...
807
+ F (s)xγs−1
808
+ ...
809
+ F (s)x0
810
+
811
+ ���������������������������
812
+ = aγ1−n−1
813
+ n
814
+ dp
815
+
816
+ ��������������������
817
+ F (0)xn−2
818
+ ...
819
+ F (0)x0
820
+ F (1)xγ1−1
821
+ ...
822
+ F (1)x0
823
+ ...
824
+ F (s)xγs−1
825
+ ...
826
+ F (s)x0
827
+
828
+ ��������������������
829
+ 2. Now we recall the following result from [6] which is the key for proving the lemma. Let G1, . . . , Gn ∈
830
+ C [x]2n−2 where C [x]2n−2 consists of all the polynomials in x with degree no greater than 2n−2. Then
831
+ dp
832
+
833
+ ���������
834
+ F (0)xn−2
835
+ ...
836
+ F (0)x0
837
+ G1
838
+ ...
839
+ Gn
840
+
841
+ ���������
842
+ =
843
+ an−1
844
+ n
845
+ ·
846
+ �������
847
+ G1(α1)
848
+ · · ·
849
+ G1(αn)
850
+ ...
851
+ ...
852
+ Gn(α1)
853
+ · · ·
854
+ Gn (αn)
855
+ �������
856
+ V (α1, . . . , αn)
857
+ (2)
858
+ 6
859
+
860
+ 3. After specializing G1, . . . , Gn in (2) with F (1)xγ1−1, . . . , F (1)x0, . . . , F (s)xγs−1, . . . , F (s)x0, respectively,
861
+ we have
862
+ D(γ) = aγ1−n−1
863
+ n
864
+ ·
865
+ an−1
866
+ n
867
+ ·
868
+ �����������������
869
+
870
+ F (1)xγ1−1�
871
+ (α1)
872
+ · · ·
873
+
874
+ F (1)xγ1−1�
875
+ (αn)
876
+ ...
877
+ ...
878
+
879
+ F (1)x0�
880
+ (α1)
881
+ · · ·
882
+
883
+ F (1)x0�
884
+ (αn)
885
+ ...
886
+ ...
887
+
888
+ F (s)xγs−1�
889
+ (α1)
890
+ · · ·
891
+
892
+ F (s)xγs−1�
893
+ (αn)
894
+ ...
895
+ ...
896
+
897
+ F (s)x0�
898
+ (α1)
899
+ · · ·
900
+
901
+ F (s)x0�
902
+ (αn)
903
+ �����������������
904
+ V (α1, . . . , αn)
905
+ which can be easily simplified into (1).
906
+ Remark 14. It is very important to note that the right hand side is a polynomial function in α1, . . . , αn,
907
+ even though written as a rational function, since the numerator is exactly divisible by the denominator.
908
+ Hence the above definition should be read as follows:
909
+ 1. Treating α1, . . . , αn as distinct indeterminates, carry out the exact division obtaining a polynomial.
910
+ 2. Treating α1, . . . , αn as numbers, evaluate the resulting polynomial.
911
+ Lemma 15 (Multiplicity discriminant in multiple roots). Let F be of degree n with m distinct roots
912
+ r1, . . . , rm, of multiplicities µ1, . . . , µm, that is µ1 + · · · + µm = n. Let γ = (γ1, . . . , γs) ∈ Γ(n). Then
913
+ we have
914
+ D(γ) =
915
+ c ·
916
+ �����������������
917
+ (F (1)xγ1−1)(0)(r1) · · · (F (1)xγ1−1)(µ1−1)(r1) · · · · · · (F (1)xγ1−1)(0)(rm) · · · (F (1)xγ1−1)(µm−1)(rm)
918
+ ...
919
+ ...
920
+ ...
921
+ ...
922
+ (F (1)x0)(0)(r1)
923
+ · · · (F (1)x0)(µ1−1)(r1)
924
+ · · · · · · (F (1)x0)(0)(rm)
925
+ · · · (F (1)x0)(µm−1)(rm)
926
+ ...
927
+ ...
928
+ ...
929
+ ...
930
+ (F (s)xγs−1)(0)(r1) · · · (F (s)xγs−1)(µ1−1)(r1) · · · · · · (F (s)xγs−1)(0)(rm) · · · (F (s)xγs−1)(µm−1)(rm)
931
+ ...
932
+ ...
933
+ ...
934
+ ...
935
+ (F (s)x0)(0)(r1)
936
+ · · · (F (s)x0)(µ1−1)(r1)
937
+ · · · · · · (F (s)x0)(0)(rm)
938
+ · · · (F (s)x0)(µm−1)(rm)
939
+ �����������������
940
+
941
+ 1≤i<j≤m
942
+ (ri − rj)µiµj
943
+ (3)
944
+ where c = ±1
945
+ � ��m
946
+ i=1
947
+ �µi−1
948
+ j=0 j!
949
+
950
+ · aγ1−2
951
+ n
952
+ .
953
+ Proof.
954
+ 1. Let F = an(x − α1) · · · (x − αn). When α1, . . . , αn are treated as numbers, without loss of generality,
955
+ we may assume that α1, . . . , αn are grouped into m sets as follows:
956
+ S1 :=
957
+ {α1
958
+ · · ·
959
+ · · ·
960
+ · · ·
961
+ · · ·
962
+ αµ1}
963
+ S2 :=
964
+ {αµ1+1
965
+ · · ·
966
+ · · ·
967
+ · · ·
968
+ αµ1+µ2}
969
+ ...
970
+ Sm :=
971
+ {αµ1+···+µm−1+1
972
+ · · ·
973
+ αµ1+···+µm}
974
+ where elements in Si are all equal to ri.
975
+ 7
976
+
977
+ 2. Recall that
978
+ D(γ) = aγ1−2
979
+ n
980
+ ·
981
+ ���������������������
982
+ (F (1)xγ1−1)(α1)
983
+ · · ·
984
+ (F (1)xγ1−1)(αn)
985
+ ...
986
+ ...
987
+ (F (1)x0)(α1)
988
+ · · ·
989
+ (F (1)x0)(αn)
990
+ ...
991
+ ...
992
+ ...
993
+ ...
994
+ (F (s)xγs−1)(α1)
995
+ · · ·
996
+ (F (s)xγs−1)(αn)
997
+ ...
998
+ ...
999
+ (F (s)x0)(α1)
1000
+ · · ·
1001
+ (F (s)x0)(αn)
1002
+ ���������������������
1003
+
1004
+ V (α1, . . . , αn)
1005
+ Next we will treat α1, . . . , αn as indeterminates and carry out the exact division so that difference
1006
+ between the collapsed αi’s do not appear in the denominator.
1007
+ 3. For the sake of simplicity, we use the follow shorthand notion:
1008
+ F :=
1009
+
1010
+ F (1)xγ1−1, . . . , F (1)x0, . . . , F (s)xγs−1, . . . , F (s)x0�T
1011
+ 4. Let P[x1, . . . , xi] denote the (i − 1)th divided difference of P ∈ C[x] at x1, . . . , xi and let
1012
+ F [α1, . . . , αi] :=
1013
+
1014
+ (F (1)xγ1−1)[α1, . . . , αi], . . . , (F (1)x0)[α1, . . . , αi],
1015
+ . . . . . . , (F (s)xγs−1)[α1, . . . , αi], . . . , (F (s)x0)[α1, . . . , αi]
1016
+ �T
1017
+ 5. It follows that
1018
+ D(γ) = aγ1−2
1019
+ n
1020
+ ·
1021
+ �� F (α1)
1022
+ · · ·
1023
+ F (αn)
1024
+ ��
1025
+ V (α1, . . . , αn)
1026
+ = aγ1−2
1027
+ n
1028
+ ·
1029
+ �� F [α1]
1030
+ · · ·
1031
+ F [αµ1]
1032
+ F [αµ1+1]
1033
+ · · ·
1034
+ F [αn]
1035
+ ��
1036
+
1037
+ αi,αj∈S1
1038
+ j−i>0
1039
+ (αi − αj)
1040
+
1041
+ αi,αj /∈S1
1042
+ j−i>0
1043
+ (αi − αj)
1044
+
1045
+ αi∈S1
1046
+ αj /∈S1
1047
+ (αi − αj)
1048
+ = ±aγ1−2
1049
+ n
1050
+ ·
1051
+ �� F [α1]
1052
+ F [α1, α2]
1053
+ · · ·
1054
+ F [αµ1−1, αµ1]
1055
+ F [αµ1+1]
1056
+ · · ·
1057
+ F [αn]
1058
+ ��
1059
+
1060
+ αi,αj∈S1
1061
+ j−i>1
1062
+ (αi − αj)
1063
+
1064
+ αi,αj /∈S1
1065
+ j−i>0
1066
+ (αi − αj)
1067
+
1068
+ αi∈S1
1069
+ αj /∈S1
1070
+ (αi − αj)
1071
+ = ±aγ1−2
1072
+ n
1073
+ ·
1074
+ �� F [α1]
1075
+ F [α1, α2]
1076
+ F [α1, α2, α3]
1077
+ · · ·
1078
+ F [αµ1−2,µ1−1, αµ1]
1079
+ F [αµ1+1]
1080
+ · · ·
1081
+ F [αn]
1082
+ ��
1083
+
1084
+ αi,αj∈S1
1085
+ j−i>2
1086
+ (αi − αj)
1087
+
1088
+ αi,αj /∈S1
1089
+ j−i>0
1090
+ (αi − αj)
1091
+
1092
+ αi∈S1
1093
+ αj /∈S1
1094
+ (αi − αj)
1095
+ ...
1096
+ = ±aγ1−2
1097
+ n
1098
+ ·
1099
+ �� F [α1]
1100
+ F [α1, α2]
1101
+ · · ·
1102
+ F [α1, . . . , αµ1]
1103
+ F (αµ1+1)
1104
+ · · ·
1105
+ F (αn)
1106
+ ��
1107
+
1108
+ αi,αj /∈S1
1109
+ j−i>0
1110
+ (αi − αj)
1111
+
1112
+ αi∈S1
1113
+ αj /∈S1
1114
+ (αi − αj)
1115
+ 8
1116
+
1117
+ 6. Repeating the procedure for αj’s in each Si for i = 2, . . . , m successively, we get
1118
+ D(γ) = ±aγ1−2
1119
+ n
1120
+ ·
1121
+ �� F [α1]
1122
+ · · ·
1123
+ F [α1, . . . , αµ1]
1124
+ · · ·
1125
+ · · ·
1126
+ F [αµ1+···+µm−1+1]
1127
+ · · ·
1128
+ F [αµ1+···+µm−1+1, . . . , αn]
1129
+ ��
1130
+
1131
+ 1≤i<j≤m
1132
+
1133
+ αp∈Si
1134
+ αq∈Sj
1135
+ (αp − αq)
1136
+ 7. Now we substitute α1 = · · · = αµ1 = r1, . . . , αµ1+···+µm−1+1 = · · · = αn = rm into D(γ) and obtain
1137
+ D(γ) = ±aγ1−2
1138
+ n
1139
+ ·
1140
+ �� F [r1]
1141
+ · · ·
1142
+ F [r1, . . . , r1]
1143
+ · · ·
1144
+ · · ·
1145
+ F [rm]
1146
+ · · ·
1147
+ F [rm, . . . , rm]
1148
+ ��
1149
+
1150
+ 1≤i<j≤m
1151
+ (ri − rj)µiµj
1152
+ (4)
1153
+ 8. Recall that for any given polynomial P ∈ C[x],
1154
+ P[ri, . . . , ri
1155
+
1156
+ ��
1157
+
1158
+ k ri’s
1159
+ ] = P (k−1)(ri)
1160
+ (k − 1)!
1161
+ Hence
1162
+ F [ri, . . . , ri
1163
+
1164
+ ��
1165
+
1166
+ k ri’s
1167
+ ] =
1168
+ �(F (1)xγ1−1)(k−1)(ri)
1169
+ (k − 1)!
1170
+ , . . . , (F (1)x0)(k−1)(ri)
1171
+ (k − 1)!
1172
+ , . . . , (F (s)xγs−1)(k−1)(ri)
1173
+ (k − 1)!
1174
+ , . . . , (F (s)x0)(k−1)(ri)
1175
+ (k − 1)!
1176
+ �T
1177
+ (5)
1178
+ 9. Substituting (5) into (4) , we have
1179
+ D(γ) = ±aγ1−2
1180
+ n
1181
+ ·
1182
+ ��������������������
1183
+ (F (1)xγ1−1)(0)(r1)
1184
+ 0!
1185
+ · · ·
1186
+ (F (1)xγ1−1)(µ1−1)(r1)
1187
+ (µ1−1)!
1188
+ · · · · · ·
1189
+ (F (1)xγ1−1)(0)(rm)
1190
+ 0!
1191
+ · · ·
1192
+ (F (1)xγ1−1)(µm−1)(rm)
1193
+ (µm−1)!
1194
+ ...
1195
+ ...
1196
+ ...
1197
+ ...
1198
+ (F (1)x0)(0)(r1)
1199
+ 0!
1200
+ · · ·
1201
+ (F (1)x0)(µ1−1)(r1)
1202
+ (µ1−1)!
1203
+ · · · · · ·
1204
+ (F (1)x0)(0)(rm)
1205
+ 0!
1206
+ · · ·
1207
+ (F (1)x0)(µm−1)(rm)
1208
+ (µm−1)!
1209
+ ...
1210
+ ...
1211
+ ...
1212
+ ...
1213
+ (F (s)xγs−1)(0)(r1)
1214
+ 0!
1215
+ · · ·
1216
+ (F (s)xγs−1)(µ1−1)(r1)
1217
+ (µ1−1)!
1218
+ · · · · · ·
1219
+ (F (s)xγs−1)(0)(rm)
1220
+ 0!
1221
+ · · ·
1222
+ (F (s)xγs−1)(µm−1)(rm)
1223
+ (µm−1)!
1224
+ ...
1225
+ ...
1226
+ ...
1227
+ ...
1228
+ (F (s)x0)(0)(r1)
1229
+ 0!
1230
+ · · ·
1231
+ (F (s)x0)(µ1−1)(r1)
1232
+ (µ1−1)!
1233
+ · · · · · ·
1234
+ (F (s)x0)(0)(rm)
1235
+ 0!
1236
+ · · ·
1237
+ (F (s)x0)(µm−1)(rm)
1238
+ (µm−1)!
1239
+ ��������������������
1240
+
1241
+ 1≤i<j≤m
1242
+ (ri − rj)µiµj
1243
+ =
1244
+ c ·
1245
+ �����������������
1246
+ (F (1)xγ1−1)(0)(r1) · · · (F (1)xγ1−1)(µ1−1)(r1) · · · · · · (F (1)xγ1−1)(0)(rm) · · · (F (1)xγ1−1)(µm−1)(rm)
1247
+ ...
1248
+ ...
1249
+ ...
1250
+ ...
1251
+ (F (1)x0)(0)(r1)
1252
+ · · · (F (1)x0)(µ1−1)(r1)
1253
+ · · · · · · (F (1)x0)(0)(rm)
1254
+ · · · (F (1)x0)(µm−1)(rm)
1255
+ ...
1256
+ ...
1257
+ ...
1258
+ ...
1259
+ (F (s)xγs−1)(0)(r1) · · · (F (s)xγs−1)(µ1−1)(r1) · · · · · · (F (s)xγs−1)(0)(rm) · · · (F (s)xγs−1)(µm−1)(rm)
1260
+ ...
1261
+ ...
1262
+ ...
1263
+ ...
1264
+ (F (s)x0)(0)(r1)
1265
+ · · · (F (s)x0)(µ1−1)(r1)
1266
+ · · · · · · (F (s)x0)(0)(rm)
1267
+ · · · (F (s)x0)(µm−1)(rm)
1268
+ �����������������
1269
+
1270
+ 1≤i<j≤m
1271
+ (ri − rj)µiµj
1272
+ where c = ±1
1273
+ � ��m
1274
+ i=1
1275
+ �µi−1
1276
+ j=0 j!
1277
+
1278
+ · aγ1−2
1279
+ n
1280
+ .
1281
+ 9
1282
+
1283
+ 4.2
1284
+ Connection between the multiplicity discriminants and multiplicity vectors
1285
+ We first decompile Theorem 9 and identify the two essential ingredients therein, which are re-stated as
1286
+ Lemmas 16 and 17 below. From now on, we will use γ to denote the conjugate of γ ∈ M(n).
1287
+ Lemma 16. Let mult(F) = µ. Then D(¯µ) ̸= 0.
1288
+ Proof. In order to convey the main underlying ideas effectively, we will show the proof for a particular case
1289
+ first. After that, we will generalize the ideas to arbitrary cases.
1290
+ Particular case: Consider the case n = 5 and mult(F) = µ = (3, 2).
1291
+ 1. Assume that r1 and r2 are the two distinct roots with multiplicities 3 and 2, respectively. In other
1292
+ words, F = a5(x − r1)3(x − r2)2.
1293
+ 2. Let γ = ¯µ. Then
1294
+ γ1 = #{µj : µj ≥ 1} = 2,
1295
+ γ2 = #{µj : µj ≥ 2} = 2,
1296
+ γ3 = #{µj : µj ≥ 3} = 1
1297
+ Thus γ = (2, 2, 1).
1298
+ 3. By Lemma 15,
1299
+ D(γ) =
1300
+ c ·
1301
+ ����������
1302
+ (F (1)x1)(0)(r1)
1303
+ (F (1)x1)(1)(r1)
1304
+ (F (1)x1)(2)(r1)
1305
+ (F (1)x1)(0)(r2)
1306
+ (F (1)x1)(1)(r2)
1307
+ (F (1)x0)(0)(r1)
1308
+ (F (1)x0)(1)(r1)
1309
+ (F (1)x0)(2)(r1)
1310
+ (F (1)x0)(0)(r2)
1311
+ (F (1)x0)(1)(r2)
1312
+ (F (2)x1)(0)(r1)
1313
+ (F (2)x1)(1)(r1)
1314
+ (F (2)x1)(2)(r1)
1315
+ (F (2)x1)(0)(r2)
1316
+ (F (2)x1)(1)(r2)
1317
+ (F (2)x0)(0)(r1)
1318
+ (F (2)x0)(1)(r1)
1319
+ (F (2)x0)(2)(r1)
1320
+ (F (2)x0)(0)(r2)
1321
+ (F (2)x0)(1)(r2)
1322
+ (F (3)x0)(0)(r1)
1323
+ (F (3)x0)(1)(r1)
1324
+ (F (3)x0)(2)(r1)
1325
+ (F (3)x0)(0)(r2)
1326
+ (F (3)x0)(1)(r2)
1327
+ ����������
1328
+ (r1 − r2)6
1329
+ where
1330
+ c = ±(0! · 1! · 2!) · (0! · 1!) · a0
1331
+ 5 = ±2
1332
+ 4. Since
1333
+ F (i)(r1)
1334
+ � = 0,
1335
+ for i = 0, 1, 2
1336
+ ̸= 0,
1337
+ for i = 3
1338
+ F (i)(r2)
1339
+ � = 0,
1340
+ for i = 0, 1
1341
+ ̸= 0,
1342
+ for i = 2
1343
+ we immediately know
1344
+ (F (1)x1)(0)(r1) = 0 (F (1)x1)(1)(r1) = 0 (F (1)x1)(2)(r1) = F (3)(r1)r1
1345
+ 1 (F (1)x1)(0)(r2) = 0 (F (1)x1)(1)(r2) = F (2)(r2)r1
1346
+ 2
1347
+ (F (1)x0)(0)(r1) = 0 (F (1)x0)(1)(r1) = 0 (F (1)x0)(2)(r1) = F (3)(r1)r0
1348
+ 1 (F (1)x0)(0)(r2) = 0 (F (1)x0)(1)(r2) = F (2)(r2)r0
1349
+ 2
1350
+ (F (2)x1)(0)(r1) = 0 (F (2)x1)(1)(r1) = F (3)(r1)r1
1351
+ 1
1352
+ (F (2)x1)(0)(r2) = F (2)(r2)r1
1353
+ 2
1354
+ (F (2)x0)(0)(r1) = 0 (F (2)x0)(1)(r1) = F (3)(r1)r0
1355
+ 1
1356
+ (F (2)x0)(0)(r2) = F (2)(r2)r0
1357
+ 2
1358
+ (F (3)x0)(0)(r1) = F (3)(r1)r0
1359
+ 1
1360
+ 5. Therefore,
1361
+ D(γ) =
1362
+ c ·
1363
+ ����������
1364
+ 0
1365
+ 0
1366
+ F (3)(r1)r1
1367
+ 1
1368
+ 0
1369
+ F (2)(r2)r1
1370
+ 2
1371
+ 0
1372
+ 0
1373
+ F (3)(r1)r0
1374
+ 1
1375
+ 0
1376
+ F (2)(r2)r0
1377
+ 2
1378
+ 0
1379
+ F (3)(r1)r1
1380
+ 1
1381
+ ·
1382
+ F (2)(r2)r1
1383
+ 2
1384
+ ·
1385
+ 0
1386
+ F (3)(r1)r0
1387
+ 1
1388
+ ·
1389
+ F (2)(r2)r0
1390
+ 2
1391
+ ·
1392
+ F (3)(r1)r0
1393
+ 1
1394
+ ·
1395
+ ·
1396
+ ·
1397
+ ·
1398
+ ����������
1399
+ (r1 − r2)6
1400
+ 10
1401
+
1402
+ 6. By rearranging the columns of the determinant in the numerator, we have
1403
+ D(γ) =
1404
+ c ·
1405
+ ����������
1406
+ F (3)(r1)r1
1407
+ 1
1408
+ F (2)(r2)r1
1409
+ 2
1410
+ F (3)(r1)r0
1411
+ 1
1412
+ F (2)(r2)r0
1413
+ 2
1414
+ F (3)(r1)r1
1415
+ 1
1416
+ F (2)(r2)r1
1417
+ 2
1418
+ ·
1419
+ ·
1420
+ F (3)(r1)r0
1421
+ 1
1422
+ F (2)(r2)r0
1423
+ 2
1424
+ ·
1425
+ ·
1426
+ F (3)(r1)r0
1427
+ 1
1428
+ ·
1429
+ ·
1430
+ ·
1431
+ ·
1432
+ ����������
1433
+ (r1 − r2)6
1434
+ =
1435
+ c ·
1436
+ ������
1437
+ M1
1438
+ M2
1439
+ ·
1440
+ M3
1441
+ ·
1442
+ ·
1443
+ ������
1444
+ (r1 − r2)6
1445
+ where
1446
+ M1 =
1447
+ � F (3)(r1)r1
1448
+ 1
1449
+ F (2)(r2)r1
1450
+ 2
1451
+ F (3)(r1)r0
1452
+ 1
1453
+ F (2)(r2)r0
1454
+ 2
1455
+
1456
+ M2 =
1457
+ � F (3)(r1)r1
1458
+ 1
1459
+ F (2)(r2)r1
1460
+ 2
1461
+ F (3)(r1)r0
1462
+ 1
1463
+ F (2)(r2)r0
1464
+ 2
1465
+
1466
+ M3 =
1467
+
1468
+ F (3)(r1)r0
1469
+ 1
1470
+
1471
+ 7. Obviously,
1472
+ D(γ) = ±c · |M1| · |M2| · |M3|
1473
+ (r1 − r2)6
1474
+ We only need to show that Mi ̸= 0 for i = 1, 2, 3. The claim follows from the following observations:
1475
+ |M1| = F (3)(r1)F (2)(r2)V (r1, r2) ̸= 0
1476
+ |M2| = F (3)(r1)F (2)(r2)V (r1, r2) ̸= 0
1477
+ |M3| = F (3)(r1)V (r1) ̸= 0
1478
+ The proof is completed.
1479
+ Arbitrary case. Now we generalize the above ideas to arbitrary cases.
1480
+ 1. Let µ = (µ1, . . . , µm). Assume that r1, . . . , rm are the m distinct roots with multiplicities µ1, . . . , µm
1481
+ respectively. In other words, F = an(x − r1)µ1 · · · (x − rm)µm.
1482
+ 2. Let γ = ¯µ = (γ1, . . . , γs). In other words, γi = #{µj : µj ≥ i}. Note that s = µ1 since µ1 ≥ · · · ≥ µm.
1483
+ 3. Recall that
1484
+ D(γ) =
1485
+ c ·
1486
+ �����������������
1487
+ (F (1)xγ1−1)(0)(r1)
1488
+ · · · (F (1)xγ1−1)(µ1−1)(r1)
1489
+ · · · · · · (F (1)xγ1−1)(0)(rm)
1490
+ · · · (F (1)xγ1−1)(µm−1)(rm)
1491
+ ...
1492
+ ...
1493
+ ...
1494
+ ...
1495
+ (F (1)x0)(0)(r1)
1496
+ · · · (F (1)x0)(µ1−1)(r1)
1497
+ · · · · · · (F (1)x0)(0)(rm)
1498
+ · · · (F (1)x0)(µm−1)(rm)
1499
+ ...
1500
+ ...
1501
+ ...
1502
+ ...
1503
+ (F (µ1)xγµ1−1)(0)(r1) · · · (F (µ1)xγµ1−1)(µ1−1)(r1) · · · · · · (F (µ1)xγµ1−1)(0)(rm) · · · (F (µ1)xγµ1−1)(µm−1)(rm)
1504
+ ...
1505
+ ...
1506
+ ...
1507
+ ...
1508
+ (F (µ1)x0)(0)(r1)
1509
+ · · · (F (µ1)x0)(µ1−1)(r1)
1510
+ · · · · · · (F (µ1)x0)(0)(rm)
1511
+ · · · (F (µ1)x0)(µm−1)(rm)
1512
+ �����������������
1513
+
1514
+ 1≤i<j≤m
1515
+ (ri − rj)µiµj
1516
+ (6)
1517
+ where c = ±1
1518
+ � ��m
1519
+ i=1
1520
+ �µi−1
1521
+ j=0 j!
