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+Caustic spin wave beams in soft, thin films: properties and classification
+Alexis Wartelle,∗ Franz Vilsmeier, Takuya Taniguchi, and Christian H. Back
+Fakult¨at fur Physik, Technische Universit¨at M¨unchen, Garching, Germany
+(Dated: January 4, 2023)
+In the context of wave propagation, caustics are usually defined as the envelope of a finite-extent
+wavefront; folds and cusps in a caustic result in enhanced wave amplitudes. Here, we tackle a related
+phenomenon, namely the existence of well-defined beams originating solely from the geometric
+properties of the corresponding dispersion relation. This directional emission, termed caustic beam,
+is enabled by a stationary group velocity direction, and has been observed first in the case of
+phonons. We propose an overview of this “focusing” effect in the context of spin waves excited
+in soft, thin ferromagnetic films. Based on an analytical dispersion relation, we provide tools for
+a systematic survey of caustic spin wave beams. Our theoretical approach is validated by time-
+resolved microscopy experiments using the magneto-optical Kerr effect. Then, we identify two cases
+of particular interest both from fundamental and applicative perspectives. Indeed, both of them
+enable broadband excitations (in terms of wave vectors) to result in narrowband beams of low
+divergence.
+I.
+INTRODUCTION
+The collective motion of magnetic moments in a ma-
+terials, referred to as spin waves, has shown remarkable
+properties from a fundamental perspective.
+Examples
+range from anisotropic dispersion in thin films [1], rel-
+evant for the field of magnonics, to Bose-Einstein con-
+densation of magnons [2], through restricted-relativity-
+like bounded domain wall velocities [3].
+Applications
+of magnetization dynamics also abound, starting with
+the infinite-wavelength ferromagnetic resonance (FMR)
+[4] and going all the way towards sub-micrometer wave-
+lengths, which are currently viewed as promising alterna-
+tive information carriers in the fields of magnonics [5]. In
+addition to the absence of Joule heating and the potential
+device downscaling (using small wavelengths), spin wave
+interference is an appealing prospect [6] as it allows logic
+operations through the design of the propagation lines.
+Several experimental techniques are readily available
+for the study of spin waves [1], especially in the case
+of thin films or patterned elements thereof.
+Among
+them, micro-/phase-resolved Brillouin Light Scattering
+(BLS) [7], Time-Resolved Magneto-Optical Kerr Effect
+(TR-MOKE) microscopy [8], and time-resolved Scan-
+ning Transmission X-ray Microscopy (TR-STXM) with
+magnetic sensitivity through X-ray Magnetic Circular
+Dichroism (XMCD [9]) [10] have demonstrated outstand-
+ing imaging capabilities. Nevertheless, the usually very
+small amplitudes of magnetization precession associated
+to spin waves as well as their attenuation lengths (typi-
+cally on the micrometer scale) pose a significant challenge
+both for fundamental investigations and for applications.
+To be of practical use, spin waves must be harnessed
+via a power-efficient strategy: some approaches like Win-
+ter’s magnons rely on channeling along domain walls [11],
+∗ alexis.wartelle@ens-lyon.org; Present address: Universit´e Greno-
+ble Alpes, CNRS, Grenoble INP, SIMaP, 38000 Grenoble, France
+others rely on careful control of spin wave scattering [12].
+Another possibility would take advantage of caustic spin
+wave beams (CSWBs), i.e.
+spin wave beams of well-
+defined propagation direction, narrow angular width and
+higher power compared to e.g. Damon-Eshbach-type [13]
+spin waves.
+Furthermore, caustics in soft, thin ferro-
+magnetic films can be very different from the well-known
+acoustical or optical caustics, which originate from inho-
+mogeneous media [14–16], : here, spin wave caustics can
+arise in perfectly homogeneous films in broad ranges of
+conditions solely because of sufficient anisotropies in their
+dispersion relation. The latter indeed allows the direc-
+tion of the group velocity to be stationary around some
+wave vectors, leading to well-defined directions of wave
+propagation associated to significantly stronger emission.
+In the context of phonon propagation, such phenomena
+have been referred to as “focussing” [17], and they have
+been observed and investigated since 1969 [17–21].
+By contrast, caustics in ferromagnetic films were re-
+ported for the first time ca. 30 years later [22]. There
+has been quite a few reports since then [23–31] but, to
+the best of our knowledge, there exists to date no sys-
+tematic survey of the properties of spin wave caustics,
+not even focusing on a certain type of systems e.g. ul-
+trathin films with perpendicular anisotropy, or soft thin
+films.
+In this work, we restrict ourselves to the latter
+and give an overview of caustics in soft thin films, as well
+as tools to further investigate them. Moreover, we high-
+light two special cases which seem particularly appealing
+notably for application in magnonics.
+II.
+MODEL
+A.
+General considerations
+Our starting point is the model derived by Kalinikos
+and Slavin [32] for spin waves in soft ferromagnetic
+thin films.
+These excitations correspond to a time-
+and space-dependent magnetization −→
+M(⃗r, t), yet its norm
+arXiv:2301.01220v1  [cond-mat.mes-hall]  3 Jan 2023
+
+2
+Ms = ||−→
+M(⃗r, t)|| the spontaneous magnetization is uni-
+form.
+As a result, it is simpler to consider the re-
+duced magnetization −→
+m(⃗r, t) = −→
+M(⃗r, t)/Ms with norm
+1.
+We focus on the linear regime i.e.
+the deviation
+δ−→
+m(⃗r, t) = −→
+m(⃗r, t) − −→
+m0(⃗r, t) from the equilibrium mag-
+netization (when no excitation is applied) −→
+m0 is such that
+||δ−→
+m|| ≪ 1. Under the assumption of negligible mode
+mixing and of a perfectly isotropic ferromagnetic mate-
+rial, one may write the dispersion relation of a thin film
+as:
+ω2 =
+�
+γ0Ha + 2Aγ0
+µ0Ms
+k2��
+γ0
+�
+Ms + Ha
+�
++ 2Aγ0
+µ0Ms
+k2�
+−γ2
+0M 2
+s · ξ(kd)
+�
+1 − ξ(kd) + Ha
+Ms
++ 2Aγ0
+µ0M 2s
+k2�
+cos2 ϕ
++γ2
+0M 2
+s · ξ(kd) · [1 − ξ(kd)]
+(1)
+where ω is the spin wave angular frequency, γ0 = µ0|γ|
+with γ = qe/(2me) the electron’s gyromagnetic ratio
+(qe = −e and me being the electron’s charge and mass,
+respectively) and µ0 the permeability of vacuum, A is the
+micromagnetic exchange constant for the soft ferromag-
+netic material of interest, Ms its spontaneous magnetiza-
+tion, k the spin wave’s wavenumber corresponding to its
+wave vector −→k , Ha = ||−→
+Ha|| the strength of the externally
+applied magnetic field −→
+Ha from which ϕ = angle
+�−→
+Ha, −→k
+�
+the wavefront angle is defined, d the film thickness, and
+ξ is the function whose values are defined as:
+ξ(u) = 1 − 1 − e−u
+|u|
+.
+(2)
+As a consequence of the ferromagnetic material’s soft-
+ness, in the absence of excitation, the equilibrium mag-
+netization configuration in our thin film is the single-
+domain state, with a corresponding reduced magnetiza-
+tion −→
+m0 exactly along the applied field. The orientations
+of −→
+m0, −→
+Ha, and −→k are illustrated in Fig. 1, which also
+highlights the natural wavelength λ0 = 2π/||−→k || of the
+spin wave as well as the unit vectors −→
+ex, −→
+ey and −→
+ez.
+Here, we focus on spin waves with no amplitude node
+across the film thickness, i.e. we do not consider per-
+pendicular standing spin waves (PSSWs). However, we
+do note that the latter may play a role in experiments
+performed on sufficiently thick films where a realistic an-
+tenna for instance could excite them due to its inhomo-
+geneous magnetic field.
+We introduce the following quantities:
+the Larmor
+angular frequencies associated to magnetization ωM =
+γ0Ms and to the applied magnetic field ωH = γ0Ha, the
+material’s dipolar-exchange length lex =
+�
+2A/(µ0M 2s ).
+We then rewrite the equation as:
+−
+→
+ex
+−
+→
+k
+−→
+Ha
+−→
+m0
+φ
+λ0= 2π
+||−
+→
+k||
+= 2π
+k
+0
+δmz
+−
+→
+ez
+−
+→
+ey
+FIG. 1. Schematic representation of a spin plane wave prop-
+agating in a soft thin film.
+The grey scale codes the local
+perpendicular component of the dynamic component of mag-
+netization, δmz.
+ω2
+ω2
+M
+=
+� ωH
+ωM
++ l2
+exk2
+� �
+1 + ωH
+ωM
++ l2
+exk2
+�
+−ξ(kd)
+�
+1 − ξ(kd) + ωH
+ωM
++ l2
+exk2�
+cos2 ϕ
++ξ(kd)
+�
+1 − ξ(kd)
+�
+(3)
+Introducing the reduced frequency ν = ω/ωM and ap-
+plied field h = ωH/ωM = Ha/Ms, and normalizing both
+the dipolar-exchange length and wavenumber to the film
+thickness d using η = lex/d and ˜k = kd, we arrive at:
+ν2 =
+�
+h + η2˜k2��
+1 + h + η2˜k2�
+−ξ(˜k)
+�
+1 − ξ(˜k) + h + η2˜k2�
+cos2 ϕ
++ξ(˜k)
+�
+1 − ξ(˜k)
+�
+(4)
+With this, it is clear that any given experiment of spin
+wave excitation corresponds to a specific value of the di-
+mensionless triplet (η, ν, h). In other words: they are
+the only independent parameters within this model.
+For a value of (η, ν, h), the solution to (4) is the pos-
+sibly empty set of accessible dimensionless wave vectors
+−→k d.
+The existence and properties of spin wave caus-
+tics depend on the geometrical characteristics of this set,
+which is why we are first going to review several of its
+general properties.
+Keeping in mind that we focus on applied fields below
+the ferromagnetic resonance field at the excitation fre-
+quency, we actually always have a non-empty solution,
+which is usually a closed curve winding around the ori-
+gin in wave-vector space. This is the so-called slowness
+curve, in reference to the fact that at fixed frequency
+k ∝ 1/||−→
+vp|| where −→
+vp is the phase velocity [33], oriented
+of course along the wave vector. Considering the parity
+of the cosine function and its antisymmetry for the re-
+flection ϕ → π − ϕ, we may restrict our analysis to only
+the quadrant ϕ ∈ [0, π/2] and deduce the others using
+mirror symmetries.
+
+3
+One can also parametrize the slowness curve using a
+curvilinear abscissa: we define it to be zero for the lowest
+dimensionless wavenumber ˜kmin at ϕ = π/2. One can in-
+deed show that the reduced wavenumber solving Eq. (4)
+at ϕ = π/2 (resp. 0) is minimum (resp. maximum) on
+the quadrant ϕ ∈ [0, π/2]. Thus, at the largest dimen-
+sionless wavenumber ˜kmax = ˜k(ϕ = 0), the correspond-
+ing curvilinear abscissa sM corresponds to the length of
+the slowness curve in the quadrant ϕ ∈ [0, π/2] i.e. one
+fourth of the whole length of this curve.
+Another important geometrical aspect of the slowness
+curve that is central to the present work is the local
+normal to it. Considering its definition as a constant-
+frequency intercept of the dispersion relation in wave-
+vector space, by nature, the frequency gradient −→
+∇−
+→
+k ω is
+perpendicular to the slowness curve. As a result, the di-
+rection of the group velocity of spin waves −→
+vg = −→
+∇−
+→
+k ω can
+be directly read from the direction of the local normal to
+the slowness curve. In our notations, we point out that:
+−→
+∇−
+→
+k ω ≡
+�
+β=x,y,z
+∂ω
+∂kβ
+· −→
+eβ
+where
+kβ = −→k · −→
+eβ.
+In the following,
+we will use the angle θV
+=
+angle(−→
+Ha, −→
+vg).
+We point out that in the present case,
+phase and group velocities need not be collinear:
+on
+the contrary, there can be differences between θV and
+ϕ much larger than in cases of light propagation through
+anisotropic media [34]. Fig. 2 illustrates this on the ex-
+ample of a slowness curve reconstructed for a vanishing
+reduced applied field.
+B.
+Distinctive features of dispersion relation
+caustics
+Typically, caustics in inhomogeneous media occur
+when a wavefront folds onto itself; in this situation, there
+exists a surface (or a line in 2D wave propagation) such
+that across it the number of rays passing through a point
+in space changes by an even number [15, 16]: this is the
+caustic. Equivalently, it can be viewed as the set of the
+local extrema of positions on the ray bundle on the wave-
+front, for all the wavefronts along the wave propagation.
+It is this extremal nature that grants these caustics large
+and localized intensities compared to other points on the
+ray bundle.
+In a geometrical optics approach, the in-
+tensity diverges as an initially finite-sized portion of the
+wavefront shrinks to a vanishing area [15]. A wave op-
+tics treatment however reveals that the intensity remains
+finite due to interferences: illumination profiles across
+caustics can in principle be determined by taking into
+account the variations of phase as a function of distance
+to the caustic [14].
+Such an approach has been used by Schneider et al.
+[26] for spin wave caustics excited by the scattering of
+a spin wave travelling in a waveguide terminating into a
+θV
+φ0
+−
+→
+k0d
+a)
+b)
+c)
+FIG. 2.
+a) Exemplary slowness curve for (ν, h, η)
+=
+(0.2873, 10−20, 0.15). As can be clearly seen in the polar plot
+of kd = ˜k(ϕ), the direction (ϕ0 =32.00°) of the phase veloc-
+ity −→
+vp and that (θV =108.9°) of the group velocity −→
+vg at the
+point −→
+k0d are very different.
+b) Radiation pattern (δmz is
+grey-coded) of a hypothetical source exciting only wavenum-
+bers very close to ||−→
+k0||d. c) Plane wave corresponding to the
+carrier wave vector −→
+k0d (red lines are guides to the eye).
+full permalloy (Ni80Fe20) film. However, this is a very
+different situation compared to the above. Indeed, the
+wavefront does not fold onto itself due to spatial varia-
+tions of medium properties, rather, its extent is deter-
+mined almost exclusively (owing to the sub-wavelength
+source size) by the characteristics of spin wave propaga-
+tion. The latter are determined by the anisotropic spin
+wave dispersion relation, which allows caustics to form
+thanks to the possibility of stationary group velocity di-
+rection i.e. a beam with a well-defined propagation direc-
+tion yet comprising a range of wave vectors in the vicinity
+of a carrier. More precisely, caustics correspond to local
+extrema of the group velocity direction; in other words, a
+caustic spin wave beam implies the existence of a caustic
+point ˜kc on the slowness curve such that:
+dθV
+d˜k
+����˜kc
+= 0.
+(5)
+The CSWB has then a carrier wavenumber ˜kc, corre-
+sponding to a central wavefront angle ϕc = ϕ(˜kc) and a
+
+90
+75°
+.09
+45°
+30°
+Ug
+15°
+kd
+0°
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+kd4
+beam direction θV,c = θV(˜kc).
+Coming back to the wavefront extent, rays from wave
+vectors not close enough to the carrier cannot play a role
+in the caustic wave amplitude simply because of differ-
+ences in propagation direction.
+More specifically, the
+experimental data presented by Schneider et al.
+sug-
+gests that beam divergences of 2° or less can be obtained.
+Thus, there seems to be a contradiction between the cu-
+bic dispersion which is assumed to define the beam pro-
+file and the measurements. The question of the CSWB’s
+profile goes however beyond the scope of this work. Nev-
+ertheless, it is clear from the low beam divergences ob-
+served in many experimental reports [23, 27, 35] that only
+small, almost straight parts of the slowness curve must
+contribute to CSWB.
+In fact, integrating the contribution of wave vectors all
+the way to infinity as done in [26] neglects the geometric
+impossibility for them to create waves travelling from the
+point source to a far-away point on the caustic. To put it
+differently: for geometrical reasons, caustics originating
+solely from anisotropies in the dispersion relation and
+excited by a point-like source naturally restrict the range
+of relevant wave vectors, in contrast to the case of caustics
+due to inhomogeneities in the propagation medium.
+We wish to emphasize the above by reminding that in
+most cases [15, 36], caustics are treated on the basis of
+wave propagation in an isotropic or weakly anisotropic
+medium. One consequence is the fact that the flow of
+power, i.e. the group velocity, is along the wave vector
+or close to parallel to it [14]. While this remains a rea-
+sonable approximation for slightly anisotropic media (as
+in usual crystal optics), in the case of perfectly soft but
+fully polarized thin ferromagnetic films this collinearity
+may break down dramatically, as was illustrated in Fig.
+2. Therefore, even small changes in wave vector may re-
+sult in drastic changes in group velocity direction. By
+contrast, large changes in wave vectors do not necessar-
+ily lead to strong variations in the apparent wavelength
+λ which we define as:
+λ = 2π · ||−→
+vg||
+−→k · −→
+vg
+=
+2π
+−→k · −→
+eg
+=
+λ0(ϕ)
+cos (θV − ϕ),
+(6)
+where we have introduced −→
+eg as a unit vector along the
+group velocity. The apparent wavelength is simply the
+spatial period measured along the beam direction. Since
+large differences θV − ϕ can easily be obtained (cf. Fig.
+2, where cos (θV − ϕ0) ≃ 0.227), and more importantly
+since the projection ˜k(ϕ) cos (θV − ϕ) may remain almost
+constant over significant portions of the slowness curve,
+one should consider notions such as propagation-induced
+phase or spectral breadth [37] of a spin wave beam care-
+fully.
+III.
+RESULTS AND DISCUSSION
+A.
+Limit of model applicability: thick films
+We start by providing an example of situation where
+the model we use cannot be fully trusted, so as to high-
+light its limitations. In Fig. 3 we show a case where the
+reconstructed slowness curve splits into two separate con-
+nected components above a certain threshold frequency.
+˜k
+FIG. 3. Slowness curves for η = 0.015, h = 10−20, and ν =
+0.331 (dashed blue line) resp. ν = 0.333 (full red line).
+Such a behaviour has been described by Kreisel et al.
+[38]: the model chosen for spin wave dispersion predicts
+a local maximum in the ω(k, ϕ = π/2) vs. wavenumber
+curve, but this extremum is not reproduced by a formal
+approach not based on the thin-film approximation [39],
+and designed to tackle the dipole-exchange regime. The
+maximum’s presence leads to an additional pair of so-
+lutions in terms of wavenumber in a certain frequency
+range, corresponding to a splitting of the slowness curve
+into two separate components.
+Clearly, results obtained within our approach about
+caustics deep in the dipole-exchange regime are not trust-
+worthy. Empirically, we see the slowness curve splitting
+into separate components for values of η up to ca. 0.075;
+for the sake of comparison, the thinnest films investigated
+by Kreisel et al. feature η < 0.035 according to literature
+data on yttrium iron garnet (YIG) [40]. Nevertheless, the
+absence of this splitting is no proof that the reconstructed
+slowness curve is accurate, and we shall remain cautious
+in discussing results concerning CSWBs with wavenum-
+bers in the dipole-exchange regime. Finally, we note that
+promising theoretical developments such as the dipole-
+exchange dispersion relations recently derived by Harms
+and Duine [39] could eventually allow a more accurate
+treatment of caustics in the dipole-exchange regime.
+
+90°
+75°
+60°
+45°
+v =0.333
+v =0.331
+30°
+15°
+5
+10
+15
+20
+25
+30
+35
+0
+kd5
+B.
+General features
+Let us have a look at a first example of frequency and
+field map of caustic properties in Fig. 4. In the presented
+graphs, the red color means that either the corresponding
+(h, ν) point was not investigated because its reduced field
+is above the reduced FMR field hFMR, or because no
+caustic points were found there.
+First of all, one can see that there is indeed a portion
+of the (h, ν) plane where no caustic points exist. This
+occurs for frequencies above a certain νm(h, η). Then,
+going down in reduced frequency, there appears to be an
+oblique boundary between two regions of the map. Above
+it, ˜kc quickly enters the dipole-exchange regime, which we
+will only present but not discuss quantitatively as it cor-
+responds to a situation where our model is less reliable.
+Below the boundary, the reduced caustic wavenumber is
+much smaller than 1. Correspondingly, a boundary which
+we will label νb(h, η) appears at the same position on the
+plot of ϕc; this angle also seems close to constant over
+much of the region below the boundary. In both cases,
+its sharpness decreases towards low h, and at vanishing
+reduced field the transitions in ˜kc or ϕc are both smooth.
+All these features are represented on a simplified repre-
+sentation of the map of ˜kc shown as inset on the ϕc map,
+including the point (hc, νc) at which the sharp boundary
+seems to end. A zoomed-in view on (hc,νc) is shown in
+the inset of Fig. 4.b).
+In the following, we will refer to the lowest reduced
+field at which this boundary is sharp as hc and denote
+νc = νb(hc, η).
+As we shall see in more details, this
+abrupt boundary corresponds to a change in the num-
+ber of caustic points by two. The lowest point (hc, νc)
+is actually a cusp in the domain of existence of the two
+additional caustic points. We point out that for all re-
+duced fields and frequencies, the maps shown in Fig. 4
+displays the lowest caustic wavenumber respectively the
+associated wavefront angle.
+Before moving on to discussing the low-frequency
+pocket, its boundary and the existence of additional caus-
+tic points, and finally the threshold frequency for the ab-
+sence of caustic points, we stress that the behaviour of
+caustics strongly depends on η. As an example, we show
+in Fig. 5 field and frequency maps for η = 0.09, 0.3, 0.6
+(from left to right). At the lowest value, the boundary
+νb extends all the way to h = 0, whereas the two other
+maps do not display such a sharp behaviour. In addition
+to the expected changes in range of values for ˜kc, one can
+see that the overall shape of the domain of existence of
+CSWBs also changes. From here on, we will call this area
+D. From η = 0.09 to 0.3, we see that D has expanded in
+the vertical direction at low h. In even thinner films, for
+η = 0.6, the average slope of νm(h, η) has not changed
+much, yet νm(0, η) has decreased; as a result, D shrinks
+vertically.
+By contrast, even if the caustic group velocity direction
+displays a similar wealth of features as the caustic wave-
+front angle and reduced wavenumber, the jumps across
+the boundary νb are much less significant when they ex-
+ist. An example of this is shown in Fig. 6, which shows
+maps for θV,c at the same values of η as in Fig. 5.
+In a certain range of reduced dipolar-exchange length,
+we find that there may actually be more than one caustic
+point on the slowness curve. Empirically, we observe that
+the additional caustic points may exist for ˜kc < 1. When
+this inequality holds, the number of caustic points is ei-
+ther equal to one or to three; two being possible but only
+on a 1D curve in the field and frequency plane; this curve
+includes the aforementioned boundary νb. Qualitatively,
+this is due to the fact that in the corresponding range
+of field and frequency, when dθV/d˜k crosses 0, it does so
+with a local behaviour somewhat reminiscent of a poly-
+nomial of the type P(˜k; a, b) = (˜k − ˜kc)3 +a·(˜k − ˜kc)+b,
+where a and b are real parameters. If a > 0, there ex-
+ists only one root, whereas if a < 0 and |b| is sufficiently
+small, there exists three distinct roots.
+The domain in the field and frequency plane with these
+three roots will be referred to as D3 from now on, by
+contrast with D1 = D \ D3 in which there is only one
+caustic point instead of three. We will now describe D3
+using the P(˜k; a, b) approximant to dθV/d˜k for the sake
+of simplicity.
+Let us start with Fig. 7, which displays the same field
+and frequency map for ˜kc as in Fig. 4 along with the
+maps for the two other reduced caustic wavenumbers.
+The two additional solutions can be shown to coincide on
+the rounded boundary of D3 to the lower left, which will
+be referred to as ∂D3,l. Entering D3 through this bound-
+ary by increasing ν corresponds to the situation where |b|
+becomes small enough to allow the two additional caustic
+points (with respect to the one with lowest ˜kc), thanks to
+a being negative enough. Increasing h on the other hand
+mostly decreases a: upon crossing ∂D3,l, a pair of caus-
+tic points with higher ˜kc’s appears. Of course, exactly on
+∂D3,l the two additional roots of dθ/d˜k are identical.
+Starting from inside D3, if one increases the reduced
+frequency, eventually the caustic point with the interme-
+diate value of ˜kc merges with the one featuring the small-
+est reduced wavenumber.
+This happens on the other
+boundary of D3, which we will call ∂D3,u from now on.
+This situation corresponds to ν = νb(h, η). Just above
+this boundary, the value of b is low enough so that only
+one root of dθ/d˜k remains. That is the reason for the dis-
+continuity in ˜kc in Fig. 4: the lowest caustic wavenumber
+jumps to what was the highest of the three ˜kc’s below
+��b. Experimentally, this could imply that SW excitation
+around this threshold wavenumber would have marked
+changes in intensity as a function of frequency.
+Based on the above, since the two boundaries other
+than ferromagnetic resonance each imply that a differ-
+ent pair of caustic points coincide, we can infer that on
+the cusped intersection of ∂D3,l and ∂D3,u, there exists a
+single caustic point corresponding to three of them coin-
+ciding on the slowness curve. This is precisely the point
+(hc, νc) from the inset in Fig. 4.
+It is important to note that while a purely math-
+
+6
+φc (◦)
+˜kc
+0
+0.15
+0.31
+0.46 0.62
+h
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0
+0.15
+0.31
+0.46 0.62
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+89.98
+83.40
+55.60
+34.37
+2.40
+4.66
+0
+0.19 0
+h
+0
+0.25
+0.48
+ν
+0.74
+1.5
+Low-
+frequency
+pocket
+νm(h, η)
+h
+ν
+0
+0.62
+νb(h, η)
+νc
+hc
+a)
+b)
+h
+1.0
+FIG. 4. Frequency and field maps for a value of η = 0.12. For high enough fields, a sharp upturn in both properties can be seen
+for reduced frequencies above ca. 0.42. We remind the reader that fields above ferromagnetic resonance are not considered.
+Only few level curves are displayed for the sake of clarity. a) Caustic wavefront angle ϕc, with a schematic representation of
+the map’s distinctive features as inset. b) Normalized wavenumber ˜kc = kcd; a zoomed-in view on the area where the upturn’s
+sharpness drastically changes.
+0
+0.21
+0.41 0.62
+h
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+˜kc
+0
+0.21
+0.41 0.62
+h
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0
+0.21
+0.41 0.62
+h
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+1.25
+2.50
+3.75
+5.00
+6.19
+0
+˜kc
+0.350
+0.700
+1.05
+1.40
+1.69
+0
+˜kc
+0.125
+0.250
+0.375
+0.500
+0.611
+0
+a)
+b)
+c)
+η = 0.09
+η = 0.3
+η = 0.6
+FIG. 5. Examples of field and frequency maps for a) η = 0.09, b) η = 0.3, and c) η = 0.6; only the reduced caustic wavenumber
+is shown.
+
+Caustic wavefront angles Φc vs. v and h
+89.98
+1.0000
+0.9000
+83.40
+0.8000
+0.7000
+0.6000
+0.5000
+0.4001
+55.60
+0.3001
+0.2001
+0.1001
+0.0001
+0.1547
+34.37
+0.0000
+0.3095
+0.4642
+0.6190
+hTNormalized wavenumber at Φc vs. v and h
+4.662
+1.0000
+0.9000
+0.8000
+0.7000
+0.6000
+2.400
+0.5000
+0.4001
+0.3001
+0.2001
+0.1001
+0.0001
+0.1547
+0.0000
+0.3095
+0.4642
+0.6190
+0.000
+h000
+0.0475
+0.0950
+0.1425
+0.1900
+7Normalized wavenumber at Φc vs. v and h
+0.9991
+1.688
+0.8992
+0.7993
+1.400
+0.6994
+1.050
+0.5995
+0.4996
+kc
+0.700
+0.3997
+0.2998
+0.350
+0.1999
+0.1000
+0.000
+0.0001
+0.0000
+0.2063
+0.4126
+0.6189
+hNormalized wavenumber at Φc vs. v and h
+0.9991
+0.6118
+0.8992
+0.7993
+0.5000
+0.6994
+0.5995
+0.3750
+0.4996
+kc
+0.2500
+0.3997
+0.2998
+0.1250
+0.1999
+0.1000
+0.0000
+0.0001
+0.0000
+0.2063
+0.4126
+0.6189
+hNormalized wavenumber at Φc vs. v and h
+0.9991
+6.189
+0.8992
+0.7993
+5.000
+0.6994
+0.5995
+3.750
+0.4996
+kc
+2.500
+0.3997
+0.2998
+1.250
+0.1999
+0.1000
+0.000
+0.0001
+0.0000
+0.2063
+0.4126
+0.6189
+h7
+ematical analysis yields well-defined, separate caustic
+points, experimentally the distinction between close caus-
+tic points may well be impossible.
+In fact, there ex-
+ists no straightforward experimental signature of dθV/d˜k
+crossing 0, and portions of the slowness curve where this
+derivative is small but non-zero can behave similarly to
+an actual caustic point, as was noted by Gallardo et al.
+[41]. Nevertheless, the presence of more than one caustic
+point constrains a slowness curve to be almost straight in
+their vicinities; this should then favour marked caustics.
+C.
+Low-frequency pocket
+The low-frequency regime is important as it corre-
+sponds to a well established domain of validity of our
+theoretical model as well as wavelengths which can still
+be excited and detected reasonably easily in experiments.
+1.
+Analytics
+As could be seen in Fig.5, the shape or even the
+existence of the low-frequency pocket strongly depends
+on the chosen value of η.
+Nevertheless, we can in-
+vestigate the behaviour of caustics there by taking the
+limit ν → 0.
+In order to remain below ferromagnetic
+resonance, we also take the limit h → 0.
+Assum-
+ing h = 0 simplifies the computation of the quantity
+tan θV = tan ϕ · [1 + f(˜k, ν, η)], where f is a function
+given in the Supplementary Materials. We can then dif-
+ferentiate this with respect to ˜k, take the limit ν → 0 and
+Taylor-expand the derivative; the details are provided in
+the Supplementary Materials. Eventually, we find that:
+˜kc(ν → 0) = 3ν2 + O(ν4).
+(7)
+It was expected that the caustic wavenumber goes to
+zero; we can furthermore show that the lowest reduced
+wavenumber on the slowness curve (still in zero applied
+field) i.e. the Damon-Eshbach wavenumber goes to zero
+as:
+˜kmin(ν → 0) = 2ν2 + O(ν4)
+(8)
+which proves that CSWBs exist down to vanishing re-
+duced frequencies, regardless of their values. In this limit,
+the associated caustic wavefront angle is such that:
+cos ϕc =
+1
+���
+3 + O(ν2).
+(9)
+From the latter, we also get the CSWB direction θV,c:
+tan θV (˜kc, h → 0, ν → 0) = −2
+√
+2 + O(ν2)
+(10)
+The strength of this result lies with its independence on
+η; this is not surprising as in the limit we are considering,
+the CSWB’s wavelength diverges which means it must
+be much larger than both the film thickness d and the
+dipolar-exchange length lex, however large they may be.
+The numerical values for the limits of ϕc and θV,c are ca.
+54.74° and 109.5°, respectively.
+2.
+Comparison with literature
+We present in Table I a comparison between experi-
+mental reports on caustics and predictions we make for
+the same conditions, focusing on the CSWB direction.
+Whenever there are three caustic points, the indicated
+predicted value for θV,c is the closest found across all
+three caustic points.
+We find a reasonable agreement in quite a few cases,
+generally for the larger values of η (i.e. for thinner films)
+with the notable exception of the report by Sebastian et
+al. [28]. However, in this case, the theoretical dispersion
+relation that we use may not be accurate any more due
+to the strong lateral confinement of spin waves.
+Furthermore, we find much larger discrepancies in sev-
+eral cases. For instance, if we consider the excitation of a
+caustic-like beam by Gieniusz et al. [43] at 4.62 GHz and
+under an induction of 98 mT in a 4.5 µm thick YIG film,
+our model predicts a caustic point at reduced wavenum-
+ber 13.2, with a beam direction 169°. However, the rele-
+vant reduced wavenumbers in this experiment are in the
+range of a few percents [43], and the measured beam di-
+rection is 128°. The origin of this strong disagreement is
+easily understood by observing the derivative dθV/d˜k in
+this case. As Fig. 8 reveals, there exists a local minimum
+at ˜k ≃ 0.0659 for dθV/d˜k deep in the dipolar-dominated
+regime. Moreover, the associated group velocity direc-
+tion is 128°, and past the next local maximum, similar
+values of dθV/d˜k are reached again only for ˜k ≃ 0.9.
+This illustrates the impossibility to distinguish a close-
+to-straight slowness curve from a true caustic point from
+measurements alone.
+Discrepancies may also arise due to the source’s non-
+ideal excitation efficiency, for instance if it is too direc-
+tional. This is illustrated by the excitation of caustic-
+like spin wave beams by K¨orner et al. [44]. One of the
+reported TR-MOKE measurements deals with a 60 nm
+thin permalloy film driven at an excitation frequency of
+16.08 GHz, under 160 mT applied induction; the authors
+observe twin beams with a wavefront angle of 65°, a beam
+direction 114°, and a reduced wavenumber of 0.314. Yet,
+the expected caustic spin wave beams in these condi-
+tions should feature a reduced wavenumber of 1.7063, a
+beam direction 138.62°, not to mention a wavefront angle
+of 53.27°. In this case, it appears that the excited spin
+waves simply correspond to the rather narrow portion of
+the slowness curve that could be excited by the authors’
+tapered coplanar waveguide segments [45]. Indeed, at the
+measured wavefront angle of 65°, in the authors’ experi-
+
+8
+TABLE I. Comparison between reports on CSWBs and our predictions for the beam direction θV,c.
+Ref. Excitation method
+Material (thickness in nm) Predicted θV,c Measured θV,c
+h
+ν
+η
+[28] Edge modes of a waveg-
+uide and nonlinearities
+Co2Mn0.6Fe0.4Si (30)
+113°
+123°
+3.81·10−2 0.287
+0.15
+[42] Corners of slotline termi-
+nation and scattering off
+a defect
+YIG (235)
+123°
+124°, 122°
+0.126
+0.427 7.36·10−2
+[35] Corners
+of
+slotline
+termination
+YIG (245)
+119°
+118°
+0.126
+0.427 7.06·10−2
+[43] Spin wave scattering off
+antidots
+YIG (4.5·103)
+169°
+128°
+0.557
+0.939 3.84·10−3
+[27] Collapsing
+spin-wave
+bullet
+YIG (5·103)
+137°
+137°
+1.040
+1.442 3.46·10−3
+[22] Spin wave scattering off
+a defect
+YIG (7·103)
+139°
+135°
+2.47·10−3 1.616 2.47·10−3
+mental conditions, the expected reduced wavenumber is
+about 0.28 (which falls rather far from zeroes in the an-
+tenna’s expected excitation efficiency [46]), and the beam
+direction 120.2°. We do not have an explanation for the
+remaining deviation in beam direction, though.
+3.
+Experimental results
+We now present results from experiments we have
+carried out in order to validate our theoretical ap-
+proach. Our aim here is to measure CSWBs and compare
+their properties with our predictions.
+In order to ac-
+cess CSWBs experimentally, the reciprocal-space Fourier
+components of its magnetic field must span a broad range
+of wave vectors. The ideal situation where all wave vec-
+tors are accessible corresponds to an unrealistic point
+source, which can obviously not correspond to any high-
+frequency antenna. As a result, we choose a compromise
+between ease of fabrication, and broad-band excitation
+efficiency, namely a half-ring shaped stripline antenna.
+This design allows for a spin wave excitation of the slow-
+ness curve within ϕ ∈ [0, π], i.e.
+twice the quadrant
+previously investigated. Of course, this excitation is not
+uniform because of the microwave antenna dimensions on
+the order of a micrometer.
+Our experiments were carried out using Time-Resolved
+Magneto-Optical Kerr Effect (TR-MOKE) microscopy.
+Here, the dynamic out-of plane component of the mag-
+netization δmz is spatially mapped in the xy-plane at a
+fixed phase between the microwave excitation frequency
+and the laser probing pulses. This enables direct imag-
+ing of the spin wave propagation in the magnetic film.
+The wavenumber resolution of the set-up lies within the
+dipolar-dominated regime. Indeed, our spatial resolution
+r is about 0.29 µm (see Supplementary Materials), so that
+for a film thickness t ∼100 nm, the largest accessible re-
+duced wavenumbers are 2π/(2r) · t ∼ 1.
+It shall be noted that the position of the microwave an-
+tenna in the resulting Kerr images is extracted from the
+topography image which is acquired simultaneously and
+is proportional to the reflectivity of the sample. Further
+information on TR-MOKE can be found in the Supple-
+mentary Materials. These experiments were performed
+on a 200 nm thick YIG film grown on a gadolinium gal-
+lium garnet (GGG) substrate using liquid phase epitaxy.
+Considering this materials’ parameters [40], if not stated
+otherwise, η = 0.087 for all measurements. On top of
+the YIG film the 2 µm to 3 µm wide microwave antenna
+was patterned by optical lithography with subsequent Ar-
+presputtering and electron-beam-induced evaporation of
+Cr(5 nm)/Au(100 nm to 220 nm). During the measure-
+ment the external bias field −→
+Ha was always kept fixed
+such that it aligned with the legs of the antenna structure
+along the x-direction. A sketch of the measurement ge-
+ometry can be found in Fig. 9. At this stage, we point out
+one complication resulting from this design. When driv-
+ing the antenna with a microwave field, the legs them-
+selves excite spin waves in the Damon-Eshbach geometry
+[13]. These modes are not of interest for the generation
+of CSWBs, but due to the relatively long attenuation
+length in YIG [35] they may propagate to the tip of the
+antenna and interfere with the spin waves excited by the
+half-ring. In order to suppress this effect, two different
+approaches where applied. Either the length of the an-
+tenna was set to 50 µm and the YIG between the legs and
+tip was etched away, or the antenna was patterned to be
+1 mm long in the first place.
+The first Kerr image shown in Fig. 10.a) was ob-
+tained at a constant microwave frequency f =1.44 GHz
+and an external field µ0Ha =5 mT.
+This corresponds
+to h = 0.028, ν = 0.292. The width of the waveguide
+was 2 µm and the distance between the legs and the tip
+was 1 mm. In the spatial map, two spin wave beams with
+well-defined propagation directions are visible; moreover,
+the phase and group velocities are clearly non-collinear
+to each other. Here, beam II stems from the waveguide
+excitation in the quadrant ϕ ∈ [π/2, π]. The beam angles
+
+9
+0
+0.21 0.41 0.62
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+a)
+b)
+c)
+0
+0.21 0.41 0.62
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0
+0.21 0.41 0.62
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+η = 0.09
+η = 0.3
+η = 0.6
+θV,c (◦)
+90.00
+95.25
+111.1
+127.0
+142.9
+153.5
+90.00
+95.25
+104.8
+114.3
+123.8
+128.1
+90.00
+95.00
+104.5
+114.0
+123.5
+128.0
+θV,c (◦)
+θV,c (◦)
+θV,c (◦)
+h
+h
+h
+FIG. 6. Examples of field and frequency maps for the CSWB direction θV,c, at a) η = 0.09, b) η = 0.3, and c) η = 0.6.
+0
+0.21
+0.41
+0.62
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+˜kc
+a)
+b)
+c)
+0
+0.21
+0.41
+0.62
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0
+0.21
+0.41
+0.62
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0
+1.17
+2.35
+3.52
+4.66
+0.150
+0.300
+0.450
+0.555
+0
+0
+0.200
+0.400
+0.600
+0.800
+0.826
+˜kc
+˜kc
+h
+h
+h
+FIG. 7. Field and frequency maps for η = 0.12, looking at the three reduced caustic wavenumbers. Note the distinct grey
+scales for each graph. a) Lowest ˜kc in the presence of several caustic points, and single value for ˜kc otherwise. b) Intermediate
+value for ˜kc if several caustic points exist. c) Largest reduced caustic wavenumber.
+
+Caustic beam directions Avc vs. v and h
+0.9991
+153.49
+0.8992
+0.7993
+142.88
+0.6994
+0.5995
+127.00
+0.4996
+0.3997
+111.12
+0.2998
+0.1999
+95.25
+0.1000
+90.00
+0.0001
+0.0000
+0.2063
+0.4126
+0.6189
+hCaustic beam directions Oyc vs. v and h
+0.9991
+128.13
+0.8992
+123.83
+0.7993
+0.6994
+114.30
+0.5995
+0.4996
+0vc
+0.3997
+104.78
+0.2998
+0.1999
+95.25
+0.1000
+90.00
+0.0001
+0.0000
+0.2063
+0.4126
+0.6189
+hCaustic beam directions Oyc vs. v and h
+0.9991
+127.96
+0.8992
+123.50
+0.7993
+0.6994
+114.00
+0.5995
+0.4996
+0vc
+0.3997
+104.50
+0.2998
+0.1999
+95.00
+0.1000
+90.00
+0.0001
+0.0000
+0.2063
+0.4126
+0.6189
+hNormalized wavenumber at Φc vs. v and h
+1.0000
+4.662
+0.9000
+0.8000
+3.525
+0.7000
+0.6000
+0.5000
+2.350
+0.4001
+0.3001
+1.175
+0.2001
+0.1001
+0.000
+0.0001
+0.0000
+0.2063
+0.4127
+0.6190
+hNormalized wavenumber at Φc vs. v and h
+1.0000
+0.5552
+0.9000
+0.8000
+0.4500
+0.7000
+0.6000
+0.3000
+0.5000
+kc
+0.4001
+0.3001
+0.1500
+0.2001
+0.1001
+0.0000
+0.0001
+0.0000
+0.2063
+0.4127
+0.6190
+hNormalized wavenumber at Φc vs. v and h
+1.0000
+0.8256
+0.8000
+0.9000
+0.8000
+0.7000
+0.6000
+0.6000
+0.5000
+0.4000
+kc 
+0.4001
+0.3001
+0.2000
+0.2001
+0.1001
+0.0000
+0.0001
+0.0000
+0.2063
+0.4127
+0.6190
+h10
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+˜k
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+dθV/d˜k
+FIG. 8. Calculated derivative of the group velocity direction
+with respect to the reduced wavenumber in the 4.62 GHz spin
+wave excitation described by Gieniusz et al. [43].