1522
+
1523
+ · aγ1−2
1524
+ n
1525
+ .
1526
+ 11
1527
+
1528
+ 4. Since for k = 1, . . . , m,
1529
+ F (i)(rk)
1530
+ � = 0,
1531
+ for i < µk
1532
+ ̸= 0,
1533
+ for i = µk
1534
+ we immediately know
1535
+ (F (1)xj)(i)(rk)
1536
+ � = 0,
1537
+ for i < µk − 1
1538
+ = F (µk)(rk)rj
1539
+ k,
1540
+ for i = µk − 1
1541
+ where µk − 1 ≥ 0
1542
+ ...
1543
+ (F (ℓ)xj)(i)(rk)
1544
+ � = 0,
1545
+ for i < µk − ℓ
1546
+ = F (µk)(rk)rj
1547
+ k,
1548
+ for i = µk − ℓ
1549
+ where µk − ℓ ≥ 0
1550
+ (7)
1551
+ ...
1552
+ (F (µ1)xj)(i)(rk)
1553
+ � = 0,
1554
+ for i < µk − µ1
1555
+ = F (µk)(rk)rj
1556
+ k,
1557
+ for i = µk − µ1
1558
+ where µk − µ1 ≥ 0
1559
+ 5. Plugging (7) into (6), we have
1560
+ D(γ) =
1561
+ c ·
1562
+ ������������������������������
1563
+ 0
1564
+ · · ·
1565
+ 0
1566
+ F (µ1)(r1)rγ1−1
1567
+ 1
1568
+ · · ·
1569
+ 0
1570
+ · · ·
1571
+ 0
1572
+ F (µi)(ri)rγ1−1
1573
+ i
1574
+ · · ·
1575
+ ...
1576
+ ...
1577
+ ...
1578
+ ...
1579
+ ...
1580
+ ...
1581
+ 0
1582
+ · · ·
1583
+ 0
1584
+ F (µ1)(r1)r0
1585
+ 1
1586
+ · · ·
1587
+ 0
1588
+ · · ·
1589
+ 0
1590
+ F (µi)(ri)r0
1591
+ i
1592
+ · · ·
1593
+ 0
1594
+ · · · F (µ1)(r1)rγ2−1
1595
+ 1
1596
+ ·
1597
+ · · ·
1598
+ 0
1599
+ · · · F (µi)(ri)rγ2−1
1600
+ i
1601
+ ·
1602
+ · · ·
1603
+ ...
1604
+ ...
1605
+ ...
1606
+ ...
1607
+ ...
1608
+ ...
1609
+ 0
1610
+ · · · F (µ1)(r1)r0
1611
+ 1
1612
+ ·
1613
+ · · ·
1614
+ 0
1615
+ · · · F (µi)(ri)r0
1616
+ i
1617
+ ·
1618
+ · · ·
1619
+ ...
1620
+ ...
1621
+ ...
1622
+ ...
1623
+ ...
1624
+ ...
1625
+ 0
1626
+ ·
1627
+ ·
1628
+ · · · F (µi)(ri)r
1629
+ γµi −1
1630
+ i
1631
+ ·
1632
+ ·
1633
+ · · ·
1634
+ ...
1635
+ ...
1636
+ ...
1637
+ ...
1638
+ ...
1639
+ ...
1640
+ 0
1641
+ ·
1642
+ ·
1643
+ · · · F (µi)(ri)r0
1644
+ i
1645
+ ·
1646
+ ·
1647
+ · · ·
1648
+ ...
1649
+ ...
1650
+ ...
1651
+ ...
1652
+ ...
1653
+ ...
1654
+ F (µ1)(r1)r
1655
+ γµ1 −1
1656
+ 1
1657
+ · · ·
1658
+ ·
1659
+ ·
1660
+ · · ·
1661
+ ·
1662
+ · · ·
1663
+ ·
1664
+ ·
1665
+ · · ·
1666
+ ...
1667
+ ...
1668
+ ...
1669
+ ...
1670
+ ...
1671
+ ...
1672
+ F (µ1)(r1)r0
1673
+ 1
1674
+ ·
1675
+ ·
1676
+ · · ·
1677
+ ·
1678
+ · · ·
1679
+ ·
1680
+ ·
1681
+ · · ·
1682
+ ������������������������������
1683
+
1684
+ 1≤i<j≤m
1685
+ (ri − rj)µiµj
1686
+ 6. By rearranging the columns of the determinant in the numerator, we have
1687
+ D(γ) = ±
1688
+ c ·
1689
+ �����������������������
1690
+ F (µ1)(r1)rγ1−1
1691
+ 1
1692
+ · · · F (µm)(rm)rγ1−1
1693
+ m
1694
+ ...
1695
+ ...
1696
+ F (µ1)(r1)r0
1697
+ 1
1698
+ · · · F (µm)(rm)r0
1699
+ m
1700
+ F (µ1)(r1)rγ2−1
1701
+ 1
1702
+ · · · F (µγ2 )(rγ2)rγ2−1
1703
+ γ2
1704
+ ·
1705
+ · · ·
1706
+ ·
1707
+ ...
1708
+ ...
1709
+ ...
1710
+ ...
1711
+ F (µ1)(r1)r0
1712
+ 1
1713
+ · · · F (µγ2 )(rγ2)r0
1714
+ γ2
1715
+ ·
1716
+ · · ·
1717
+ ·
1718
+ · · · · ·
1719
+ ·
1720
+ · · ·
1721
+ ·
1722
+ ·
1723
+ · · ·
1724
+ ·
1725
+ ...
1726
+ ...
1727
+ ...
1728
+ ...
1729
+ ...
1730
+ ...
1731
+ · · · · ·
1732
+ ·
1733
+ · · ·
1734
+ ·
1735
+ ·
1736
+ · · ·
1737
+ ·
1738
+ F (µ1)(r1)r
1739
+ γµ1 −1
1740
+ 1
1741
+ · · · F
1742
+ (µγµ1
1743
+ )(rγµ1 )r
1744
+ γµ1 −1
1745
+ γµ1
1746
+ · · · · ·
1747
+ ·
1748
+ · · ·
1749
+ ·
1750
+ ·
1751
+ · · ·
1752
+ ·
1753
+ ...
1754
+ ...
1755
+ ...
1756
+ ...
1757
+ ...
1758
+ ...
1759
+ ...
1760
+ ...
1761
+ F (µ1)(r1)r0
1762
+ 1
1763
+ · · · F
1764
+ (µγµ1
1765
+ )(rγµ1 )r0
1766
+ γµ1
1767
+ · · · · ·
1768
+ ·
1769
+ · · ·
1770
+ ·
1771
+ ·
1772
+ · · ·
1773
+ ·
1774
+ �����������������������
1775
+
1776
+ 1≤i<j≤m
1777
+ (ri − rj)µiµj
1778
+ 12
1779
+
1780
+ = ±
1781
+ c ·
1782
+ �����������
1783
+ M1
1784
+ M2
1785
+ ·
1786
+ ...
1787
+ ·
1788
+ Mµ1
1789
+ · · ·
1790
+ ·
1791
+ ·
1792
+ �����������
1793
+
1794
+ 1≤i<j≤m
1795
+ (ri − rj)µiµj
1796
+ where
1797
+ Mi =
1798
+
1799
+ ��
1800
+ F (µ1)(r1)rγi−1
1801
+ 1
1802
+ · · ·
1803
+ F (µγi)(rγi)rγi−1
1804
+ γi
1805
+ ...
1806
+ ...
1807
+ F (µ1)(r1)r0
1808
+ 1
1809
+ · · ·
1810
+ F (µγi)(rγi)r0
1811
+ γi
1812
+
1813
+ ��
1814
+ for i = 1, . . . , µ1. Then
1815
+ D(γ) = ±
1816
+ c · |M1| · · ·
1817
+ ��Mµ1
1818
+ ��
1819
+
1820
+ 1≤i<j≤m
1821
+ (ri − rj)µiµj
1822
+ 7. It only remains to show that Mi ̸= 0. The claim follows from the following observations:
1823
+ |Mi| =
1824
+
1825
+
1826
+ γi
1827
+
1828
+ j=1
1829
+ F (µj)(rj)
1830
+
1831
+ � V (r1, . . . , rγi) ̸= 0 for i = 1, . . . , µ1
1832
+ The proof is completed.
1833
+ Lemma 17. Let mult(F) = µ. Then D(λ) = 0 for any λ such that ¯µ ≺lex λ.
1834
+ Proof. In order to convey the main underlying ideas, we will show the proof for a particular case first. After
1835
+ that, we will generalize the ideas to arbitrary cases.
1836
+ Particular case: Consider the case n = 5 and mult(F) = µ = (3, 2). Thus ¯µ = (2, 2, 1). Let λ = (3, 1, 1).
1837
+ Obviously, ¯µ ≺lex λ. We will show that D(λ) = 0.
1838
+ 1. Assume that r1 and r2 are the two distinct roots with multiplicities 3 and 2, respectively. In other
1839
+ words, F = a5(x − r1)3(x − r2)2.
1840
+ 2. By Lemma 15,
1841
+ D(λ) =
1842
+ c ·
1843
+ ����������
1844
+ (F (1)x2)(0)(r1)
1845
+ (F (1)x2)(1)(r1)
1846
+ (F (1)x1)(2)(r1)
1847
+ (F (1)x2)(0)(r2)
1848
+ (F (1)x2)(1)(r2)
1849
+ (F (1)x1)(0)(r1)
1850
+ (F (1)x1)(1)(r1)
1851
+ (F (1)x1)(2)(r1)
1852
+ (F (1)x1)(0)(r2)
1853
+ (F (1)x1)(1)(r2)
1854
+ (F (1)x0)(0)(r1)
1855
+ (F (1)x0)(1)(r1)
1856
+ (F (1)x0)(2)(r1)
1857
+ (F (1)x0)(0)(r2)
1858
+ (F (1)x0)(1)(r2)
1859
+ (F (2)x0)(0)(r1)
1860
+ (F (2)x0)(1)(r1)
1861
+ (F (2)x0)(2)(r1)
1862
+ (F (2)x0)(0)(r2)
1863
+ (F (2)x0)(1)(r2)
1864
+ (F (3)x0)(0)(r1)
1865
+ (F (3)x0)(1)(r1)
1866
+ (F (3)x0)(2)(r1)
1867
+ (F (3)x0)(0)(r2)
1868
+ (F (3)x0)(1)(r2)
1869
+ ����������
1870
+ (r1 − r2)6
1871
+ where
1872
+ c = ±(0! · 1! · 2!) · (0! · 1!) · a1
1873
+ 5 = ±2a5
1874
+ 3. Since
1875
+ F (i)(r1)
1876
+ � = 0,
1877
+ for i = 0, 1, 2
1878
+ ̸= 0,
1879
+ for i = 3
1880
+ F (i)(r2)
1881
+ � = 0,
1882
+ for i = 0, 1
1883
+ ̸= 0,
1884
+ for i = 2
1885
+ 13
1886
+
1887
+ we immediately know
1888
+ (F (1)x2)(0)(r1) = 0 (F (1)x2)(1)(r1) = 0 (F (1)x2)(2)(r1) = F (3)(r1)r2
1889
+ 1 (F (1)x2)(0)(r2) = 0 (F (1)x2)(1)(r2) = F (2)(r2)r2
1890
+ 2
1891
+ (F (1)x1)(0)(r1) = 0 (F (1)x1)(1)(r1) = 0 (F (1)x1)(2)(r1) = F (3)(r1)r1
1892
+ 1 (F (1)x1)(0)(r2) = 0 (F (1)x1)(1)(r2) = F (2)(r2)r1
1893
+ 2
1894
+ (F (1)x0)(0)(r1) = 0 (F (1)x0)(1)(r1) = 0 (F (1)x0)(2)(r1) = F (3)(r1)r0
1895
+ 1 (F (1)x0)(0)(r2) = 0 (F (1)x0)(1)(r2) = F (2)(r2)r0
1896
+ 2
1897
+ (F (2)x0)(0)(r1) = 0 (F (2)x0)(1)(r1) = F (3)(r1)r0
1898
+ 1
1899
+ (F (2)x0)(0)(r2) = F (2)(r2)r0
1900
+ 2
1901
+ (F (3)x0)(0)(r1) = F (3)(r1)r0
1902
+ 1
1903
+ Therefore,
1904
+ D(λ) =
1905
+ c ·
1906
+ ����������
1907
+ 0
1908
+ 0
1909
+ F (3)(r1)r2
1910
+ 1
1911
+ 0
1912
+ F (2)(r2)r2
1913
+ 2
1914
+ 0
1915
+ 0
1916
+ F (3)(r1)r1
1917
+ 1
1918
+ 0
1919
+ F (2)(r2)r1
1920
+ 2
1921
+ 0
1922
+ 0
1923
+ F (3)(r1)r0
1924
+ 1
1925
+ 0
1926
+ F (2)(r2)r0
1927
+ 2
1928
+ 0
1929
+ F (3)(r1)r0
1930
+ 1
1931
+ ·
1932
+ F (2)(r2)r0
1933
+ 2
1934
+ ·
1935
+ F (3)(r1)r0
1936
+ 1
1937
+ ·
1938
+ ·
1939
+ ·
1940
+ ·
1941
+ ����������
1942
+ (r1 − r2)6
1943
+ 4. By rearranging the columns of the determinant in the numerator, we have
1944
+ D(λ) = ±
1945
+ c ·
1946
+ ����������
1947
+ F (3)(r1)r2
1948
+ 1
1949
+ F (2)(r2)r2
1950
+ 2
1951
+ F (3)(r1)r1
1952
+ 1
1953
+ F (2)(r2)r1
1954
+ 2
1955
+ F (3)(r1)r0
1956
+ 1
1957
+ F (2)(r2)r0
1958
+ 2
1959
+ F (3)(r1)r0
1960
+ 1
1961
+ F (2)(r2)r0
1962
+ 2
1963
+ ·
1964
+ ·
1965
+ F (3)(r1)r0
1966
+ 1
1967
+ ·
1968
+ ·
1969
+ ·
1970
+ ·
1971
+ ����������
1972
+ (r1 − r2)6
1973
+ = ±
1974
+ c ·
1975
+ ��������
1976
+ M1
1977
+ M2
1978
+ ·
1979
+ M3
1980
+ ·
1981
+ ·
1982
+ ��������
1983
+ (r1 − r2)6
1984
+ where
1985
+ M1 =
1986
+
1987
+
1988
+ F (3)(r1)r2
1989
+ 1
1990
+ F (2)(r2)r2
1991
+ 2
1992
+ F (3)(r1)r1
1993
+ 1
1994
+ F (2)(r2)r1
1995
+ 2
1996
+ F (3)(r1)r0
1997
+ 1
1998
+ F (2)(r2)r2
1999
+
2000
+
2001
+ M2 =
2002
+
2003
+ F (3)(r1)r0
2004
+ 1
2005
+ F (2)(r2)r2
2006
+
2007
+ M3 =
2008
+
2009
+ F (3)(r1)r0
2010
+ 1
2011
+
2012
+ 5. We repartition the columns so that the reverse diagonal consists of two square matrices and obtain the
2013
+ following:
2014
+ D(λ) =
2015
+ c ·
2016
+ ����������
2017
+ F (3)(r1)r2
2018
+ 1
2019
+ F (2)(r2)r2
2020
+ 2
2021
+ F (3)(r1)r1
2022
+ 1
2023
+ F (2)(r2)r1
2024
+ 2
2025
+ F (3)(r1)r0
2026
+ 1
2027
+ F (2)(r2)r0
2028
+ 2
2029
+ F (3)(r1)r0
2030
+ 1
2031
+ F (2)(r2)r0
2032
+ 2
2033
+ ·
2034
+ ·
2035
+ F (3)(r1)r0
2036
+ 1
2037
+ ·
2038
+ ·
2039
+ ·
2040
+ ·
2041
+ ����������
2042
+ (r1 − r2)6
2043
+ =
2044
+ c ·
2045
+ ����
2046
+ T
2047
+ B
2048
+ ·
2049
+ ����
2050
+ (r1 − r2)6
2051
+ 14
2052
+
2053
+ where the size of the square matrix T is λ1 = 3, namely,
2054
+ T =
2055
+ � 0
2056
+ M1
2057
+
2058
+ where 0 is the λ1 × (λ1 − γ1) matrix.
2059
+ 6. Since λ1 − γ1 = 3 − 2 > 0, the first column of T is all zeros. Hence |T| = 0 and in turn D(λ) = 0
2060
+ Arbitrary case. Now we generalize the above ideas to arbitrary cases.
2061
+ 1. Let µ = (µ1, . . . , µm).
2062
+ Assume that r1, . . . , rm are the m distinct roots of F with multiplicities
2063
+ µ1, . . . , µm respectively. In other words, F = an(x − r1)µ1 · · · (x − rm)µm.
2064
+ 2. Let γ = ¯µ = (γ1, . . . , γs). By the definition of conjugate, γi = #{µj : µj ≥ i}. Note that s = µ1 since
2065
+ µ1 ≥ · · · ≥ µm.
2066
+ 3. Consider λ = (λ1, . . . , λt) ∈ M(n) such that γ ≺lex λ. By Lemma 15, we have
2067
+ D(λ) =
2068
+ c ·
2069
+ �����������������
2070
+ (F (1)xλ1−1)(0)(r1) · · · (F (1)xλ1−1)(µ1−1)(r1) · · · · · · (F (1)xλ1−1)(0)(rm) · · · (F (1)xλ1−1)(µm−1)(rm)
2071
+ ...
2072
+ ...
2073
+ ...
2074
+ ...
2075
+ (F (1)x0)(0)(r1)
2076
+ · · · (F (1)x0)(µ1−1)(r1)
2077
+ · · · · · · (F (1)x0)(0)(rm)
2078
+ · · · (F (1)x0)(µm−1)(rm)
2079
+ ...
2080
+ ...
2081
+ ...
2082
+ ...
2083
+ (F (t)xλt−1)(0)(r1) · · · (F (t)xλt−1)(µ1−1)(r1) · · · · · · (F (t)xλt−1)(0)(rm) · · · (F (t)xλt−1)(µm−1)(rm)
2084
+ ...
2085
+ ...
2086
+ ...
2087
+ ...
2088
+ (F (t)x0)(0)(r1)
2089
+ · · · (F (t)x0)(µ1−1)(r1)
2090
+ · · · · · · (F (t)x0)(0)(rm)
2091
+ · · · (F (t)x0)(µm−1)(rm)
2092
+ �����������������
2093
+
2094
+ i<j(ri − rj)µiµj
2095
+ (8)
2096
+ where c = ±1
2097
+ � ��m
2098
+ i=1
2099
+ �µi−1
2100
+ j=0 j!
2101
+
2102
+ · aλ1−2
2103
+ n
2104
+ .
2105
+ 4. Plugging (7) into (8), we have
2106
+ D(γ) =
2107
+ c ·
2108
+ ������������������������������
2109
+ 0
2110
+ · · ·
2111
+ 0
2112
+ F (µ1)(r1)rλ1−1
2113
+ 1
2114
+ · · ·
2115
+ 0
2116
+ · · ·
2117
+ 0
2118
+ F (µi)(ri)rλ1−1
2119
+ i
2120
+ · · ·
2121
+ ...
2122
+ ...
2123
+ ...
2124
+ ...
2125
+ ...
2126
+ ...
2127
+ 0
2128
+ · · ·
2129
+ 0
2130
+ F (µ1)(r1)r0
2131
+ 1
2132
+ · · ·
2133
+ 0
2134
+ · · ·
2135
+ 0
2136
+ F (µi)(ri)r0
2137
+ i
2138
+ · · ·
2139
+ 0
2140
+ · · · F (µ1)(r1)rλ2−1
2141
+ 1
2142
+ ·
2143
+ · · ·
2144
+ 0
2145
+ · · · F (µi)(ri)rλ2−1
2146
+ i
2147
+ ·
2148
+ · · ·
2149
+ ...
2150
+ ...
2151
+ ...
2152
+ ...
2153
+ ...
2154
+ ...
2155
+ 0
2156
+ · · · F (µ1)(r1)r0
2157
+ 1
2158
+ ·
2159
+ · · ·
2160
+ 0
2161
+ · · · F (µi)(ri)r0
2162
+ i
2163
+ ·
2164
+ · · ·
2165
+ ...
2166
+ ...
2167
+ ...
2168
+ ...
2169
+ ...
2170
+ ...
2171
+ 0
2172
+ ·
2173
+ ·
2174
+ · · · F (µi)(ri)r
2175
+ λµi −1
2176
+ i
2177
+ ·
2178
+ ·
2179
+ · · ·
2180
+ ...
2181
+ ...
2182
+ ...
2183
+ ...
2184
+ ...
2185
+ ...
2186
+ 0
2187
+ ·
2188
+ ·
2189
+ · · · F (µi)(ri)r0
2190
+ i
2191
+ ·
2192
+ ·
2193
+ · · ·
2194
+ ...
2195
+ ...
2196
+ ...
2197
+ ...
2198
+ ...
2199
+ ...
2200
+ F (µ1)(r1)r
2201
+ λµ1 −1
2202
+ 1
2203
+ · · ·
2204
+ ·
2205
+ ·
2206
+ · · ·
2207
+ ·
2208
+ · · ·
2209
+ ·
2210
+ ·
2211
+ · · ·
2212
+ ...
2213
+ ...
2214
+ ...
2215
+ ...
2216
+ ...
2217
+ ...
2218
+ F (µ1)(r1)r0
2219
+ 1
2220
+ ·
2221
+ ·
2222
+ · · ·
2223
+ ·
2224
+ · · ·
2225
+ ·
2226
+ ·
2227
+ · · ·
2228
+ ������������������������������
2229
+
2230
+ 1≤i<j≤m
2231
+ (ri − rj)µiµj
2232
+ 15
2233
+
2234
+ 5. By rearranging the columns of the determinant in the numerator, we have
2235
+ D(λ) = ±
2236
+ c ·
2237
+ �����������������������
2238
+ F (µ1)(r1)rλ1−1
2239
+ 1
2240
+ · · · F (µγ1 )(rγ1)rλ1−1
2241
+ γ1
2242
+ ...
2243
+ ...
2244
+ F (µ1)(r1)r0
2245
+ 1
2246
+ · · · F (µγ1 )(rγ1)r0
2247
+ γ1
2248
+ F (µ1)(r1)rλ2−1
2249
+ 1
2250
+ · · · F (µγ2 )(rγ2)rλ2−1
2251
+ γ2
2252
+ ·
2253
+ · · ·
2254
+ ·
2255
+ ...
2256
+ ...
2257
+ ...
2258
+ ...
2259
+ F (µ1)(r1)r0
2260
+ 1
2261
+ · · · F (µγ2 )(rγ2)r0
2262
+ γ2
2263
+ ·
2264
+ · · ·
2265
+ ·
2266
+ · · · · ·
2267
+ ·
2268
+ · · ·
2269
+ ·
2270
+ ·
2271
+ · · ·
2272
+ ·
2273
+ ...
2274
+ ...
2275
+ ...
2276
+ ...
2277
+ ...
2278
+ ...
2279
+ · · · · ·
2280
+ ·
2281
+ · · ·
2282
+ ·
2283
+ ·
2284
+ · · ·
2285
+ ·
2286
+ F (µ1)(r1)rλt−1
2287
+ 1
2288
+ · · · F (µγs )(rγs)rλt−1
2289
+ γs
2290
+ · · · · ·
2291
+ ·
2292
+ · · ·
2293
+ ·
2294
+ ·
2295
+ · · ·
2296
+ ·
2297
+ ...
2298
+ ...
2299
+ ...
2300
+ ...
2301
+ ...
2302
+ ...
2303
+ ...
2304
+ ...
2305
+ F (µ1)(r1)r0
2306
+ 1
2307
+ · · · F (µγs )(rγs)r0
2308
+ γs
2309
+ · · · · ·
2310
+ ·
2311
+ · · ·
2312
+ ·
2313
+ ·
2314
+ · · ·
2315
+ ·
2316
+ �����������������������
2317
+
2318
+ 1≤i<j≤m
2319
+ (ri − rj)µiµj
2320
+ = ±
2321
+ c ·
2322
+ �����������
2323
+ M1
2324
+ M2
2325
+ ·
2326
+ ...
2327
+ ·
2328
+ Mµ1
2329
+ · · ·
2330
+ ·
2331
+ ·
2332
+ �����������
2333
+
2334
+ 1≤i<j≤m
2335
+ (ri − rj)µiµj
2336
+ where Mi is λi by γi.
2337
+ 6. Since γ ≺lex λ, there exists ℓ such that γj = λj for j < ℓ and γℓ < λℓ. Thus
2338
+ γ1 + · · · + γℓ < λ1 + · · · + λℓ
2339
+ 7. We repartition the numerator matrix so that the reverse diagonal consists of two square matrices T
2340
+ and B as follows.
2341
+ D(λ) = ±
2342
+ c ·
2343
+ ����
2344
+ T
2345
+ B
2346
+ ����
2347
+
2348
+ 1≤i<j≤m
2349
+ (ri − rj)µiµj
2350
+ where the size of the square matrix T is λ1 + · · · + λℓ, namely,
2351
+ T =
2352
+
2353
+ ��
2354
+ M1
2355
+ ...
2356
+ ...
2357
+ 0
2358
+ Mℓ
2359
+ · · ·
2360
+ ·
2361
+
2362
+ ��
2363
+ where 0 is the γℓ × p and p = (λ1 + · · · + λℓ) − (γ1 + · · · + γℓ).