+YIG film
+2 µm
+−
+→
+k
+x
+y
+z
+−→
+Ha
+FIG. 9. Schematic of the measurement geometry. The half-
+ring shaped antenna excites spin wave propagation within a
+broad angular spectrum.
+of beams I and II with respect to the positive x direction
+are found to be 119.00◦ (beam I) and 64.28◦ (beam II)
+which results in effective beam directions of θI = 119.00◦
+and θII = 180◦ − 64.28◦ = 115.72◦, respectively.
+The
+discrepancy between θI and θII simply originates from
+a small misalignment of the external field with respect
+to the waveguide legs. Since −→
+Ha is not fully parallel to
+the x-axis, the slowness curve is rotated by a small an-
+gle αH = (θI − θII)/2 ≈ 1.64◦ in our frame of reference.
+Keeping this in mind, we extract an average beam direc-
+tion θV,e = 117.36◦, a wavefront angle ϕe = 50.66◦ and
+a reduced wavenumber ˜ke = 0.211. These experimental
+findings are in good agreement with our theoretical ap-
+proach; indeed, values of θV,c = 115.05◦, ϕc = 51.29◦ and
+˜kc = 0.223 are predicted for a CSWB in our experimental
+conditions.
+We can obtain further insight in reciprocal space with
+30
+20
+10
+10
+20
+30
+x (µm)
+y (µm)
+0
+0
+˜kx
+0
+0.2
+-0.2
+0
+˜ky
+|FT(δmz)|2
+(arb. u.)
+δmz (arb. u.)
+0
+0.4
+0.8
+-0.4
+-0.8
+a)
+b)
+FIG. 10. Measurement data obtained for η = 0.087, h = 0.028
+and ν = 0.292.
+a) Kerr image acquired from TR-MOKE.
+Two spin wave beams highlighted in yellow and red propa-
+gate from the tip of the antenna.
+b) Squared modulus of
+the Fourier transform (FT) of the Kerr image and expected
+slowness curve (blue).
+The yellow and red points and ar-
+rows indicate the expected caustic points and their respective
+group velocity directions. Caustic points I and II correspond
+to beams I and II in the Kerr image.
+the Fourier-transformed (FT) data shown in Fig. 10.b).
+Generally speaking, the FT data allows for a direct ob-
+servation of the slowness curve in ˜k-space. In order to re-
+duce spectral leakage, a Hanning windowing was applied;
+the latter provides a good trade-off between frequency
+and amplitude accuracy. We see that the chosen antenna
+structure indeed excites a wide range of wave vector di-
+rections. The gaps in the spectrum arise from the finite
+antenna dimensions, as previously mentioned. We find a
+good agreement between the slowness curve (blue curve)
+derived from our model (and corrected by the external
+field angle αH). More importantly, this graph confirms
+that the antenna structure grants access to the expected
+caustic points (yellow and red points) since the Fourier
+magnitude is still sufficiently large in that region. To con-
+clude, caustic points I and II can be assigned to beams I
+and II from the Kerr image.
+We may now turn to the additional caustic points pre-
+dicted by our model. The chosen triplet (η, h, ν) is an ele-
+ment of the D3 set, and we would expect two further caus-
+tic points θV,c,2 = 113.74◦, ϕc,2 = 33.00◦, ˜kc,2 = 0.662
+and θV,c,3 = 114.02◦, ϕc,3 = 28.78◦, ˜kc,3 = 1.227. These
+reduced wave vectors could actually be resolved by our
+experimental set-up where ˜kres ≈ 2.2.
+The reciprocal
+space image in Fig. 10.b), however, displays a very low
+amplitude for ˜k ≳ 0.55 meaning that the microwave an-
+tenna cannot excite the other caustic points very effi-
+ciently.
+Hence, only the low frequency pocket can be
+accessed.
+Further Kerr images were taken for the same ν, but
+
+I
+II11
+for different h values.
+The h values were chosen such
+that they lie beneath the expected FMR field hFMR ≈
+0.078778. A selection of the resulting Kerr images is il-
+lustrated in the upper part of Fig. 11. In each of them,
+twin spin wave beams are apparent. An overview of all
+the beam properties for the corresponding h values is
+plotted in the lower part of Fig. 11. Here, the relevant pa-
+rameters from every individual beam are extracted with
+image processing and bootstrapping least squares regres-
+sion procedures. An example on how one set of experi-
+mental data points is obtained can be found in the Sup-
+plementary Materials. The reasonable, sometimes even
+very good agreement between predicted and experimen-
+tal values of θV,c and ˜kc strongly suggests true CSWBs.
+The deviation of the beam directions is mostly within the
+range of the external field angle. The larger discrepancy
+between predicted and measured wavefront angles ϕc is
+attributed to the narrowness of the CSWB.
+b1)
+b2)
+b3)
+a1)
+a2)
+a3)
+h = 0.0341
+h = 0.0398
+h = 0.0511
+Theory
+Experiment
+θV,c (◦)
+φc (◦)
+˜kc
+30
+42
+54
+0.10
+0.18
+0.26
+110
+113
+116
+119
+122
+0.025
+0.030
+0.035
+0.040
+0.045
+0.050
+0.055
+0.060
+h
+30
+20
+10
+0
+10
+20
+30
+x (µm)
+y (µm)
+30
+20
+10
+0
+x (µm)
+30
+20
+10
+0
+x (µm)
+0
+0
+0
+0
+FIG. 11. Measurement data obtained for η = 0.087 and ν =
+0.292. Upper part: acquired Kerr images for reduced fields of
+a1) h = 0.0341, a2) h = 0.0398, and a3) h = 0.0511,. b1-3)
+comparison between experiment and theoretical predictions
+of caustic point properties θV,c, ˜kc, ϕc. The error bars are
+the standard deviations from a bootstrapping fit procedure.
+Beam-like features which do not coincide with a caus-
+tic point were detected as well. This time, the measure-
+ments were conducted with the 50 µm antennna struc-
+ture and partially etched film. The width of the antenna
+was 3 µm.
+The resulting Kerr map for f =1.84 GHz
+(ν = 0.372) and µ0Ha =5 mT (h = 0.028) is shown
+in the left upper half of Fig. 12.
+In this geometry, a
+Damon Eshbach-like mode propagating from the YIG
+edge could not be fully suppressed; it is visible as a
+plane wave background. Our procedure to analyze spin
+wave beams yields θV,e = 136.33◦, ϕe = 68.97◦ and
+˜ke = 0.522, whereas our model predicts a caustic point
+with θV,c = 121.39◦, ϕc = 35.84◦ and ˜kc = 1.564.
+The origin of the experimentally observed beams may
+be twofold.
+Firstly, a close-to-straight slowness curve
+similar to the case of Gieniusz et al. [43] is predicted to
+exist within relatively close distance to ˜ke. The dθV/d˜k
+plot in Fig. 12.b) displays almost a constant behaviour
+between 0.6 ≲ ˜k ≲ 1.2 (marked with green dashed lines).
+The proximity of the experimental caustic point to a
+straight-to-close slowness curve is also illustrated in the
+FT data in the lower part of Fig. 12. Here, the dashed
+green semicircle represents the lower bound of ˜k = 0.6
+and the extracted beam points are highlighted in yellow.
+For this portion of the slowness curve, group velocity
+directions of up to 121.39◦ are predicted. This beam di-
+rection, however, is still in stark contrast with the mea-
+surement result. Moreover, the calculated slowness curve
+(blue curve) deviates significantly from the FT data. The
+difference between reciprocal space image and our model
+may show the limit of the model applicability, since a film
+with η = 0.087 may not be considered a thin film any-
+more. This results in predictions which are less reliable
+at higher ν values. A second possible origin of the beams
+is the excitation efficiency of the microwave antenna as
+there are many gaps in the FFT spectrum. The beams
+appear to be located close to some of them, and hence,
+may correspond to the excitation of only a small portion
+of the slowness curve within this region.
+|FT(δmz)|2
+(arb. u.)
+c)
+Theory
+Fit data
+˜kx
+0
+1.2
+0.8
+0
+˜ky
+0.8
+-0.8
+-1.6
+-1.6
+0.4
+30
+20
+10
+10
+20
+30
+y (µm)
+0
+0
+δmz (arb. u.)
+0
+a)
+b)
+dθV/d˜k
+0
+0
+0.2
+0.4
+0.6
+0.8
+1.6
+x (µm)
+˜k
+FIG. 12.
+a) Kerr image with twin beams obtained with
+η = 0.087 h = 0.028 and ν = 0.372. b) Calculated derivative
+of the group velocity direction with respect to the reduced
+wavenumber. Dashed green lines highlight close-to-straight
+slowness curve.
+c) FT of Kerr image.
+The experimentally
+observed beam parameters are depicted in yellow, the calcu-
+lated slowness curve in blue and the calculated caustic points
+in red. Dashed green semicircle illustrates lower limit of close-
+to-straight portion of slowness curve.
+
+12
+D.
+Caustic point of higher order
+Based on the conclusions from section III B, we know
+that the intersection of ∂D3,l and ∂D3,u there exists a sin-
+gle caustic point on the slowness curve; in the schematic
+discussion from the above based on the approximant
+P(˜k; a, b), it corresponds to a = 0 and b = 0, which
+means that dθV/d˜k ∼ (˜k − ˜kc)3 around this point. To
+put it differently: at this intersection, corresponding to
+the cusp seen in Fig. 7, the caustic point is not a simple
+extremum for θV on the slowness curve but an undulation
+point, in the vicinity of which θV − θV,c ∼ (˜k − ˜kc)4.
+The existence of such an undulation point is of par-
+ticular interest since the higher order in the dependence
+of θV on ˜k implies a flatter extremum in group veloc-
+ity direction and therefore the possibility of larger por-
+tions of the slowness curve contributing to the CSWB.
+Moreover, as was discussed in Sec. II.II B, this does not
+necessarily mean an increase in spectral breadth of the
+CSWB since the latter depends on the apparent wave-
+length. In order to evidence this, we show in Fig. 13
+how the group velocity direction as well as the natural
+and apparent wavelengths vary around a caustic point
+very close to one of higher order, here the one such that
+its corresponding critical field hc is zero. The considered
+slowness curve corresponds to h = h1 = 1.15 · 10−21,
+ν = ν1 = 0.315279504, η = η1 = 0.10253664614147.
+Let us briefly outline how the coordinates νc,0 =
+νc(hc = 0) and ηc,0 = ηc(hc = 0) were found with a
+good accuracy. More details can be found in the Sup-
+plementary Materials. The starting point was a rough,
+hand-performed search for a value of η bringing the cusp
+of D3 to lie on the ordinate axis in a field and fre-
+quency map. This yielded a starting point of η(0)
+c,0 = 0.10
+and ν(0)
+c,0 = 0.31.
+In these conditions, a caustic point
+was found for ˜k(0)
+c,0 ≃ 0.73.
+We then began an itera-
+tive procedure using appropriate Taylor expansions of
+the dispersion relation and of an exact expression for
+θV(h = 0, η, ν, ˜k, ϕ). Updating these at each step with
+the new solutions found by looking for the undulation
+point allows to converge to numerical values which we
+assimilate to the intersection of ∂D3,l and ∂D3,u.
+Over three iterations, the relative changes in the esti-
+mates steadily decrease in absolute value, from at most
+5% in the first step to at most 5 · 10−6 in the last one,
+which provides the following guesses : ˜k(g)
+c,0 = 0.731717,
+η(g)
+c,0 = 0.1025366, ν(g)
+c,0 = 0.3152796. The latter can be
+compared with e.g. the hand-refined values used for Fig.
+13: ν = ν1 = 0.315279504, η = η1 = 0.10253664614147,
+corresponding to ˜kc = 0.725904. It must be noted that
+the somewhat larger relative difference in terms of ˜kc,0 is
+due to the very steep dependence of ˜kc(ν, η, h → 0) on η.
+We do emphasize that the exact location (νc,0,ηc,0) is nec-
+essarily different from (ν1, η1) but close enough to high-
+light the qualitatively different behaviour of several char-
+acteristics of the slowness curve. Finally, we note that for
+˜k
+˜k
+˜k
+φ
+˜kc
+φc
+0
+1
+2
+3
+4
+0 ◦
+15 ◦
+30 ◦
+45 ◦
+60 ◦
+75 ◦
+90 ◦
+b)
+0.04
+0.02
+0
+-0.02
+-0.04
+−0.04
+0.5
+2.0
+0
+0.5
+1.5
+1.0
+2.0
+0.04
+θV/θV,c−1
+λ0/λ0,c−1
+λ/λc−1
+a)
+FIG. 13.
+a) Plots of the relative deviations from the fol-
+lowing caustic point properties as a function of ˜k: its group
+velocity direction θV, its natural wavelength λ0 = 2π/˜k
+and its apparent wavelength λ = 2π/[˜k cos (θV − ϕ)]. Main
+graph:
+h = h1 = 1.15 · 10−21, ν = ν1 = 0.315279504,
+η = η1 = 0.10253664614147, which are extremely close to
+the values of νc and η for which hc = 0. Inset: same h and
+η = η1, ν = 0.95 · ν1 = 0.2995155288. b) Slowness curve for
+ν1, η1 and h1; ˜kc ≃ 0.7259. The slowness curve at ν2 is not
+shown for clarity, as it is very similar to the other one.
+
+90°
+75°
+60°
+45°
+30°
+15°
+0
+2
+3
+1
+4
+kd13
+the parameters from Fig. 13, θV,c =118.36°, ϕc ≃42.75°,
+λ0,c = 84.41lex = 8.655d, and λc ≃ 339.7lex = 34.83d.
+We now examine the properties of the caustic point of
+higher order in more detail. From Fig. 13, the depen-
+dence of θV,c and the apparent wavelength λ on ˜k (in
+blue and green, respectively) clearly appears to be quar-
+tic rather than quadratic around the caustic point, which
+is where the deviations in natural wavelength (in red) go
+through 0. Its much steeper behaviour is easily under-
+stood by looking at the corresponding slowness curve in
+Fig. 13.b): around ˜kc it is not only almost straight but
+the angle γ between
+−→˜k and d
+−→˜k /ds is low, γ ≃14.38°.
+Hence, since d(˜k2)/ds is large, λ0 ∝ 1/˜k varies fast.
+By contrast, one can show that in the Taylor expansion
+of λ in (s−sc)/˜kc around λc, the first coefficient is always
+exactly zero at a caustic point. We stress again that this
+is caused by an unchanging projection of −→k on −→
+eg across
+the caustic point. If it is of higher order, it may be shown
+(see Supplementary Materials) that in this term, the con-
+tributions due to the second- and third-order variations
+of ϕ and to those of ˜k cancel out. To put it differently, the
+projection k · cos (θV − ϕ) is now constant up to fourth
+order in (s − sc)/˜kc. On the other hand, if the consid-
+ered caustic point is a regular extremum for θV, the term
+∝ (s − sc)2 will be non-zero.
+To summarize the above paragraph: for geometrical
+reasons, the caustic point of higher order suppresses the
+quadratic and cubic variations of the apparent wave-
+length around λc. Hence, λ has then a markedly quartic
+behaviour at a caustic point of higher order. Further-
+more, we point out that even a small offset in frequency
+makes it display a clearly quadratic behaviour. This is
+shown in the inset of Fig. 13, showing the same relative
+variations for the slowness curve at h = h1 = 1.15·10−21,
+η = η1 = 0.10253664614147, but ν = 0.95 · ν1 =
+0.2995155288.
+We have thus shown that in a sufficiently close vicin-
+ity of a higher-order caustic point, a broadband excita-
+tion in terms of wavenumber can result in a narrowband
+CSWB with a very well-defined direction. As a result,
+this phenomenon is expected to be extremely favourable
+in experiments, since any realistic antenna cannot have
+an arbitrarily narrow excitation efficiency as a function
+of wavenumber. Provided that its design yields AC mag-
+netic fields with Fourier components in the (broad) range
+of interest and with phases in a given interval of width
+< π, all the corresponding spin waves will coherently add
+in a beam with very small spectral breadth.
+In other
+words: in such a situation, counter-intuitively, exciting
+additional wave vectors with different wavenumbers does
+not average out the carrier wave’s amplitude but rather
+increase it. This naturally prompts the question of how
+much stronger the emission from a caustic point of higher
+order would be with respect to that of a regular caustic
+point, and more generally, of the spin wave amplitude
+enhancement due to the caustics. This, however, goes
+beyond the scope of the present manuscript.
+To conclude this section, we point out that the reduced
+field hc(η) corresponding to the caustic point of higher
+order decreases as a function of reduced dipolar-exchange
+length. Thus, this feature is expected to exist only for
+η < ηc,0 ≃ 0.1025366.
+E.
+Merged caustic spin wave beams
+We now move on to the topic of the threshold fre-
+quency νm(h, η) corresponding to the upper boundary
+of D, i.e. above which there are no caustic points any
+more.
+As was shown in Fig.
+6, the CSWB direction
+θV,c goes to π/2 as ν → νm(h, η).
+This is illustrated
+in Fig. 14, where we show a slowness curve for η1, h1,
+and ν2 = 0.71836419052. We stress again that νm(h, η)
+is strictly speaking an infinitely narrow boundary and
+therefore ν2 ̸= νm(h1, η1), but in these conditions, we
+find a unique caustic point on the slowness curve, with
+π/2 − ϕc ≃0.32 µrad, and θV,c is equal to π/2 (within
+numerical precision). Moreover, at ν′
+2 = ν2 + δν, where
+δν = 1 · 10−11, we do not find any caustic point on the
+slowness curve.
+As a result, we take the slowness curve at (ν2, h1, η1)
+to be assimilable to the one at (νm(h1, η1), h1, η1). Its
+very straight aspect around ϕ = π/2 is somewhat rem-
+iniscent of the one seen in the discussion of the caus-
+tic point of higher order. To illustrate this in more de-
+tail, Fig. 14.b) displays the relative deviations in group
+velocity direction θV, natural wavelength and apparent
+wavelength around the caustic point at ϕc.
+We point
+out that in the present case, the deviations are plotted
+against the curvilinear abscissa s normalized to the slow-
+ness curve’s length sM instead of ˜k as in Fig. 13. This
+choice is motivated by (i) the fact that in this case, to
+lowest order ˜k − ˜kc = O(s2) instead of O(s − sc) as be-
+fore, and (ii) the much smaller relative difference between
+the smallest and largest normalized wavenumbers ˜km re-
+spectively ˜kM: ˜km ≃ 5.17 and ˜kM ≃ 7.91, compared
+to ˜km ≃ 0.240 and ˜kM ≃ 5.66 before. (i) implies that
+for (ν2, h1, η1), ˜k cannot serve as a meaningful abscissa
+along the curve since d˜k/ds = 0, which was not the case
+for (ν1, h1, η1), while (ii) shows that the slowness curve
+for (ν2, h1, η1) is much closer to a fourth of a circle than
+that for (ν1, h1, η1); as a matter of fact, for (ν2, h1, η1),
+we find that 1 − [π/2 · (˜km + ˜kM)/2]/sM = 3.7%. There-
+fore, s/sM provides a better feeling for how much of the
+slowness curve contributes to the CSWB.
+From the graph, it seems that the apparent wavelength
+has once more a quartic behaviour around the caustic
+point. We show in the Supplementary Materials that this
+is indeed the case: in the conditions where ν = νm(h, η),
+to the lowest non-zero order, θV(s → 0) − π/2 varies
+with an s3 dependence around s = 0, and the lowest-
+order variations in ˜k and ϕ (around ˜km and π/2) cancel
+each other out in the projection ˜k · cos (θV − ϕ).
+As a result, a caustic point at νm(h, η) is such that
+
+14
+0
+-0.02
+-0.04
+-0.06
+-0.08
+-0.10
+-0.12
+-0.14
+0
+0.1
+0.2
+0.3
+0.4
+s/sM
+φ
+0 ◦
+15 ◦
+30 ◦
+45 ◦
+60 ◦
+75 ◦
+90 ◦
+˜k
+0
+2
+4
+8
+6
+a)
+b)
+θV/θV,c−1
+λ0/λ0,c−1
+λ/λc−1
+FIG. 14. a) Slowness curve at (ν2 = 0.71836419052, h1, η1).
+b) Relative deviations in group velocity direction θV (blue),
+natural wavelength λ0 (red) and apparent wavelength λ
+(green), as a function of curvilinear abscissa along the slow-
+ness curve normalized by its total length sM.
+an excitation from a suitable, moderately directional an-
+tenna would be effectively narrowband, and weakly di-
+vergent around the group velocity direction θV,c = π/2.
+This orientation is itself also advantageous in practice:
+as long as the used antenna can excite sufficiently high
+wavenumbers, the CSWB direction becomes in this case
+simply perpendicular to the applied field. Moreover, ow-
+ing to the symmetries of the dispersion relation, the
+CSWB benefits from the part of the slowness curve at
+ϕ ≳ π/2, which also feature θV ≃ π/2. That is why large
+spin wave amplitudes can be expected, as effectively two
+CSWB have merged at this particular frequency. We note
+that this merging phenomenon has already been observed
+in simulations by Kim et al. [29] in perpendicularly mag-
+netized ultrathin films and by Gallardo et al. [41] in syn-
+thetic antiferromagnets. For the sake of completeness, let
+us comment on what happens from an analytical point of
+view when the two caustic points just below and above
+ϕ = π/2 coincide. It must be kept in mind that they
+respectively correspond to a maximum and a minimum
+for θV, temporarily considered for ϕ ∈ [0, π]. Thus, when
+they do coincide at ϕc = π/2, strictly speaking there is
+no caustic point any more.
+To put it differently: be-
+low νm(h, η), over ϕ ∈ [0, π], θV increases up to the first
+caustic point where it reaches θV,c > π/2, decreases until
+the second one (for ϕ > π/2) where it reaches π − θV,c,
+then increases again to reach π when ϕ = π. Exactly
+at νm(h, η), it is monotonously increasing with an inflex-
+ion point, and above νm(h, η), it is strictly monotonously
+increasing.
+In order to go beyond the particular case presented
+here, we now investigate the evolution of νm(h → 0, η) =
+νm,0(η) as a function of reduced dipolar-exchange length
+η.
+Similarly to the caustic point of higher order, the
+evolutions as a function of reduced field quickly become
+cumbersome. This is why we focus on the νm,0(η), which
+is both the lowest frequency at which CSWBs merge and
+a threshold frequency that is easier to reach in experi-
+ments owing to the vanishing applied field, provided that
+the studied film is soft enough.
+We do keep in mind that below a certain limit in terms
+of reduced dipolar-exchange length, the model we use
+loses its validity.
+However, it has been shown that at
+sufficiently high frequency [38], the analytical dispersion
+relation derived by Kalinikos and Slavin describes spin
+waves once more with a good accuracy.
+Fig.
+15 displays the numerically determined depen-
+dence of νm0 on η, as well as that of λm/lex the wave-
+length of the corresponding CSWB, normalized by the
+dipolar-exchange length. The procedure to find first a
+coarse estimate of this curve (before refining it with ac-
+tual field and frequency maps) is described in the Supple-
+mentary Materials. We point out that in the case of the
+merged CSWBs, the apparent and natural wavelengths
+are equal since θV,c = ϕc = π/2. The minimum value
+of η in these graphs corresponds to the smallest one we
+used such that the slowness curve (in vanishing fields)
+has only one connected component. While we may not
+expect our findings to hold at the lowest η’s, we do expect
+their accuracy to improve as η increases; it should be suf-
+ficient at least for η > 1 since in this case the considered
+ferromagnetic film can truly be considered thin.
+If we think about searching for the merged CSWBs,
+Fig. 15.b) indicates that for realistic values of η = lex/d,
+the CSWBs’ apparent wavelengths λ are only about one
+order of magnitude larger than the material’s dipolar-
+exchange length, typically λ ≲ 25lex. This is in stark
+contrast with the case of the caustic point of higher
+order in vanishing field, where the natural wavelength
+was λ0,HO ≃ 84lex, and the apparent wavelength λHO ≃
+334lex. As a result, it seems that while caustic points
+of higher order may readily be excited by antennas cre-
+ated with even conventional electron beam lithography,
+in the case of the merged CSWB achieving a sufficient
+excitation efficiency at the proper wavevectors should
+
+...15
+a)
+b)
+FIG. 15. a) νm,0(η) as a function of reduced dipolar-exchange
+length η.
+b) The corresponding reduced wavelength ˜λm =
+(2π/˜k)/η = λ/lex = λ0/lex. Here, natural and apparent wave-
+lengths coincide as phase and group velocities are collinear.
+prove quite challenging.
+For instance, even the low-
+magnetization, low-damping and rather soft ferrimagnet
+YIG features lex =17.3 nm [40], meaning that high-end
+antennas with a characteristic periodicity down to about
+200 nm would be required in this easiest of cases.
+IV.
+CONCLUSIONS
+We have focused on some properties displayed by spin
+wave caustics in soft, thin ferromagnetic films. On the
+theoretical side, our approach relied on the analytical
+dispersion relation established by Kalinikos and Slavin.
+We could show that many reports on CSWBs in the lit-
+erature can be interpreted within this frame, although
+the absence of characteristic signs of a true CSWB may
+still cause some ambiguity. Following up on most stud-
+ies, we have performed time-resolved magneto-optical
+Kerr-effect-based microscopy on samples designed for the
+study of CSWBs. Despite the large thickness of the fer-
+romagnetic material, our measurements are in very good
+agreement with our predictions, thus validating the ap-
+proach. Furthermore, we have specifically highlighted the
+large misalignment between phase and group velocities in
+this case, and succeeded in observing narrow CSWBs.
+Just at the boundary of the dipolar-dominated regime
+accessible in our experiments, we have predicted the ex-
+istence of a special caustic point. We refer to it as caustic
+point of higher order because it corresponds to an undu-
+lation point for the group velocity direction rather than
+a quadratic extremum.
+This configuration was shown
+to be of particular interest because the apparent wave-
+length also featured a quartic behaviour, which implies
+a low spectral breadth for the CSWB even in the case
+of a broadband excitation. Although we focused on the
+special value ηc of reduced dipolar-exchange length such
+that the caustic point of higher order occurs at vanish-
+ing applied fields, we stress that this phenomenon would
+appear at non-zero fields for η < ηc, as long as the dis-
+persion relation we use is valid.
+Finally, we have investigated the merging of CSWBs.
+Once again, we have studied in detail the case of van-
+ishing applied fields, yet the merging may occur for any
+field value, provided that the excitation frequency is large
+enough. In terms of model validity, it must be recalled
+that while vanishing values of η are problematic for the
+chosen dispersion relation, the merging always occurs
+at frequencies close to the exchange-dominated regime.
+The discrepancies between the actual spin wave disper-
+sion and the model by Kalinikos and Slavin decrease in
+this frequency range [39]. As a result, our claim is that
+the merging frequencies νm0 obtained for low η may be
+slightly inaccurate yet the phenomenology should remain
+the same as for larger η, where we expect our predictions
+to be more reliable. As the CSWBs merge, a very sig-
+nificant portion of the slowness curve contributes to spin
+wave emission around θV = ϕ = π/2. Therefore, this
+configuration appears promising in terms of channelling
+strong spin wave beams with short wavelengths, as low
+as ∼ 15lex.
+One of the most important questions remaining un-
+addressed so far concerns the quantification and predic-
+tion of the enhancement of amplitude associated with
+CSWBs. More precisely, the crucial distinction between
+natural and apparent wavelength as well as the inad-
+equacy of the usual Huygens-Fresnel approach (due to
+the strong non-collinearity between phase and group ve-
+locities) in the construction of CSWBs calls for alterna-
+tive evaluations of their amplitudes. We intend to clar-
+ify these points and to go beyond the usually described
+amplitude divergence so as to reconcile the theoretically
+vanishing curvature (on the slowness curve) and experi-
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+

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+Impact of electron correlations on the k-resolved electronic structure of PdCrO2 revealed by
+Compton scattering
+A. D. N. James,1 D. Billington,2 and S. B. Dugdale1
+1H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, United Kingdom
+2School of Physics and Astronomy, Cardiff University, Queen’s Building, The Parade, Cardiff, CF24 3AA, United Kingdom
+(Dated: January 6, 2023)
+Delafossite PdCrO2 is an intriguing material which displays nearly-free electron and Mott insulating be-
+haviour in different layers. Both angle-resolved photoemission spectroscopy (ARPES) and Compton scattering
+measurements have established a hexagonal Fermi surface in the material’s paramagnetic phase. However, the
+Compton experiment detected an additional structure in the projected occupancy which was originally inter-
+preted as an additional Fermi surface feature not seen by ARPES. Here, we revisit this interpretation of the
+Compton data. State-of-the-art density functional theory (DFT) with dynamical mean field theory (DMFT), the
+so-called DFT+DMFT method, predicts the Mott insulating state along with a single hexagonal Fermi surface in
+excellent agreement with ARPES and Compton. However, DFT+DMFT fails to predict the intensity of the ad-
+ditional spectral weight feature observed in the Compton data. We infer that this discrepancy may arise from the
+DFT+DMFT not being able to correctly predict certain features in the shape and dispersion of the unoccupied
+quasiparticle band near the Fermi level. Therefore, a theoretical description beyond our DFT+DMFT model
+is needed to incorporate vital electron interactions, such as inter-layer electron coupling interactions which for
+PdCrO2 gives rise to the Kondo-like so-called intertwined excitation.
+I.
+INTRODUCTION
+Interest has grown over the last few decades in layered
+triangular-lattice delafossite materials with chemical formula
+ABO2 (A = Pt, Pd, Ag or Cu, and B = Cr, Co, Fe, Rh,
+Al, Ga, Sc, In or Tl). This interest in metallic delafossites
+was sparked by reports from Tanaka et al. [1, 2] of strongly
+anisotropic conductivity in their PdCoO2 and PtCoO2 single
+crystals. Previous measurements of PtCoO2 displayed an ex-
+tremely low in-plane room temperature resistivity of 3 µΩcm,
+a value comparable to elemental Cu [3, 4]. This led to the
+emergence of a new field of research into these materials [4].
+The PdCrO2 compound also has the anticipated anisotropic
+conductivity [5], but displays an antiferromagnetic phase be-
+low its N´eel temperature, TN = 37.5 K, above which the lo-
+cal Cr3+ (S = 3/2) electron spins in the CrO2 layers are
+frustrated. Within the antiferromagnetic phase this frustration
+is relieved, resulting in the local spins ordering with a rota-
+tion of 120◦ between adjacent sites [5–8]. The observation of
+this ordered state offers an opportunity to study the coupling
+between nearly-free electrons and (frustrated) local electron
+spins in a frustrated antiferromagnet. This interest in PdCrO2
+has led to further experimental characterisation of this mate-
+rial, leading to the discovery of an unconventional anomalous
+Hall effect [7, 9], and a reconstructed Fermi surface within the
+(smaller) antiferromagnetic Brillouin zone, measured by both
+angle resolved photoemission spectroscopy (ARPES) [10–12]
+and quantum oscillations [13, 14]. For PdCrO2, it has been
+implicitly assumed that electron correlations are the driving
+force for the antiferromagnetic state, and hence why the be-
+haviour of the CrO2 layer has been described with the concept
+of local moments [4].
+Within the paramagnetic phase, both ARPES [10, 11] and
+Compton scattering [8] measurements were performed to de-
+termine the Fermi surface geometry. ARPES measures the
+energies of the emitted photoelectrons from the sample sur-
+face together with their angle of emission such that the quasi-
+particle energy and its dispersion with (crystal) momentum
+up to and including the Fermi energy can be extracted. The
+ARPES spectra show both the ground and excited states of
+the electronic structure and measurements are sensitive to the
+surface and matrix element effects [15]. Compton scattering
+experiments probe the bulk ground-state electronic structure
+through its electron momentum distribution [16] by measur-
+ing so-called Compton profiles which are the doubly projected
+electron momentum densities (EMDs) [17]. The EMD is the
+electron density distribution in real momentum, p, which can
+be directly related to the electron occupancy by folding the
+EMD back into the first Brillouin zone (the Lock-Crisp-West
+(LCW) theorem [18]) to recover the full translational sym-
+metry of the reciprocal lattice. This folded EMD is now a
+function of the crystal momentum, k. Electron occupancy in
+k-space is influenced by temperature, site disorder, and many-
+body electron correlations.
+The step changes in the occu-
+pancy can be used to determine the Fermi wave-vectors (and
+hence Fermi surface) even in materials which are either inac-
+cessible by or challenging for other techniques. Such mate-
+rials include highly chemically-disordered alloys [19] which
+have short electronic mean-free paths. Evidently, ARPES and
+Compton scattering probe different aspects of the electronic
+structure. The Fermi surface may be extracted from either the
+k-resolved photoelectron dispersion around the Fermi energy
+measured by ARPES, or the changes in the occupation derived
+from the Compton data.
+Both ARPES and Compton scattering confirmed the pres-
+ence of the hexagonal Fermi surface, but the Compton ex-
+periments clearly showed an additional contribution to the
+projected electron occupancy around the corners of the (pro-
+jected) Brillouin zone. In an effort to understand their re-
+sult, Billington et al. [8] performed density functional theory
+(DFT) calculations from which they concluded that at least
+two Fermi surface sheets were required to describe all the fea-
+tures in the k-space occupancy. This led to speculation by
+arXiv:2301.02143v1  [cond-mat.str-el]  5 Jan 2023
+
+2
+Billington et al. that what appeared to be an additional Fermi
+surface sheet observed in the bulk-sensitive Compton exper-
+iments but not in the ARPES might be due to some combi-
+nation of the surface not being representative of the bulk or
+unfavourable matrix elements. Although Ong et al. [20] had
+shown that the DFT magnetic structure of PdCrO2 was three
+dimensional, at the time of the study by Billington et al. there
+were no published DFT calculations of non-magnetic PdCrO2
+in opposition to their two Fermi surface model.
+This interpretation of the Compton data was subsequently
+critically examined by Mackenzie [4] who argued that it did
+not take into account the existence of the Mott insulating state
+in the CrO2 layers (the existence of which is supported by
+several experiments). However, these arguments do not ex-
+plain the extra features in the occupation number measured by
+the Compton scattering. Recent calculations combining DFT
+with dynamical mean field theory (DFT+DMFT) [12, 21, 22]
+showed that the Mott insulating state in the CrO2 layers is a
+natural consequence of the inclusion of the local dynamical
+electron correlations. Also, DFT+DMFT naturally includes
+paramagnetic electron correlations within the DMFT part [23]
+which is vital for this frustrated antiferromagnetic material.
+Therefore, the interpretation of the results from the Compton
+experiment warrants further investigation in order to reconcile
+it with the picture of local moments within the Mott insulating
+CrO2 layers and to help resolve the inconsistent conclusions
+about the Fermiology from the different measurements.
+In light of the recent PdCrO2 DFT+DMFT calculations, it
+is necessary to first reproduce them in order to then deter-
+mine the DFT+DMFT EMD using the recent technique im-
+plemented by James et al. [24]. From such calculations, a
+comparison with the Compton scattering experiment, primar-
+ily the projected occupation, could be made. With respect to
+a non-interacting prediction, the inclusion of many-body cor-
+relations (such as that predicted by Fermi liquid theory [25])
+generally leads to a redistribution and apparent smearing in
+k-space of the occupation around the Fermi wave-vector. The
+presence of the Mott insulating CrO2 layers in the previous
+DFT+DMFT predictions lead to significant changes to the
+shape and dispersion of the quasiparticle bands which also
+means that there would be significant changes to the occupa-
+tion, which the Compton scattering will be sensitive to. Hence
+it is important to use DFT+DMFT to determine whether the
+predicted electronic structure with these Mott insulating CrO2
+layers are compatible with the electron occupancy as mea-
+sured by Compton scattering.
+In this study, we revisit the interpretation of the Compton
+scattering experimental results and compare them with the
+corresponding quantities calculated from the non-magnetic
+DFT and paramagnetic DFT+DMFT methods.
+Here, we
+see that the size and shape of the predicted DFT+DMFT
+hexagonal Fermi surface is in excellent agreement with the
+ARPES [10–12], quantum oscillations [13, 14], and Comp-
+ton measurements [8].
+However, there are still discrep-
+ancies between the experimental Compton data and the
+DFT+DMFT calculations around the corners of the Brillouin
+zone.
+These discrepancies can be reduced (but not elimi-
+nated) in the DFT+DMFT calculation by artificially (and un-
+physical) broadening the unoccupied quasiparticle band just
+above the Fermi level around the corners of the Brillouin
+zone. This suggests that changes to both the shape and disper-
+sion of that quasiparticle band are required, most likely driven
+by certain electron correlation effects which theories beyond
+our DFT+DMFT would possibly capture, such as inter-layer
+electron coupling interactions which gives rise to the previ-
+ously observed (Kondo-like) so-called intertwined excitation
+in Ref. [12] which is a convolution of the charge spectrum of
+the metallic layer and the spin susceptibility of the Mott insu-
+lating layer.
+II.
+METHODS
+We have used the full potential augmented plane-wave
+plus local orbitals (APW+lo) ELK code [26] in combina-
+tion with the toolbox for research on interacting quantum
+systems (TRIQS) library [27]. This so-called ELK+TRIQS
+DFT+DMFT framework is described in Ref. [28]. Further
+discussion of interfacing the APW+lo DFT basis with the
+DMFT Anderson’s impurity model is found in Ref. [29]. The
+PdCrO2 delafossite structure is shown in Fig. 1 (a) and the
+lattice parameters of the conventional (hexagonal) unit cell
+are a = 2.929 ˚A, c = 18.093 ˚A [7] with the Pd–O distance,
+dPd−O = 0.11c.
+The DFT calculation used the Perdew-
+Burke-Ernzerhof (PBE) generalized gradient approximation
+(GGA) for the exchange-correlation functional [30] and was
+converged on a 32 × 32 × 16 Monkhorst-Pack k-mesh of
+2601 irreducible k-points in the irreducible Brillouin zone.
+We used the all-electron full-potential APW+lo DFT method
+instead of the pseudo-potential plane-wave approach used in
+Refs. [21, 22].
+The DFT outputs were interfaced to the
+TRIQS/DFTTools application of the TRIQS library [31] by
+constructing Wannier projectors, as described in Ref. [28],
+for all the Cr 3d-states within a correlated energy window of
+[−8.5, 3] eV relative to the Fermi level.
+The paramagnetic DMFT part of the DFT+DMFT calcu-
+lation was implemented using the continuous-time quantum
+Monte Carlo (CT-QMC) solver within the TRIQS/CTHYB
+application [32] with the Slater interaction Hamiltonian pa-
+rameterised by the Hubbard interaction U = 3.0 eV and Hund
+exchange interaction J = 0.7 eV, unless otherwise specified.
+These U and J values are similar to those used in previous
+calculations of PdCrO2 [12, 21, 22], and other CrO2 com-
+pounds [22, 33]. We approximated the double counting in the
+fully localised limit in line with the previous DFT+DMFT cal-
+culations [12, 21, 22]. Our DFT+DMFT approach slightly dif-
+fers from previous DFT+DMFT calculations where different
+correlated energy windows were used and either the Hubbard-
+Kanamori interaction Hamiltonian [21, 22] or the Hubbard I
+approximation for the impurity solver [12] were chosen. We
+note that our DFT+DMFT calculations are paramagnetic with
+no overall ordered moment. The Cr 3d orbitals were diag-
+onalised from the complex spherical harmonic basis into the
+diagonal trigonal basis (obtained by diagonalising the orbital
+density matrix) resulting in the three sets of non-degenerate
+orbitals, namely the two doubly degenerate e′
+g and eg or-
+
+3
+Pd
+CrO2
+(a)
+(c)
+(b)
+(d)
+FIG. 1. (a) The PdCrO2 delafossite structure showing the triangular-lattice Pd and CrO2 layers. (b) The logarithm of the DFT+DMFT spectral
+spectral function A(k, ω) overlaid with the DFT band structure (blue solid lines). These have been evaluated along a path connecting points
+within the kz = 0 plane of the Brillouin zone of PdCrO2. The points Γ and K are high-symmetry points, with K on the Brillouin zone
+boundary of the primitive rhombohedral cell. While M is not a high-symmetry point in that Brillouin zone, it is used here with reference to the
+equivalent point in a simple hexagonal Brillouin zone, as in previous work [12, 21, 22]. Here, the changes to the Fermi surface between these
+theoretical methods are most prominent. (c) The three DFT Fermi surface sheets in the rhombohedral Brillouin zone and (d) the DFT+DMFT
+hexagonal Fermi surface (given by the spectral function evaluated at the Fermi level where ω = 0 eV) in the same kz plane as described in
+(b). Note that there is distinguishable spectral weight at K with respect to the Γ and M points.
+bitals along with the single a1g orbital, in agreement with
+Ref. [21]. We used the fully-charge-self-consistent (FCSC)
+DFT+DMFT method with a total of 8.4 × 107 Monte Carlo
+sweeps within the impurity solver for each DMFT cycle. An
+inverse temperature β = 40 eV−1 (∼ 290 K) was used which
+is similar to the (room) temperature of the Compton scattering
+experiments. The spectral functions were calculated by ana-
+lytically continuing the DMFT self-energy obtained from the
+LineFitAnalyzer technique of the maximum entropy analytic
+continuation method implemented within the TRIQS/Maxent
+application [34].