2364
+ 8. Obviously,
2365
+ D(λ) = ±
2366
+ c · |T| · |B|
2367
+
2368
+ 1≤i<j≤m
2369
+ (ri − rj)µiµj
2370
+ 9. Since p > 0, the first column of T is all zeros. Hence |T| = 0, which implies that D(λ) = 0.
2371
+ 16
2372
+
2373
+ 4.3
2374
+ Proof of Theorem 9
2375
+ Now we are ready to prove Theorem 9.
2376
+ Proof of Theorem 9.
2377
+ The result of Theorem 9 is equivalent to the following claim: let
2378
+ δ =
2379
+ max
2380
+ γ∈M(n)
2381
+ D(γ)̸=0
2382
+ γ
2383
+ where max is with respect to the lexicographic ordering ≺lex. Then mult(F) = δ.
2384
+ Next we will show the correctness of the claim.
2385
+ 1. Assume that mult(F) = µ. We will show µ = δ by disproving µ ≺lex δ and δ ≺lex µ.
2386
+ 2. If µ ≺lex δ, then δ ≺lex µ. By the condition for determining δ, we immediately have D(µ) = 0, leading
2387
+ to a contradiction with Lemma 16.
2388
+ 3. If δ ≺lex µ, then µ
2389
+ ≺lex δ. By Lemma 17, D(δ) = 0. However, it contradicts the condition for
2390
+ determining δ.
2391
+ 4. Therefore, the only possibility is µ = µ′.
2392
+ 5
2393
+ Comparison
2394
+ In this section, we compare the multiplicity discriminant condition given by Theorem 9 (mentioned as HY22
2395
+ hereinafter) and that given by a complex root version of YHZ’s condition [11] as well as the one given by
2396
+ the authors in [6, Theorem 6] (mentioned as HY21 hereinafter). In particular, we will make comparison on
2397
+ the forms and the maximum degrees of discriminants appearing in the conditions.
2398
+ 5.1
2399
+ Form of discriminants
2400
+ We will illustrate the forms of conditions generated by the three methods for a fixed µ. For example, we
2401
+ consider the polynomial F = a5x5 + a4x4 + a3x3 + a2x2 + a1x + a0 and µ = (2, 2, 1). The condition for F
2402
+ having the multiplicity structure µ is given as follows:
2403
+ 1. YHZ’s condition: P1 = 0 ∧ P2 = 0 ∧ P3 ̸= 0 where
2404
+ P1 =
2405
+ ������������������
2406
+ a5
2407
+ a4
2408
+ a3
2409
+ a2
2410
+ a1
2411
+ a0
2412
+ a5
2413
+ a4
2414
+ a3
2415
+ a2
2416
+ a1
2417
+ a0
2418
+ a5
2419
+ a4
2420
+ a3
2421
+ a2
2422
+ a1
2423
+ a0
2424
+ a5
2425
+ a4
2426
+ a3
2427
+ a2
2428
+ a1
2429
+ a0
2430
+ 5a5
2431
+ 4a4
2432
+ 3a3
2433
+ 2a2
2434
+ a1
2435
+ 5a5
2436
+ 4a4
2437
+ 3a3
2438
+ 2a2
2439
+ a1
2440
+ 5a5
2441
+ 4a4
2442
+ 3a3
2443
+ 2a2
2444
+ a1
2445
+ 5a5
2446
+ 4a4
2447
+ 3a3
2448
+ 2a2
2449
+ a1
2450
+ 5a5
2451
+ 4a4
2452
+ 3a3
2453
+ 2a2
2454
+ a1
2455
+ ������������������
2456
+ P2 =
2457
+ ��������������
2458
+ a5
2459
+ a4
2460
+ a3
2461
+ a2
2462
+ a1
2463
+ a0
2464
+ a5
2465
+ a4
2466
+ a3
2467
+ a2
2468
+ a1
2469
+ a0
2470
+ a5
2471
+ a4
2472
+ a3
2473
+ a2
2474
+ a1
2475
+ 5a5
2476
+ 4a4
2477
+ 3a3
2478
+ 2a2
2479
+ a1
2480
+ 5a5
2481
+ 4a4
2482
+ 3a3
2483
+ 2a2
2484
+ a1
2485
+ 5a5
2486
+ 4a4
2487
+ 3a3
2488
+ 2a2
2489
+ a1
2490
+ 5a5
2491
+ 4a4
2492
+ 3a3
2493
+ 2a2
2494
+ ��������������
2495
+ 17
2496
+
2497
+ P3 =
2498
+ ����������������������������������
2499
+ ����������
2500
+ a5
2501
+ a4
2502
+ a3
2503
+ a2
2504
+ a1
2505
+ a5
2506
+ a4
2507
+ a3
2508
+ a2
2509
+ 5a5
2510
+ 4a4
2511
+ 3a3
2512
+ 2a2
2513
+ a1
2514
+ 5a5
2515
+ 4a4
2516
+ 3a3
2517
+ 2a2
2518
+ 5a5
2519
+ 4a4
2520
+ 3a3
2521
+ ����������
2522
+ ����������
2523
+ a5
2524
+ a4
2525
+ a3
2526
+ a2
2527
+ a0
2528
+ a5
2529
+ a4
2530
+ a3
2531
+ a1
2532
+ 5a5
2533
+ 4a4
2534
+ 3a3
2535
+ 2a2
2536
+ 5a5
2537
+ 4a4
2538
+ 3a3
2539
+ a1
2540
+ 5a5
2541
+ 4a4
2542
+ 2a2
2543
+ ����������
2544
+ ����������
2545
+ a5
2546
+ a4
2547
+ a3
2548
+ a2
2549
+ a5
2550
+ a4
2551
+ a3
2552
+ a0
2553
+ 5a5
2554
+ 4a4
2555
+ 3a3
2556
+ 2a2
2557
+ 5a5
2558
+ 4a4
2559
+ 3a3
2560
+ 5a5
2561
+ 4a4
2562
+ a1
2563
+ ����������
2564
+ 2
2565
+ ����������
2566
+ a5
2567
+ a4
2568
+ a3
2569
+ a2
2570
+ a1
2571
+ a5
2572
+ a4
2573
+ a3
2574
+ a2
2575
+ 5a5
2576
+ 4a4
2577
+ 3a3
2578
+ 2a2
2579
+ a1
2580
+ 5a5
2581
+ 4a4
2582
+ 3a3
2583
+ 2a2
2584
+ 5a5
2585
+ 4a4
2586
+ 3a3
2587
+ ����������
2588
+ ����������
2589
+ a5
2590
+ a4
2591
+ a3
2592
+ a2
2593
+ a0
2594
+ a5
2595
+ a4
2596
+ a3
2597
+ a1
2598
+ 5a5
2599
+ 4a4
2600
+ 3a3
2601
+ 2a2
2602
+ 5a5
2603
+ 4a4
2604
+ 3a3
2605
+ a1
2606
+ 5a5
2607
+ 4a4
2608
+ 2a2
2609
+ ����������
2610
+ 2
2611
+ ����������
2612
+ a5
2613
+ a4
2614
+ a3
2615
+ a2
2616
+ a1
2617
+ a5
2618
+ a4
2619
+ a3
2620
+ a2
2621
+ 5a5
2622
+ 4a4
2623
+ 3a3
2624
+ 2a2
2625
+ a1
2626
+ 5a5
2627
+ 4a4
2628
+ 3a3
2629
+ 2a2
2630
+ 5a5
2631
+ 4a4
2632
+ 3a3
2633
+ ����������
2634
+ ����������
2635
+ a5
2636
+ a4
2637
+ a3
2638
+ a2
2639
+ a0
2640
+ a5
2641
+ a4
2642
+ a3
2643
+ a1
2644
+ 5a5
2645
+ 4a4
2646
+ 3a3
2647
+ 2a2
2648
+ 5a5
2649
+ 4a4
2650
+ 3a3
2651
+ a1
2652
+ 5a5
2653
+ 4a4
2654
+ 2a2
2655
+ ����������
2656
+ ����������������������������������
2657
+ 2. HY21’s condition: Q1 = 0 ∧ Q2 = 0 ∧ Q3 ̸= 0 ∧ Q4 ̸= 0 where
2658
+ Q1 =
2659
+ ������������������
2660
+ a5
2661
+ a4
2662
+ a3
2663
+ a2
2664
+ a1
2665
+ a0
2666
+ a5
2667
+ a4
2668
+ a3
2669
+ a2
2670
+ a1
2671
+ a0
2672
+ a5
2673
+ a4
2674
+ a3
2675
+ a2
2676
+ a1
2677
+ a0
2678
+ a5
2679
+ a4
2680
+ a3
2681
+ a2
2682
+ a1
2683
+ a0
2684
+ 5a5
2685
+ 4a4
2686
+ 3a3
2687
+ 2a2
2688
+ a1
2689
+ 5a5
2690
+ 4a4
2691
+ 3a3
2692
+ 2a2
2693
+ a1
2694
+ 5a5
2695
+ 4a4
2696
+ 3a3
2697
+ 2a2
2698
+ a1
2699
+ 5a5
2700
+ 4a4
2701
+ 3a3
2702
+ 2a2
2703
+ a1
2704
+ 5a5
2705
+ 4a4
2706
+ 3a3
2707
+ 2a2
2708
+ a1
2709
+ ������������������
2710
+ Q2 =
2711
+ ��������������
2712
+ a5
2713
+ a4
2714
+ a3
2715
+ a2
2716
+ a1
2717
+ a0
2718
+ a5
2719
+ a4
2720
+ a3
2721
+ a2
2722
+ a1
2723
+ a0
2724
+ a5
2725
+ a4
2726
+ a3
2727
+ a2
2728
+ a1
2729
+ 5a5
2730
+ 4a4
2731
+ 3a3
2732
+ 2a2
2733
+ a1
2734
+ 5a5
2735
+ 4a4
2736
+ 3a3
2737
+ 2a2
2738
+ a1
2739
+ 5a5
2740
+ 4a4
2741
+ 3a3
2742
+ 2a2
2743
+ a1
2744
+ 5a5
2745
+ 4a4
2746
+ 3a3
2747
+ 2a2
2748
+ ��������������
2749
+ Q3 =
2750
+ ����������
2751
+ a5
2752
+ a4
2753
+ a3
2754
+ a2
2755
+ a1
2756
+ a5
2757
+ a4
2758
+ a3
2759
+ a2
2760
+ 5a5
2761
+ 4a4
2762
+ 3a3
2763
+ 2a2
2764
+ a1
2765
+ 5a5
2766
+ 4a4
2767
+ 3a3
2768
+ 2a2
2769
+ 5a5
2770
+ 4a4
2771
+ 3a3
2772
+ ����������
2773
+ Q4 =
2774
+ ������������������
2775
+ a5
2776
+ a4
2777
+ a3
2778
+ a2
2779
+ a1
2780
+ a0
2781
+ a5
2782
+ a4
2783
+ a3
2784
+ a2
2785
+ a1
2786
+ a0
2787
+ a5
2788
+ a4
2789
+ a3
2790
+ a2
2791
+ a1
2792
+ a0
2793
+ a5
2794
+ a4
2795
+ a3
2796
+ a2
2797
+ a1
2798
+ a0
2799
+ 10a5
2800
+ 6a4
2801
+ 3a3
2802
+ a2
2803
+ 10a5
2804
+ 6a4
2805
+ 3a3
2806
+ a2
2807
+ 10a5
2808
+ 6a4
2809
+ 3a3
2810
+ a2
2811
+ 10a5
2812
+ 6a4
2813
+ 3a3
2814
+ a2
2815
+ 5a5
2816
+ 4a4
2817
+ 3a3
2818
+ 2a2
2819
+ a1
2820
+ ������������������
2821
+ +
2822
+ ������������������
2823
+ a5
2824
+ a4
2825
+ a3
2826
+ a2
2827
+ a1
2828
+ a0
2829
+ a5
2830
+ a4
2831
+ a3
2832
+ a2
2833
+ a1
2834
+ a0
2835
+ a5
2836
+ a4
2837
+ a3
2838
+ a2
2839
+ a1
2840
+ a0
2841
+ a5
2842
+ a4
2843
+ a3
2844
+ a2
2845
+ a1
2846
+ a0
2847
+ 10a5
2848
+ 6a4
2849
+ 3a3
2850
+ a2
2851
+ 10a5
2852
+ 6a4
2853
+ 3a3
2854
+ a2
2855
+ 10a5
2856
+ 6a4
2857
+ 3a3
2858
+ a2
2859
+ 5a5
2860
+ 4a4
2861
+ 3a3
2862
+ 2a2
2863
+ a1
2864
+ 10a5
2865
+ 6a4
2866
+ 3a3
2867
+ a2
2868
+ ������������������
2869
+ 18
2870
+
2871
+ ��������������������
2872
+ a5
2873
+ a4
2874
+ a3
2875
+ a2
2876
+ a1
2877
+ a0
2878
+ a5
2879
+ a4
2880
+ a3
2881
+ a2
2882
+ a1
2883
+ a0
2884
+ a5
2885
+ a4
2886
+ a3
2887
+ a2
2888
+ a1
2889
+ a0
2890
+ a5
2891
+ a4
2892
+ a3
2893
+ a2
2894
+ a1
2895
+ a0
2896
+ 10a5
2897
+ 6a4
2898
+ 3a3
2899
+ a2
2900
+ 10a5
2901
+ 6a4
2902
+ 3a3
2903
+ a2
2904
+ 5a5
2905
+ 4a4
2906
+ 3a3
2907
+ 2a2
2908
+ a1
2909
+ 10a5
2910
+ 6a4
2911
+ 3a3
2912
+ a2
2913
+ 10a5
2914
+ 6a4
2915
+ 3a3
2916
+ a2
2917
+ ��������������������
2918
+ +
2919
+ ������������������
2920
+ a5
2921
+ a4
2922
+ a3
2923
+ a2
2924
+ a1
2925
+ a0
2926
+ a5
2927
+ a4
2928
+ a3
2929
+ a2
2930
+ a1
2931
+ a0
2932
+ a5
2933
+ a4
2934
+ a3
2935
+ a2
2936
+ a1
2937
+ a0
2938
+ a5
2939
+ a4
2940
+ a3
2941
+ a2
2942
+ a1
2943
+ a0
2944
+ 10a5
2945
+ 6a4
2946
+ 3a3
2947
+ a2
2948
+ 5a5
2949
+ 4a4
2950
+ 3a3
2951
+ 2a2
2952
+ a1
2953
+ 10a5
2954
+ 6a4
2955
+ 3a3
2956
+ a2
2957
+ 10a5
2958
+ 6a4
2959
+ 3a3
2960
+ a2
2961
+ 10a5
2962
+ 6a4
2963
+ 3a3
2964
+ a2
2965
+ ������������������
2966
+ ������������������
2967
+ a5
2968
+ a4
2969
+ a3
2970
+ a2
2971
+ a1
2972
+ a0
2973
+ a5
2974
+ a4
2975
+ a3
2976
+ a2
2977
+ a1
2978
+ a0
2979
+ a5
2980
+ a4
2981
+ a3
2982
+ a2
2983
+ a1
2984
+ a0
2985
+ a5
2986
+ a4
2987
+ a3
2988
+ a2
2989
+ a1
2990
+ a0
2991
+ 5a5
2992
+ 4a4
2993
+ 3a3
2994
+ 2a2
2995
+ a1
2996
+ 10a5
2997
+ 6a4
2998
+ 3a3
2999
+ a2
3000
+ 10a5
3001
+ 6a4
3002
+ 3a3
3003
+ a2
3004
+ 10a5
3005
+ 6a4
3006
+ 3a3
3007
+ a2
3008
+ 10a5
3009
+ 6a4
3010
+ 3a3
3011
+ a2
3012
+ ������������������
3013
+ 3. HY22’s condition: R1 = 0 ∧ R2 = 0 ∧ R3 ̸= 0 where
3014
+ R1 = 1
3015
+ a5
3016
+ ������������������
3017
+ a5
3018
+ a4
3019
+ a3
3020
+ a2
3021
+ a1
3022
+ a0
3023
+ a5
3024
+ a4
3025
+ a3
3026
+ a2
3027
+ a1
3028
+ a0
3029
+ a5
3030
+ a4
3031
+ a3
3032
+ a2
3033
+ a1
3034
+ a0
3035
+ a5
3036
+ a4
3037
+ a3
3038
+ a2
3039
+ a1
3040
+ a0
3041
+ 5a5
3042
+ 4a4
3043
+ 3a3
3044
+ 2a2
3045
+ a1
3046
+ 5a5
3047
+ 4a4
3048
+ 3a3
3049
+ 2a2
3050
+ a1
3051
+ 5a5
3052
+ 4a4
3053
+ 3a3
3054
+ 2a2
3055
+ a1
3056
+ 5a5
3057
+ 4a4
3058
+ 3a3
3059
+ 2a2
3060
+ a1
3061
+ 5a5
3062
+ 4a4
3063
+ 3a3
3064
+ 2a2
3065
+ a1
3066
+ ������������������
3067
+ R2 = 1
3068
+ a5
3069
+ ��������������
3070
+ a5
3071
+ a4
3072
+ a3
3073
+ a2
3074
+ a1
3075
+ a0
3076
+ a5
3077
+ a4
3078
+ a3
3079
+ a2
3080
+ a1
3081
+ a0
3082
+ a5
3083
+ a4
3084
+ a3
3085
+ a2
3086
+ a1
3087
+ a0
3088
+ 5a5
3089
+ 4a4
3090
+ 3a3
3091
+ 2a2
3092
+ a1
3093
+ 5a5
3094
+ 4a4
3095
+ 3a3
3096
+ 2a2
3097
+ a1
3098
+ 5a5
3099
+ 4a4
3100
+ 3a3
3101
+ 2a2
3102
+ a1
3103
+ 20a5
3104
+ 12a4
3105
+ 6a3
3106
+ 2a2
3107
+ ��������������
3108
+ R3 = 1
3109
+ a5
3110
+ ��������������
3111
+ a5
3112
+ a4
3113
+ a3
3114
+ a2
3115
+ a1
3116
+ a0
3117
+ a5
3118
+ a4
3119
+ a3
3120
+ a2
3121
+ a1
3122
+ a0
3123
+ 5a5
3124
+ 4a4
3125
+ 3a3
3126
+ 2a2
3127
+ a1
3128
+ 5a5
3129
+ 4a4
3130
+ 3a3
3131
+ 2a2
3132
+ a1
3133
+ 5a5
3134
+ 4a4
3135
+ 3a3
3136
+ 2a2
3137
+ a1
3138
+ 20a5
3139
+ 12a4
3140
+ 6a3
3141
+ 2a2
3142
+ 20a5
3143
+ 12a4
3144
+ 6a3
3145
+ 2a2
3146
+ ��������������
3147
+ From the above conditions, we make the following observations which are also true in general.
3148
+ 1. YHZ’s discriminant involves a nested determinant;
3149
+ 2. HY21’s discriminant involves a sum of several determinants;
3150
+ 3. HY22’s discriminant involves a non-nested determinant.
3151
+ 19
3152
+
3153
+ 5.2
3154
+ Maximum degree of discriminants
3155
+ For the sake of simplicity, we use the following short-hands:
3156
+ • dYHZ : the maximum of the degrees of the polynomials appearing in YHZ’s conditions ([11])
3157
+ • dHY21 : the maximum of the degrees of the polynomials appearing in HY21’s conditions ([6] )
3158
+ • dHY22 : the maximum of the degrees of the polynomials appearing in the new conditions (Theorem 9).
3159
+ Lemma 18. Let dYHZ(µ),dHY21(µ) and dHY22(µ) denote the maximum degrees of the polynomials appear-
3160
+ ing in YHZ’s condition, HY21’s condition and HY22’s condition for a given µ = (µ1, . . . , µm) ∈ M(n),
3161
+ respectively. Then we have:
3162
+ 1. Under some minor and reasonable assumption (see [5, Assumption 2]),
3163
+ dYHZ(µ) =
3164
+
3165
+
3166
+
3167
+
3168
+
3169
+
3170
+
3171
+ µ2−1
3172
+
3173
+ j=0
3174
+ (2 mj − 1)
3175
+
3176
+
3177
+
3178
+ 1
3179
+ if
3180
+ µ1 = µ2
3181
+ 1 +
3182
+ 2
3183
+ 2mµ2−1−1
3184
+ if
3185
+ µ1 = µ2 + 1
3186
+ (2 (µ1 − µ2) − 1)
3187
+ if
3188
+ µ1 > µ2 + 1
3189
+
3190
+ 2n + 3µ2 − 4µ2,
3191
+ for m > 1
3192
+ 2n − 1,
3193
+ for m = 1
3194
+ where mi is the largest k such that µk > i;
3195
+ 2. dHY21(µ) = 2n − 1;
3196
+ 3. dHY22(µ) = 2n − 2.
3197
+ Proof.
3198
+ 1. When m = 1, µ = (n). In this case, the condition for the polynomial having multiplicity structure µ
3199
+ is given by the 0-th,. . . ,(n − 1)-th subdiscriminants. Thus the maximum degree dYHZ(µ) is 2n − 1,
3200
+ achieved at the 0-th subdiscriminant.
3201
+ When m > 1, see [5, Appendix] for a detailed proof.
3202
+ 2. Recall that HY21’s condition consists of two parts: (i) the 0-th,. . . ,(n − m)-th subdiscriminants whose
3203
+ highest degree is 2n − 1; (ii) the multiplicity discriminant given by
3204
+
3205
+ σ∈Sp
3206
+ dp
3207
+
3208
+ ���������
3209
+ xn−µm−1F
3210
+ ...
3211
+ x0F
3212
+ xn−1F(σ1)/σ1!
3213
+ ...
3214
+ x0F (σn)/σn!
3215
+
3216
+ ���������
3217
+ where p = (µ1, . . . , µ1
3218
+
3219
+ ��
3220
+
3221
+ µ1
3222
+ , . . . , µm, . . . , µm
3223
+
3224
+ ��
3225
+
3226
+ µm
3227
+ ) and Sp is the set of all permutations of p. It is easy to see
3228
+ that the degree of the multiplicity discriminant is 2n − µm. Hence the maximum degree of the above
3229
+ discriminants is 2n − 1.
3230
+ 20
3231
+
3232
+ 3. HY22’s condition only consists of the multiplicity discriminants given by
3233
+ D (γ) = 1
3234
+ an
3235
+ dp
3236
+
3237
+ ��������������������
3238
+ F (0)xγ0−1
3239
+ ...
3240
+ F (0)x0
3241
+ F (1)xγ1−1
3242
+ ...
3243
+ F (1)x0
3244
+ ...
3245
+ F (s)xγs−1
3246
+ ...
3247
+ F (s)x0
3248
+
3249
+ ��������������������
3250
+ where γ = (γ1, . . . , γs) ranges over (n) ≻lex · · · ≻lex µ. Note that the highest degree is achieved when
3251
+ γ = (n) and γ0 = γ1 − 1. In this case, the degree of D (γ) is 2n − 2.
3252
+ Remark 19. It is noted that in HY21’s condition, the multiplicity discriminant is always divisible by the
3253
+ leading coefficient an and thus with this division carried out, the degree can be made smaller by 1.
3254
+ By Lemma 18, the maximum degree in YHZ’s condition grows exponentially with respect to n while the
3255
+ maximum degrees in HY21 and HY22’s conditions grow linearly. Below we show a comparison with examples
3256
+ where n < 10.
3257
+ n
3258
+ dYHZ
3259
+ dHY21
3260
+ dHY22
3261
+ 3
3262
+ 5
3263
+ 5
3264
+ 4
3265
+ 4
3266
+ 9
3267
+ 7
3268
+ 6
3269
+ 5
3270
+ 15
3271
+ 9
3272
+ 8
3273
+ 6
3274
+ 27
3275
+ 11
3276
+ 10
3277
+ 7
3278
+ 45
3279
+ 13
3280
+ 12
3281
+ 8
3282
+ 81
3283
+ 15
3284
+ 14
3285
+ 9
3286
+ 135
3287
+ 17
3288
+ 16
3289
+ 0
3290
+ 20
3291
+ 40
3292
+ 60
3293
+ 80
3294
+ 100
3295
+ 120
3296
+ 140
3297
+ 160
3298
+ 3
3299
+ 4
3300
+ 5
3301
+ 6
3302
+ 7
3303
+ 8
3304
+ 9
3305
+ YHZ
3306
+ HY21
3307
+ HY22
3308
+ max deg
3309
+ n
3310
+ Acknowledgements. The second author’s work was supported by National Natural Science Foundation of
3311
+ China (Grant Nos.: 12261010 and 11801101).
3312
+ References
3313
+ [1] S. Basu, R. Pollack, and M.-F. Roy. Algorithms in real algebraic geometry. Springer-Verlag, Berlin-
3314
+ Heidelberg, 2006.
3315
+ [2] W. Brown and J. Traub.
3316
+ On Euclid’s algorithm and the theory of subresultants.
3317
+ Journal of the
3318
+ Association for Computing Machinery, 18:505–514, 1971.
3319
+ [3] G. Collins. Subresultants and Reduced Polynomial Remainder Sequences. Journal of the Association
3320
+ for Computing Machinery, 14:128–142, 1967.
3321
+ 21
3322
+
3323
+ [4] L. Gonz´alez-Vega, T. Recio, H. Lombardi, and M.-F. Roy. Sturm-Habicht Sequences, Determinants and
3324
+ Real Roots of Univariate Polynomials. In Quantifier Elimination and Cylindrical Algebraic Decomposi-
3325
+ tion. Texts and Monographs in Symbolic Computation (A Series of the Research Institute for Symbolic
3326
+ Computation, Johannes-Kepler-University, Linz, Austria), pages 300–316. Springer, 1998.
3327
+ [5] H. Hong and J. Yang.
3328
+ A condition for multiplicity structure of univariate polynomials.
3329
+ CoRR,
3330
+ abs/2001.02388, 2020.