+For the DFT and DFT+DMFT EMD calculations, we
+used the method of Ernsting et al. [35] together with the
+DFT+DMFT L¨owdin-type basis electron wave functions and
+occupation numbers determined by the method described in
+Ref. [24]. A maximum momentum cut-off of 16.0 a.u. was
+used. We emphasise that the EMD related results do not use
+analytic continuation so they do not suffer from its associ-
+ated complications. We concentrate on the projected EMD
+for comparisons with the experimental Compton data.
+To
+compare with the experimental 2D occupancy in Ref. [8],
+which directly relates to the electron occupation, the calcu-
+lated EMDs were first projected along the kz-axis (parallel to
+the c-axis of the conventional unit cell) and this projected 2D
+EMD was then convoluted with a 2D Gaussian function with a
+full-width-at-half-maximum of 0.106 a.u. approximating the
+effect of the finite Compton scattering experimental momen-
+tum resolution [8]. These convoluted EMDs are subsequently
+folded back into the first Brillouin zone, via the LCW theo-
+rem, producing the theoretical 2D projected occupancy. The
+Compton profiles, J(pz), which are double-projections of the
+EMD, were evaluated along the experimental scattering vec-
+tors (which for convenience are conventionally referred to as
+being along pz in the local coordinate system),
+J(pz) =
+��
+ρ(p)dpxdpy,
+(1)
+where ρ(p) is the 3D EMD. The so-called directional differ-
+ences, which are the differences between Compton profiles
+resolved along different crystallographic directions, were cal-
+culated so that they could be compared to the experimental
+ones.
+III.
+RESULTS
+Fig. 1 (b) shows the DFT band structure and DFT+DMFT
+spectral function plotted along the high symmetry directions
+in the kz = 0 plane.
+The DFT and DFT+DMFT results
+show good agreement with previous studies [12, 21, 22]. We
+note that our spin-orbit coupling DFT calculation differs to
+that presented in Ref. [8], even though those previously pub-
+lished results are reproducible with the same version (2.2.9)
+of ELK. The lack of reproducibility of the Ref. [8] ground
+state with the current version of ELK suggests that there was
+some problem with that calculation in version 2.2.9 (which
+has been fixed in later versions) which coincidentally gave
+convincing agreement between the reported electronic struc-
+ture and experimental Compton data. In agreement with the
+other previously reported DFT and DFT+DMFT predictions,
+the hybridised Pd 4d and Cr 3d DFT bands which lie around
+the Fermi level and which contribute to the DFT Fermi sur-
+face shown in Fig. 1 (c) drastically redistribute, with the Cr
+3d dominant bands now insulating in DFT+DMFT due to the
+
+4
+FIG. 2. The DFT+DMFT (fixed U = 3.0 eV and J = 0.7 eV)
+spectral function A(k, ω) plotted in the style of ARPES energy dis-
+tribution curves (EDCs). This shows the spectral function dispersion
+around the Fermi level along a portion of the path of Fig. 1 (b) fo-
+cusing on the Pd quasiparticle conduction band crossing the Fermi
+level (ω = 0 eV). The inset reveals structure in the spectral func-
+tion evaluated at a k-point between M to Γ which is highlighted in
+red in the EDCs. The axes of the inset are the same as the main fig-
+ure. The Pd quasiparticle conduction band centre is just above the
+Fermi level, but there is spectral weight from this Lorentzian-like
+quasiparticle band spectral function crossing the Fermi level and this
+occupied spectral weight will contribute to the occupation distribu-
+tion. This occupied weight is referred to as spectral weight spillage
+across the Fermi level.
+formation of a Mott insulating state within the CrO2 layers
+which arises from the strong local electron correlations on the
+Cr site. The remaining quasiparticle band which crosses the
+Fermi level in DFT+DMFT A(k, ω) is now predominantly Pd
+4d in character and forms the hexagonal Fermi surface sheet
+shown in Fig. 1 (d), in excellent agreement with that observed
+in the paramagnetic ARPES [11] measurements. There are
+also incoherent, non-dispersive Hubbard-like bands, shown in
+Fig. 1 (b) centred around ±1.5 eV, which arise from the Mott
+insulating Cr states. We note that the DFT+DMFT spectral
+function in Fig. 1 (d) shows significant spectral weight around
+the K point which is also seen in previous DFT+DMFT cal-
+culations by Lechermann [21].
+To help illustrate certain concepts which link the spectral
+function to the occupation distribution (required for subse-
+quent discussions), we have included the DFT+DMFT spec-
+tral function around the Fermi level in Fig. 2, plotted in the
+style of ARPES energy distribution curves (EDCs). The spec-
+tral function of the Pd dominant quasiparticle conduction band
+is seen to be broader and have a smaller amount of spectral
+weight than the inverted parabolic quasiparticle band around
+M which peaks at about −0.5 eV (which is also shown in the
+inset of Fig. 2). The inset shows that at a particular k-point
+between M and Γ the Pd quasiparticle conduction band cen-
+tre is just above the Fermi level which of course means that
+there is no Fermi surface at this wave-vector. However, owing
+to the finite width of the spectral function around the quasi-
+particle peaks (which arises from the finite lifetime linked to
+the imaginary part of the DMFT self-energy), there is a por-
+tion of the spectral function tail which crosses the Fermi level
+and is consequently occupied. This occupied portion of the
+Pd quasiparticle conduction band contributes to the EMD and
+will be seen in the electron occupancy measured by Comp-
+ton scattering. Conversely, if the band centre (quasiparticle
+peak) were below the Fermi level, but the higher energy tail
+crosses the Fermi level, then that quasiparticle band will have
+a reduced contribution to the occupation at that k-point with
+respect to a fully occupied quasiparticle band. We refer to the
+spectral weight from the quasiparticle band tails crossing the
+Fermi level as spectral weight spillage. The spectral weight
+spillage will be dependent on factors which influence the fi-
+nite width (inverse lifetime) of the (quasiparticle) peaks in
+the spectral function. In the DFT picture within the Green’s
+function formalism, the typical DFT spectral function would
+be a series of Lorentzian-like functions corresponding to the
+DFT bands and most likely have small widths relating to the
+temperature used in the calculation. The corresponding oc-
+cupation distribution will therefore have contributions from
+the fully occupied spectral function below the Fermi level and
+from spectral weight spillage. The common consequence of
+spectral weight spillage contribution in DFT is the apparent
+smearing of the occupation distribution in (crystal) momen-
+tum around the Fermi wave-vector (which is temperature de-
+pendent because of the temperature dependence of the spectral
+weight spillage). The effects of spectral weight spillages on
+the occupation distribution are often less prominent in DFT
+but have been seen for DFT bands grazing the Fermi level
+such as in ZrZn2 [36] and in highly compositionally disor-
+dered systems [19].
+The 2D projected occupancy (along the projected bulk high
+symmetry path used in Ref. [8]) determined from the DFT
+and DFT+DMFT calculated EMDs, together with the the ex-
+perimental 2D occupancy, are shown in Fig. 3. Here, we see
+that the agreement in the DFT+DMFT (U = 3.0 eV, J = 0.7
+eV) 2D projected occupancy significantly improves along the
+Γ to M direction compared to the DFT results, with there
+being a single step along this direction in the DFT+DMFT
+compared to the smoothed shoulder predicted by the DFT.
+The location of this single step along Γ to M gives the Fermi
+wave-vector of the hexagonal Fermi surface sheet along this
+direction. We can also extract the Fermi wave-vector of the
+hexagonal Fermi surface sheet along the Γ to K from the lo-
+cation of the largest change in the projected occupation. The
+DFT+DMFT projected occupation which relates to hexago-
+nal Fermi surface sheet along with the region it encompasses
+(see Fig. 4) is in excellent agreement with the Compton data.
+We find the occupied fraction of the Brillouin zone associ-
+ated with DFT+DMFT hexagonal Fermi surface is approxi-
+mately equal to one half, which is in excellent agreement with
+both the occupation fraction expected from the Fermi surface
+of a monovalent metal and the experimental fractions deter-
+mined from Compton [8], ARPES [10], and quantum oscilla-
+tions [13].
+The DFT projected occupancy has some similarities to the
+experiment around K.
+This feature in the DFT relates to
+
+5
+M
+K
+M
+K
+min
+max
+occupancy
+DFT
+J=0.25 eV
+J=0.30 eV
+J=0.50 eV
+J=0.70 eV
+experiment
+FIG. 3. The 2D occupancy (projected along the kz-axis) plotted along the projected bulk high-symmetry directions (denoted with overlines)
+for DFT and DFT+DMFT with different values of the Hund exchange J at fixed Hubbard U = 3.0 eV. The theoretical projected EMDs were
+convoluted with a two dimensional Gaussian (full-width-at-half-maximum = 0.106 a.u.) to approximate the effect of the finite experimental
+momentum resolution prior to calculating the occupancy. The experimental data are from Ref. [8]. Varying J explores the changes to the
+electronic structure passing through the Mott transition of the Cr states, with the Mott insulating state occurring for J > 0.25 eV.
+the Cr DFT band crossing the Fermi level near K (and the
+corresponding points along the kz-axis) resulting in an elec-
+tron Fermi surface pocket around K (see Fig. 1 (b) and (c)).
+However, the agreement at K significantly worsens in the
+DFT+DMFT (at J = 0.7 eV) as there is no contribution from
+the Cr band as it is now below the Fermi level and hence fully
+occupied (insulating). Interestingly, however, there is a small
+contribution at K in the DFT+DMFT (J = 0.7 eV) projected
+occupations which arises from the spectral weight of the Pd
+quasiparticle conduction band spilling across the Fermi level
+(such as that seen around K in Fig. 1 (d)) which then becomes
+occupied, similar to that seen in Refs. [19, 36] as discussed
+previously. This additional spectral weight is small relative
+to the background (i.e., relative to the Γ point) which would
+likely mean that this feature might be difficult for the ARPES
+to distinguish within the experimental and statistical error. It
+should be noted that the projected occupation from Compton
+presented here relates to the energy integral of the occupied
+part of the spectral function which is then integrated along
+the kz-axis. Consequently, the accumulation of this feature
+around K seen in the spectral function in Fig. 1 (d) becomes
+more prominent in the projected occupation at K. This is seen
+in the DFT+DMFT (J = 0.7 eV) projected occupation fea-
+ture around K in Fig. 3. We note that the DFT+DMFT spec-
+tral plot along the same in-plane path as Fig. 1 (b) but with a
+shift of 0.5 reciprocal lattice units along the kz-axis shows a
+similar dispersion to kz = 0 plane which is expected for this
+quasi-2D system. Therefore, this DFT+DMFT feature at K
+will have contributions from all the spectral weight spillage
+along the kz-axis centred at K owing to the projected nature
+of the Compton occupation data.
+Also shown in Fig. 3 are several DFT+DMFT calculations
+of the 2D projected occupancy plotted for different J but with
+U fixed to 3.0 eV. These show the evolution of the 2D pro-
+jected occupancy (and by inference, the electronic structure)
+as a function of the size of the Hund exchange interaction J as
+the CrO2 layer transitions from the metallic (low J) to Mott
+insulating state (high J), where the Mott insulating state oc-
+curs for J > 0.25 eV. The result of increasing J causes the
+smoothed double-step feature prominent in the DFT projected
+occupancy along Γ to M to transform into a single step due to
+the spectral weight from the previously conducting Cr quasi-
+particle bands shifting below the Fermi level and becoming
+fully occupied. On the other hand, increasing J suppresses
+the 2D occupancy contribution around K as the Cr quasipar-
+ticle bands transition into being Mott insulating. There are no
+optimal DFT+DMFT U and J parameters which are able to
+simultaneously capture the 2D projected occupancy features
+from Γ to K and the single step along Γ to M. The hexagonal
+Fermi surface sheet is a robust feature in all the Fermi surface
+measurements and is clearly captured by the DFT+DMFT pre-
+dictions with Mott insulating CrO2 layers.
+To get a better perspective of the agreement between the
+different calculations with the experimental data, Fig. 4 shows
+
+6
+DFT
+experiment
+DFT+DMFT
+K
+M
+min
+max
+occupancy
+FIG. 4. The 2D (projected along the kz-axis) occupancy in the 2D
+hexagonal Brillouin zone. The left hand side shows the experimental
+data, whereas each quadrant on the right hand side represents a differ-
+ent calculation, as indicated. The theoretical two dimensional EMDs
+were convoluted as described in the Fig. 3 caption. The DFT quan-
+drant includes the Brillouin zone boundary as well as the projected
+2D high symmetry points (denoted with overlines). The experimen-
+tal data are from Ref. [8].
+the 2D projected occupancy of the different calculations and
+experiment. The DFT results give good agreement in cer-
+tain regions, but is overall worse than the DFT+DMFT as ex-
+pected. The size of DFT+DMFT hexagonal occupancy weight
+around Γ is in excellent agreement with the experimental 2D
+projected occupancy, as previously established. However, it
+is clear that the DFT+DMFT is unable to predict the signifi-
+cant additional occupation feature surrounding the hexagonal
+region which gives rise to elongated black ellipsoidal region
+centred around M, with the major axis of this ellipsoid along
+the M—K path. Next, we present the comparison of the di-
+rectional differences along the different measured (crystallo-
+graphic) directions in Fig. 5 for the DFT, DFT+DMFT and the
+experimental data. It is clear that the DFT+DMFT results are
+superior in agreement with the experiment compared with the
+DFT.
+Thus far, the origin of the features measured in the ex-
+perimental techniques has been discussed from a theoretical
+perspective. However, the discrepancy between experimen-
+tally measured features by ARPES and Compton still needs to
+be addressed. Both of these experiments were performed at
+different temperatures with the Compton being at room tem-
+perature, whereas the ARPES was measured at 100 K. There
+have been no reported signatures which could be related to
+a temperature-dependent Lifshitz transition in the transport
+measurements [5] which could have explained this extra fea-
+ture in the Compton data at K being related to the Fermi sur-
+face. However, it should be noted that the spectral weight of
+the Pd quasiparticle conduction band would be more broadly
+distributed in energy in the room temperature Compton data
+than the 100 K ARPES data meaning more spectral weight
+0.2
+0.1
+0.0
+0.1
+0.2
+J(pz) (a. u.
+1)
+M 
+ K
+DFT
+DFT+DMFT
+experiment
+0.2
+0.1
+0.0
+0.1
+0.2
+J(pz) (a. u.
+1)
+M 
+ 22.5
+0.2
+0.1
+0.0
+0.1
+0.2
+J(pz) (a. u.
+1)
+M 
+ 15
+0
+1
+2
+3
+4
+5
+6
+pz (a.u.)
+0.2
+0.1
+0.0
+0.1
+0.2
+J(pz) (a. u.
+1)
+M 
+ 7.5
+FIG. 5.
+The directional differences ∆J(pz) (i.e., the difference
+between two Compton profiles measured along different crystallo-
+graphic directions) as specified at the bottom right of each panel
+where the angle refers to the rotation away from the ΓM direc-
+tion towards ΓK. These differences are of the DFT, DFT+DMFT,
+and the experiment. The theoretical Compton profiles were convo-
+luted with a one dimensional (1D) Gaussian of full-width-at-half-
+maximum = 0.106 a.u. to represent the experimental momentum
+resolution. The experimental data are from Ref. [8].
+from the tail of that quasiparticle band would likely be oc-
+cupied.
+It would be strange if the ARPES spectra would
+miss a Fermi surface feature at K due to cross-section ef-
+fects as it is very unlikely for ARPES not measure the same
+band in different regions of the Brillouin zone. It is also un-
+likely that the ARPES matrix elements effects are suppressing
+a Fermi surface feature originating from the Pd quasiparticle
+band, although ARPES matrix elements effects do cause some
+changes in the measured intensity [12]. The reduced dimen-
+sionality at the surface may enhance the electron correlation
+effects within the Mott insulating CrO2 layers at the surface,
+similar to that seen in SrVO3 [37–42]. On the other hand,
+there is unlikely any notable contribution from surface states
+in the ARPES as these would give additional features [10], not
+remove some.
+Returning to the experimental feature at K, one possible ex-
+planation is that this may actually arise from the DFT+DMFT
+Pd conduction quasiparticle band at K (and the positions dis-
+
+7
+0.5
+0.0
+0.5
+1.0
+ (eV)
+(a)
+K
+M
+min
+max
+occupancy
+(b)
+ = 0.0 eV
+ = 0.1 eV
+ = 1.0 eV
+ = 5.0 eV
+FIG. 6. (a) The logarithm of DFT+DMFT spectral function with an
+additional artificial broadening term (in energy) along the same path
+and colour scale as in Fig. 1 (b). This broadening is only applied
+to the Pd quasiparticle conduction band centred around K up to the
+dashed boundaries. This broadening term varies quadratically from
+zero at the dashed boundaries to a maximum of δ (here it is equal to
+1 eV) at K. There is no physical significance to the relation between
+the additional broadening and its k-dependence, it just ensures a con-
+tinuous change in the broadening. (b) The occupancy along this path
+obtained from integrating the artificially broadened spectral function
+up to the Fermi level for different maximum δ values given in the
+legend. Both panels help to show how the spectral function (mea-
+sured by ARPES) and occupancy (measured by Compton scattering)
+are related to each other, along with the different features of the elec-
+tronic structure ARPES and Compton scattering would probe.
+placed along kz) being broader and/or closer to the Fermi level
+than predicted in the DFT+DMFT with the feature arising
+from the spectral weight spillage. A computationally inexpen-
+sive way to gain some insight into the contribution that this
+part of the quasiparticle band would make to the occupancy
+is to add an artificial (and arbitrary) broadening term (in en-
+ergy) to this DFT+DMFT Pd quasiparticle conduction band
+around the K point as shown in Fig. 6 (a). It is clear how dis-
+persive this makes the quasiparticle band around K resulting
+in additional spectral weight spillage crossing the Fermi level
+which gives rise to a more prominent occupancy feature at K
+in Fig. 6 (b). The occupancy in Fig. 6 (b) is calculated by
+integrating the real-frequency-dependent broadened spectral
+function up to the Fermi level. This feature grows as a func-
+tion of increasing δ, which is the maximum of the additional
+broadening as explained in Fig. 6 (b), until exceeding δ = 5
+eV. The occupation for the unbroadened (δ = 0.0 eV) spectral
+function is very similar to the DFT+DMFT (J = 0.7 eV) 2D
+projected occupancy in Fig. 3, with the additional smearing in
+that occupancy coming from the convolution with the exper-
+imental momentum resolution function. This similarity is to
+be expected as this is a quasi-2D electronic structure. We note
+that the additional occupation from this broadening violates
+charge conservation and as such, the Fermi level would need
+to move to compensate for this.
+This broadened spectral function serves to illustrate how
+the feature at K in the experimental Compton data may arise
+from this quasiparticle band. However, even with the unphys-
+ical arbitrary broadening, it is still not enough to fully agree
+with the experimental Compton data. This would suggest that
+the shape of this quasiparticle band may need to change with
+the dip around K likely being closer to the Fermi level, but its
+band centre must remain above the Fermi level to agree with
+the established single hexagonal Fermi surface. We emphasise
+Fig. 6 illustrates how ARPES and Compton probe the elec-
+tronic structure differently, in this case around K. For δ = 1
+eV, Compton scattering would probe a distinct occupation fea-
+ture around K, but the spectral function at the Fermi level
+around K is relatively small in magnitude which may make
+it difficult to distinguish in ARPES. We note that Lecher-
+mann [21] showed that the introduction of relatively large
+(electron) doping results in a downward shift in energy of the
+Pd quasiparticle band around K. This will likely give a more
+prominent feature in the occupancy feature around K, for the
+reasons previously discussed. However, the PdCrO2 single-
+crystal sample measured by Billington et al. were grown by
+H. Takatsu as described in Ref. [43] and were of similar high
+purity and quality to those measured by ARPES [10–12] and
+quantum oscillations [13, 14]. Therefore, it is highly unlikely
+that the measured feature at K in the projected occupation
+comes solely from naturally occurring doping effects, but their
+contributions cannot be fully ruled out.
+The Cr 3d DMFT self-energy significantly influences the
+Pd quasiparticle conduction band around K due to coupling
+between the layers of the localised Cr and itinerant Pd elec-
+trons, as discussed in detail by Lechermann [21]. This type of
+coupling is similar to the Kondo effect, but here the localised
+spins in PdCrO2 originate from a Mott mechanism which sup-
+presses the electron hopping between sites. The inclusion of
+the DMFT self-energy brings the Pd quasiparticle conduction
+band closer to the Fermi level around K and redistributes a
+significant amount of the Cr 3d contribution to the spectral
+function away from this quasiparticle band peak and into the
+Hubbard-like bands. The disagreement in the occupancy may
+stem from inadequacies in the description of the hybridisation
+between the Cr 3d and Pd 4d states (which relates to the inter-
+layer electron coupling) at the DFT level. This can be very
+sensitive to the exchange-correlation functional used at the
+DFT level, as seen for group V and VI elements [44]. Con-
+sidering that the Pd states are primarily treated on the DFT
+level, higher order electron correlation contributions may in-
+fluence the Pd conduction quasiparticle band dispersion and
+impact the inter-layer electron coupling. The correct descrip-
+tion of this inter-layer coupling may cause the shape and
+broadening of the Pd quasiparticle band to change to give the
+(2D projected) occupancy feature at K revealed by Comp-
+ton scattering while also being potentially difficult to distin-
+guish in ARPES. We note that the reduced dimensionality at
+the surface could influence the inter-layer electron coupling
+(and other electron correlation effects), which may alter the
+Pd quasiparticle conduction band shape and dispersion which
+ARPES (potentially) would measure in comparison to what
+the Compton scattering bulk-probe measures. The results of
+DFT+DMFT calculations performed with an additional im-
+purity site for the Pd 4d orbitals (with the Cr and Pd DMFT
+impurities are treated independently) do not significantly alter
+
+8
+the presented DFT+DMFT results which suggests that the lo-
+cal Pd electron correlations are insignificant when it comes to
+explaining the origin of the missing feature around K in the
+projected occupations.
+There is increasing amount of experimental evidence show-
+ing significant inter-layer electron coupling. Transport mea-
+surements in Ref. [5] show that the frustrated Cr spins affect
+the out-of-plane and in-plane motion of the conduction elec-
+trons in the Pd layer. The interpretation of the magnetother-
+mopower measurements in Ref. [45] also point to there being
+significant coupling between the itinerant Pd electrons with
+the short-range electron spin-correlations of the Cr electron
+spins well above TN. The short-range electron spin correla-
+tions which persisted above TN were also measured by single-
+crystal neutron scattering in Ref. [8]. Further transport mea-
+surements have shown the effect of the short-range order on
+the Hall and Nernst effects [46]. Raman and electron spin res-
+onance (ESR) measurements [47] have also shown evidence
+for inter-layer hoppings along the c-axis and a reconstruction
+of electronic bands on approaching TN. Recent ARPES [12]
+measurements in the antiferromagnetic phase showed that the
+measured spectra can be explained by an intertwined excita-
+tion consisting of a convolution of the charge spectrum of the
+metallic Pd layer and the spin susceptibility of the Mott insu-
+lating CrO2 layer. This excitation arises from an inter-layer
+Kondo-like coupling. The authors of Ref. [12] draw parallels
+with the results of the doping calculations of the Mott layer
+calculated in Ref. [21] which, as already discussed, signifi-
+cantly affects the shape and dispersion of the Pd quasiparti-
+cle band at K. They emphasise that the results of their mea-
+surements and the doped DFT+DMFT calculations reflect the
+fact that in a coupled Mott-itinerant system, the itinerant layer
+will support charge excitations [12]. As the short-range elec-
+tron spin correlations persist beyond TN, our interpretation of
+the Compton results with respect to the Pd quasiparticle band
+ties in with the experimental evidence of the inter-layer elec-
+tron coupling, and may be linked to the intertwined excitation.
+Therefore, electron correlation effects which contribute to the
+inter-layer electron coupling, such as those in the models used
+in Refs. [12, 48], beyond those included in our DFT+DMFT
+calculations, seem to be significant. To confirm that the Pd
+conduction quasiparticle band is indeed broader and closer
+to the Fermi level than that predicted, the experimental k-
+resolved dispersion of that band could, for example, be mea-
+sured by pump-probe ARPES or k-resolved inverse photoe-
+mission spectroscopy (KRIPES) experiments which can probe
+the unoccupied part of the band structure.
+IV.
+CONCLUSION
+We have shown that the paramagnetic DFT+DMFT theo-
+retical description of the electronic structure of PdCrO2 is su-
+perior to DFT as it gives excellent agreement with the features
+relating to the hexagonal Fermi surface sheet measurement by
+all the Fermi surface experimental data, all of which agrees
+with the picture of the Mott insulating CrO2 layers [4]. How-
+ever, there are still discrepancies between the paramagnetic
+DFT+DMFT results and the Compton data measured within
+the paramagnetic phase. We found that there is no combina-
+tion of U and J around the Mott insulator transition (in the
+CrO2 layers) in DFT+DMFT which agrees with the presence
+of both the hexagonal Fermi surface and the feature around K
+as measured by the Compton. By adding an unphysical broad-
+ening term (in energy) to the DFT+DMFT the Pd quasiparti-
+cle conduction band around K, more spectral weight spills
+across the Fermi level which gives rise to a more prominent
+feature in the occupancy. However, this is still not enough to
+agree with the measured projected occupancy feature in the
+Compton data, so a change in both the broadening and shape
+of this quasiparticle band is needed while keeping its band
+centre above the Fermi level to avoid any changes to the es-
+tablished Fermi surface topology. Overall, our DFT+DMFT
+results help to clarify the origin of features in the Compton
+data.
+From the available experimental and theoretical evidence
+thus far, the feature in the projected electron occupancy mea-
+sured at K by Compton scattering is likely from the spectral
+weight of the Pd conduction quasiparticle band spilling across
+the Fermi level and becoming occupied. The ARPES may not
+measure this proposed spectral weight spillage if the Pd quasi-
+particle band is very dispersive around K (and the positions
+displaced along kz) and if the surface influences the electron
+correlation effects, such as the inter-layer electron coupling,
+which may then alter the quasiparticle band shape and disper-
+sion. As the DFT+DMFT model used does not predict the
+measured projected occupation feature at K, theories beyond
+our DFT+DMFT are required to establish the exact origin of
+this feature, which likely relates to the inter-layer electron
+coupling between the Pd and CrO2 layers which gives rise
+to new Kondo-like physics such as the previously observed
+intertwined excitation [12]. The discrepancy with the Comp-
+ton data gives motivation to experimentally measure the dis-
+persion of the unoccupied part of the Pd quasiparticle con-
+duction band to determine if it is indeed closer to the Fermi
+level and much more smeared in energy than predicted by our
+DFT+DMFT calculations. Evidently, Compton scattering is a
+powerful probe of many-body electron correlation effects.
+V.
+ACKNOWLEDGEMENTS
+A.D.N.J. acknowledges the Doctoral Prize Fellowship
+funding and support from the Engineering and Physical Sci-
+ences Research Council (EPSRC). We are grateful for the use-
+ful discussions with J. Laverock, M. Favaro-Bedford, Wenhan
+Chen, and C. Mackellar. Calculations were performed using
+the computational facilities of the Advanced Computing Re-
+search Centre, University of Bristol (http://bris.ac.uk/acrc/).
+The VESTA package (https://jp-minerals.org/vesta/en/) has
+been used in the preparation of some figures.
+
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+

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+Scalable Quantum Error Correction for Surface Codes using FPGA
+Namitha Liyanage, Yue Wu, Alexander Deters and Lin Zhong
+Department of Computer Science, Yale University, New Haven, CT
+Abstract
+A fault-tolerant quantum computer must decode and correct
+errors faster than they appear. The faster errors can be cor-
+rected, the more time the computer can do useful work. The
+Union-Find (UF) decoder is promising with an average time
+complexity slightly higher than O(d3). We report a distributed
+version of the UF decoder that exploits parallel computing re-
+sources for further speedup. Using an FPGA-based implemen-
+tation, we empirically show that this distributed UF decoder
+has a sublinear average time complexity with regard to d,
+given O(d3) parallel computing resources. The decoding time
+per measurement round decreases as d increases, a first time
+for a quantum error decoder. The implementation employs
+a scalable architecture called Helios that organizes parallel
+computing resources into a hybrid tree-grid structure. Using
+Xilinx’s cycle-accurate simulator, we present cycle-accurate
+decoding time for d up to 15, with the phenomenological
+noise model with p = 0.1%. We are able to implement d
+up to 7 with a Xilinx ZC106 FPGA, for which an average
+decoding time is 120 ns per measurement round. Since the
+decoding time per measurement round of Helios decreases
+with d, Helios can decode a surface code of arbitrarily large
+d without a growing backlog.
+1
+Introduction
+The high error rates of quantum devices pose a significant ob-
+stacle to the realization of a practical quantum computer. As a
+result, the development of effective quantum error correction
+(QEC) mechanisms is crucial for the successful implementa-
+tion of a fault-tolerant quantum computer.
+One promising approach for implementing QEC is the use
+of surface codes [1–3] in which information of a single qubit
+(called a logical qubit) is redundantly encoded across many
+physical data qubits, with a set of ancillary qubits interacting
+with the data qubits. By periodically measuring the ancillary
+qubits, one can detect and potentially correct errors in physical
+qubits.
+Once the presence of errors has been detected through
+the measurement of ancillary qubits, a classical algorithm, or
+decoder, guesses the underlying error pattern based on the
+measurement results. The faster errors can be corrected, the
+more time a quantum computer can spend on useful work.
+Due to the error rate of the state of the art qubits, very large
+surface codes (d > 25) are necessary to achieve fault-tolerant
+quantum computing [2, 4, 5]. See §2 for more background.
+As surveyed in §3, previously reported decoders capable
+of decoding errors as fast as measured, or backlog-free, either
+exploit limited parallelism [6, 7], or sacrifice accuracy [8, 9].
+The largest d reported for any backlog-free implementations
+is 5 [6], based on a design that is physically infeasible beyond
+d = 5.
+In this paper we report a distributed Union-Find (UF) de-
+coder (§4) and its FPGA implementation called Helios (§5).
+Given O(d3) parallel resources, our decoder achieves sublin-
+ear average time complexity according to empirical results
+for d up to 15, the first to the best of our knowledge. No-
+tably, adding more parallel resources will not reduce the time
+complexity of the decoder, due to the inherent nature of error
+patterns. Our decoder is a distributed design of and logically
+equivalent to the UF decoder first proposed in [10]. We im-
+plement the distributed UF decoder with Helios, a scalable
+architecture for organizing the parallel computation units.
+Helios is the first architecture of its kind that can scale to
+arbitrarily large surface codes by exploiting parallelism at
+the vertex level of the model graph. In §6, we report experi-
+mental validations of the distributed UF decoder and Helios
+with a ZCU106 FPGA board [11] which is capable of run-
+ning surface codes up to d = 7. For d = 7 the decoder has
+an average decoding time of 120 ns per measurement round,
+faster than any existing decoder. We validate our design for
+surface codes of d > 7 by using Xilinx Vivado cycle accurate
+simulator [12]. These validations successfully demonstrate,
+for the first time, a decoder design with decreasing average
+time per measurement round when d increases. This shows
+evidence that the decoder can scale to arbitrarily large surface
+codes without a growing backlog.
+arXiv:2301.08419v1  [quant-ph]  20 Jan 2023
+
+2
+Background
+2.1
+Qubit and Errors
+Qubit is the basic unit of quantum computing which is rep-
+resented as |ψ⟩ = α|0⟩ + β|1⟩. Here α and β are complex
+numbers such that |α|2 + |β|2 = 1 and |0⟩ and |1⟩ are the
+basis states of a qubit.
+Unlike classical bits, qubits are highly susceptible to er-
+rors. A qubit can unintentionally interact with its surrounding
+resulting in a change of its quantum state. Even the latest
+quantum computers still have an error rate of 10−3 [4] which
+is significantly worse than classical computers which have
+error rates lower than 10−18. In contrast a useful quantum
+application requires an error rate of 10−15 or below necessi-
+tating error correction. Errors in qubits can be modeled as
+bit flip errors and phase flip errors. A bit flip is marked by
+the X operator, i.e., X|ψ⟩ = β|0⟩+α|1⟩, while a phase flip is
+marked by Z operator, i.e., Z|ψ⟩ = α|0⟩−β|1⟩ .
+2.2
+Error Correction and Surface Code
+Quantum Error Correction (QEC) is more challenging than
+classical error correction due to the nature of Quantum bits.
+First, qubits cannot be copied to achieve redundancy due to
+the no-cloning theorem. Second, the value of the qubits cannot
+be directly measured as measurements perturb the state of
+qubits. Therefore QEC is achieved by encoding the logical
+state of a qubit, as a highly entangled state of many physical
+qubits. Such an encoded qubit is called a logical qubit.
+The surface code is the widely used error correction code
+for quantum computing due to its high error correction capa-
+bility and the ease of implementation due to only requiring
+connectivity between adjacent qubits. A distance d surface
+code is a topological code made out of a (2d −1)×(2d −1)
+array of qubits as shown in Figure 1. A key feature of surface
+codes is that a larger d can exponentially reduce the rate of
+logical errors making them advantageous. For example, even
+if the physical error rate is 10 times below the threshold, d
+should be greater than 17 to achieve a logical error rate below
+10−10 [2].
+A surface code contains two types of qubits, namely data
+qubits and ancilla qubits. The data qubits collectively encode
+the logical state of the qubit. The ancilla qubits (called X-type
+and Z-type) entangle with the data qubits and by periodically
+measuring the ancilla qubits, physical errors in all qubits can
+be discovered and corrected. An X error occurring in a data
+qubit will flip the measurement outcome of Z ancilla qubits
+connected with the data qubit and Z error will flip the X ancilla
+qubits likewise. Such a measurement outcome is called non-
+trivial measurement value. Because ancilla qubits themselves
+could also suffer from physical qubit errors, multiple rounds
+of measurements are necessary. Figure 2 shows some example
+physical qubit errors occurring in a surface code and how
+Z
+Z
+Z
+Z
+Z
+Z
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+Z
+Z
+Z
+Z
+Z
+Z
+Z
+Z
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+Z
+Z
+Z
+Z
+Z
+Z
+d = 3
+d = 3
+(a)
+Z
+Z
+Z
+Z
+S
+A
+B
+C
+D
+A
+B
+C
+D
+S
+|0
+Z
+(b)
+X
+X
+X
+X
+S
+A
+B
+C
+D
+A
+B
+C
+D
+S
+|+
+X
+(c)
+Figure 1: (a) : CSS surface code (d = 3), a commonly used type of surface
+code. The white circles are data qubits and the black the Z-type and X-type
+ancillas. (b) and (c) : Measurement circuit of Z-type and X-type ancillas.
+Excluding the ancillas in the border, each Z-type and X-type ancilla interacts
+with 4 adjacent data qubits.
+X
+(a)
+Z
+(b)
+X
+X
+X
+(c)
+X
+X
+X
+(d)
+Round 1
+Round 2
+Round 3
+X
+X
+M
+M
+time
+(e)
+(f)
+Figure 2: (a) to (d) : Various error patterns on d = 3 surface code. X and Z
+mark the corresponding physical qubit errors. Ancillas reporting non trivial
+measurements are shown in red. The red lines are to visualize error chains.
+(a) isolated X error (b) isolated Z error (c) error chain of three X errors (d)
+error chain introducing a logical error which has no non-trivial measurements.
+Note that even though (a) and (c) are different error patterns, they produce
+the same syndrome. (e) Error patterns spread across multiple measurement
+rounds. Here single X and Z errors can also spread across two rounds and
+error chains can include measurement errors (indicated by ‘M’) as well. (f)
+Decoding graph with vertices with nontrivial measurement marked red for
+the error pattern in (e).
+they are detected by ancilla qubits. We show X and Z errors
+separately because they can be independently dealt with in
+the same way. The outcomes from these multiple rounds of
+measurements of ancilla qubits constitute a syndrome.
+A syndrome can be conveniently represented by a graph
+called decoding graph in which a vertex represents a measure-
+ment outcome of an ancilla and an edge a data qubit. Vertices
+of nontrivial measurement outcome are specially marked. The
+weight of edge is determined by the probability of error in
+the corresponding data qubit or measurement. For distance
+d surface code, there are d ×(d −1) vertices. This decoding
+graph can be extended to three dimensional in which multi-
+ple identical planar layers are stacked on each other. Each
+layer represents a round of measurement. The total number of
+rounds is usually the same as the distance of the surface code.
+Corresponding vertices in adjacent layers are connected by
+edges which represent the probability of measurement error
+of the corresponding ancilla. That is, there are d ×d ×(d −1)
+vertices in this three-dimensional graph. Figure 2f shows the
+decoding graph for a syndrome from d = 3 surface code.
+2
+
+2.3
+Error Decoders
+Given a syndrome, an error decoder identifies the underlying
+error pattern, which will be used to generate a correction
+pattern. As multiple error patterns can generate the same
+syndrome, the decoder has to make a probabilistic guess of
+the underlying physical error. The objective is that when the
+correction pattern is applied, the chance of the surface code
+entering a different logical state (i.e a logical error) will be
+minimized.
+Metrics
+The two important aspects of decoders are accu-
+racy and speed. A decoder must correct errors faster than
+syndromes are produced to avoid a backlog. A faster decoder
+also allows more time for the quantum hardware to do actual
+useful work. The average decoding time per measurement
+round is a widely used criteria for speed.
+A decoder must make careful tradeoff between speed and
+accuracy. A faster decoder with lower accuracy requires a
+larger d to achieve any given logical error rate, which may
+require more computation overall.
+Union-Find (UF) Decoder
+The UF decoder is a fast sur-
+face code decoder design first described by Delfosse and
+Nickerson [10]. According to [13], it can be viewed as an
+approximation to the blossom algorithm that solves minimum-
+weight perfect matching (MWPM) problems. It has a worst
+case time complexity of O(d3α(d)), where α is the inverse
+of Ackermann’s function, a slow growing function that is less
+than three for any practical code distances. Based on our anal-
+ysis, it has an average case time complexity slightly higher
+than O(d3).
+algorithm 1 describes the UF decoder. It takes a decoding
+graph G(V,E) as input. Each edge e ∈ E has a weight and a
+growth, denoted by e.w and e.g, respectively. e.g is initialized
+with 0 and the decoder may grow e.g until it reaches e.w.
+When that happens, we say the edge is fully grown.
+The decoder maintains a set of odd clusters, denoted by
+L. L is initialized to include all {v} that v ∈ V is non-trivial
+(L81). Each cluster C keeps track of whether its cardinality is
+odd or even as well as its root element.
+The UF decoder iterates over growing and merging the
+odd cluster list until there are no more odd clusters (inside
+the while loop of algorithm 1). Each iteration has two stages:
+Growing and Merging. In the Growing stage, each odd cluster
+“grows” by increasing the growth of the edges incidental to its
+boundary. This process creates a set of fully grown edges F
+(L86 to L95). The Growing stage is the more time-consuming
+step as it requires traversing all the edges in the boundary of
+all the odd clusters and updating the global edge table. Since
+the number of edges is O(d3), the UF decoder is not scalable
+for surface codes with large d.
+In the Merging stage, the decoder goes through each fully-
+grown edge to merge the two clusters connected by the edge.
+Algorithm 1: Union Find Decoder
+input :A decoding graph G(V,E) with X (or Z) syndrome
+output :A correction pattern
+77 % Initialization
+78 for each v ∈ V do
+79
+if v is non-trivial then
+80
+Create a cluster {v}
+81
+end
+82 end
+83 while there is an odd cluster do
+84
+% Growing
+85
+F ← /0
+86
+for each odd cluster C do
+87
+for each e =< u,v >, u ∈ C,v ̸∈ C do
+88
+if e.growth < e.w then
+89
+e.growth ← e.growth+1
+90
+if e.growth = e.w then
+91
+F ← F ∪{e}
+92
+end
+93
+end
+94
+end
+95
+end
+96
+% Merging
+97
+for each e =< u,v >∈ F do
+98
+UNION(u, v)
+99
+end
+100 end
+101 Build correction within each cluster
+When two clusters merge, the new cluster may become even.
+When there is no more odd cluster, the decoder finds a
+correction within each cluster and combines them to produce
+the correction pattern (L101).
+3
+Related Work
+There is a large body of literature on fast QEC decoding, e.g.,
+[14–16]. The most related are solutions that leverage parallel
+compute resources.
+Fowler [17] describes a method for decoding at the rate of
+measurement (O(d)). The proposed design divides the decod-
+ing graph among specialized hardware units arranged in a grid.
+Each unit contains a subset of vertices and can independently
+decode error chains contained within it. The design is based
+on the observation that large error patterns spanning multiple
+units are exponentially rare, so inter-unit communication is
+not frequently required. It, however, paradoxically assumes
+that the number of vertices per unit is “sufficient large” and
+a unit can find an MWPM for its vertices within half the
+measurement time on average. Not surprisingly, to date, no
+implementation or empirical data have been reported for this
+work. Our approach distributes computation to a vertex-level
+and leverages the same observation that communication be-
+tween distant vertices is infrequent.
+NISQ+[8] and QECOOL[9] parallelize computation at the
+ancilla level, where all vertices in the decoding graph repre-
+senting measurements of one ancilla are handled by a single
+3
+
+compute unit. This results in an increase in decoding time
+per measurement round as d increases. In contrast we allo-
+cate a processing element per each vertex, which results in
+decreasing decoding time per measurement round with d at
+the expense of number of parallel units growing O(d3). Fur-
+thermore, they both implement the same greedy decoding
+algorithm that has much lower accuracy than the UF decoder
+used in this work. QECOOL has an accuracy that is approx-
+imately four orders of magnitude lower than that of a UF
+decoder [7] and NISQ+ ignores measurement errors further
+lowering its accuracy than QECOOL.