3331
+ [6] H. Hong and J. Yang. A condition for multiplicity structure of univariate polynomials. Journal of
3332
+ Symbolic Computation, 104:523–538, 2021.
3333
+ [7] S. Liang and D. J. Jeffrey. An algorithm for computing the complete root classification of a para-
3334
+ metric polynomial. In J. Calmet, T. Ida, and D. Wang, editors, Artificial Intelligence and Symbolic
3335
+ Computation, pages 116–130, Berlin, Heidelberg, 2006. Springer Berlin Heidelberg.
3336
+ [8] S. Liang, D. J. Jeffrey, and M. M. Maza. The complete root classification of a parametric polynomial
3337
+ on an interval. In International Symposium on Symbolic and Algebraic Computation, 2008.
3338
+ [9] S. Liang and J. Zhang. A complete discrimination system for polynomials with complex coefficients and
3339
+ its automatic generation. Science in China Series E: Technological Sciences, 42:113–128, 1999.
3340
+ [10] R. Loos. Generalized Polynomial Remainder Sequences, pages 115–137. Springer Vienna, Vienna, 1983.
3341
+ [11] L. Yang, X. Hou, and Z. Zeng. A complete discrimination system for polynomials. Science in China
3342
+ (Series E), 39(6):628–646, 1996.
3343
+ 22
3344
+
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1
+ Type II multiferroic order in two-dimensional transition metal halides from first principles
2
+ spin-spiral calculations
3
+ Joachim Sødequist and Thomas Olsen∗
4
+ Computational Atomic-Scale Materials Design (CAMD), Department of Physics,
5
+ Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
6
+ (Dated: January 13, 2023)
7
+ We present a computational search for spin spiral ground states in two-dimensional transition metal halides
8
+ that are experimentally known as van der Waals bonded bulk materials. Such spin spirals break the rotational
9
+ symmetry of the lattice and lead to polar ground states where the axis of polarization is strongly coupled to
10
+ the magnetic order (type II multiferroics). We apply the generalized Bloch theorem in conjunction with non-
11
+ collinear density functional theory calculations to find the spiralling vector that minimizes the energy and then
12
+ include spin-orbit coupling to calculate the preferred orientation of the spin plane with respect to the spiral
13
+ vector. We find a wide variety of magnetic orders ranging from ferromagnetic, stripy anti-ferromagnetic, 120◦
14
+ non-collinear structures and incommensurate spin spirals. The latter two introduce polar axes and are found in
15
+ the majority of materials considered here. The spontaneous polarization is calculated for the incommensurate
16
+ spin spirals by performing full supercell relaxation including spinorbit coupling and the induced polarization
17
+ is shown to be strongly dependent on the orientation of the spiral planes. We also test the effect of Hubbard
18
+ corrections on the results and find that for most materials LDA+U results agree qualitatively with LDA. An
19
+ exception is the Mn halides, which are found to exhibit incommensurate spin spiral ground states if Hubbard
20
+ corrections are included whereas bare LDA yields a 120◦ non-collinear ground state.
21
+ I.
22
+ INTRODUCTION
23
+ The recent discovery of ferromagnetic order in two-
24
+ dimensional (2D) CrI3 [1] has initiated a vast interest in 2D
25
+ magnetism [2–4]. Several other materials have subsequently
26
+ been demonstrated to preserve magnetic order in the mono-
27
+ layer limit when exfoliated from magnetic van der Waals
28
+ bonded compounds and the family of 2D magnets is steadily
29
+ growing. A crucial requirement for magnetic order to persist
30
+ in the 2D limit is the presence of magnetic anisotropy that
31
+ breaks the spin rotational symmetry that would otherwise ren-
32
+ der magnetic order at finite temperatures impossible by the
33
+ Mermin-Wagner theorem [5]. This is exemplified by the cases
34
+ of 2D CrBr3 [6, 7] and CrCl3 [8, 9], which are isostructural
35
+ to CrI3 and while the former remains ferromagnetic in the
36
+ atomic limit due to easy-axis anisotropy (like CrI3) the lat-
37
+ ter has a weak easy plane that forbids proper long range or-
38
+ der. Other materials with persisting ferromagnetic order in
39
+ the 2D limit include the metallic compounds Fe3/4/5GeTe2
40
+ [10–12] and the anisotropic insulator CrSBr [13], which has
41
+ an easy-axis aligned with the atomic plane. Finally, FePS3
42
+ [14] and MnPS3 [15] constitute examples of in-plane anti-
43
+ ferromagnets that preserve magnetic order in the monolayer
44
+ limit due to easy-axis anisotropy, whereas the magnetic order
45
+ is lost in monolayers of the isostructural easy-plane compound
46
+ NiPS3 [16]. The 2D materials mentioned above all consti-
47
+ tute examples of rather simple collinear magnets. However,
48
+ the ground state of three-dimensional magnetic materials of-
49
+ ten exhibit complicated non-collinear order that gives rise to a
50
+ range of interesting properties [17]. Such materials, are so far
51
+ largely lacking from the field of 2D magnetism and the discov-
52
+ ery of new non-collinear 2D magnets would greatly enhance
53
+ ∗ tolsen@fysik.dtu.dk
54
+ the possibilities of constructing versatile magnetic materials
55
+ using 2D magnets as building blocks [18].
56
+ The ground state of the classical isotropic Heisenberg
57
+ model can be shown to be a planar spin spiral characterised by
58
+ a propagation vector Q [19] and such spin configurations thus
59
+ comprise a broad class of states that generalise the concept of
60
+ ferromagnetism and anti-ferromagnetism. In fact, spin spiral
61
+ order is rather common in layered van der Waals bonded ma-
62
+ terials [20] and it is thus natural to investigate the ground state
63
+ order of the corresponding monolayers for spin spiral order.
64
+ Moreover, for non-bipartite magnetic lattices the concept of
65
+ anti-ferromagnetism is not unique. This is exemplified by the
66
+ abundant example of the triangular lattice where one may con-
67
+ sider the cases of anti-aligned ferromagnetic stripes or 120◦
68
+ non-collinear order, which can be represented as spin spirals
69
+ of Q = (1/2,0) and Q = (1/3,1/3) respectively [21, 22]. The
70
+ concept of spin spirals thus constitute a general framework for
71
+ specifying the magnetic order, which may or may not be com-
72
+ mensurate with the crystal lattice.
73
+ Finite spin spiral vectors typically break symmetries inher-
74
+ ent to the crystal lattice and may thus induce physical prop-
75
+ erties that are predicted to be absent if one only considers the
76
+ crystal symmetries. In particular, the spin spiral may yield a
77
+ polar axis that lead to ferroelectric order [23]. Such materials
78
+ are referred to as type II multiferroics and examples include
79
+ MnWO4 [24], CoCr2O4 [25], LiCu2O2 [26] and LiCuVO4
80
+ [27] as well as the triangular magnets CuFeO2 [28], CuCrO2
81
+ [28], AgCrO2 [29] and MnI2 [30]. In addition to these ma-
82
+ terials, 2D NiI2 has recently been shown to host a spin spiral
83
+ ground state that induces a spontaneous polarization [31] and
84
+ 2D NiI2 thus comprises the first example of a 2D type II mul-
85
+ tiferroic.
86
+ The prediction of new materials with certain desired prop-
87
+ erties can be vastly accelerated by first principles simulations.
88
+ In general, the search for materials with spin spiral ground
89
+ states is complicated by the fact that the magnetic order re-
90
+ arXiv:2301.05107v1 [cond-mat.mtrl-sci] 12 Jan 2023
91
+
92
+ 2
93
+ quires large super cells in the simulations. However, if one
94
+ neglects spinorbit coupling, spin spirals of arbitrary wavevec-
95
+ tors can be represented in the chemical unit cell by utilising
96
+ the generalized Bloch theorem that encodes the spiral in the
97
+ boundary conditions [32, 33]. This method has been applied
98
+ in conjunction with density functional theory (DFT) to a wide
99
+ range of materials and typically produces results that are in
100
+ good agreement with experiments [34–38].
101
+ In the present work we use DFT simulations in the frame-
102
+ work of the generalized Bloch theorem to investigate the mag-
103
+ netic ground state of monolayers derived from layered van der
104
+ Waals magnets. We then calculate the preferred orientation of
105
+ the spiral plane by adding a single component of the spinorbit
106
+ coupling in the normal direction of various trial spiral planes.
107
+ This yields a complete classification of the magnetic ground
108
+ state for these materials under the assumption that higher or-
109
+ der spin interactions can be neglected. On the other hand, the
110
+ effect of higher order spin interactions can be quantified by
111
+ deviations between spin spiral energies in the primitive unit
112
+ cell and a minimal super cell. The results for all compounds
113
+ are discussed and compared with existing knowledge from ex-
114
+ periments on the parent bulk materials. Finally, we analyse the
115
+ spontaneous polarization in all cases where an incommensu-
116
+ rate ordering vector is predicted.
117
+ The paper is organised as follows. In Sec. II we summarise
118
+ the theory used to obtain spin spiral ground states based on the
119
+ generalized Bloch theorem and briefly outline the implemen-
120
+ tation. In Sec. III we present the results and summarise the
121
+ magnetic ground states of all the investigated materials. Sec.
122
+ IV provides a conclusion and outlook.
123
+ II.
124
+ THEORY
125
+ A.
126
+ Generalized Bloch’s Theorem
127
+ The Heisenberg model plays a prominent role in the the-
128
+ ory of magnetism and typically gives an accurate account of
129
+ the fundamental magnetic excitations as well as the thermo-
130
+ dynamic properties of a given material. In the isotropic case
131
+ it can be written as
132
+ H = −1
133
+ 2 ∑
134
+ i j
135
+ Ji jSi ·Sj,
136
+ (1)
137
+ where Si is the spin operator for site i and Ji j is the exchange
138
+ coupling between sites i and j. In a classical treatment, the
139
+ spin operators are replaced by vectors of fixed magnitude and
140
+ it can be shown that the classical energy is minimised by a
141
+ planar spin spiral [19]. Such a spin configuration is charac-
142
+ terised by a wave vector Q, which is determined by the set of
143
+ exchange parameters Ji j. The spin at site i is rotated by an an-
144
+ gle Q · Ri with respect to the origin and the wave vector may
145
+ or may not be commensurate with the lattice.
146
+ In a first principles framework it is thus natural to search for
147
+ planar spin spiral ground states that give rise to periodically
148
+ modulated magnetisation densities satisfying
149
+ mq(r+Ri) = Uq,Rimq(r).
150
+ (2)
151
+ Here Ri is a lattice vector (of the chemical unit cell) and Uq,Ri
152
+ is a rotation matrix that rotates the magnetisation by an an-
153
+ gle q · Ri around the normal of the spiral plane. In the ab-
154
+ sence of spinorbit coupling we are free to perform a global
155
+ rotation of the magnetisation density and we will fix the spi-
156
+ ral plane to the xy-plane from hereon. In the framework of
157
+ DFT, the magnetisation density (2) gives rise to an exchange-
158
+ correlation magnetic field satisfying the same symmetry un-
159
+ der translation. If spinorbit coupling is neglected the Kohn-
160
+ Sham Hamiltonian thus commutes with the combined action
161
+ of translation (by a lattice vector) and a rotation of spinors by
162
+ the angle q·Ri. This implies that the Kohn-Sham eigenstates
163
+ can be written as
164
+ ψq,k(r) = eik·rU†
165
+ q(r)
166
+
167
+ u↑
168
+ q,k(r)
169
+ u↓
170
+ q,k(r)
171
+
172
+ (3)
173
+ where u↑
174
+ q,k(r) and u↓
175
+ q,k(r) are periodic in the chemical unit cell
176
+ and the spin rotation matrix is given by
177
+ Uq(r) =
178
+
179
+ eiq·r/2
180
+ 0
181
+ 0
182
+ e−iq·r/2
183
+
184
+ (4)
185
+ This is known as the generalized Bloch Theorem (GBT) and
186
+ the Kohn-Sham equations can then be written as
187
+ HKS
188
+ q,kuq,k = εq,kuq,k
189
+ (5)
190
+ where the generalized Bloch Hamiltonian:
191
+ HKS
192
+ q,k = e−ik·rUq(r)HKSU†
193
+ q(r)eik·r
194
+ (6)
195
+ is periodic in the unit cell. Here k is the crystal momentum,
196
+ q is the spiral wave vector and HKS is the Kohn-Sham Hamil-
197
+ tonian, which couples to the spin degrees of freedom through
198
+ the exchange-correlation magnetic field.
199
+ In the present work, we will not consider constraints be-
200
+ sides the boundary conditions defined by Eq. (2). For a given
201
+ q we can thus obtain a unique total energy Eq and the mag-
202
+ netic ordering vector is determined as the point where Eq has
203
+ a minimum (denoted by Q) when evaluated over the entire
204
+ Brillouin zone. However, if the chemical unit cell contains
205
+ more than one magnetic atom there may be different local ex-
206
+ trema corresponding to different intracell alignments of mag-
207
+ netic moments. In order ensure that the correct ground state
208
+ is obtained it is thus pertinent to perform a comparison be-
209
+ tween calculations that are initialised with different relative
210
+ magnetic moments. As a simple example of this, one may
211
+ consider a honeycomb lattice of magnetic atoms where the
212
+ ferromagnetic and anti-ferromagnetic configurations both cor-
213
+ respond to q = 0, but are distinguished by different intracell
214
+ orderings of the local magnetic moments. We will discuss this
215
+ in the context of CrI3 in section III C.
216
+ We also note that the true magnetic ground state is not nec-
217
+ essarily representable by the ansatz (2) and one is therefore
218
+ not guaranteed to find the ground state by searching for spin
219
+ spirals based on the minimal unit cell. In figure 1 we show
220
+ four examples of possible magnetic ground states of the tri-
221
+ angular lattice. Three of these correspond to spin spirals of
222
+
223
+ 3
224
+ Q = (1/3, 1/3)
225
+ Q = (1/2, 0)
226
+ Q = (0.14, 0.14)
227
+ Q = (0, 1/2)
228
+ (a)
229
+ Γ
230
+ M/S
231
+ K
232
+ X
233
+ Y
234
+ (b)
235
+ FIG. 1. (a) Examples of magnetic structures in the triangular lattice. The Q = (1/3,1/3) (corresponding to the high symmetry point K)
236
+ is the classical ground state in the isotropic Heisenberg model with nearest neighbour antiferromagnetic exchange and is degenerate with
237
+ Q = (−1/3,−1/3). The stripy antiferromagnetic Q = (1/2,0) (corresponding to the high symmetry point M) is only found for CoI2 in
238
+ the present study and is degenerate with Q = (0,1/2) and Q = (1/2,1/2). The incommensurate spiral with Q = (0.14,0.14) corresponds
239
+ to the prediction of NiI2 in the present work. The rectangular cell with Q = (0,1/2) is a bicollinear antiferromagnet that corresponds to
240
+ superpositions of (0, ±1/4) states in the primitive cell. (b) Brillouin zone of the hexagonal (blue) and rectangular (orange) unit cell. The high
241
+ symmetry band paths used to sample the spiral ordering vectors are shown in black.
242
+ the minimal unit cell while the fourth - a bicollinear antifer-
243
+ romagnet - requires a larger unit cell. The bicollinear state
244
+ may arise as a consequence of higher order exchange interac-
245
+ tions, which tend to stabilize linear combinations of degener-
246
+ ate single-q states.
247
+ B.
248
+ Spinorbit coupling
249
+ In the presence of spinorbit coupling, the spin spiral plane
250
+ will have a preferred orientation and the magnetic ground state
251
+ is thus characterised by a normal vector ˆn0 of the spiral plane
252
+ as well as the spiral vector Q. Spinorbit coupling is, however,
253
+ incompatible with application of the GBT and has to be ap-
254
+ proximated in a post processing step when working with the
255
+ spin spiral representation in the chemical unit cell. It can be
256
+ shown that first order perturbation theory only involves contri-
257
+ butions from the spinorbit components orthogonal to the plane
258
+ [39]
259
+ ⟨ψq,ˆn|L·S|ψq,ˆn⟩ = ⟨ψq,ˆn|(L· ˆn)(S· ˆn)|ψq,ˆn⟩,
260
+ (7)
261
+ and this term is thus expected to yield the most important con-
262
+ tribution to the spinorbit coupling. Since (L · ˆn)(S · ˆn) com-
263
+ mutes with a spin rotation around the axis ˆn, the spin spi-
264
+ ral wavefunctions remain eigenstates when such a term is in-
265
+ cluded in HKS. This approach was proposed by Sandratskii
266
+ [40] and we will refer to it as the projected spinorbit coupling
267
+ (PSO). For the spin spiral calculations in the present work we
268
+ include spinorbit coupling non-selfconsistently by performing
269
+ a full diagonalization of the HKS
270
+ q,k including the PSO. The mag-
271
+ netic ground state is then found by evaluating the total energy
272
+ at all normal vectors ˆn, which will yield ˆn0 as the normal vec-
273
+ tor that minimizes the energy.
274
+ C.
275
+ Computational Details
276
+ The GBT has been implemented in the electronic structure
277
+ software package GPAW [41], which is based on the projector
278
+ augmented wave method (PAW) and plane waves. The im-
279
+ plementation uses a fully non-collinear treatment within the
280
+ local spin density approximation where both the interstitial
281
+ and atom-centered PAW regions are handled non-collinearly.
282
+ Spinorbit coupling is included non-selfconsistently [42] as de-
283
+ scribed in Section II B. The implementation is described in de-
284
+ tail in Appendix V A and benchmarked for fcc Fe in Appendix
285
+ V B. We find good agreement with previous results from the
286
+ literature and we also assert that results from spin spiral cal-
287
+ culations within the GBT agree exactly with supercell calcu-
288
+ lations without spinorbit in the case of bilayer CoPt. Finally,
289
+ we compare the results of the PSO approximations with full
290
+ inclusion of spinorbit coupling for both supercells and GBT
291
+ spin spirals of the CoPt bilayer. We find exact agreement be-
292
+ tween the PSO in the supercell and GBT spin spiral and the
293
+ approximation only deviates slightly compared to full spinor-
294
+ bit coupling for the supercell calculations.
295
+ All calculations have been carried out with a plane wave
296
+ cutoff of 800 eV, a k-point density of 14 ˚A and a Fermi smear-
297
+ ing of 0.1 eV. The structures and initial magnetic moments
298
+ are taken from the Computational Materials Database (C2DB)
299
+ [43, 44].In order to find the value of Q, which describes the
300
+ ground state magnetic order, we calculate Eq along a represen-
301
+ tative path connecting high symmetry points in the Brillouin
302
+ zone. While the true value of Q could be situated away from
303
+ such high symmetry lines we deem this approach sufficient
304
+ for the present study.
305
+ III.
306
+ RESULTS
307
+ A comprehensive review on the magnetic properties of lay-
308
+ ered transition metal halides was provided in Ref. [20]. Here
309
+ we present spin spiral calculations and extract the magnetic
310
+ properties of the corresponding monolayers. In addition to the
311
+ magnetic moments, the properties are mainly characterised by
312
+ a spiral ordering vector Q and the normal vector to the spin
313
+ spiral plane ˆn0. The materials either have AB2 or AB3 stoi-
314
+ chiometries and we will discuss these cases separately below.
315
+
316
+ 4
317
+ Q
318
+ Emin [meV] (θ,ϕ)
319
+ Exp. IP order BW [meV] PSO BW [meV] mΓ [µB] ∆εQ [eV]
320
+ TiBr2
321
+ (1/3, 1/3)
322
+ -78.12
323
+ (90,90)
324
+ -
325
+ 78.1
326
+ 0.6
327
+ 1.5
328
+ 0.0
329
+ TiI2
330
+ (1/3, 1/3)
331
+ -44.33
332
+ (90,90)
333
+ -
334
+ 44.3
335
+ 1.0
336
+ 1.9
337
+ 0.0
338
+ NiCl2
339
+ (0.06, 0.06) -0.81
340
+ (90,31)
341
+ FM ∥
342
+ 45.2
343
+ 0.0
344
+ 2.0
345
+ 0.81
346
+ NiBr2
347
+ (0.11, 0.11) -8.62
348
+ (44,0)
349
+ FM ∥, HM
350
+ 50.7
351
+ 0.3
352
+ 2.0
353
+ 0.62
354
+ NiI2
355
+ (0.14, 0.14) -28.48
356
+ (64,0)
357
+ HM
358
+ 68.3
359
+ 4.1
360
+ 1.8
361
+ 0.28
362
+ VCl2
363
+ (1/3, 1/3)
364
+ -60.07
365
+ (90,0)
366
+ 120◦
367
+ 60.1
368
+ 0.1
369
+ 3.0
370
+ 0.96
371
+ VBr2
372
+ (1/3, 1/3)
373
+ -36.21
374
+ (90,18)
375
+ 120◦
376
+ 36.2
377
+ 0.1
378
+ 3.0
379
+ 0.9
380
+ VI2
381
+ (0.14, 0.14) -4.43
382
+ (6,0)
383
+ stripe
384
+ 9.8
385
+ 0.7
386
+ 3.0
387
+ 0.96
388
+ MnCl2 (1/3, 1/3)
389
+ -20.48
390
+ (90,15)
391
+ stripe or HM 20.5
392
+ 0.0
393
+ 5.0
394
+ 1.92
395
+ MnBr2 (1/3, 1/3)
396
+ -20.13
397
+ (90,15)
398
+ stripe ∥
399
+ 20.1
400
+ 0.1
401
+ 5.0
402
+ 1.76
403
+ MnI2
404
+ (1/3, 1/3)
405
+ -21.32
406
+ (0,0)
407
+ HM
408
+ 21.3
409
+ 1.1
410
+ 5.0
411
+ 1.41
412
+ FeCl2
413
+ (0, 0)
414
+ 0.0
415
+ (0, 0)∗
416
+ FM ⊥
417
+ 115.2
418
+ 0.5∗
419
+ 4.0
420
+ 0.0
421
+ FeBr2
422
+ (0, 0)
423
+ 0.0
424
+ (0, 0)∗
425
+ FM ⊥
426
+ 81.3
427
+ 0.8∗
428
+ 4.0
429
+ 0.0
430
+ FeI2
431
+ (0, 0)
432
+ 0.0
433
+ (0, 0)∗
434
+ stripe ⊥
435
+ 36.5
436
+ 1.9∗
437
+ 4.0
438
+ 0.0
439
+ CoCl2
440
+ (0, 0)
441
+ 0.0
442
+ (90,90)∗ FM ∥
443
+ 46.0
444
+ 1.2∗
445
+ 3.0
446
+ 0.0
447
+ CoBr2 (0.03, 0.03) -0.04
448
+ (0,0)
449
+ FM ∥
450
+ 21.2
451
+ 0.1
452
+ 3.0
453
+ 0.0
454
+ CoI2
455
+ (1/2, 0)
456
+ -20.95
457
+ (90,90)
458
+ HM
459
+ 41.7
460
+ 5.6
461
+ 1.2
462
+ 0.0
463
+ TABLE I. Summary of magnetic properties of the AB2 compounds. The ground state ordering vector is denoted by Q and Emin is the ground
464
+ state energy relative to the ferromagnetic state. The normal vector of the spiral plane is defined by the angles θ and ϕ (see text). We also
465
+ display the experimental in-plane order of the parent layered compound (Exp. IP order). In addition we state the spin spiral band width
466
+ BW, the magnetic moment per unit cell in the ferromagnetic state mΓ and the band gap at the ordering vector ∆εQ. For the case of NiI2, mΓ
467
+ deviates from an integer value because the ferromagnetic state is metallic in LDA (whereas the spin spiral ground state has a gap). The cases
468
+ of FeX2, CoCl2 and CoBr2 are half metals, which enforces integer magnetic moment despite the metallic ground state. The asterisks indicate
469
+ ferromagnets where full spinorbit coupling was included and the angles then refer to the direction of the spins rather that the spiral plane
470
+ normal vector.
471
+ We have performed LDA and LDA+U calculations for all
472
+ materials. In most cases, the Hubbard corrections does not
473
+ make any qualitative difference although the spiral ordering
474
+ vector does change slightly and we will not discuss these cal-
475
+ culations further here. The Mn halides comprise an exception
476
+ to this where LDA+U calculations differ significantly from
477
+ those of bare LDA and the LDA+U calculations will be dis-
478
+ cussed separately for these materials below.
479
+ For the AB2 materials, we find 12 that exhibit a spiral or-
480
+ der that breaks the crystal symmetry and yields a ferroelec-
481
+ tric ground state. For six of these compounds we have calcu-
482
+ lated the spontaneous polarization by performing full relax-
483
+ ation (including self-consistent spinorbit coupling) in super-
484
+ cells hosting the spiral order.
485
+ A.
486
+ Magnetic ground state of AB2 materials
487
+ The AB2 materials all have space group P¯3m1 correspond-
488
+ ing to monolayers of the CdI2 (or CdCl2) prototype. The mag-
489
+ netic lattice is triangular and a few representative possibilities
490
+ for the magnetic order is illustrated in figure 1. The magnetic
491
+ properties of all the considered compounds are summarized
492
+ in table I. In addition to the ordering vector Q we provide
493
+ the angles θ and φ, which are the polar and azimuthal an-
494
+ gles of ˆn0 with respect to the out-of-plane direction and the
495
+ ordering vector respectively. It will be convenient to consider
496
+ three limiting cases of the orientation of spin spiral planes:
497
+ the proper screw (θ = 90,ϕ = 0), the out-of-plane cycloid
498
+ (�� = 90,ϕ = 90) and the in-plane cycloid (θ = 0,ϕ = 0).
499
+ We also provide the ground state energy relative to the fer-
500
+ romagnetic configuration (Q = (0,0)), the band gap, the spin
501
+ spiral band width, which reflects the strength of the magnetic
502
+ interactions and the PSO band width, which is the energy dif-
503
+ ference between the easy and hard orientations of the spiral
504
+ plane. The magnetic moments are calculated as the total mo-
505
+ ment in the unit cell using the ferromagnetic configurations
506
+ without spinorbit interaction and thus yields an integer num-
507
+ ber of Bohr magnetons for insulators. The magnitude of the
508
+ local magnetic moments (obtained by integrating the magne-
509
+ tization density over the PAW spheres) in the ground state are
510
+ generally found to be very close to the moments in the ferro-
511
+ magnetic configuration, unless explicitly mentioned. The spin
512
+ spiral energy dispersions are provided for all AB2 materials in
513
+ the supporting information. The different classes of materials
514
+ are described in detail below.
515
+ NiX2
516
+ The nickel halides all have ground states with incommen-
517
+ surate spiral vectors between Γ and K. Experimentally, both
518
+ NiI2 and NiBr2 in bulk form have been determined to have in-
519
+ commensurate spiral vectors [45–47] in qualitative agreement
520
+ with the LDA results. The case of NiCl2, however, have been
521
+ found to have ferromagnetic intra-layer order whereas we find
522
+ a rather small spiral vector of Q = (0.06,0.06).
523
+ In bulk NiI2 the experimental ordering vector Qexp =
524
+ (0.1384,0,1.457) has an in-plane component in the ΓM-
525
+ direction with a magnitude of roughly 1/7 of a recipro-
526
+ cal lattice vector, while for the monolayer we find Q =
527
+ (0.14,0.14,0), which is in the ΓK-direction. Evaluating the
528
+ spin spiral energy in the entire Brillouin zone, however, re-
529
+ veals a nearly degenerate ring encircling the Γ-point with a
530
+
531
+ 5
532
+ Γ
533
+ M
534
+ K
535
+ Γ
536
+ −20
537
+ 0
538
+ 20
539
+ 40
540
+ E(q) [meV]
541
+ (a)
542
+ K
543
+ G
544
+ M
545
+ 28
546
+ 18
547
+ 8
548
+ 0
549
+ 10
550
+ 20
551
+ 30
552
+ 40
553
+ Energy [meV]
554
+ (b)
555
+ IP
556
+ Screw
557
+ OoP
558
+ IP
559
+ −44
560
+ −42
561
+ −40
562
+ Esoc(θ, ϕ) [meV]
563
+ θ
564
+ ϕ
565
+ θ
566
+ (c)
567
+ FIG. 2. Spin spiral energy of NiI2. Left: the spin spiral energy as a function of q without spinorbit coupling. Center: Spin spiral energy in
568
+ evaluated in entire Brillouin zone. Right: spiral energy as a function of spiral plane orientation evaluated at the minimum Q = (0.14,0.14).
569
+ The spiral plane orientation is parameterized in terms of the polar angle θ and azimuthal angle ϕ (measured from Q) of the spiral plane normal
570
+ vector.
571
+ radius of roughly 1/5 of a reciprocal lattice vector. The point
572
+ qM = (0.21,0) thus comprises a very shallow saddle point
573
+ with an energy that exceeds the minimum by merely 2 meV.
574
+ This is illustrated in figure 2. We also show a scan of the
575
+ spin spiral energy (within the PSO approximation) as a func-
576
+ tion of orientation of the spin spiral plane on a path that con-
577
+ nects the limiting cases of in-plane cycloid, out-of-plane cy-
578
+ cloid and proper screw. An unconstrained spin spiral calcu-
579
+ lation using the rectangular unit cell of figure 1 does not re-
580
+ veal any new minima in the energy, which implies that the
581
+ ground state is well represented by a single-q spiral and that
582
+ higher order exchange interactions are neglectable in NiI2.
583
+ The normal vector of the spiral makes an angle of 64◦ with
584
+ the out-of-plane direction. This orientation is in good agree-
585
+ ment with the experimental assignment of a proper screw
586
+ (along Qexp = (0.1384,0,1.457)), which corresponds to a tilt
587
+ of 55◦±10◦ with respect to the c-axis [47], but disagrees with
588
+ the model proposed in Ref. [31] where the spiral was found
589
+ to be a proper screw.
590
+ At low temperatures NiBr2 has been reported to exhibit
591
+ Qexp = (x,x,3/2) where x changes continuously from 0.027
592
+ at 4.2 K to 0.09 at 22.8 K and then undergoes first order transi-
593
+ tion at 24 K to intra-layer ferromagnetic order [48]. The struc-
594
+ ture predicted here is close to the one observed in bulk at 22.8
595
+ K. The discrepancy could be due to the magnetoelastic defor-
596
+ mation [49] that has been associated with the modulation of
597
+ the spiral vector. This effect could in principle be captured by
598
+ relaxing the structure in supercell calculations, but the small
599
+ wavelength spirals require prohibitively large supercells and
600
+ are not easily captured by first principles methods. It is also
601
+ highly likely that LDA is simply not accurate enough to de-
602
+ scribe the intricate exchange interactions that define the true
603
+ ground state in this material.
604
+ Bulk NiCl2 is known to be an inter-layer antiferromag-
605
+ net with ferromagnetically ordered layers [50]. We find the
606
+ ground state to be a long wavelength incommensurate spin
607
+ spiral with Q = (0.06,0.06), which is in rather close prox-
608
+ imity to ferromagnetic order. The ground state energy is less
609
+ than 1 meV lower than the ferromagnetic state, but we cannot
610
+ say at present whether this is due to inaccuracies of LDA or if
611
+ the true ground state indeed exhibits spiral magnetic order in
612
+ the monolayer limit.
613
+ VX2
614
+ The three vanadium halides are insulators and whereas
615
+ VCl2 and VBr2 are found to form Q = (1/3,1/3) spiral struc-
616
+ tures, VI2 has an incommensurate ground state with Q =
617
+ (0.14,0.14). The magnetic ground state of VCl2 and VBr2 is
618
+ in good agreement with experiments on bulk materials where
619
+ both have been found to exhibit out-of-plane 120◦ order [51].
620
+ This structure is expected to arise from strong nearest neigh-
621
+ bour anti-ferromagnetic interactions between the V atoms.
622
+ The case of VI2 has a significantly smaller spiral band width,
623
+ signalling weaker exchange interactions compared to VCl2
624
+ and VBr2. A collinear energy mapping based on the Perdew-
625
+ Burke-Ernzerhof (PBE) exchange-correlation functional [44]
626
+ yields a weakly ferromagnetic nearest neighbour interaction
627
+ for VI2 and strong anti-ferromagnetic interactions for VCl2
628
+ and VBr2. This is in agreement with the present result, which
629
+ indicate that the magnetic order of VI2 is not dominated by
630
+ nearest neighbour interactions.
631
+ Experimentally [52], the bulk VI2 magnetic order has been
632
+ found to undergo a phase transition at 14.4 K from a 120◦ state
633
+ to a bicollinear state with Q = (1/2,0), where the spins are
634
+ perpendicular to Q and tilted by 29◦ from the z-axis. Such a
635
+ bicollinear state implies that the true ground state is a double-
636
+ q state stabilized by higher order spin interactions and cannot
637
+ be represented as a spin spiral in the primitive unit cell. To
638
+ check whether LDA predicts the experimental ground state we
639
+ have therefore performed spiral calculations in the rectangular
640
+ cell shown in figure 1. The result is shown in figure 3 along
641
+ with the spiral calculation in the primitive cell and we do not
642
+ find any new minima in the super cell calculation. We have
643
+ initalized angles in the super cell caluculation such that they
644
+ corresponds to bicollinear order and the angles are observed
645
+ to relax to the single-q spin spiral of the primitive cell. It
646
+ is likely that LDA is insufficient to capture the subtle higher
647
+ order exchange interactions in this material, but it is possible
648
+ that the monolayer simply has a magnetic order that differs
649
+
650
+ 6
651
+ Γ
652
+ K
653
+ M
654
+ Γ
655
+ X
656
+ S
657
+ Y
658
+ Γ
659
+ S
660
+ −4
661
+ −2
662
+ 0
663
+ 2
664
+ 4
665
+ E(q) [meV/uc]
666
+ FIG. 3. Spin spiral energies of VI2 obtained from the primitive cell
667
+ (black) and the rectangular super cell (blue). The dashed lines repeat
668
+ the primitive cell results on the corresponding super cell path.
669
+ from the individual layers in the bulk material.
670
+ In the PSO approximation we find that VCl2 and VBr2 pre-
671
+ fer out-of-plane spiral planes. The energy is rather insensitive
672
+ to ϕ forming a nearly degenerate subspace of ground states
673
+ with a slight preference of the proper screw. The ground state
674
+ of VI2 is found to be close to the in-plane cycloid with a nor-
675
+ mal vector to the spiral plane forming a 6◦ angle with Q. The
676
+ spinorbit corrections in VI2 are also found to be the smallest
677
+ compared to other iodine based transition metal halides stud-
678
+ ied here and the ground state energy only deviates by 0.7 meV
679
+ per unit cell from the out-of-plane cycloid, which constitutes
680
+ the orientation of the spin plane with highest energy.
681
+ MnX2
682
+ The manganese halides are all found to form 120◦ ground
683
+ states, which is in agreement with previous theoretical studies
684
+ [53] using PBE. In contrast to the other insulators studied in
685
+ the present work, however, we find that the results are qualita-
686
+ tively sensitive to the inclusion of Hubbard corrections. This
687
+ was also found in Ref. [54], where the sign of the nearest
688
+ neighbour exchange coupling was shown to change sign when
689
+ a Hubbard U parameter was included in the calculations. With
690
+ U = 3.8 eV we find that all three compounds has spiral ground
691
+ states with incommensurate spiral vector Q = (0.11,0.11,0).
692
+ Moreover, spin spiral band width in the LDA+U calculations
693
+ decrease by more than an order of magnitude compared to the
694
+ bare LDA calculations.
695
+ The experimental magnetic structure of the manganese
696
+ halides are rather complicated, exhibiting several magnetic
697
+ phase transitions in a range of 0.1 K below the initial order-
698
+ ing temperature. In particular MnI2 (MnBr2) has been found
699
+ to have three (two) complex non-collinear phases [55], and
700
+ MnCl2 has two complex phases that are possibly collinear
701
+ [56].
702
+ The experimental ground state of bulk MnCl2 not unam-
703
+ biguously known, but under the assumption of collinearity a
704
+ possible ground state contains 15 Mn atoms in an extended
705
+ stripy pattern [56]. Due to the weak and subtle nature of mag-
706
+ netic interactions in the manganese compounds, however, it is
707
+ not unlikely that the ground state in the monolayers can dif-
708
+ fer from that of bulk. This is corroborated by an experimental
709
+ study of MnCl2 intercalated by graphite where a helimagnetic
710
+ ground state with Qexp = (0.153,0.153) was found [57]. This
711
+ is rather close to our predicted ordering vector obtained from
712
+ LDA+U.
713
+ Experimentally, bulk MnBr2 is found to exhibit a stripy
714
+ bicollinear uudd order at low temperatures [58]. The order
715
+ cannot be represented by a spiral in the minimal cell, but re-
716
+ quires calculations in rectangular unit cells with spiral order
717
+ Q = (0,1/2) similar to VI2 discussed above. We have calcu-
718
+ lated the high symmetry band path required to show this order
719
+ and do not find any new minima. It is likely that the situation
720
+ resembles MnCl2 where a single-q spiral has been observed
721
+ for decoupled monolayers in agreement with our calculations.
722
+ FeX2
723
+ We find all the iron halides to have ferromagnetic ground
724
+ states. For FeCl2 and FeBr2 this is in agreement with the
725
+ experimentally determined magnetic order for the bulk com-
726
+ pounds [59]. In contrast, FeI2 has been reported to exhibit a
727
+ bicollinear antiferromagnetic ground state [60] similar to the
728
+ case of MnBr2 discussed above. It is again possible that the
729
+ ground state of the monolayer (calculated here) could differ
730
+ from the magnetic ground state of the bulk compound as has
731
+ been found for MnCl2.
732
+ LDA predict the three compounds to be half metals, mean-
733
+ ing that the majority spin bands are fully occupied and only
734
+ the minority bands have states at the Fermi level. This en-
735
+ forces an integer number of Bohr magnetons (four) per unit
736
+ cell at any q-vector in the spin spiral calculations. Thus longi-
737
+ tudinal fluctuations are expected to be strongly suppressed in
738
+ iron halides and it is likely that these materials can be accu-
739
+ rately modelled by Heisenberg Hamiltonians despite the itin-
740
+ erant nature of the electronic structure.
741
+ The projected spin orbit coupling is not applicable to
742
+ collinear structures and we therefore include full spin orbit
743
+ coupling, which is compatible with the Q = (0,0) ground
744
+ state. We find that all the iron compounds have an out-of-
745
+ plane easy axis, which is in agreement with experiments. The
746
+ bandwidth provided in table I then simply corresponds to the
747
+ magnetic anisotropy energy which is smallest for FeCl2 and
748
+ increases for the heavier Br and I compounds as expected.
749
+ CoX2
750
+ We predict CoCl2 to have an in-plane ferromagnetic ground
751
+ state in agreement with the experimentally determined mag-
752
+ netic order of the bulk compound [59]. CoBr2 is found to have
753
+ a long wavelength spin spiral with Q = (0.03,0.03). The spi-
754
+ ral energy in the vicinity of the Γ-point is, however, extremely
755
+ flat with almost vanishing curvature and the ground state en-
756
+ ergy is merely 0.04 meV lower than the ferromagnetic state.
757
+ We regard this as being in agreement with the experimental re-
758
+ port of intra-layer ferromagnetic order in the bulk compound
759
+ [59].
760
+
761
+ 7
762
+ The case of CoI2 deviates substantially from the other two
763
+ halides. CoCl2 and CoBr2 are half-metals with m = 3 µB per
764
+ unit cell, whereas CoI2 is an ordinary metal with m ≈ 1.2 µB
765
+ per unit cell. We find the magnetic ground state of CoI2 to
766
+ be stripy anti-ferromagnetic with Q = (1/2,0), whereas ex-
767
+ periments on the bulk compound have reported helimagnetic
768
+ in-plane order with Qexp = (1/6,1/8,1/2) in the rectangular
769
+ cell [61]. We note, however, that the calculated local mag-
770
+ netic moments vary strongly with q (up to 0.5 µB) in the spin
771
+ spiral calculations, which signals strong longitudinal fluctu-
772
+ ations. This could imply that the material comprises a rather
773
+ challenging case for DFT and LDA may be insufficient to treat
774
+ this material properly.
775
+ B.
776
+ Spontaneous polarization of AB2 materials
777
+ The materials in table I that exhibit spin spiral ground states
778
+ are expected to introduce a polar axis due to spinorbit cou-
779
+ pling and thus allow for spontaneous electric polarization.
780
+ The stripy antiferromagnet with Q = (1/2,0) preserves a site-
781
+ centered inversion center and remains non-polar. In addition,
782
+ the case of Q = (1/3,1/3) with in-plane orientation of the spi-
783
+ ral plane breaks inversion symmetry, but retains the three-fold
784
+ rotational symmetry (up to translation of a lattice vector) and
785
+ therefore cannot acquire components of in-plane polarization.
786
+ To investigate the effect of symmetry breaking we have
787
+ constructed 7×1 supercells of VI2 and the Ni halides and per-
788
+ formed a full relaxation of the q = (1/7,0) spin spiral com-
789
+ mensurate with the supercell. This is not exactly the spin spi-
790
+ rals found as the ground state from LDA, but we will use these
791
+ to get a rough estimate of the spontaneous polarization. We
792
+ note that this is very close to the in-plane component of Qexp
793
+ for bulk NiI2, which is found to be nearly degenerate with
794
+ the predicted ground state (see figure 2). The other materi-
795
+ als exhibit similar near-degeneracies, but the calculated polar-
796
+ ization could be sensitive to which spiral ordering vector is
797
+ used. We have chosen to focus on the incommensurate spi-
798
+ rals, but note that all the Q = (1/3,1/3) materials of table II
799
+ are expected to introduce a spontaneous polarization as well.
800
+ Besides the incommensurate spirals we thus only include the
801
+ cases of MnBr2 and MnI2 where the Q = (1/3,1/3) spirals
802
+ may be represented in
803
+
804
+ 3 ×
805
+
806
+ 3 supercells. The former case
807
+ represents an example of a proper screw while the latter is an
808
+ in-plane cycloid. The experimental order in the Mn halides
809
+ materials is complicated, and our LDA+U calculations yield
810
+ an ordering vector that differs from that of LDA. However,
811
+ here we mostly consider these examples for comparison and
812
+ to check the symmetry constraints on the polarization in the
813
+ Q = (1/3,1/3) spirals.
814
+ In order to calculate the spontaneous polarization we relax
815
+ the atomic positions in the super cells both with and without
816
+ spinorbit coupling (included self-consistently) and calculate
817
+ the 2D polarization from the Berry phase formula [44]. The
818
+ results are summarized in Tab. II. We can separate the effect
819
+ of relaxation from the pure electronic contribution by calcu-
820
+ lating the polarization (including spin-orbit) of the structures
821
+ that were relaxed without spinorbit coupling. These numbers
822
+ are stated in brackets in table II as well as the total polarization
823
+ (including relaxation) and the angles that define the orienta-
824
+ tion of the spiral plane with respect to Q. The self-consistent
825
+ calculations yield the optimal orientations of the spiral planes
826
+ without the PSO approximations and it is reassuring that the
827
+ orientation roughly coincides with the results of the GBT and
828
+ the PSO approximation.
829
+ The magnitude of polarization largely scales with the
830
+ atomic number of ligands (as expected from the strength of
831
+ spinorbit coupling) and the iodide compounds thus produce
832
+ the largest polarization. The in-plane cycloid in MnI2 only
833
+ give rise to out-of-plane polarization as expected from sym-
834
+ metry and the Q = (1/3,1/3) proper screw in MnBr2 has po-
835
+ larization that is strictly aligned with Q. The latter results is
836
+ expected for any proper screw in the ΓK-direction because Q
837
+ then coincides with a two-fold rotational axis and the ground
838
+ state remains invariant under the combined action of this ro-
839
+ tation and time-reversal symmetry. Since the polarization is
840
+ not affected by time-reversal it must be aligned with the two-
841
+ fold axis. The polarization vectors of the remaining materials
842
+ (except for NiCl2) are roughly aligned with the intersection
843
+ between the spiral plane and the atomic plane.
844
+ It is interesting to note that the calculated magnitudes of to-
845
+ tal polarization are 5-10 times larger than the prediction from
846
+ the pure electronic contribution where the atoms were not re-
847
+ laxed with spinorbit coupling. We also tried to calculate the
848
+ polarization by using the Born effective charge tensors (with-
849
+ out spin-orbit) and the atomic deviations from the centrosym-
850
+ metric positions. However, this approximation severely un-
851
+ derestimates the polarization and even produces the wrong
852
+ sign of the polarization in the case of NiBr2 and NiI2. To
853
+ obtain reliable values for the polarization it is thus crucial to
854
+ include the relaxation effects and take the electronic contribu-
855
+ tion properly into account (going beyond the Born effective
856
+ charge approximation). In Ref. [31] a value of 141 fC/m was
857
+ predicted in 2D NiI2 from the gKNB model [62] and this is
858
+ comparable to the values found in table II without relaxation
859
+ effects. When relaxation is included we find a magnitude of
860
+ 1.9 pC/m for NiI2, which is an order of magnitude larger com-
861
+ pared to the previous prediction. The results are, however, not
862
+ directly comparable since Ref. [31] considered a spiral along
863
+ the ΓK direction whereas the present result is for a spiral along
864
+ ΓM. We note that Ref. [31] finds the polarization to be aligned
865
+ with Q in agreement with the symmetry considerations above.
866
+ Finally, the values for the spontaneous polarization in table II
867
+ may be compared with those of ordinary 2D ferroelectrics,
868
+ which are typically on the order of a few hundred pC/m for
869
+ in-plane ferroelectrics and a few pC/m for out-of-plane ferro-
870
+ electrics [63] .
871
+ In all of these type II multiferroics, the orientation of the
872
+ induced polarization depends on the direction of the ordering
873
+ vector, which may thus be switched by application of an ex-
874
+ ternal electric field. We have checked explicitly that the sign
875
+ of polarization is changed if we relax a right-handed instead
876
+ of a left-handed spiral (corresponding to a reversed ordering
877
+ vector). The small values of spontaneous polarization in these
878
+ materials implies that rather modest electric fields are required
879
+ for switching the ordering vector and thus comprise an in-
880
+
881
+ 8
882
+ (θ,ϕ)
883
+ P∥
884
+ P⊥
885
+ Pz
886
+ VI2
887
+ (11, 0)
888
+ -0.6 (-31)
889
+ 290 (96)
890
+ 0.05 (0.11)
891
+ NiCl2 (90, -30) -37 (-1.4)
892
+ 76 (15)
893
+ 3.5(-5.1)
894
+ NiBr2 (69, -10)
895
+ 12 (-6)
896
+ 340 (32)
897
+ 26 (37)
898
+ NiI2
899
+ (70, 0)
900
+ -8 (-48)
901
+ 1890 (400) -0.18 (12)
902
+ MnBr2
903
+ (90, 0)
904
+ 430 (38)
905
+ 0 (0.02)
906
+ 0 (0)
907
+ MnI2
908
+ (0, 0)
909
+ 0 (0.6)
910
+ 0.3 (-7)
911
+ -260 (-105)
912
+ TABLE II. Orientation of spin planes, and 2D polarization (in fC/m)
913
+ of selected transition metal halides. P∥ denotes the polarization along
914
+ Q, while P⊥ denotes the polarization in the atomic plane orthogonal
915
+ to Q and Pz is the polarization orthogonal to the atomic plane. The
916
+ numbers in brackets are the polarization values obtained prior to re-
917
+ laxation of atomic positions. We have used 7×1 supercells for the V
918
+ and Ni halides and
919
+
920
+
921
+
922
+ 3 supercells for the Mn halides. All calcu-
923
+ lations are set up with left-handed spirals. The numbers in brackets
924
+ state the spontaneous polarization without relaxation effects.
925
+ teresting alternative to standard multiferroics such as BiFeO3
926
+ and YMnO3, where the coercive electric fields are orders of
927
+ magnitude larger.
928
+ C.
929
+ Magnetic ground state of AB3 materials
930
+ The AB3 materials all have space group P¯3m1 correspond-
931
+ ing to monolayers of the BI3 (or AlCl3) prototype. The mag-
932
+ netic lattice is the honeycomb motif, thus hosting two mag-
933
+ netic ions in the primitive cell. Several materials of this pro-
934
+ totype have been characterized experimentally, but here we
935
+ only present results for the Cr compounds. This is due to the
936
+ fact that experimental data of in-plane order is missing for all
937
+ but CrX3, FeCl3 and RuCl3. Moreover, all magnetic com-
938
+ pounds were found to have a simple ferromagnetic ground
939
+ state. RuCl3 is a well known insulator with stripy antiferro-
940
+ magnetic in-plane order. However, bare LDA finds a metallic
941
+ state and both Hubbard corrections and self-consistent spinor-
942
+ bit coupling are required to obtain the correct insulating state
943
+ [64]. The latter is incompatible with the GBT approach and
944
+ we have not pursued this further here. Bulk FeCl3 is known to
945
+ be an insulating helimagnet with Q = ( 4
946
+ 15, 1
947
+ 15, 3
948
+ 2) [65], while
949
+ we find the monolayer to be a metallic ferromagnet.
950
+ For CrI3 we compare the spin spiral dispersion to the spiral
951
+ energy determined by a third nearest neighbour energy map-
952
+ ping procedure. The prototype thus serves as a testing ground
953
+ for applying unconstrained GBT to materials with multiple
954
+ magnetic atoms in the unit cell. We analyse the intracell an-
955
+ gle between the Cr atoms of CrI3 and provide an expression
956
+ for generating good initial magnetic moments for GBT calcu-
957
+ lations. We finally discuss the observed deviations from the
958
+ classical Heisenberg model and to what extend the flat spiral
959
+ spectrum can be used to obtain the magnon excitation spec-
960
+ trum.
961
+ CrX3
962
+ The chromium trihalides are of considerable interest due
963
+ to the versatile properties that arise across the three different
964
+ halides. Monolayer CrI3 was the first 2D monolayer that were
965
+ demonstrated to host ferromagnetic order below 45 K [1] and
966
+ has spurred intensive scrutiny in the physics of 2D magnetism.
967
+ The magnetic order is governed by strong magnetic easy-axis
968
+ anisotropy, which is accurately reproduced by first principles
969
+ simulations [66, 67]. In contrast, monolayers of CrCl3 exhibit
970
+ ferromagnetic interactions as well, but no proper long range
971
+ order due easy-plane anisotropy. Instead, these monolayers
972
+ exhibit Kosterlitz-Thouless physics, which give rise to quasi
973
+ long range order below 13 K [9].
974
+ The GBT is not really necessary to find the ground state of
975
+ the monolayer chromium halides. They are all ferromagnetic
976
+ and insulating and only involve short range exchange interac-
977
+ tions that are readily obtained from collinear energy mapping
978
+ methods [66–68]. Nevertheless, the gap between the acoustic
979
+ and optical magnons in bulk CrI3 has been proposed to arise
980
+ from either (second neighbor) Dzyalosinskii-Moriya interac-
981
+ tions [69] or Kitaev interactions [70, 71]. The former could in
982
+ principle be extracted directly from planar spin-spiral calcu-
983
+ lations [40], while the latter requires conical spin spirals. The
984
+ origin of this gap is, however, still subject to debate [72] and
985
+ here we will mainly focus on the magnetic interactions that do
986
+ not rely on spinorbit coupling. In the following we will focus
987
+ on CrI3 as a representative member of the family.