+Skoric et al. [18] propose a method of using measurement
+round-level parallelism, in which a decoder waits for a large
+number of measurement rounds to be completed and then
+decodes multiple blocks of measurement rounds in parallel.
+By using sufficient parallel resources this method can achieve
+a rate of decoding faster than the rate of measurement. How-
+ever, the latency of this approach grows with the number of
+measurement rounds the decoder needs to batch to achieve
+a throughput equal to the rate of measurement. In contrast,
+our approach exploits vertex-level parallelism and completes
+decoding of every d rounds of measurements with an average
+latency that grows sublinearly with d.
+Pipelining can be considered a special form of using com-
+pute resources in parallel, i.e., in different pipeline stages.
+AFS [7] is a UF decoder architected in three pipeline stages.
+The authors estimate the decoder will have a 42 ns latency
+for d = 11 surface code, which is three times lower than what
+we report based on implementation and measurement. The
+authors assume a specialized hardware that is capable of run-
+ning at 4 GHz and as a result, the decoding latency will be
+dominated by memory access. However, no implementation
+or cycle-accurate simulation is known for this decoder. Im-
+portantly, pipelining is limited in how much parallelism it can
+leverage: the number of pipeline stages. In contrast, paral-
+lelism of our decoder grows along d3, which enables us to
+achieve a sublinear average case latency.
+LILLIPUT [6] is a three stage look-up-table based decoder
+similar to AFS. Look-up-table based decoders can achieve
+fast decoding but are not scalable beyond d = 5 as the size
+of the look-up table grows O(2d3). For d = 7 surface code
+with 7 measurement rounds, it would require a memory of
+2168 Bytes, which is infeasible in any foreseeable future.
+4
+Distributed UF Decoder Design
+Our goal to build a QEC decoder is scalability to the number
+of qubits. As surface codes can exponentially reduce logical
+error rate with respect to d, larger surface codes with hundreds
+or even thousands of qubits are necessary for fault-tolerant
+quantum computing. Therefore, the average decoding time
+per measurement round should not grow with d, to avoid
+exponential backlog for any larger d.
+We choose the UF decoder for two reasons. First, it has
+much lower time complexity than the MWPM algorithm. Al-
+though in general the UF decoder achieve lower decoding
+accuracy than MWPM decoders, it is as accurate in many
+interesting surface codes and noise models [13]. Second, the
+UF decoder maintains much less intermediate states, which
+makes it easier to implement in a distributed manner. We
+observe that growing stage from L86 to L95 in algorithm 1
+operates on each vertex independently without dependencies
+from other vertices. A vertex requires only the parity of the
+cluster it is a part of for the growing stage. Second, during
+the merging stage, a vertex only needs to interact with its
+immediate neighbors (L98).
+Like the original UF decoder, our distributed UF decoder
+is also based on the decoding graph. Logically, the distributed
+decoder associates a processing element (PE) with each ver-
+tex in the decoding graph. Therefore, When describing the
+distributed decoder, we often use PE and vertex in an inter-
+exchangeable manner. PEs operate with the same algorithm,
+specified by algorithm 2. The PE algorithm iterates over three
+stages.
+4.1
+PE States
+A PE has direct read access to its local states and some states
+of incident PEs. A PE can only modify its local states.
+Thanks to the decoding graph, a PE has immediate access
+to the following objects.
+• v, the vertex it is associated with.
+• v.E, the set of edges incident to v.
+• v.U, the set of vertices that are incident to any e ∈ v.E. We
+say these vertices are adjacent to v.
+The algorithm augments the data structures of vertex and
+edge of the decoding graph, according to the UF decoder
+design [10]. For each vertex v ∈ V, the following information
+is added
+• id : a unique identity number which ranges from 1 to n
+where n = |V|. id is statically assigned and never changes.
+• m is a binary indicating whether the measurement outcome
+is trivial (false) or not (true). m is initialized according
+to the syndrome.
+• cid: a unique integer identifier for the cluster to which v
+belongs to, and is equal to the lowest id of all the vertices
+inside the cluster. The vertex with this lowest id is called
+the cluster root. v.cid is initialized to be v.id. That is, each
+vertex starts with its own single-vertex cluster. When cid =
+id, the vertex is a root of a cluster.
+• odd is a binary indicating whether the cluster is odd. odd
+is initialized to be m.
+• codd is a copy of odd.
+• stage indicates the stage the PE currently operates in
+4
+
+• busyis a binary indicating whether the PE has any pending
+operations.
+For each edge e ∈ E, the decoder maintains e.growth, which
+indicates the growth of the edge, in addition to e.w, the weight.
+e.growth is initialized as 0. The decoder grows e.growth
+until it reaches e.w and e becomes fully grown.
+For clarity of exposition, we introduce a mathematical
+shorthand v.nb, the set of vertices connected with v by full-
+grown edges, i.e., v.nb={u|e = ⟨v,u⟩ ∈ v.E & e.growth=
+e.w}. We call these vertices the neighbors of v. Note neigh-
+bors are always adjacent but not all adjacent vertices are neigh-
+bors.
+4.2
+Shared memory based communication
+We use coherent shared memory for shared state that has a
+single writer. For all shared memories, given the coherence,
+a read always returns the most recently written value. Like
+ordinary memory, we also assume both read and write are
+atomic.
+• memory read/write for PE (v) and read-only for adjacent
+PEs, i.e., ∀u ∈ v.U. v.cid and v.odd reside in this memory
+(S1).
+• memory read/write for PE (v) and read-only for the con-
+troller. The PE local states, v.codd, v.stage and v.busy
+reside in this memory (S2).
+• memory for e.growth, which can be written by incident
+PEs (S3).
+• memory read/write for the controller and read-only for all
+PEs. The controller state global_stage is stored in this
+memory (S4).
+4.3
+Message based communication
+Only instance in our decoder where a PE needs to commu-
+nicate with a distant PE is when a PE needs to notify the
+root when joining a new cluster (L32). Implementing this
+using shared memory is costly because the PE is not neces-
+sarily adjacent to the root. As there is one type of message in
+our decoder, each message M contains only the destination of
+the message. The destination take value from 1 to n, which
+represents the vertex identifier.
+For the correctness of the decoder we only assume guaran-
+teed delivery of messages and do not assume a time bound
+for message delivery.
+4.4
+PE Algorithm
+All PEs iterate over three stages of operation. Within each
+stage, they operate independently but transit from one stage to
+the next when the controller updates global_stage. When a
+PE enters a stage, it sets v.stage accordingly and keep v.busy
+Algorithm 2: Algorithm for vertex v in the distributed
+UF decoder.
+1 v.cid ← v.id; v.odd ← v.m
+2 while true do
+3
+if global_stage =terminate then
+4
+return
+5
+end
+6
+growing(v)
+7
+merging(v)
+8
+syncing(v)
+9 end
+as true until it finishes all work in the stage. The controller
+uses these two pieces of information from all PEs to determine
+if a stage has started and completed, respectively (See §4.5).
+We next describe the three stages of the PE algorithm.
+In the Growing stage, vertices at the boundary of an odd
+cluster increase e.growth for boundary edges (L16). As PEs
+perform Growing simultaneously, two adjacent PEs may com-
+pare e.w and e.growth and update e.growth for the same e.
+Such compare-and-update operations must be atomic to avoid
+data race.
+In the Merging stage, two clusters connected through a
+fully-grown edge merge by adopting the lower cluster id (cid)
+of theirs. To achieve this each PE compares its cid with PEs
+connected through fully-grown edges (L31). If the other in-
+cident vertex of a fully grown edge has a lower cid the PE
+adopts the lower cid as its own (L31). Merging process con-
+tinues until every PE in the cluster have the same cid which
+is the lowest v.id of the cluster. This procedure is related to
+leader election in a distributed systems: vertices in a newly
+formed cluster must adopt the lowest id. The Merging stage
+also calculates the parity of the cluster. Each PE representing
+a non-trivial measurement (m is true) messages the root of
+the cluster it joins (L32). Likewise, the root updates its parity
+when it receives a message from a PE (L38).
+In the Syncing stage, a root broadcasts its v.odd to all PEs
+in its cluster, which is necessary for the next Growing stage.
+We achieve this using a modified version of the flooding
+algorithm, which uses shared memory instead of message
+passing. Every non-root node initially set its v.odd as false
+and continues comparing v.odd with PEs with fully connected
+edges. If any of the PEs connected with a fully grown edge has
+v.odd as true the PE set its v.odd as true (L53). If a cluster
+has v.odd as truein the root, this results in propagating true
+to all vertices in the cluster similar to a flooding algorithm.
+4.5
+Controller Algorithm
+The controller moves all PEs and itself along the three stages.
+In each stage, it checks for v.busy signals and in addition
+in merging stage it checks for outstanding messages. The
+controller determines completion of a stage when all PEs
+have v.busy as false and there are no outstanding messages.
+5
+
+Algorithm 3: Vertex growing algorithm
+10 function growing(vertex v)
+11
+Wait until global_stage=growing
+12
+v.busy← true; v.stage← growing
+13
+if v.odd then
+14
+for each e = ⟨u,v⟩ ∈ v.E atomic do
+15
+if e.growth< e.w and u.cid ̸= v.cid then
+16
+e.growth← e.growth+1
+17
+end
+18
+end
+19
+end
+20
+v.busy← false;
+21 end
+Algorithm 4: Vertex merging algorithm
+22 function merging(vertex v)
+23
+Wait until global_stage=merging
+24
+v.busy← true; v.stage← merging
+25
+26
+while true do
+27
+if global_stage ̸=merging then return
+28
+29
+if ∃u ∈ v.nb s.t. u.cid < v.cid then
+30
+v.busy← true
+31
+v.cid ← MIN(u.cid|u ∈ v.nb)
+32
+if v.m then send M(v.cid)
+33
+else if ∀u ∈ v.nb,u.cid = v.cid then
+34
+v.busy← false
+35
+end
+36
+37
+for each received message M do
+38
+v.odd ← ¬v.odd
+39
+end
+40
+end
+41 end
+Algorithm 5: Vertex syncing algorithm
+42 function syncing(vertex v)
+43
+v.busy← true; v.stage← syncing
+44
+if v.cid ̸= v.id then v.odd ← false
+45
+v.codd ← v.odd
+46
+47
+while true do
+48
+if global_stage ̸=syncing then return
+49
+50
+if ∀u ∈ v.nb,u.odd = v.odd then
+51
+v.busy← false
+52
+else
+53
+v.odd ← true
+54
+v.busy← true
+55
+end
+56
+end
+57 end
+Upon completion, the controller updates the global_stage
+variable to move to the next stage and the PEs acknowledge
+this update by updating their own v.stage variable.
+The controller also calculates the presence of odd clusters.
+At the end of the syncing stage, it reads the v.odd value of
+Algorithm 6: The controller coordinates all PEs along
+stages and detects the presence of odd clusters.
+58 while true do
+59
+global_stage← growing
+60
+wait until ∀v ∈ V,v.stage= growing
+61
+wait until ∀v ∈ V,v.busy= false
+62
+63
+global_stage← merging
+64
+wait until ∀v ∈ V,v.stage= merging
+65
+wait until ∀v ∈ V,v.busy= false
+66
+wait until no outstanding messages in the system
+67
+68
+global_stage← syncing
+69
+wait until ∀v ∈ V,v.syncing= growing
+70
+wait until ∀v ∈ V,v.busy= false
+71
+72
+if ∀v ∈ V,v.codd = false then
+73
+global_stage← terminate
+74
+return
+75
+end
+76 end
+each vertex. If any vertex has v.odd = true, the controller
+updates the global stage variable to Growing to continue the
+algorithm. Otherwise, it updates it to Terminate to end the
+algorithm.
+4.6
+Time Complexity Analysis
+The worst case time complexity of our distributed UF decoder
+is O(d3). The worst case occurs when parallelism is maxi-
+mally lost in the system; all vertices are non-trivial and merge
+into a single cluster and the root must process all incoming
+messages from all other vertices (L38). However, the occur-
+rence of the worst case scenario is extremely rare as larger
+clusters are exponentially unlikely to occur. Empirical results
+reported in §6 show that average time grows sublinearly with
+d.
+The time complexity of the controller depends on the im-
+plementation of the shared memory for v.busy and checking
+for outstanding messages in the system. As both checks are
+logical OR operators of individual PE information, the most
+efficient implementation is a logical tree of OR operations
+which results in a time complexity of O(log(d)). Thus, the
+overhead of coordination is significantly smaller than the
+worst case time complexity.
+PE Communication Complexity
+The communication
+complexity of the shared memory based communication is
+O(d3). The leader election in the Merging stage and the broad-
+casting of v.odd in the Syncing stage are implemented using a
+shared memory based flooding algorithm. The time complex-
+ity of a flooding operation is O(D), where D is the diameter of
+the cluster. Therefore, in the worst case the time complexity
+of flooding messages is O(d3).
+6
+
+The communication complexity of the message based com-
+munication is O(d6). Messages from each trivial measure-
+ment to the root of the cluster is proportional to the number
+of trivial vertices in the cluster and number of changes of cid
+of each vertex. Thus in the worst case there would be O(d6)
+messages and the time complexity will be O(d3).
+5
+Helios Architecture and Implementation
+We next describe Helios, the architecture for the distributed
+UF decoder.
+5.1
+Overview
+Helios organizes PEs and controller in a custom topology that
+combines a 3-D grid and a B+ tree as illustrated by Figure 3
+and explained below.
+• PEs are organized according to the position of vertices
+they represent in the model graph. We assign v.id sequen-
+tially, starting with 1 from bottom left corner and continuing
+in row-major order for each measurement round. Shared
+memory S1 (v.cid and v.odd) and S2 (v.codd, v.stage, and
+v.busy) are added alongside each PE.
+• Shared memory S3 (e.growth) is added to the incident PE
+with the lower id.
+• A link between every two adjacent PEs to read from each
+other’s S1 and for the one with the higher id to read the
+other’s S4. This results in a network of links in a 3-D grid
+topology. As a PE represents a vertex in the model graph,
+a link represents an edge. Broad pink lines in Figure 3
+represent these links.
+• A directional link between two adjacent PEs and between
+PEs with consecutive v.id values for message passing (L32).
+These links are directed from the PE with higher v.id to the
+other and are buffered. They are represented by blue arrows
+in Figure 3.
+• The controller, realized as a tree of control nodes (§5.3).
+The leaf control nodes of the tree contain shared memory
+S4.
+• A link between each PE and the controller for the controller
+to read from S2 and for the PEs to read from S4. Dashed
+orange lines in Figure 3 represent these links.
+5.2
+Message-passing between PEs
+To implement the vertex merging algorithm (algorithm 4), a
+PE may send and receive messages from another PE, which
+is not necessarily adjacent. Helios implements this with the
+directional links and allows a PE to forward messages over
+directional links. The forward logic is trivially simple because
+PE 5
+PE 1
+PE 2
+PE 6
+PE 13
+PE 17
+Control 
+node 
+Control 
+node
+Root 
+control 
+node 
+Control 
+node 
+Controller
+PE 3
+PE 4
+PE 9
+PE 11
+PE 15
+PE 14
+PE 18
+PE 16
+PE 8
+PE 12
+PE 10
+PE 7
+Figure 3: Helios architecture for d=3 surface code for 3 measurement rounds.
+As d=3 surface code has 6 (3 by 2) ancilla qubits, Helios contains of a 3x2x3
+PE array. PE n indicates PE with v.id = n.
+S3
+growth
+grow
+logic_busy
+S3
+mem
+logic
+ PE 1 
+ FIFO 
+growth
+grow
+logic_busy
+S3
+growth
+grow
+logic_busy
+S2
+stage
+codd
+busy
+mem
+logic
+ FIFO 
+mem
+logic
+ FIFO 
+ nonempty 
+growth, odd, cid
+odd, cid
+growth 
+odd 
+cid
+odd 
+cid
+ PE 2 
+odd, cid
+growth, odd, cid
+nonempty
+nonempty
+To/from controller
+ PE 3 
+ PE 7 
+codd 
+busy
+stage
+Figure 4: The bottom left corner of the PE array shown in Figure 3. Only part
+of the logic and memory inside PE 1 is shown : growth (S3) is per edge and
+is stored in the PE with lower id. grow logic (in pink) calculates the updated
+growth value (Figure 5). logic_busy(in green) (Figure 6) is per adjacent PE
+and is used to calculate the busy signal.
+(1) a PE only messages another PE with a lower id per al-
+gorithm 4 and (2) the links are directional from a PE with a
+higher id to that with a lower one.
+We note that the directional links consist of the 3-D grid
+structure, from the edges of the model graph, and additional
+links between PEs with consecutive v.id values, i.e., the “di-
+agonal” ones in Figure 3. The 3-D grid topology is optimal
+for exchanging messages between nearby PEs, which is fre-
+quent. The additional “diagonal” links prevent deadlocks by
+breaking potential circular dependency amongst several PEs,
+e.g., PE 1 to PE 4 in Figure 3.
+7
+
+The directional links between PEs are buffered because a
+PE can receive multiple messages at a time. Because these
+buffers have a finite size, the sending PE can stall if a buffer
+is full. In §6.2, we show empirical evidence that stall rarely
+happens.
+5.3
+Controller
+Helios implements the controller as a tree of nodes to avoid
+the scalability bottleneck. The controller requires four pieces
+of information from each PE: v.codd, v.stage, v.busy and
+the presence of outstanding messages of the system. Each
+leaf node of the tree is directly connected with a subset of
+PEs. We can consider these PEs as the children of the leaf
+node. Each node in the tree gathers vertex information from
+its children and reports it to the parent. With information
+from all vertices, the root node runs algorithm 6 and decides
+whether to advance the stage.
+We leave height, branching factor and the subset of PEs
+connected to each leaf node as implementation choices. The
+necessary requirement is that the controller should not slow
+down the overall design.
+5.4
+FPGA Implementation
+We next describe an implementation of Helios targeting a
+single FPGA. We choose FPGA for two reasons. It supports
+massively parallel logic, which is essential as the number of
+PEs grows proportional to d3 in our distributed UF design.
+Moreover, it allows deterministic latency for each operation,
+which facilitates synchronizing all the PEs.
+Figure 4 shows a minimal diagram of a PE and a controller
+in the FPGA implementation.
+Controller: Since we only use a single FPGA and evaluate
+with d below 20, a single node controller suffices.
+Directional links: We implement the directional links as
+first-in-fist-out (FIFO) buffers, which are mapped by Xilinx
+Vivado to LUT based RAMs. We choose the buffer size of four
+because our evaluations in §6.2 show that increasing the buffer
+size beyond four does not improve decoding time. Reducing
+the buffer size below four slightly increases decoding time
+(by 0.01%) while using the same number of LUTs as memory
+as a buffer of size four (up to 32).
+Shared memory:
+We implement all shared memories as
+FPGA registers, i.e., reg in Verilog. FPGA registers by de-
+sign guarantee that a read returns the last written value. In
+order to ensure that the S4 memory has a single writer, we
+modify the PE logic as shown in Figure 5. Compare and up-
+date operation (L15) is implemented in the PE that the S3
+memory resides in, and the PE increases e.growth by two if
+both endpoints of the edge have v.odd as true.
+Detecting outstanding messages:
+Each PE updates its
+busy state based on pending messages in addition to condi-
+tions in L33 and L50 as shown in the code snippet in Figure 6.
+Adder
+odd[0]
+odd[1]
+Min
+w
+2x1 
+Mux
+==
+stage
+growing
+ grow 
+D
+Q
+Q
+growth
+clk
+reg growth;
+always@(posedge clk)
+if(stage == growing)
+growth <= ‘MIN(growth
++ odd[0] + odd[1], w);
+Figure 5: Circuit diagram of grow sub-module and Verilog implementation.
+This implements the atomic compare and update operation in L15 as part
+of the PE module. odd[0] and odd[1] represents the odd state of the two
+incident PEs of the edge.
+==
+w
+==
+cid[0]
+u∈nb
+cid[i]
+==
+stage
+merging
+==
+odd[0]
+odd[i]
+==
+stage
+syncing
+growth[i]
+assign logic_busy[i] =
+(growth[i] == w &&
+stage == merging
+&& cid[i] != cid[0]) ||
+(growth[i] == w &&
+stage == syncing
+&& odd[i] != odd[0]);
+Figure 6: Circuit diagram of logic_busy sub-module and Verilog imple-
+mentation. The sub-module is implemented per each adjacent PE which are
+indexed from 1 to the number of edges. The variables odd[0] and cid[0]
+represent the odd and cid of the PE, while odd[i], cid[i] and growth[i]
+represent the corresponding values for the ith adjacent PE and the edge
+connecting them.
+The sub-circuit logic_busychecks for the conditions in L33
+and L50 for each incident edge. In our FPGA implementation
+of FIFO buffers, when a value is written to a FIFO (using
+the we signal), nonempty state of the FIFO will be true in
+the next cycle. This results in at least one PE having busy
+as true when there are outstanding messages in the system.