988
+ The honeycomb lattice contains two magnetic atoms per
989
+ unit cell and the magnetic moments at the two sites will in
990
+ general differ by an angle ξ. Since we do not impose any con-
991
+ straints except for the boundary conditions specified by q, the
992
+ angle will be relaxed to its optimal value when the Kohn-Sham
993
+ equations are solved self-consistently. The convergence of ξ,
994
+ may be a tedious process since the total energy has a rather
995
+ weak dependence on ξ. For a given q the classical energy of
996
+ the model (1) is minimized by the angle ξ 0 given by
997
+ tanξ 0 = −ImJ12(q)
998
+ ReJ12(q),
999
+ (8)
1000
+ where
1001
+ J12(q) = ∑
1002
+ i
1003
+ J12
1004
+ 0i e−iq·Ri
1005
+ (9)
1006
+ is the Fourier transform of the inter-sublattice exchange cou-
1007
+ pling. If one assumes nearest neighbour interactions only, ξ 0
1008
+ becomes independent of exchange parameters and the result-
1009
+ ing expression thus comprises a suitable initial guess for the
1010
+ inter-sublattice angle. We note that the classical spiral en-
1011
+ ergy is independent of ξ (in the absence of spinorbit coupling)
1012
+ when J12(q) = 0 and the angle may be discontinuous at such
1013
+ q-points. This occurs for example in the magnetic honeycomb
1014
+ lattice at the K-point (q = (1/3,1/3)). In general, Eq. (8) has
1015
+ two solutions that differ by π and only one of these minimzes
1016
+ the energy while the other maximizes it. The maximum en-
1017
+ ergy constitutes an ”optical” spin spiral branch, which is if
1018
+ interest if one wishes to extract the exchange coupling con-
1019
+ stants.
1020
+ The spiral energies of CrI3 (with optimized intracell angles)
1021
+ are shown in figure 4, where we show both the ferromagnetic
1022
+ (ξ = 0) and the antiferromagnetic (ξ = π) results on the ΓK
1023
+
1024
+ 9
1025
+ Γ
1026
+ M
1027
+ K
1028
+ Γ
1029
+ 0
1030
+ 10
1031
+ 20
1032
+ 30
1033
+ E(q) [meV]
1034
+ Mapping
1035
+ FM
1036
+ AFM
1037
+ Γ
1038
+ M
1039
+ K
1040
+ Γ
1041
+ 2.92
1042
+ 2.94
1043
+ m [µB]
1044
+ Γ
1045
+ M
1046
+ K
1047
+ Γ
1048
+ 0
1049
+ 90
1050
+ 180
1051
+ ξ [o]
1052
+ FIG. 4. (Left: spin spiral energies of CrI3 compared to third nearest neighbour energy mapping. Right: angles beteen the two magnetic
1053
+ moments. The spin spirals are initialised with angles determined by Eq. (8) which are shown in black. The moments are collinear on the ΓK
1054
+ path and so the AFM solution is also quasi-stable in DFT. Center: the magnitude of local magnetic moments along the spiral path.
1055
+ path. We also show the spiral energy obtained from the model
1056
+ (1) with exchange parameters calculated from a collinear en-
1057
+ ergy mapping using four differnet spin configurations. We get
1058
+ J1 = 2.47 meV, J2 = 0.682 meV and J3 = −0.247 meV for the
1059
+ first, second and nearest neighbour interactions respectively,
1060
+ which is in good agreement with previous LDA calculations
1061
+ [73]. The model spiral energy is seen to agree very well with
1062
+ that obtained from the GBT, which largely validates such a
1063
+ three-parameter model (when spinorbit is neglected). We do,
1064
+ however, find a small deviation in the regions between high-
1065
+ symmetry points. This is likely due to higher order exchange
1066
+ interaction, which will deviate in the two approaches. For
1067
+ example, a biquadratic exchange term [38], will cancel out
1068
+ in any collinear mapping, but will influence the energies ob-
1069
+ tained from the GBT. Biquadratic exchange parameters could
1070
+ thus be extracted from the deviation between the two calcula-
1071
+ tions.
1072
+ In figure 4 we also show the calculated values of ξ and
1073
+ the magnitude of the local magnetic moment at the Cr sites
1074
+ along the path. The self-consistent intracell angles are found
1075
+ to match very well with the initial guess, except for a slight
1076
+ deviation on the Brillouin zone boundary. This corroborates
1077
+ the fact that exchange couplings beyond second neighbours
1078
+ are insignificant (the second nearest neighbor coupling is an
1079
+ intra-sublattice interaction and does not influence the angle).
1080
+ It is also rather instructive to analyze the variation in the
1081
+ magnitude of local magnetic moments. In general, the map-
1082
+ ping of electronic structure problems to Heisenberg types of
1083
+ models like (1) rests on an adiabatic assumption where it is
1084
+ assumed that the magnitude of the moments are fixed. How-
1085
+ ever, the present variation in the magnitude of moments does
1086
+ not imply a breakdown of the adiabatic assumption, but re-
1087
+ flects that DFT should be mapped to a quantum mechanical
1088
+ Heisenberg model rather than a classical model. In particu-
1089
+ lar, the ratio of spin expectation values between the ferromag-
1090
+ netic ground state and the (anti-ferromagnetic) state of highest
1091
+ energy is approximately ⟨Si⟩AFM/⟨Si⟩FM = 0.83 in the quan-
1092
+ tized model [74]. While this ratio is somewhat smaller than
1093
+ the difference between ferromagnetic and anti-ferromagnetic
1094
+ moments found here, the result does imply that the magni-
1095
+ tude of moments should depend on q. And the fact that the
1096
+ q = 0 anti-ferromagnetic moments are smaller than the ferro-
1097
+ magnetic ones in a self-consistent treatments reflects that DFT
1098
+ captures part of the quantum fluctuations inherent to the model
1099
+ (1).
1100
+ We note that the spin spiral energy Eq calculated from the
1101
+ isotropic Heisenberg model using the optimal angle given by
1102
+ Eq. (8) is related to the dynamical excitations (magnon ener-
1103
+ gies) by ω±
1104
+ q = E±
1105
+ q /S and the spiral energies thus comprise a
1106
+ simple method to get the magnetic excitation spectrum. How-
1107
+ ever, even if a model like (1) fully describes a magnetic mate-
1108
+ rial (no anisotropy or higher order terms) there will be a sys-
1109
+ tematic error in the extracted exchange parameters (and re-
1110
+ sulting magnon spectrum) if the parameters are extracted by
1111
+ mapping to the classical model. The reason is, that the clas-
1112
+ sical energies correspond to expectation values of spin con-
1113
+ figurations with fixed magnitude of the spin, which is not ac-
1114
+ commodated in a self-consistent approach. This error is di-
1115
+ rectly reflected by the variation of the magnitude of moments
1116
+ in figure 4. The true exchange parameters can only be ob-
1117
+ tained either by mapping to eigenstates of the model [74] or
1118
+ by considering infinitesimal rotations of the spin, which may
1119
+ be handled non-selfconsistently using the magnetic force the-
1120
+ orem [37, 75–78]. Nevertheless, the deviations between ex-
1121
+ change parameters obtained from classical and quantum me-
1122
+ chanical energy mapping typically deviates by less than 5 %
1123
+ [74] and for insulators it is a good approximation to extract
1124
+ the magnon energies from planar spiral calculations although
1125
+ the mapping is only strictly valid in the limit of small q.
1126
+ IV.
1127
+ CONCLUSION AND OUTLOOK
1128
+ In conclusion, we have demonstrated the abundance of spi-
1129
+ ral magnetic order in 2D transition metal dichalcogenides
1130
+ from first principles calculations. The calculations imply that
1131
+ type II multiferroic order is rather common in these materials
1132
+ and we have calculated the spontaneous polarization in a se-
1133
+ lected subset of these using fully relaxed structures in super
1134
+ cells. While the super cell calculations does not correspond to
1135
+ the exact spirals found from the GBT, the calculations show
1136
+ that relaxation effects plays a crucial role for the induced po-
1137
+ larization and should be taken into account in any quantitative
1138
+ analysis. The spontaneous polarization in type II multifer-
1139
+ roics is in general rather small compared to what is found in
1140
+ ordinary 2D ferroelectrics and could imply that the chirality
1141
+
1142
+ 10
1143
+ of spirals are switchable by small electric fields. It would be
1144
+ highly interesting to calculate the coercive field for switch-
1145
+ ing in these materials, but due to the importance of relaxation
1146
+ effects and spin-orbit coupling this is a non-trivial computa-
1147
+ tion that cannot simply be obtained from the Born effective
1148
+ charges and force constant matrix.
1149
+ The GBT comprises a powerful framework for extract-
1150
+ ing the magnetic properties of materials from first princi-
1151
+ ples. In addition to the single-q states considered here, one
1152
+ may use super cells to extract the importance of higher order
1153
+ exchange interactions and unravel the possibility of having
1154
+ multi-q ground states. In addition, for non-centrosymmetric
1155
+ materials, the PSO approach may be readily applied to ob-
1156
+ tain the Dzyaloshinskii-Moriya interactions, which may lead
1157
+ to Skyrmion lattice ground states or stabilize other multi-q
1158
+ states.
1159
+ V.
1160
+ APPENDIX
1161
+ A.
1162
+ Implementation
1163
+ In the PAW formalism we expand the spiral spinors using the standard PAW transformation [79]
1164
+ ψq,k(r) = ˆ
1165
+ T ˜ψq,k(r) = ˜ψq,k(r)+∑
1166
+ a ∑
1167
+ i
1168
+ (φ a
1169
+ i (r)− ˜φ a
1170
+ i (r))
1171
+
1172
+ dr[ ˜pa
1173
+ i (r)]∗ ˜ψq,k(r),
1174
+ (10)
1175
+ where ˜ψq,k(r) is a smooth (spinor) pseudo-wavefunction that coincides with ψq,k(r) outside the augmentation spheres and devi-
1176
+ ates from ψq,k(r) by the second term inside the augmentation spheres. The all-electron wavefunction ψq,k(r) is thus expanded
1177
+ in terms of (spinor) atomic orbitals φ a
1178
+ i inside the PAW spheres and the expansion coefficients are given by the overlap between
1179
+ the pseudowavefunction and atom-centered spinor projector functions ˜pa
1180
+ i . Using Eq. (3) we may write this as
1181
+ ψq,k(r) = eik·rU†
1182
+ q(r) ˜uq,k(r)+∑
1183
+ a ∑
1184
+ i
1185
+ (φ a
1186
+ i (r)− ˜φ a
1187
+ i (r))
1188
+
1189
+ dr[ ˜pa
1190
+ i (r)]∗eik·rU†
1191
+ q(r) ˜uq,k(r)
1192
+ = eik·rU†
1193
+ q(r) ˜uq,k(r)+∑
1194
+ a ∑
1195
+ i
1196
+ (φ a
1197
+ i (r)− ˜φ a
1198
+ i (r))
1199
+
1200
+ dr[e−ik·rUq(r) ˜pa
1201
+ i (r)]∗ ˜uq,k(r)
1202
+ = eik·rU†
1203
+ q(r) ˜uq,k(r)+∑
1204
+ a ∑
1205
+ i
1206
+ (φ a
1207
+ i (r)− ˜φ a
1208
+ i (r))
1209
+
1210
+ dr[ ˜pa
1211
+ i,q,k(r)]∗ ˜uq,k(r)
1212
+ ≡ Tq,k ˜uq,k(r),
1213
+ (11)
1214
+ where Uq(r) was given in Eq. (4) and we defined
1215
+ ˜pa
1216
+ i,q,k(r) = e−ik·rUq(r) ˜pa
1217
+ i (r).
1218
+ (12)
1219
+ The PAW transformed Kohn-Sham equations then read
1220
+ ˜Hq,k ˜uq,k(r) = εq,kSq,k ˜uq,k(r),
1221
+ (13)
1222
+ with
1223
+ ˜Hq,k = T †
1224
+ q,kHTq,k,
1225
+ Sq,k = T †
1226
+ q,kTq,k.
1227
+ (14)
1228
+ Calculations in the framework of the GBT thus requires two modifications compared to the approach for solving the ordinary
1229
+ Kohn-Sham equations in the PAW formalism. 1) The k-dependence of the standard Bloch Hamiltonian is replaced by k →
1230
+ k ∓ q/2 for spin-up and spin down components respectively. 2) Different spin dependent projector functions has to be applied
1231
+ when calculating the projector overlaps with the spin-up and spin-down components of the psudowavefunctions (see Eq. (12)).
1232
+ B.
1233
+ Benchmark
1234
+ The LDA implementation of the GBT have been tested by
1235
+ checking that our results agree with similar calculations from
1236
+ the literature and by verifying internal consistency by compar-
1237
+ ing with super cell calculations. The case of fcc Fe has been
1238
+ found to have a spin spiral ground state [81] and the calcu-
1239
+ lation of the ordering vector Q has been become a standard
1240
+ benchmark for spin spiral implementations [82]. In previous
1241
+ simulations the ordering vector was found to be rather sensi-
1242
+ tive to the lattice constant and in figure 6 we show the spin spi-
1243
+ ral energies along the ΓXW path using the experimental lattice
1244
+ constant as well as the lattice constant which has been found
1245
+ to reproduce the experimental ordering vector [80]. The cal-
1246
+ culated value of Q is in good agreement with previous reports
1247
+ in both cases [83]. We also confirm a similar low energy bar-
1248
+
1249
+ 11
1250
+ −0.4
1251
+ −0.2
1252
+ 0.0
1253
+ 0.2
1254
+ 0.4
1255
+ qx
1256
+ 0.0
1257
+ 0.1
1258
+ 0.2
1259
+ E − E0 [eV]
1260
+ Spiral
1261
+ Supercell
1262
+ FIG. 5. Comparison between GBT spin spiral calculations and su-
1263
+ percell calculations without spinorbit coupling in monolayer CoPt.
1264
+ 100
1265
+ 120
1266
+ 140
1267
+ 3.58 ˚A (exp)
1268
+ 3.50 ˚A
1269
+ Γ
1270
+ X
1271
+ W
1272
+ −20
1273
+ −10
1274
+ 0
1275
+ E(q) [meV]
1276
+ FIG. 6. Spin spiral energies of fcc Fe for the experimental lattice
1277
+ constant (red) and a strained latice constant, which is known to re-
1278
+ produce the experimental spin spiral order in (blue). The dashed
1279
+ vertical lines indicate the minima found in Ref. [80].
1280
+ rier between the two local minima, as is expected from LDA
1281
+ [84].
1282
+ In order to check internal consistency we have investigated
1283
+ the case of monolayer CoPt [40] where we compare spin spi-
1284
+ ral energies calculated using the GBT with energies calculated
1285
+ from super cells. We thus construct a 16x1 super cell of the
1286
+ CoPt monolayer and consider spirals with qc = ( n
1287
+ 16) in units of
1288
+ reciprocal lattice vectors. This allows us to extract 16 different
1289
+ spiral energies in the supercell using standard non-collinear
1290
+ DFT. In order to compare the two methods we have used a
1291
+ k-point grid of 16×16×1 for the GBT and 1×16×1 for the
1292
+ supercell and a plane wave cutoff of 700 eV for both calcu-
1293
+ lations. In Fig. 5 we compare the results without spinorbit
1294
+ coupling and find excellent agreement between supercell and
1295
+ GBT calculations. We note that when spinorbit coupling is
1296
+ neglected one has Eq = E−q.
1297
+ Since spinorbit coupling is incompatible with the GBT one
1298
+ has to resort to approximate schemes to include it in the cal-
1299
+ culations. In the present work we have used the PSO method
1300
+ proposed by Sandratskii [40]. In Fig. 7 we compare spin spi-
1301
+ ral calculations with supercell calculations where the spinorbit
1302
+ coupling has been included either fully or by the PSO method.
1303
+ The PSO method is fully compatible with the GBT and we
1304
+ −0.5
1305
+ −0.25
1306
+ 0
1307
+ 0.25
1308
+ 0.5
1309
+ qx
1310
+ 0.00
1311
+ 0.05
1312
+ 0.10
1313
+ 0.15
1314
+ 0.20
1315
+ E − E0 [eV]
1316
+ Spiral Proj. SOC
1317
+ Spiral Full SOC
1318
+ SC Full SOC
1319
+ SC Projected SOC
1320
+ FIG. 7. Comparison between GBT spin spiral calculations and super-
1321
+ cell calculations with projected and full spinorbit coupling in mono-
1322
+ layer CoPt.
1323
+ find excellent agreement between the spin spiral energies cal-
1324
+ culated with the GBT and with supercells. The PSO approach
1325
+ is, however an approximation and the correct result can only
1326
+ be obtained from the supercell using the full spinorbit cou-
1327
+ pling. We see that the PSO calculations are in good agreement
1328
+ with those obtained from full spinorbit coupling but overesti-
1329
+ mates the energies at the Brillouin zone boundary by a few
1330
+ percent. In contrast, if one tries to include the full spinor-
1331
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1332
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1333
+ without spinorbit coupling) the energies are severely under-
1334
+ estimated with respect to the exact result (from the supercell
1335
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1336
+ bit coupling shows a slight asymmetry between points at q and
1337
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1338
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+ [70] C. Xu, J. Feng, H. Xiang, and L. Bellaiche, Interplay be-
1616
+ tween Kitaev interaction and single ion anisotropy in ferromag-
1617
+ netic CrI3 and CrGeTe3 monolayers, npj Comput. Mater. 4, 57
1618
+ (2018).
1619
+ [71] I. Lee, F. G. Utermohlen, D. Weber, K. Hwang, C. Zhang, J. van
1620
+ Tol, J. E. Goldberger, N. Trivedi, and P. C. Hammel, Fundamen-
1621
+ tal Spin Interactions Underlying the Magnetic Anisotropy in the
1622
+ Kitaev Ferromagnet CrI3, Phys. Rev. Lett. 124, 017201 (2020).
1623
+ [72] S.-H. Do, J. A. M. Paddison, G. Sala, T. J. Williams, K. Kaneko,
1624
+ K. Kuwahara, A. F. May, J. Yan, M. A. McGuire, M. B. Stone,
1625
+ M. D. Lumsden, and A. D. Christianson, Gaps in topological
1626
+ magnon spectra: Intrinsic versus extrinsic effects, Phys. Rev. B
1627
+ 106, L060408 (2022).
1628
+ [73] T. Olsen, Unified Treatment of Magnons and Excitons in Mono-
1629
+ layer CrI3 from Many-Body Perturbation Theory, Physical Re-
1630
+ view Letters 127, 166402 (2021).
1631
+ [74] D. Torelli and T. Olsen, First principles Heisenberg models of
1632
+ 2D magnetic materials: the importance of quantum corrections
1633
+ to the exchange coupling, Journal of Physics: Condensed Mat-
1634
+ ter 32, 335802 (2020).
1635
+ [75] A. Liechtenstein, M. Katsnelson, V. Antropov, and V. Gubanov,
1636
+ Local spin density functional approach to the theory of ex-
1637
+ change interactions in ferromagnetic metals and alloys, Journal
1638
+ of Magnetism and Magnetic Materials 67, 65 (1987).
1639
+ [76] P. Bruno, Exchange Interaction Parameters and Adiabatic Spin-
1640
+ Wave Spectra of Ferromagnets: A “ Renormalized Magnetic
1641
+ Force Theorem”, Physical Review Letters 90, 087205 (2003),
1642
+ 0407739.
1643
+ [77] S. V. Halilov, H. Eschrig, A. Y. Perlov, and P. M. Oppeneer, Adi-
1644
+ abatic spin dynamics from spin-density-functional theory: Ap-
1645
+ plication to Fe, Co, and Ni, Physical Review B 58, 293 (1998).
1646
+ [78] F. L. Durhuus, T. Skovhus, and T. Olsen, Plane wave imple-
1647
+ mentation of the magnetic force theorem for magnetic exchange
1648
+ constants: Application to bulk fe, co and ni, Journal of Physics:
1649
+ Condensed Matter (2022).
1650
+ [79] P. E. Bl¨ochl, Projector augmented-wave method, Phys. Rev. B
1651
+ 50, 17953 (1994).
1652
+ [80] M. Marsman and J. Hafner, Broken symmetries in the crys-
1653
+ talline and magnetic structures of γ-iron, Physical Review B
1654
+ 66, 224409 (2002).
1655
+ [81] Y. Tsunoda, N. Kunitomi, and R. M. Nicklow, Magnetic struc-
1656
+ ture of γ-fe precipitates in a cu matrix, Journal of Physics F:
1657
+ Metal Physics 17, 2447 (1987).
1658
+ [82] P. Kurz, F. F¨orster, L. Nordstr¨om, G. Bihlmayer, and S. Bl¨ugel,
1659
+ Ab initio treatment of noncollinear magnets with the full-
1660
+ potential linearized augmented plane wave method, Physical
1661
+ Review B 69, 024415 (2004).
1662
+ [83] V. Garc´ıa-Su´arez, C. Newman, C. J. Lambert, J. Pruneda, and
1663
+ J. Ferrer, First principles simulations of the magnetic and struc-
1664
+ tural properties of iron, The European Physical Journal B-
1665
+ Condensed Matter and Complex Systems 40, 371 (2004).
1666
+ [84] K. Kn¨opfle, L. Sandratskii, and J. K¨ubler, Spin spiral ground
1667
+ state of γ-iron, Physical Review B 62, 5564 (2000).
1668
+
1669
+ 15
1670
+ SUPPLEMENTARY INFORMATION - TYPE II MULTIFERROIC ORDER IN TWO-DIMENSIONAL TRANSITION METAL
1671
+ HALIDES FROM FIRST PRINCIPLES SPIN-SPIRAL CALCULATIONS
1672
+ Joachim Sødequist1 and Thomas Olsen1,∗
1673
+ 1CAMD, Department of Physics, Technical University of Denmark, 2820 Kgs. Lyngby Denmark
1674
+ I.
1675
+ SPIN SPIRAL DISPERSIONS
1676
+ The entire spin spiral dispersion carry more information than just the energy minima was reported we reported in the main text,
1677
+ these are shown here for in figures 2 and 4. One can find not only the stability with respect to the ferromagnetic configuration,
1678
+ but also compare to any other configuration in the energy landscape. Additionally, we can observe whether the remain magnetic
1679
+ moments are unchanged during self-consistent field cycle, and we find this is generally true except for CoI2 and perhaps the
1680
+ titanium compounds. We note that the local magnetic moments found here are the integrated inside the PAW spheres of the
1681
+ respective atoms, thus these local moments does not integrate to the total moments reported in the main text since interstitial
1682
+ magnetization density is neglected. We also provide the projected spin orbit energies in figure 3 for the lowest energy state,
1683
+ naturally the shape will depend very much on the specific spiral, in some cases such as the q = K we find quite similar energies in
1684
+ the out-of-plane orientations, whereas incommensurate spirals tend to have more well defined minima. The in-plane orientations
1685
+ reported here are related by a 90◦ phase shift, but the dashed line highlight that they are indeed degenerate as expected.
1686
+ II.
1687
+ SPIN SPIRAL CONVERGENCE
1688
+ An example of convergence of a intracell angle in a rectangular supercell of the hexagonal VI2 system is represented in 1. We
1689
+ find that for all calculations which reach the convergence criteria on particularly the density, all converge the angle within some
1690
+ narrow region of the true angle. We observe that the number of iterations required increase dramatically, when the initial guess
1691
+ is further away from the true angle, hence highlighting the importance of choosing initial conditions according to Eq. (8) in the
1692
+ main text.
1693
+ FIG. 1. Convergence of the intracell angle ξ in spin spiral ground state calculation of VI2 at the spiral vector q = (1/4,0,0) at varying different
1694
+ initial conditions. The calculations shown in red, did not reach the convergence criteria on the density within the time-wall of the calculation,
1695
+ while the blue were considered converged. The black horizontal line is the expected angle for a smooth spin spiral as it if it was an equivalent
1696
+ spin spiral in the primitive unit cell.
1697
+
1698
+ Vl2convergence,g=(1/4,0,0),rect
1699
+ 175
1700
+ 150
1701
+ 125
1702
+ 5-angle
1703
+ 100
1704
+ 75
1705
+ 50
1706
+ 25
1707
+ 0
1708
+ 0
1709
+ 200
1710
+ 400
1711
+ 600
1712
+ 800
1713
+ 1000
1714
+ SCF-count16
1715
+ FIG. 2. Spin spiral energies for AB2 magnets and the local magnetic moments of the magnetic atoms and ligands. For ferromagnetic refer to
1716
+ Fig. 4.
1717
+
1718
+ TiBr2
1719
+ Wave length ^ [A]
1720
+ inf
1721
+ 11.2
1722
+ 5.6
1723
+ 6.5
1724
+ 12.9
1725
+ inf
1726
+ 0
1727
+ []
1728
+ Energy
1729
+ -10
1730
+ Ti magmom
1731
+ 1.2
1732
+ [meV]
1733
+ I norm magnetic moment
1734
+ -20
1735
+ Br magmom
1736
+ 1.0
1737
+ Spin spiral energy [
1738
+ -30
1739
+ 0.8
1740
+ 40
1741
+ 0.6
1742
+ 50
1743
+ 0.4
1744
+ -60
1745
+ S
1746
+ -70
1747
+ 0.2
1748
+ ocal
1749
+ -80
1750
+ Lo
1751
+ 0.0
1752
+ K
1753
+ M
1754
+ q vector [A-1]Til2
1755
+ Wave length ^ [A]
1756
+ inf
1757
+ 12.3
1758
+ 6.1
1759
+ 7.1
1760
+ 14.1
1761
+ inf
1762
+ 0
1763
+ n magnetic moment [lμsl]
1764
+ Energy
1765
+ 1.4
1766
+ Ti magmom
1767
+ M
1768
+ -10
1769
+ [me]
1770
+ I magmom
1771
+ 1.2
1772
+ Spin spiral energy
1773
+ -20
1774
+ 0.8
1775
+ 0.6
1776
+ -30
1777
+ ocal norm
1778
+ 0.4
1779
+ -40
1780
+ 0.2
1781
+ Lo
1782
+ 0.0
1783
+ K
1784
+ M
1785
+ q vector [A-1]VCI2
1786
+ Wave length ^ [A]
1787
+ inf
1788
+ 11.0
1789
+ 5.5
1790
+ 6.3
1791
+ 12.7
1792
+ inf
1793
+ 0
1794
+
1795
+
1796
+ Energy
1797
+ 2.5
1798
+ -10
1799
+ V magmom
1800
+ [meV]
1801
+ moment
1802
+ Cl magmom
1803
+ 2.0
1804
+ -20
1805
+ Spin spiral energy
1806
+ 1.5
1807
+ 5
1808
+ -30
1809
+ 40
1810
+ 1.0
1811
+ -50
1812
+ 0.5
1813
+ -60
1814
+ 0.0
1815
+ K
1816
+ M
1817
+ q vector [A-1]VBr2
1818
+ Wave length ^ [A]
1819
+ inf
1820
+ 11.5
1821
+ 5.8
1822
+ 6.7
1823
+ 13.3
1824
+ inf
1825
+ 0
1826
+
1827
+ : moment [lμl]
1828
+ Energy
1829
+ 2.5
1830
+ V magmom
1831
+ [meV]
1832
+ Br magmom
1833
+ -10
1834
+ 2.0
1835
+ Spin spiral energy
1836
+ -15
1837
+ 1.5
1838
+ -20
1839
+ 1.0
1840
+ 25
1841
+ 30
1842
+ 0.5
1843
+ -35
1844
+ 0.0
1845
+ K
1846
+ M
1847
+ q vector [A-1]V12
1848
+ Wave length 入 [A]
1849
+ inf
1850
+ 12.4
1851
+ 6.2
1852
+ 7.1
1853
+ 14.3
1854
+ inf
1855
+
1856
+ Energy
1857
+ 2.5
1858
+ V magmom
1859
+ [meV]
1860
+ 4
1861
+ I magmom
1862
+ 2.0
1863
+ I energy
1864
+ 2
1865
+ Local norm magnetic
1866
+ 1.5
1867
+ Spin spiral
1868
+ 0
1869
+ 1.0
1870
+ 0.5
1871
+ 0.0
1872
+ K
1873
+ M
1874
+ q vector [A-1]MnCI2
1875
+ Wave length ^ [A]
1876
+ inf
1877
+ 11.2
1878
+ 5.6
1879
+ 6.4
1880
+ 12.9
1881
+ inf
1882
+ 0
1883
+ Energy
1884
+ Mn magmom
1885
+ Local norm magnetic moment [
1886
+ Spin spiral energy [meV]
1887
+ Cl magmom
1888
+ -10
1889
+ 2
1890
+ 15
1891
+ -20
1892
+ K
1893
+ M
1894
+ q vector [A-1]MnBr2
1895
+ Wave length ^ [A]
1896
+ inf
1897
+ 11.7
1898
+ 5.8
1899
+ 6.7
1900
+ 13.5
1901
+ inf
1902
+ 0.0
1903
+ [|μB|]
1904
+ Energy
1905
+ -2.5
1906
+ Mn magmom
1907
+ [meV]
1908
+ -5.0
1909
+ Br magmom
1910
+ 7.5
1911
+ I energy
1912
+ Spin spiral
1913
+ 2
1914
+ -12.5
1915
+ -20.0
1916
+ 0
1917
+ K
1918
+ M
1919
+ q vector [A-1]Mn12
1920
+ Wave length ^ [A]
1921
+ inf
1922
+ 12.5
1923
+ 6.2
1924
+ 7.2
1925
+ 14.4
1926
+ inf
1927
+ 0
1928
+ Energy
1929
+ Mn magmom
1930
+ Spin spiral energy [meV]
1931
+ I magmom
1932
+ 10
1933
+ 15
1934
+ -20
1935
+ K
1936
+ M
1937
+ q vector [A-1]CoBr2
1938
+ Wave length 入 [A]
1939
+ inf
1940
+ 11.2
1941
+ 5.6
1942
+ 6.5
1943
+ 12.9
1944
+ inf
1945
+ Energy
1946
+ Co magmom
1947
+ [meV]
1948
+ 20
1949
+ 2.0
1950
+ Br magmom
1951
+ Spin spiral energy
1952
+ 15
1953
+ 1.5
1954
+ Local norm magnetic i
1955
+ 10
1956
+ 1.0
1957
+ 5
1958
+ 0.0
1959
+ 0
1960
+ K
1961
+ M
1962
+ q vector [A-1]Col2
1963
+ Wave length ^ [A]
1964
+ inf
1965
+ 11.7
1966
+ 5.8
1967
+ 6.7
1968
+ 13.5
1969
+ inf
1970
+ Energy
1971
+ 20
1972
+ Co magmom
1973
+ 1.2
1974
+ Spin spiral energy [meV]
1975
+ I magmom
1976
+ 10
1977
+ 1.0
1978
+ 0.8
1979
+ 0.6
1980
+ 0.4
1981
+ -10
1982
+ 0.2
1983
+ -20
1984
+ 0.0
1985
+ K
1986
+ M
1987
+ q vector [A-1]NiCI2
1988
+ Wave length 入 [A]
1989
+ inf
1990
+ 10.5
1991
+ 5.3
1992
+ 6.1
1993
+ 12.1
1994
+ inf
1995
+ 50
1996
+ Energy
1997
+ 1.4
1998
+ Ni magmom
1999
+ Spin spiral energy [meV]
2000
+ 1.2
2001
+ 40
2002
+ Cl magmom
2003
+ 1.0
2004
+ 30
2005
+ 0.8
2006
+ Local norm magnetic
2007
+ 20
2008
+ 0.6
2009
+ 0.4
2010
+ 10
2011
+ 0.2
2012
+ 0
2013
+ 0.0
2014
+ K
2015
+ M
2016
+ q vector [A-1]NiBr2
2017
+ Wave length 入 [A]
2018
+ inf
2019
+ 11.1
2020
+ 5.5
2021
+ 6.4
2022
+ 12.8
2023
+ inf
2024
+ 1.4
2025
+ Energy
2026
+ n magnetic moment [lμbl
2027
+ 40
2028
+ Ni magmom
2029
+ 1.2
2030
+ [meV]
2031
+ Br magmom
2032
+ 1.0
2033
+ 30
2034
+ Spin spiral energy
2035
+ 0.8
2036
+ 20
2037
+ 0.6
2038
+ 10
2039
+ 0.4
2040
+ Local norm
2041
+ 0.2
2042
+ 0
2043
+ 0.0
2044
+ K
2045
+ M
2046
+ q vector [A-1]Nil2
2047
+ Wave length ^ [A]
2048
+ inf
2049
+ 11.9
2050
+ 6.0
2051
+ 6.9
2052
+ 13.8
2053
+ inf
2054
+ 1.2
2055
+ 40
2056
+ Energy
2057
+ magnetic moment [lμ]
2058
+ Ni magmom
2059
+ [meV]
2060
+ 1.0
2061
+ 30
2062
+ I magmom
2063
+ 20 -
2064
+ 0.8
2065
+ T energy
2066
+ 10 -
2067
+ 0.6
2068
+ Spin spiral
2069
+ 0
2070
+ 0.4
2071
+ norm
2072
+ -10
2073
+ 0.2
2074
+ -20
2075
+ ocal
2076
+ 0.0
2077
+ -30 -
2078
+ K
2079
+ M
2080
+ q vector [A-1]17
2081
+ FIG. 3. Projected spin orbit energies of the ground state found in Fig. 2
2082
+
2083
+ TiBr2
2084
+ -5.1
2085
+ -
2086
+ -
2087
+ -
2088
+ -
2089
+ -5.2 -
2090
+ -
2091
+ -
2092
+ -
2093
+ -
2094
+ -
2095
+ -
2096
+ -
2097
+ -
2098
+ [meV]
2099
+ -5.3
2100
+ -
2101
+ -
2102
+ -
2103
+ -
2104
+ -
2105
+ -
2106
+ -
2107
+ -
2108
+ -
2109
+ -5.4
2110
+ -
2111
+ -
2112
+ E
2113
+ -
2114
+ -
2115
+ -
2116
+ -5.5
2117
+ -
2118
+ -
2119
+ -5.6 -
2120
+ -
2121
+ -
2122
+ -
2123
+ IP
2124
+ Screw
2125
+ OoP
2126
+ IP
2127
+ theta, phiTil2
2128
+ -
2129
+ -25.2
2130
+ -
2131
+ -
2132
+ -25.4
2133
+ -
2134
+ -
2135
+ -
2136
+ [meV]
2137
+ 25.6
2138
+ -25.8
2139
+ -26.0
2140
+ -
2141
+ -
2142
+ -
2143
+ -
2144
+ IP
2145
+ Screw
2146
+ OoP
2147
+ IP
2148
+ theta, phiVCI2
2149
+ -0.68
2150
+ -0.69
2151
+ -0.70
2152
+ [meV]
2153
+ -0.71
2154
+ -0.72
2155
+ -0.73
2156
+ -0.74
2157
+ IP
2158
+ Screw
2159
+ OoP
2160
+ IP
2161
+ theta, phiVBr2
2162
+ -5.28
2163
+ -
2164
+ -
2165
+ -5.29
2166
+ -
2167
+ -5.30
2168
+ -
2169
+ -
2170
+ -
2171
+ -5.31
2172
+ [meV]
2173
+ -5.32
2174
+ Jos:
2175
+ -5.33
2176
+ E
2177
+ -5.34
2178
+ -5.35
2179
+ -
2180
+ -
2181
+ -
2182
+ -5.36
2183
+ IP
2184
+ Screw
2185
+ OoP
2186
+ IP
2187
+ theta, phiVI2
2188
+ -25.2
2189
+ -25.3
2190
+ -25.4
2191
+ 25.5
2192
+ 25.6
2193
+ E
2194
+ -25.7
2195
+ -25.8
2196
+ -25.9
2197
+ IP
2198
+ Screw
2199
+ OoP
2200
+ IP
2201
+ theta, phiMnCI2
2202
+ -1.780
2203
+ -
2204
+ -1.785
2205
+ -
2206
+ -
2207
+ -
2208
+ -
2209
+ -
2210
+ -
2211
+ [meV]
2212
+ 790
2213
+ -
2214
+ -
2215
+ -
2216
+ Jos:
2217
+ -
2218
+ -
2219
+ E
2220
+ -1.795
2221
+ -
2222
+ -
2223
+ -1.800
2224
+ -
2225
+ -
2226
+ -1.805
2227
+ IP
2228
+ Screw
2229
+ OoP
2230
+ IP
2231
+ theta, phiMnBr2
2232
+ -
2233
+ -6.17
2234
+ -6.18
2235
+ 6.19
2236
+ -6.20
2237
+ -6.21
2238
+ -
2239
+ -
2240
+ -
2241
+ -
2242
+ -
2243
+ -6.22
2244
+ IP
2245
+ Screw
2246
+ OoP
2247
+ IP
2248
+ theta, phiMn12
2249
+ -27.4
2250
+ -
2251
+ -
2252
+ -27.6
2253
+ -
2254
+ -
2255
+ -
2256
+ -27.8
2257
+ [meV]
2258
+ -28.0
2259
+ -28.2
2260
+ -28.4
2261
+ IP
2262
+ Screw
2263
+ OoP
2264
+ IP
2265
+ theta, phiCoBr2
2266
+ -8.38
2267
+ -
2268
+ -8.40
2269
+ [meV]
2270
+ -8.42
2271
+ soc
2272
+ E
2273
+ -8.44
2274
+ -8.46
2275
+ -8.48
2276
+ IP
2277
+ Screw
2278
+ OoP
2279
+ IP
2280
+ theta, phiCol2
2281
+ -40
2282
+ -41
2283
+ [meV]
2284
+ -42
2285
+ soc
2286
+ E
2287
+ -43
2288
+ -44 -
2289
+ -45 -
2290
+ -
2291
+ IP
2292
+ Screw
2293
+ OoP
2294
+ IP
2295
+ theta, phiNiCI2
2296
+ -3.050
2297
+ -3.055
2298
+ [meV]
2299
+ -3.060
2300
+ -3.065
2301
+ -3.070
2302
+ IP
2303
+ Screw
2304
+ OoP
2305
+ IP
2306
+ theta, phiNiBr2
2307
+ -
2308
+ -8.65
2309
+ -8.70
2310
+ [meV]
2311
+ -8.75
2312
+ -8.80
2313
+ E
2314
+ -8.85
2315
+ -8.90
2316
+ -
2317
+ IP
2318
+ Screw
2319
+ OoP
2320
+ IP
2321
+ theta, phiNil2
2322
+ -
2323
+ -40
2324
+ -
2325
+ -41 -
2326
+ [meV]
2327
+ -42
2328
+ -
2329
+ -43
2330
+ -44 :
2331
+ -
2332
+ -
2333
+ IP
2334
+ Screw
2335
+ OoP
2336
+ IP
2337
+ theta, phi18
2338
+ FIG. 4. Spin spiral energies for ferromagnetic AB2 magnets and the local magnetic moments on the atoms
2339
+
2340
+ FeCI2
2341
+ Wave length 入 [A]
2342
+ inf
2343
+ 10.2
2344
+ 6.0
2345
+ 6.2
2346
+ 12.4
2347
+ inf
2348
+ 3.5
2349
+ Energy
2350
+ : moment [lμb]
2351
+ 120
2352
+ Fe magmom
2353
+ [meV]
2354
+ 3.0
2355
+ Cl magmom
2356
+ 100
2357
+ 2.5
2358
+ Spin spiral energy
2359
+ 80
2360
+ 2.0
2361
+ Local norm magnetic
2362
+ 60
2363
+ 1.5
2364
+ 40
2365
+ 1.0
2366
+ 0.5
2367
+ 20
2368
+ 0.0
2369
+ 0
2370
+ M
2371
+ K
2372
+ q vector [A-1]FeBr2
2373
+ Wave length 入 [A]
2374
+ inf
2375
+ 10.8
2376
+ 6.4
2377
+ 6.5
2378
+ 13.1
2379
+ inf
2380
+ 3.5
2381
+ Energy
2382
+ moment [lμBl]
2383
+ 80
2384
+ Fe magmom
2385
+ 3.0
2386
+ Spin spiral energy [meV]
2387
+ Br magmom
2388
+ 2.5
2389
+ 60
2390
+ 2.0
2391
+ Local norm magnetic
2392
+ 40
2393
+ 1.5
2394
+ 1.0
2395
+ 0>
2396
+ 0.5
2397
+ 0.0
2398
+ M
2399
+ K
2400
+ q vector [A-1]Fe12
2401
+ Wave length 入 [A]
2402
+ inf
2403
+ 11.6
2404
+ 6.8
2405
+ 7.0
2406
+ 14.1
2407
+ inf
2408
+ 3.5
2409
+ 40
2410
+ Energy
2411
+ [8l]
2412
+ 35
2413
+ Fe magmom
2414
+ 3.0
2415
+ Spin spiral energy [meV]
2416
+ Local norm magnetic moment [
2417
+ I magmom
2418
+ 30
2419
+ 2.5
2420
+ 25
2421
+ 2.0
2422
+ 20
2423
+ 1.5
2424
+ 15
2425
+ 1.0
2426
+ 10
2427
+ 0.5
2428
+ 5
2429
+ 0.0
2430
+ 0
2431
+ M
2432
+ K
2433
+ q vector [A-1]CoCI2
2434
+ Wave length 入 [A]
2435
+ inf
2436
+ 10.6
2437
+ 5.3
2438
+ 6.1
2439
+ 12.3
2440
+ inf
2441
+ 2.5
2442
+ 50
2443
+ Energy
2444
+ Local norm magnetic moment [lμbl
2445
+ Co magmom
2446
+ Spin spiral energy [meV]
2447
+ Cl magmom
2448
+ 2.0
2449
+ 40
2450
+ 1.5
2451
+ 30
2452
+ 1.0
2453
+ 20
2454
+ 0.5
2455
+ 10
2456
+ 0.0
2457
+ K
2458
+ M
2459
+ q vector [A-1]
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1
+ Primordial black holes in loop
2
+ quantum gravity: The effect on the
3
+ threshold
4
+ Theodoros Papanikolaoua
5
+ aNational Observatory of Athens, Lofos Nymfon, 11852 Athens, Greece
6
+ E-mail: papaniko@noa.gr
7
+ Abstract. Primordial black holes form in the early Universe and constitute one of the
8
+ most viable candidates for dark matter. The study of their formation process requires
9
+ the determination of a critical energy density perturbation threshold δc, which in general
10
+ depends on the underlying gravity theory.
11
+ Up to now, the majority of analytic and
12
+ numerical techniques calculate δc within the framework of general relativity.
13
+ In this
14
+ work, using simple physical arguments we estimate semi-analytically the PBH formation
15
+ threshold within the framework of quantum gravity, working for concreteness within
16
+ loop quantum gravity (LQG), which constitutes a non-perturbative and background-
17
+ independent quantization of general relativity. In particular, for low mass PBHs formed
18
+ close to the quantum bounce, we find a reduction in the value of δc up to 50% compared to
19
+ the general relativistic regime quantifying for the first time to the best of our knowledge
20
+ how quantum effects can influence PBH formation within a quantum gravity framework.
21
+ Finally, by varying the Barbero-Immirzi parameter γ of LQG we show its effect on
22
+ the value of δc while using the observational/phenomenological signatures associated to
23
+ ultra-light PBHs, namely the ones affected by LQG effects, we propose the PBH portal
24
+ as a novel probe to constrain the potential quantum nature of gravity.
25
+ Keywords: primordial black holes, quantum gravity, loop quantum gravity
26
+ arXiv:2301.11439v1 [gr-qc] 26 Jan 2023
27
+
28
+ Contents
29
+ 1
30
+ Introduction
31
+ 1
32
+ 2
33
+ The fundamentals of loop quantum gravity
34
+ 2
35
+ 2.1
36
+ The classical dynamics
37
+ 2
38
+ 2.2
39
+ The quantum dynamics
40
+ 4
41
+ 3
42
+ The threshold of primordial black hole formation in loop quantum
43
+ gravity
44
+ 5
45
+ 4
46
+ Results
47
+ 8
48
+ 5
49
+ Conclusions
50
+ 10
51
+ 1
52
+ Introduction
53
+ PBHs, firstly proposed in early ’70s [1–3], form in the early universe, typically during
54
+ the Hot Big Bang (HBB) radiation-dominated (RD) era out of the gravitational collapse
55
+ of enhanced cosmological perturbations.
56
+ According to recent arguments, PBHs can
57
+ naturally account for a part or the totality of dark matter [4, 5], seed the large-scale
58
+ structures through Poisson fluctuations [6–9] as well as the primordial magnetic fields
59
+ through the presence of disks around them [10, 11]. At the same time, they are associated
60
+ with a plethora of gravitational-wave (GW) signals from black-hole merging events [12–
61
+ 16] up to primordial scalar induced GWs [17–22] (for a recent review see [23]).
62
+ In
63
+ particular, through the aforementioned GW portal, PBHs can act as well as a novel
64
+ probe shedding light on the underlying gravity theory [24, 25]. Other hints in favor of
65
+ PBHs can be found here [26].
66
+ In the standard PBH formation scenario, where PBHs form from the collapse of lo-
67
+ cal overdensity regions, the PBH formation threshold δc depends in general on the shape
68
+ of the energy density perturbation profile of the collapsing overdensity [27–30] as well
69
+ as on the equation-of-state parameter at the time of PBH gravitational collapse [30–33].
70
+ This critical threshold value is very important since it can affect significantly the abun-
71
+ dance of PBHs, a quantity which is constrained by numerous observational probes [34].
72
+ From a historic perspective, after a first analytic calculation of δc by B. Carr and
73
+ S. Hawking in 1975 [31, 35], δc was studied mostly through numerical hydrodynamic
74
+ simulations by [36–40]. Within the last decade, there has been witnessed a remarkable
75
+ progress regarding the determination δc both at the analytic as well as at the numerical
76
+ level.
77
+ In particular, at the analytic level, T.Harada, C-M. Yoo & K. Kohri (HYK)
78
+ in 2013 [32] refined the PBH formation threshold value obtained by Carr in 1975 by
79
+ comparing the time at which the pressure sound wave crosses the overdensity collapsing
80
+ – 1 –
81
+
82
+ to a PBH with the onset time of the gravitational collapse. Their expression for δc in
83
+ the uniform Hubble gauge reads as:
84
+ δc = sin2
85
+ � π√w
86
+ 1 + 3w
87
+
88
+ .
89
+ (1.1)
90
+ At this point, it is very important to stress that very recently there was exhibited a
91
+ rekindled interest in the scientific community regarding the effect of non-linearities [41–
92
+ 45] and non-Gaussianities [46–50] on the value of δc. In addition, some first research
93
+ works were also performed regarding the dependence of the PBH formation threshold on
94
+ non sphericities [51, 52], on anisotropies [53], on the velocity dispersion of the collapsing
95
+ matter [54] as well as within the context of modified theories of gravity [55].
96
+ In this work, we study semi-analytically the effect of the potential quantumness of
97
+ spacetime on the determination of the PBH formation threshold by using simple physical
98
+ arguments studying whether the PBH portal can act as a novel way to probe the quantum
99
+ nature of gravity. For concreteness, we work within the framework of loop quantum
100
+ gravity (LQG) [56, 57] which constitutes a nonperturbative and background-independent
101
+ quantization of general relativity. Very interestingly, LQG is able to solve the problem of
102
+ past and future singularities [58] and provide the initial conditions for inflation, solving
103
+ in this way naturally the flatness and the horizon cosmological problems [59]. It can also
104
+ account for the large scale structure formation [60] as well as for the currently observed
105
+ cosmic acceleration [61–63]. PBHs were studied firstly within the context of LQG in [64]
106
+ where the PBH evolution was explored accounting for the effects of Hawking radiation
107
+ and accretion in a LQG background. In the present work, we investigate the effect of
108
+ LQG on the PBH formation process and in particular at the level of the determination
109
+ of the PBH formation threshold.
110
+ The paper is organized as follows: In Sec. 2 we revise the basics of loop quantum
111
+ gravity. Then, in Sec. 3 we determine semi-analytically the PBH formation threshold δc
112
+ by comparing the gravity and the sound wave pressure forces. Followingly, in Sec. 4 we
113
+ present our results while Sec. 5 is devoted to conclusions.
114
+ 2
115
+ The fundamentals of loop quantum gravity
116
+ Loop quantum gravity brings conceptually together the two fundamental pillars of mod-
117
+ ern physics, namely General Relativity (GR) and Quantum Mechanics (QM). It consti-
118
+ tutes actually a non-perturbative and background-independent quantization of general
119
+ relativity [65, 66]. In particular, it is based on a connection-dynamical formulation of
120
+ GR defined on a spacetime manifold M = R × Σ, where Σ stands for the 3D spatial
121
+ manifold.
122
+ 2.1
123
+ The classical dynamics
124
+ Working within the Hamiltonian framework, the classical phase space consists of the
125
+ Ashtekar-Barbero variables which are actually the two canonically conjugate variables
126
+ – 2 –
127
+
128
+ of the theory. These variables are the densitized triad Ea
129
+ i and the Ashtekar connection
130
+ Ai
131
+ a defined as follows [65–68]:
132
+ Ea
133
+ i = |det(eb
134
+ j)|−1ea
135
+ i ,
136
+ (2.1)
137
+ Ai
138
+ a
139
+ = Γi
140
+ a + γKi
141
+ a,
142
+ (2.2)
143
+ where ea
144
+ i is the triad field, Γi
145
+ a is the spin connection, Ki
146
+ a is the extrinsic curvature and
147
+ γ is the so-called Barbero-Immirzi parameter which allows the quantisation procedure
148
+ to be performed on a compact group. Such a setup is based on a 3+1 decomposition of
149
+ the metric written in the following form:
150
+ ds2 = N2dt2 − qab(dxa + Nadt)(dxb + Nbdt),
151
+ (2.3)
152
+ where qab = ei
153
+ aeib is the spatial metric, N is the lapse function and Ni is the shift
154
+ vector. This metric choice simplifies the quantisation process and is chosen for conve-
155
+ nience. However, as said before, the LQG background equations will not depend on
156
+ the choice of the spacetime metric. This independence of the background on the choice
157
+ of the spacetime foliation is associated to some constraints. Firstly, one should require
158
+ the diffeomorphism constraint which renders the theory independent of the choice of
159
+ the spatial geometry, i.e. shift vector, and secondly the Hamiltonian constraint which
160
+ ensures the theory to be invariant under the choice of temporal coordinates1, i.e. lapse
161
+ function. These two constraints conserve the general spacetime covariance of the theory.
162
+ Thirdly, one imposes the Gaussian constraint which makes the theory invariant under
163
+ any rotations of the triad fields.
164
+ At the classical level, the two canonically conjugate variables Ea
165
+ i and Ai
166
+ a will be
167
+ related with the following non-vanishing Poisson bracket:
168
+ {Ai
169
+ a(x), Ea
170
+ i (y)} = 8πGγδb
171
+ aδi
172
+ jδ(3)(x − y),
173
+ (2.4)
174
+ while the dynamics of the theory will be governed by the following Hamiltonian acting
175
+ on the canonical variables [69, 70]:
176
+ H[N] =
177
+ 1
178
+ 8πG
179
+
180
+ Σ
181
+ d3xN
182
+
183
+ F j
184
+ ab − (1 + γ2)ϵjmnKm
185
+ a Kn
186
+ b
187
+ � ϵjklEa
188
+ kEb
189
+ l
190
+ √q
191
+ ,
192
+ (2.5)
193
+ where F j
194
+ ab is the curvature of the Ashtekar connection defined as F j
195
+ ab ≡ ∂aAj
196
+ b − ∂bAj
197
+ a +
198
+ ϵijkAj
199
+ aAk
200
+ b.
201
+ Working now within the spatially flat Friedman-Lemaˆıtre-Robertson-Walker (FLRW)
202
+ model, one introduces a fiducial cell V connected to a fiducial metric oqab and a fiducial
203
+ orthonormal triad and co-triad (oea
204
+ i ,o ωi
205
+ a) such as oqab =o ωi
206
+ a
207
+ oωi
208
+ b. At the end, the reduced
209
+ Ashtekar connection and densitized triad read as [67]
210
+ Ai
211
+ a = cV −1/3
212
+ 0
213
+ oωi
214
+ a,
215
+ Eb
216
+ i = pV −2/3
217
+ 0
218
+
219
+ det(oq) oeb
220
+ i,
221
+ (2.6)
222
+ 1Here, we conventionally denote as temporal coordinates the ones perpendicular to the 3D spatial
223
+ slices. This notation is convenient but does not preassume a preferred time. As a consequence, general
224
+ covariance is conserved.
225
+ – 3 –
226
+
227
+ where V0 is the fiducial volume as measured by the fiducial metric oqab and c, p are
228
+ functions of the cosmic time t.
229
+ In order now, to identify an internal clock of our theory, we introduce a dynamical
230
+ massless scalar field described by the Hamiltonian:
231
+ Hφ =
232
+ p2
233
+ φ
234
+ 2|p|3/2 .
235
+ (2.7)
236
+ At the end, the cosmological classical phase space is composed of two congugate pairs
237
+ (c, p) and (φ, pφ) which obey the following Poisson brackets:
238
+ {c, p} = 8πG
239
+ 3
240
+ γ,
241
+ {φ, pφ} = 1.
242
+ (2.8)
243
+ with |p| = a2V 2/3
244
+ 0
245
+ and c = γ ˙aV 1/3
246
+ 0
247
+ . Finally, using he Hamiltonian constraint one obtains
248
+ the usual Friedmann equation within GR for a flat FRLW model,
249
+ H2 = 8πG
250
+ 3
251
+ ρ.
252
+ (2.9)
253
+ 2.2
254
+ The quantum dynamics
255
+ Working now at the quantum level, the classical phase space variables and the clas-
256
+ sical Hamiltonian will be promoted to quantum operators while the Poisson brackets
257
+ will be replaced by commutation relations. However, within quantum field theory, the
258
+ commutation algebra of quantum operators requires integration over the 3D space, thus
259
+ assuming a well pre-defined background.
260
+ Nevertheless, this setup cannot be applied
261
+ within the framework of LQG since we want a background independent theory. For this
262
+ reason, the quantisation process is performed at the level of two new canonical variables,
263
+ namely the holonomy of the Ashtekar connection he(A) along a curve e ⊂ Σ and the
264
+ flux of the densitized triad FS(E) along a 2-surface S defined as [67]
265
+ he(A) ≡ Pexp
266
+ ��
267
+ e
268
+ τiAi
269
+ adxa
270
+
271
+ ,
272
+ FS(E) ≡
273
+
274
+ S
275
+ τiEi
276
+ anad2y,
277
+ (2.10)
278
+ where τ = −iσi/2 (σi are the Pauli matrices) with [τi, τj] = ϵijkτ k, na is the unit vector
279
+ vertical to the surface S and P is a path-ordering operator. These functions constitute
280
+ non-trivial SU(2) variables satisfying a unique holonomy-flux Poisson algebra [71–74].
281
+ Working within this representation one can then construct a kinematical Hilbert
282
+ space for the gravity sector which is actually the space of the square integrable func-
283
+ tions on the Bohr compactification of the real line, i.e. Hgrav
284
+ kin ≡ L2(RBohr, dµBohr) [67].
285
+ Regarding the matter sector, the respective kinematical Hilbert space is defined like in
286
+ the standard Shrondigner picture as Hmatter
287
+ kin
288
+ ≡ L2(R, dµ). Thus, the whole kinematical
289
+ Hilbert space of the theory is defined as Hkin ≡ Hgrav
290
+ kin ⊗ Hmatter
291
+ kin
292
+ .
293
+ Focusing now on the homogeneous and isotropic FLRW model, usually called
294
+ as Loop Quantum Cosmology (LQC) and following the conventional quantisation ¯µ
295
+ scheme [75] one introduces two new conjugate variables defined as follows:
296
+ u ≡ 2
297
+
298
+ 3 sgn(p)/¯µ3,
299
+ b ≡ ¯µc,
300
+ (2.11)
301
+ – 4 –
302
+
303
+ where ¯µ =
304
+
305
+ ∆/|p| and ∆ = 4
306
+
307
+ 3 πγGℏ being the minimum nonzero eigenvalue of the
308
+ area operator [76].
309
+ Finally, one can show that the new variables obey the following
310
+ Poisson bracket:
311
+ {b, u} = 2
312
+ ℏ,
313
+ (2.12)
314
+ and that in Hgrav there are two elementary operators, namely �
315
+ eib/2 and ˆu related to
316
+ the holonomy and the flux conjugate variables.
317
+ In particular, it turns out that the
318
+ eigenstates |u⟩ of ˆu form an orthonormal basis in Hgrav
319
+ kin
320
+ and the actions of these two
321
+ operators in this basis can read as
322
+
323
+ eib/2 |u⟩ = |u + 1⟩ ,
324
+ ˆu |u⟩ = u |u⟩ .
325
+ (2.13)
326
+ Letting now |φ⟩ being the orthonormal bases in Hmatter
327
+ kin
328
+ one can define |u, φ⟩ ≡ |u⟩ ⊗ |φ⟩
329
+ as the generalized basis of the whole kinematic Hilbert space Hkin. Thus, after defining
330
+ the relevant Hilbert space and the associated to it orthonormal basis, one can promote
331
+ the Hamiltonian to a quantum operator. In particular, it is possible to define a quasi-
332
+ classical sharped initial state living in Hkin, which can be viewed as wavepacket around
333
+ a classical trajectory. Consequently, expressing the Hamiltonian (2.5) in terms of fluxes
334
+ and holonomies one can derive the expectation value of the Hamiltonian operator over
335
+ the initial semi-classical sharped state which at the end will contain first order quantum
336
+ corrections. Finally, accounting only for the holonomy corrections (since flux or inverse
337
+ volume corrections face the issue of a fiducial cell dependence [75]) one obtains the
338
+ following modified Friedmann equation [75, 77, 78]:
339
+ H2 = 8πG
340
+ 3
341
+ ρ
342
+
343
+ 1 − ρ
344
+ ρc
345
+
346
+ ,
347
+ (2.14)
348
+ where ρc = 2
349
+
350
+ 3 M4
351
+ Pl
352
+ γ3
353
+ . As it can be seen from Eq. (2.14) for ρ > ρc there is no physical evo-
354
+ lution since H2 < 0. One then finds that the effect of holonomies leads to a non-singular
355
+ evolution where the classical Big Bang singularity is replaced by a non-singular quantum
356
+ bounce where ρ = ρc and H = 0. This bouncing point constitutes a transitioning point
357
+ between a contracting (H < 0) and an expanding phase (H > 0).
358
+ 3
359
+ The threshold of primordial black hole formation in loop quantum
360
+ gravity
361
+ Having introduced before the fundamentals of LQG, we estimate in this section the PBH
362
+ formation threshold δc accounting for effects of loop quantum gravity at the level of the
363
+ background cosmic evolution.
364
+ To do so, we assume that the collapsing overdensity region is described by a homo-
365
+ geneous core (closed Universe) described by the following fiducial metric:
366
+ ds2 = −dt2 + a2(t)
367
+
368
+ dχ2 + sin2 χdΩ2�
369
+ ,
370
+ (3.1)
371
+ – 5 –
372
+
373
+ where dΩ2 is the line element of a unit two-sphere and a(t) is the scale factor of the
374
+ perturbed overdensity region.
375
+ For this type of close homogeneous and isotropic spacetime foliations one can show
376
+ that following the procedure as described in Sec. 2 the modified Friedmann equation in
377
+ k = 1 LQC accounting only for the holonomy corrections will read as [79, 80]:
378
+ H2 =
379
+ � ˙a
380
+ a
381
+ �2
382
+ =
383
+ 1
384
+ 3M2
385
+ Pl
386
+ (ρ − ρ∗)
387
+
388
+ 1 − ρ − ρ∗
389
+ ρc
390
+
391
+ ,
392
+ (3.2)
393
+ where ρ is the energy density of the overdense region and ρ∗ = ρc
394
+
395
+ (1 + γ2)D2 + sin2 D
396
+
397
+ with D ≡ λ(2π2)1/3/v1/3, λ2 = 4
398
+
399
+ 3 πγℓ2
400
+ Pl [81], ℓPl being the Planck length and v =
401
+ 2π2a3 being the physical volume of the unit sphere spatial manifold [81]. Since v increases
402
+ with time one can expand Eq. (3.2) in the limit u ≫ 1 [79]. At the end, keeping terms
403
+ up to O(1/v2/3) one can show that Eq. (3.2) takes the following form:
404
+ H2 =
405
+ � ˙a
406
+ a
407
+ �2
408
+ =
409
+ 1
410
+ 3M2
411
+ Pl
412
+
413
+ ρ
414
+
415
+ 1 − ρ
416
+ ρc
417
+
418
+ − 3M2
419
+ Pl
420
+ a2
421
+
422
+ 1 − 2 ρ
423
+ ρc
424
+ ��
425
+ = F(ρ)
426
+ 3M2
427
+ Pl
428
+ − G(ρ)
429
+ a2 ,
430
+ (3.3)
431
+ where F(ρ) = ρ(1 − ρ/ρc) and G(ρ) = 1 − 2ρ/ρc. In the limit where γ = 0, ρc → ∞ and
432
+ one recovers the standard GR k = 1 Friedmann equation H2 =
433
+ ρ
434
+ 3M2
435
+ Pl − 1
436
+ a2 .
437
+ As regards now the background, the latter it will behave as the standard homoge-
438
+ neous and isotropic FLRW background whose fiducial metric reads as
439
+ ds2 = −dt2 + a2
440
+ b(t)
441
+
442
+ dr2 + r2dΩ2�
443
+ (3.4)
444
+ and whose modified Friedmann equation within LQC will read as
445
+ H2
446
+ b =
447
+ � ˙ab
448
+ ab
449
+ �2
450
+ =
451
+ ρb
452
+ 3M2
453
+ Pl
454
+
455
+ 1 − ρ
456
+ ρc
457
+
458
+ .
459
+ (3.5)
460
+ In this setup, the collapsing overdense region corresponds to the region where 0 ≤
461
+ χ ≤ χa and the areal radius at the edge of the ovedensity will read as
462
+ Ra = a sin χa.
463
+ (3.6)
464
+ At this point, we need to stress that the characteristic size of the overdensity is
465
+ initially super-horizon and will reenter the cosmological horizon when the areal radius
466
+ of the overdensity becomes equal to the cosmological horizon H−1, i.e.
467
+ 1
468
+ Hhc
469
+ = ahc sin χa,
470
+ (3.7)
471
+ where the index “hc” denotes quantities at the horizon crossing time. Writing now the
472
+ energy density of the overdensity as ρ = ρb(1 + δ), where δ ≡ δρ
473
+ ρb , one can plug ρ into
474
+ Eq. (3.3) and working within the uniform Hubble gauge where H = Hb they can recast
475
+ Eq. (3.7) as
476
+ sin2 χa =
477
+ 1
478
+
479
+ hc
480
+ �F δ
481
+ hc
482
+ Fhc
483
+ − 1
484
+
485
+ ,
486
+ (3.8)
487
+ – 6 –
488
+
489
+ where F δ
490
+ hc = F [ρb,hc(1 + δ)] and Gδ
491
+ hc = G [ρb,hc(1 + δ)].
492
+ Once then the overdensity region crosses the cosmological horizon will initially
493
+ follow the cosmic expansion and at some point it will detach from it starting to collapse
494
+ to form a black hole horizon. This basically happens at the time of maximum expansion
495
+ of the overdensity, when the Hubble parameter in Eq. (3.5) becomes zero, i.e. Hm = 0,
496
+ or equivalently when
497
+ am = 3M2
498
+ PlGm
499
+ Fm
500
+ ,
501
+ (3.9)
502
+ with the subscript “m” denoting quantities at the maximum expansion time.
503
+ Having derived above the horizon crossing time and the time at maximum expan-
504
+ sion we establish below a criterion for PBH formation by investigating the necessary
505
+ conditions for the triggering of the gravitational collapse process. Doing so, we confront
506
+ the gravitational force which pushes matter inwards and enhances in this way the black
507
+ hole gravitational collapse with the sound wave pressure force which pushes matter out-
508
+ wards, thus disfavoring the collapse of the overdensity. In particular, the criterion which
509
+ we adopt is the requirement that the time at which the pressure sound wave crosses
510
+ the radius of the overdensity region should be larger than the time at the maximum
511
+ expansion, which is actually the time of the onset of the gravitational collapse. Thus,
512
+ the sound pressure force will not have time to disperse the collapsing fluid matter to
513
+ the background medium and prevent in this way the collapsing process. Equivalently,
514
+ we require that the proper size of the overdensity χa is larger than the sound crossing
515
+ distance by the time of maximum expansion χs, i.e.
516
+ χa > χs.
517
+ (3.10)
518
+ To compute now the sound crossing distance by the time of maximum expansion we
519
+ assume matter in terms of a perfect fluid characterized by a constant equation-of-state
520
+ (EoS) parameter w, defined as the ratio between the pressure p and the energy density
521
+ ρ of the fluid, w ≡ p/ρ. Within this framework, the sound wave propagation equation
522
+ reads as
523
+ adχ
524
+ dt = √w ,
525
+ (3.11)
526
+ where we used the fact that for a perfect fluid with a constant EoS parameter the square
527
+ sound wave c2
528
+ s is equal to w, i.e. c2
529
+ s = w. At the end, χs can be recast in following form:
530
+ χs = √w
531
+ � tm
532
+ tini
533
+ dt
534
+ a =
535
+ √w
536
+ 3
537
+ � ρini
538
+ ρb,m
539
+ dρb
540
+ (1 + w)ρb
541
+ �� ρb,m
542
+ ρb
543
+
544
+ 2
545
+ 3(1+w) GmF(ρb)
546
+ Fm
547
+ − G(ρb)
548
+ ,
549
+ (3.12)
550
+ where we have assumed that for a perfect fluid ρb ∝ a−3(1+w) and used Eq. (3.9) to
551
+ express am in terms of Fm and Gm.
552
+ At the end, using Eq. (3.8) the criterion for PBH formation reads as
553
+ 1
554
+
555
+ hc
556
+ �F δ
557
+ hc
558
+ Fhc
559
+ − 1
560
+
561
+ > sin2 χs.
562
+ (3.13)
563
+ – 7 –
564
+
565
+ To determine therefore the PBH formation threshold, one can follow the following
566
+ procedure: From Eq. (3.8), one should firstly determine the the ratio ρhc/ρm for a given
567
+ value of χa and then solve numerically the inequality Eq. (3.13) in order to extract the
568
+ value of the critical energy density contrast δc required for the overdensity region to
569
+ collapse and form a PBH.
570
+ Practically, one should compute χs from Eq. (3.12) for a given value of ρm and
571
+ equate χs with χa. Then, solving Eq. (3.8) they will extract the ratio ρhc/ρm and then
572
+ plugging it into Eq. (3.13) they can extract numerically δc. At the end, given the fact
573
+ that ρb,hc < ρc, i.e. PBHs form after the quantum bounce, and that δ < 1 since we want
574
+ to be within the perturbative regime, one can show from Eq. (3.8) that
575
+ δc ≃ sin2 χs,
576
+ (3.14)
577
+ with χs given by Eq. (3.12).
578
+ At this point, we should stress that the above expression for the value of δc is a
579
+ lower bound estimate of its true value since it assumes the homogeneity of the collapsing
580
+ overdensity region which in general is not the case when one is met with strong pressure
581
+ gradients. Thus, it is strictly valid for regimes where w ≪ 1. However, PBH formation
582
+ was never studied before in a rigorous way within the context of LQG through numerical
583
+ simulations. To that end, Eq. (3.14) provides a reliable estimate for the value of δc.
584
+ 4
585
+ Results
586
+ Following the procedure described above, we calculate here the PBH formation threshold
587
+ δc within the framework of LQG and we compare it with its value in GR. In particular,
588
+ in the left panel of Fig. 1 we show the PBH formation threshold as a function of the
589
+ energy density at the time of maximum expansion ρm by fixing the EoS parameter
590
+ w = 1/3 since we study PBH formation during the RD era and the value of the Barbero-
591
+ Immirzi parameter γ = 0.2375 obtained from the computation of the entropy of black
592
+ holes [82]. Interestingly, we see a deviation from GR for high energies at the time of
593
+ maximum expansion which correspond to very small mass PBHs forming close to the
594
+ quantum bounce. This behavior is somehow expected since in this high energy regime,
595
+ one expects to see a quantifiable effect of the quantum nature of gravity. In particular,
596
+ we observe a drastic reduction of the value of δc in this region of high values of ρm up
597
+ to 50% compared the GR case.
598
+ This reduction of δc should be related with a smaller cosmological/sound horizon
599
+ in LQC compared to GR as it can be speculated from Eq. (3.14). To see this, let us find
600
+ the necessary conditions to get a cosmological horizon in LQC smaller than that in GR.
601
+ Doing so, one should require that
602
+ H2
603
+ LQC > H2
604
+ GR ⇔
605
+ ρ
606
+ 3M2
607
+ Pl
608
+
609
+ ρ − ρ
610
+ ρc
611
+
612
+ − 1
613
+ a2
614
+
615
+ 1 − 2 ρ
616
+ ρc
617
+
618
+ >
619
+ ρ
620
+ 3M2
621
+ Pl
622
+ − 1
623
+ a2 ⇔ ρ < 6M2
624
+ Pl
625
+ a2 .
626
+ (4.1)
627
+ For ρ = ρm and a = am as given by Eq. (3.9) one can verify that the inequality 4.1 is
628
+ identically satisfied. Thus, indeed the cosmological horizon in LQC is smaller than in
629
+ that GR leading to a reduction of δc compared to its GR value.
630
+ – 8 –
631
+
632
+ 10−13
633
+ 10−11
634
+ 10−9
635
+ 10−7
636
+ 10−5
637
+ 10−3
638
+ 10−1
639
+ 101
640
+ ρm/M 4
641
+ Pl
642
+ 0.30
643
+ 0.35
644
+ 0.40
645
+ 0.45
646
+ 0.50
647
+ 0.55
648
+ 0.60
649
+ δc
650
+ w = 1/3
651
+ LQG
652
+ GR
653
+ 10−5
654
+ 10−3
655
+ 10−1
656
+ 101
657
+ 103
658
+ 105
659
+ 107
660
+ MPBH in grams
661
+ 0.30
662
+ 0.35
663
+ 0.40
664
+ 0.45
665
+ 0.50
666
+ 0.55
667
+ 0.60
668
+ δc
669
+ w = 1/3
670
+ γ=10−1
671
+ γ=1
672
+ γ=101
673
+ γ=102
674
+ γ=103
675
+ Figure 1: Left Panel:The PBH formation threshold in the uniform Hubble gauge in the
676
+ radiation-dominated era (w = 1/3) as a function of the energy density at the onset of the
677
+ PBH gravitational collapse in LQG (green curve) and in GR (Eq. (1.1)) (black dashed
678
+ curve).
679
+ Right Panel: The PBH formation threshold in the uniform Hubble gauge in
680
+ the radiation-dominated era (w = 1/3) as a function of the primordial black hole mass
681
+ MPBH in LQG for different values of the Barbero-Immirzi parameter γ. The vertical
682
+ black dashed line corresponds to MPBH = MPl.
683
+ Then, we consider the Barbero-Immirzi parameter γ as a free parameter of the
684
+ underlying quantum theory in the context of LQG. In particular, despite the fact that
685
+ the Bekenstein-Hawking entropy has been standardly used as a way to fix the value of γ,
686
+ the dependence of the entropy calculation on γ is controversial, and the value γ ≃ 0.2375,
687
+ calculated using thermodynamical arguments, is not broadly accepted [83–85]. In fact,
688
+ the choice to vary this parameter is motivated by the fact that γ is actually a coupling
689
+ constant with a topological term in the gravitational action, with no consequence at the
690
+ level of the classical equations of motion [86–91]. Thus, we vary the Barbero-Immirzi
691
+ parameter within the range of 0.1 < γ < 1000 accounting for observational constraints
692
+ for the duration of inflation after a quantum bounce [92, 93]. At the end, we plot in
693
+ the right panel of Fig. 1 the PBH formation threshold δc as a function of the PBH
694
+ mass for different values of the parameter γ within the observationally allowed range
695
+ γ ∈ [0.1, 1000]. We set the lower bound on the PBH mass equal to the Planck mass as
696
+ predicted within the quantum gravity approach [94] (See vertical black dashed line in
697
+ the right panel of Fig. 1).
698
+ In order to get the PBH mass, we account for the fact that the PBH mass is of
699
+ the order of the cosmological horizon mass at horizon crossing time. Solving at the end
700
+ numerically Eq. (3.8) we found that ρhc/ρm ∼ 10. This corresponds to
701
+ N = ln
702
+ � am
703
+ ahc
704
+
705
+ = 1
706
+ 4 ln
707
+ �ρhc
708
+ ρm
709
+
710
+ ∼ 0.6 e − folds
711
+ (4.2)
712
+ passing from horizon crossing time up to the onset of the gravitational collapse process at
713
+ – 9 –
714
+
715
+ ρ = ρm, thus being in agreement with the results from PBH numerical simulations [95].
716
+ As expected, when we increase the value of γ the overall mass range moves to higher
717
+ masses given the fact that higher values of γ are equivalent with lower values of ρc, thus
718
+ the quantum bounce happens at later times. Consequently, PBHs if formed will form at
719
+ later times, thus will acquire larger masses.
720
+ Interestingly, independently on the value of the Barbero-Immirzi variable δc is re-
721
+ duced on the low mass region, which for γ < 1000 corresponds to masses MPBH < 103g.
722
+ In particular, this reduction in δc in this very small PBH mass range will entail an
723
+ enhancement in their abundances with tremendous consequences on the associated to
724
+ them phenomenology. Indicatively, we mention here that these ultra-light PBHs can
725
+ trigger early PBH-matter dominated eras [20, 21, 96] before BBN and reheat the Uni-
726
+ verse through their evaporation [97] while at the same time they can account for the
727
+ Hubble tension through the injection to the primordial plasma of light dark radiation
728
+ degrees of freedom [98, 99] while at the same time they can produce naturally the baryon
729
+ assymetry through CP violating out-of-equilibrium decays of their Hawking evaporation
730
+ products [100, 101].
731
+ Consequently, one can constrain the above mentioned observa-
732
+ tional/phenomenological signatures by studying PBH formation within the context of
733
+ LQG while vice-versa given the above mentioned phenomenology one can constrain the
734
+ Barbero-Immirzi parameter γ which is the fundamental parameter within LQG. In this
735
+ way, PBHs are promoted as a novel probe to constrain the potential quantum nature of
736
+ gravity.
737
+ 5
738
+ Conclusions
739
+ PBHs firstly introduced in ’70s are of great significance, since they can naturally account
740
+ for a part or all of the dark matter sector, while at the same time they might seed the
741
+ formation of large-scale structures through Poisson fluctuations. Moreover, they can also
742
+ offer the seeds for the progenitors of the black-hole merging events recently detected by
743
+ LIGO/VIRGO as well as for the supermassive black holes present in the galactic centers.
744
+ Their formation was mainly studied within the context of general relativity using both
745
+ analytic and numerical techniques.
746
+ In this work, we studied PBH formation within the context of LQG by investigating
747
+ the impact of the potential quantum character of spacetime on the critical PBH forma-
748
+ tion threshold δc, whose value can crucially affect the abundance of PBHs, a quantity
749
+ which is constrained by numerous observational probes. In particular, by comparing
750
+ the gravitational force with the sound wave pressure force during the process of the
751
+ gravitational collapse we obtained a reliable estimate on the value of δc.
752
+ Interestingly, we found that for low mass PBHs formed close to the quantum
753
+ bounce, the value of δc is drastically reduced up to 50% compared to the general rel-
754
+ ativistic regime with tremendous consequences for the observational/phenomenological
755
+ footprints of such small PBH masses. In this way, we quantified for the first time to the
756
+ best of our knowledge how quantum effects can influence PBH formation in the early
757
+ Universe within a quantum gravity framework.
758
+ – 10 –
759
+
760
+ Finally, by treating the Barbero-Immirzi parameter γ as the free parameter of LQG
761
+ we varied its value by studying its effect on the value of the PBH formation threshold.
762
+ As expected, we found an overall shift of the PBH masses affected by the choice of γ.
763
+ Very interestingly, we showed as well that using the observational and phenomenological
764
+ signatures associated to ultra-light PBHs, namely the ones affected by LQG effects, one
765
+ can constrain the quantum parameter γ. At this point, we should highlight the fact that
766
+ our formalism can be applied to any quantum theory of gravity giving an explicit form
767
+ for the equations of the background cosmic evolution establishing in this way the PBH
768
+ portal as a novel probe to constrain the potential quantum nature of gravity.
769
+ Acknowledgments
770
+ The author acknowledges financial support from the Foundation for Education and
771
+ European Culture in Greece as well the contribution of the COST Action CA18108
772
+ “Quantum Gravity Phenomenology in the multi-messenger approach”.
773
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774
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