+The controller reads busy every clock cycle to identify the
+completion of a stage.
+In total, our implementation contains approximately 6000
+lines of Verilog code. The code is available at [19].
+On the ZCU106 FPGA development board [11], we are
+able to support the distributed UF decoder with d up to 7,
+due to resource limits. Table 1 shows the resource usage for
+various d. While the numbers of vertices and edges grow
+by O(d3), the resource usage grows faster for the following
+reasons. First, resource usage by a PE grows due to the in-
+crease of bitwidth required for v.id, and v.cid. A PE for d = 7
+with six adjacent PEs requires 182 LUTs and a similar PE for
+d = 3 requires only 127 LUTs. Second, PEs on the surface
+of the three-dimensional array as shown in Figure 3 use less
+resources than those inside because the latter have more in-
+cident edges. When d increases a higher portion of PEs are
+inside the array.
+We find that LUTs are the most critical resource in the
+FPGA for our design. It may be possible to run a design with
+d = 15 on a Xilinx VU19 FPGA [20], which currently has
+the highest number of LUTs among commercially available
+FPGAs at the time of this writing.
+Existing commercial FPGAs like ZCU106 often dedicate
+a lot of silicon to digital signal processing (DSP) units and
+block RAMs (BRAMs). However, our design does not use
+8
+
+Table 1: Resource usage of Helios on ZCU106 FPGA board for various code
+distances
+d
+# of LUTs
+# of
+registers
+as logic
+as memory (FIFOs)
+3
+2419
+608
+1187
+5
+18655
+3236
+7189
+7
+61793
+12636
+27664
+any DSPs because it only requires comparison operators and
+fixed point additions. Our design does not use any BRAMs
+because the FIFOs have a depth of four and can be efficiently
+implemented using LUTs. Each BRAM tile in Xilinx has a
+default size of 18 Kbits and using BRAM for FIFOs would re-
+sult in significant unused space in each BRAM tile. Therefore,
+an ideal FPGA designed to run our distributed UF decoder
+would be simpler than current large FPGAs, as it would only
+need a large number of LUTs, no DSP units and a limited
+amount of BRAM.
+6
+Evaluation
+The main objective of our evaluation is to assess the scalability
+of our distributed UF implementation. To that end, we first
+describe our methodology and then show that the latency of
+our implementation grows sub-linearly with respect to the
+surface code size d.
+6.1
+Methodology
+For speed, we measure the number of cycles required to de-
+code a syndrome. To evaluate correctness, we compare the
+result of clustering generated by our distributed UF decoder
+with the clustering generated by the original UF decoder. We
+compare clusters because the original UF decoder and ours
+only differ in the clustering process. This shows that both
+decoders generate identical clusters in all cases tested, con-
+firming the correctness of our decoder. In the rest of our
+evaluation, we will focus only on the speed of the distributed
+UF decoder and not on the accuracy of its results.
+Experimental Setup
+We use two setups to evaluate our
+FPGA implementation. The primary setup is a Xilinx
+ZCU106 FPGA development board [11], which is capable
+of handling surface codes with d up to 7. As an alternative
+setup, we run our implementation on the Xilinx Vivado simu-
+lator [12], which emulates the behavior of FPGA in a cycle-
+accurate manner, allowing us to evaluate the performance of
+our implementation for surface codes of any size. We simu-
+lated up to d = 15 as this is the upper bound of d possible in
+the largest FPGA currently available [20].
+We also compare the results obtained from the Vivado
+simulator with those obtained from the FPGA development
+board for surface code sizes 7 and smaller, to gain confidence
+in the correctness of the simulator itself.
+Noise Model
+We use the phenomenological noise model [1]
+that accounts for errors in both data and ancilla qubits. As
+decoding for X-errors and Z-errors are independent and iden-
+tical, we only focus on decoding X-errors in the evaluation.
+To emulate noise, we independently flip each qubit with
+a probability of p (the physical error rate) between every
+two measurement rounds. This is a widely used approach
+by prior QEC decoders [7, 8, 18]. We then generate the
+syndrome from the physical errors and provides it as input to
+our decoder.
+For most of our experiments, we use as default p = 0.001,
+like other works [7]. This value is reasonable for surface
+codes, as p should be sufficiently below the threshold (at least
+ten times lower) to exponentially reduce errors. We note that
+the UF decoder has a threshold of p = 0.026, calculated by
+Delfosse and Nickerson [10].
+6.2
+Decoding Time
+We experimentally show how average time for decoding
+grows with the size of the surface code. Additionally, we
+show the effect of noise and buffer size on the average time.
+Average time
+To demonstrate the scalability of our algo-
+rithm with respect to the size of the surface code, we plot
+the average time for decoding against the size of the surface
+code. In Figure 7 (left) the y-axis shows the average FPGA
+clock cycle count and the x-axis shows the distance (d) of the
+surface code. We obtained these values from running the dis-
+tributed UF decoder on the Vivado simulator where each data
+point represents the average of 1000 trials. We see that for all
+3 physical error rates we tested, average decoding time grows
+sub-linearly with respect to the surface code size, which sat-
+isfies the scalability criteria to avoid an exponential backlog.
+This implies that the average time to decode a measurement
+round reduces with increasing d as shown in Figure 7 (right).
+Distribution of decoding time
+To understand the growth
+of decoding time with respect to the code distance, in Fig-
+ure 8a we plot the distribution of decoding time for different
+code distances. The y-axis shows the FPGA clock cycle count
+and the x-axis shows the distance (d) of the surface code. We
+ran both our test setups for this experiment and the distribu-
+tion of FPGA clock cycle count for each surface code size is
+shown in green, while the distribution of clock cycle count
+on the Vivado simulator is shown in gray. The average cycle
+count is indicated with ×.
+Due to resource limitations on the ZCU106 FPGA, we
+are unable to run surface codes with d > 7 on the FPGA.
+9
+
+ 40
+ 60
+ 80
+ 100
+ 120
+ 140
+ 160
+ 180
+ 200
+ 220
+ 240
+ 1
+ 3
+ 5
+ 7
+ 9
+ 11
+ 13
+ 15
+ 17
+decoding time
+code distance (d)
+ p = 0.0005
+p = 0.001
+p = 0.005
+ 6
+ 8
+ 10
+ 12
+ 14
+ 16
+ 18
+ 20
+ 22
+ 1
+ 3
+ 5
+ 7
+ 9
+ 11
+ 13
+ 15
+ 17
+time per measurement round
+code distance (d)
+ p = 0.0005
+p = 0.001
+p = 0.005
+Figure 7: Average decoding time scales sub-linearly with d. We measure the average decoding time for 3 different noise levels using the Vivado simulator.
+(Left) The average decoding time in FPGA clock cycles. (Right) The average decoding time per measurement round in FPGA clock cycles. Average time per
+measurement round reducing continuously justifies that our decoder is scalable for large surface codes. We show the distributions separately in Figure 8a
+For d = 3,5 and 7, the results from the FPGA and those
+from the Vivado simulator agree. The statistical parameters
+such as mean, median, and percentile values(P25, P75, P90)
+differ between running on the FPGA and using the simulator
+by less than 1%. Only noticeable difference is the higher
+maximum observed value on the FPGA, which is caused
+by exponentially unlikely long error chains appearing when
+running for 108 trials in the FPGA. This justifies the use of
+the Vivado simulator to obtain results for large surface codes
+that cannot be mapped to the ZCU 106 FPGA board due to
+resource limitations.
+The key factor determining the decoding time is the number
+of iterations of growing, merging and syncing the distributed
+UF decoder requires. The peaks in the probability distribution
+for each distance in Figure 8a correspond to the number of
+iterations. The variation around each peak is caused by the
+delay due to routing messages. The number of iterations is
+related to the size of the largest cluster, which in turn corre-
+lates with the size of the longest error chain in the syndrome.
+As the size of the surface code increases, the probability of a
+longer error chain also increases, resulting in the probability
+distribution shifting to the right.
+Furthermore, as seen in Figure 8a, the distribution for each
+surface code size is right-skewed. For example, for d = 7,
+90% of trials required two iterations or fewer, which were
+completed within 140 cycles. In the same test, 99.99% of
+trials were completed within 237 cycles. Only a very small
+number of error patterns require long decoding times, corre-
+sponding to syndromes with long error chains. Since such
+syndromes occur rarely and have poor decoding accuracy
+even if the decoding time is bounded, the impact on accuracy
+will be minimal.
+Effect of physical error rate
+To understand the effect of
+the physical error rate on decoding time, in Figure 8b we plot
+the distribution of latency for three different noise levels. We
+obtained this distribution by running on the ZCU106 FPGA
+with 108 trials. The y-axis shows the FPGA clock cycle count
+and the x-axis shows the physical error rate.
+As the noise level increases, the probability distribution
+of latency shifts to the right. This is caused by the increased
+probability of a longer error chain when the physical error rate
+increases, which in turn requires more iterations to decode. As
+a result, the average decoding time increases with the physical
+error rate.
+Effect of buffer size
+To measure the impact of the buffer
+size on decoding time, we varied the buffer size and analyzed
+the latency distribution. In Figure 8c, the x-axis shows the cy-
+cle count and the y-axis shows the cumulative distribution of
+the latency. We varied the buffer size from 1 to 32. Our results
+showed that there was no noticeable difference in latency with
+respect to the buffer size. The obtained results were identical
+for all buffer sizes above 4 and showed a slowdown of less
+than 0.01% for buffer sizes of 1 and 2. This indicates that
+the communication overhead in our design is minimal for the
+average case
+We can explain this result using statistics on the number of
+messages generated. For example, when the physical error rate
+is 0.001 and d = 7, 97.7% of trials are statistically unaffected
+by the buffer size. This includes 46% of trials resulting in
+fully non-trivial syndromes, 47.6% of trials resulting in a
+single qubit error in each cluster, and 4.1% of trials resulting
+in a chain of two qubit errors. In all of these cases, at most a
+single message is generated in each cluster, making the buffer
+size irrelevant. In the remaining 2.3% of trials, the buffer size
+will only affect the results if error chains occur close to each
+other and share a common link in their message paths. In our
+experiments, such congestion occurred in less than 0.1% of
+runs. Therefore, the buffer size can be reduced without any
+significant impact on average decoding time.
+10
+
+ 0
+ 50
+ 100
+ 150
+ 200
+ 250
+ 300
+ 350
+ 400
+ 1
+ 3
+ 5
+ 7
+ 9
+ 11
+ 13
+ 15
+ 17
+decoding time
+distance (d)
+(a) Simulator and implementation results agree
+ 0
+ 100
+ 200
+ 300
+ 400
+ 500
+ 600
+ 700
+0.0005
+0.001
+0.005
+decoding time
+physical error rate (p)
+(b) Decoding time grows with physical error rate.
+ 0
+ 50
+ 100
+ 150
+ 200
+ 250
+ 300
+ 350
+ 400
+1
+2
+4
+8
+16
+decoding time
+buffer size
+(c) Buffer size does not matter for decoding time.
+Figure 8: Distribution of decoding time with the average marked with ×. For
+each error rate we ran 108 trials. Results from implementation with Xilinx
+ZCU 106 FPGA are in green; those from Xilinx Vivado simulator gray. By
+default d = 7, p = 0.001.
+6.3
+Comparison with related work
+Our empirical results as shown in Figure 8a suggest that He-
+lios has a lower asymptotic complexity than any existing
+MWPM or UF implementation for which asymptotic com-
+plexities are available, e.g., [10, 17]. Indeed, the empirical
+results suggest that our decoder has a sub-linear time complex-
+ity: the decoding time per round decreases with the number
+of measurement rounds, which has never been achieved be-
+fore. This implies that Helios can support arbitrarily large
+d as rate of decoding will always be faster than the rate of
+measurement.
+Das et al [7] calculate an average latency for their AFS de-
+coder based on memory access cycles and assuming a design
+running at 4 GHz. As the number of memory access cycles
+grows quadratically with d, the average decoding time per
+measurement round of AFS grows O(d2). Similarly, Ueno et
+al [9] estimate the decoding time of QECOOL from d = 5
+to d = 13 based on SPICE-level simulations with a clock
+frequency of 5 GHz. For the given range of d the decoding
+time per measurement round increases quadratically with d.
+In comparison, the decoding time of Helios decreases per
+measurement round.
+We should like to point out that AFS and QECOOL assume
+very high clock frequencies, which is key to their estimated
+low latency. For example, for d = 11, AFS and QECOOL
+respectively report latencies of 42 ns and 8.32 ns per measure-
+ment round. Helios, in contrast, requires 107 ns per measure-
+ment round with a 100 MHz clock. In terms of clock cycles,
+Helios requires on average 10.7 cycles for d = 11 surface
+code, lower than both AFS (168 cycles) and QECOOL (41
+cycles).
+To the best of our knowledge, LILLIPUT [6] is the only
+hardware decoder in literature that provides implementation-
+based results, for d = 5. The decoder has an average time of
+21 ns per measurement round, which is shorter than that of
+Helios for d = 5, i.e., 126 ns. However, as analyzed in §3,
+LILLIPUT is not scalable for d > 5. Our work, in contrast,
+has successfully demonstrated the implementation of a d = 7
+surface code on a ZCU106 FPGA with 120 ns per measure-
+ment round. The architecture of Helios can potentially support
+larger d using a larger FPGA, for example d = 15 for Xilinx
+VU19P [20], and even larger d using a network of FPGAs.
+7
+Conclusion
+We describe a distributed design of the Union Find decoder for
+quantum error-correcting surface codes and present Helios, a
+system architecture for realizing it. We report an FPGA-based
+implementation Helios. Using Xilinx Vivaldo cycle-accurate
+simulator, we demonstrate empirically that the average decod-
+ing time of Helios grows sub-linearly with d. Using a ZCU106
+FPGA, we implement the fastest decoding of distance 7 sur-
+face codes, which achieves 120ns average decoding time per
+measurement round. Helios is faster and more scalable than
+any reported implementation of surface code decoder. Our
+results suggest that by leveraging parallel hardware resources,
+Helios can avoid a growing backlog of syndrome measure-
+ments for arbitrarily large surface codes.
+Acknowledgments
+This work was supported in part by Yale University and NSF
+MRI Award #2216030.
+11
+
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+12
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+Astronomy & Astrophysics manuscript no. aanda
+©ESO 2023
+January 16, 2023
+New upper limits on low-frequency radio emission
+from isolated neutron stars with LOFAR
+I. Pastor-Marazuela1, 2, S. M. Straal3, J. van Leeuwen2, and V. I. Kondratiev2
+1 Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, PO Box 94249, 1090 GE Amsterdam, The
+Netherlands
+e-mail: ines.pastormarazuela@uva.nl
+2 ASTRON, the Netherlands Institute for Radio Astronomy, PO Box 2, 7790 AA Dwingeloo, The Netherlands
+3 NYU Abu Dhabi, PO Box 129188, Abu Dhabi, United Arab Emirates
+January 16, 2023
+ABSTRACT
+Neutron stars that show X-ray and γ-ray pulsed emission must, somewhere in the magnetosphere, generate electron-positron pairs.
+Such pairs are also required for radio emission, but then why do a number of these sources appear radio quiet? Here, we carried out
+a deep radio search towards four such neutron stars that are isolated X-ray/γ-ray pulsars but for which no radio pulsations have been
+detected yet. These sources are 1RXS J141256.0+792204 (Calvera), PSR J1958+2846, PSR J1932+1916 and SGR J1907+0919.
+Searching at lower radio frequencies, where the radio beam is thought to be wider, increases the chances of detecting these sources,
+compared to the earlier higher-frequency searches. We thus carried a search for periodic and single-pulse radio emission with the
+LOFAR radio telescope at 150 MHz. We used the known periods, and searched a wide range of dispersion measures, as the distances
+are not well constrained. We did not detect pulsed emission from any of the four sources. However, we put very constraining upper
+limits on the radio flux density at 150 MHz, of ≲ 1.4 mJy.
+Key words. Stars: neutron – pulsars: general
+1. Introduction
+Through their spin and magnetic field, neutron stars act as pow-
+erful cosmic dynamos that can generate a wide variety of electro-
+magnetic emission. There thus exist many subclasses of neutron
+stars, with different observed behavior. The evolutionary links
+between some of the classes are established, while for others
+these connections are currently unknown. The largest group in
+this varied population is formed by the regular rotation-powered
+radio pulsars. The fast spinning, high magnetic field influx to this
+group are the young pulsars. These show a high spin-down en-
+ergy loss rate ˙E, and a number of energetic phenomena such as
+radio giant pulse (GP) emission. The most extreme of these fast-
+spinning and/or high-field sources could potentially also power
+Fast Radio Bursts (FRBs; e.g. Pastor-Marazuela et al. 2022).
+On the long-period outskirts of the P- ˙P diagram, slowly-rotating
+pulsars (e.g. Young et al. 1999; Tan et al. 2018) and magnetars
+(e.g. Caleb et al. 2022; Hurley-Walker et al. 2022) sometimes
+continue to shine.
+Some neutron stars, however, only shine intermittently at ra-
+dio frequencies. The rotating radio transients (RRATs) burst very
+irregularly, and in the P- ˙P diagram most are found near the death
+line (Keane et al. 2011), between the canonical radio pulsars and
+magnetars. The exact evolutionary connection between RRATs
+and the steadily radiating normal pulsars is unclear, but studies
+suggest the presence of an evolutionary link between these dif-
+ferent classes (e.g. Burke-Spolaor 2012).
+Finally, populations of neutron stars exist that appear to not
+emit in radio at all: radio-quiet magnetars such as most anoma-
+lous X-ray pulsars (AXPs) and soft gamma repeaters (SGRs),
+X-ray dim isolated neutron stars (XDINSs; Haberl 2007), and
+γ-ray pulsars (e.g. Abdo et al. 2013). These are able to produce
+high-energy emission but are often radio quiet. (Gençali & Er-
+tan 2018) proposed RRATs can evolve into XDINSs through a
+fallback accretion disk, thus becoming radio quiet. However, the
+magnetar SGR 1935+2154 was recently seen to emit a bright ra-
+dio burst bridging the gap in radio luminosities between regular
+pulsars and FRBs (CHIME/FRB Collaboration 2020; Bochenek
+et al. 2020; Maan et al. 2022b). This suggests magnetars could
+explain the origin of some, if not all, extragalactic FRBs.
+Potentially, some of these could produce radio emission only
+visible at low radio frequencies. Detections of radio pulsations of
+the γ and X-ray pulsar Geminga, PSR J0633+1746, have been
+claimed at and below the 100 MHz observing frequency range
+(Malofeev & Malov 1997; Malov et al. 2015; Maan 2015), al-
+though a very deep search using the low frequency array (LO-
+FAR van Haarlem et al. 2013) came up empty (Ch. 6 in Coenen
+2013). Such low-frequency detections offer an intriguing possi-
+bility to better understand the radio emission mechanism of these
+enigmatic objects. Radio detections of a magnetar with LOFAR,
+complementary to higher-frequency studies such as Camilo et al.
+(2006) and Maan et al. (2022a) for XTE J1810−197, could of-
+fer insight into emission mechanisms and propagation in ultra-
+strong magnetic fields.
+XDINSs feature periods that are as long as those in magne-
+tars, but they display less extreme magnetic field strength. The
+XDINSs form a small group of seven isolated neutron stars that
+show thermal emission in the soft X-ray band. Since their dis-
+covery with ROSAT in the 1990s, several attempts were made
+to detect these sources at radio frequencies, but they were un-
+successful (e.g. Kondratiev et al. 2009). As those campaigns
+operated above 800 MHz, a sensitive lower-frequency search
+Article number, page 1 of 7
+arXiv:2301.05509v1  [astro-ph.HE]  13 Jan 2023
+
+A&A proofs: manuscript no. aanda
+could be opportune. It has been proposed (e.g. Komesaroff 1970;
+Cordes 1978) and observed (e.g. Chen & Wang 2014) that pul-
+sar profiles are usually narrower at higher frequencies and be-
+come broader at lower radio frequencies. This suggests the radio
+emission cone is broader at low frequencies, and sweeps across
+a larger fraction of the sky as seen from the pulsar. Additionally,
+radio pulsars often present negative spectral indices, and are thus
+brighter at lower frequencies (Bilous et al. 2016). If all neutron
+star radio beams are broader and brighter at lower frequencies,
+chances of detecting radio emission from γ and X-ray Isolated
+Neutron Stars (INSs) increase at the lower radio frequencies of-
+fered through LOFAR. The earlier observations that resulted in
+non-detections could then have just missed the narrower high-
+frequency beam, where the wider lower-frequency beam may, in
+contrast, actually enclose Earth. In that situation, LOFAR could
+potentially detect the source.
+Recently, a number of radio pulsars were discovered that
+shared properties with XDINSs and RRATs, such as soft
+X-ray thermal emission, a similar position in the P- ˙P dia-
+gram, and a short distance to the solar system. These sources,
+PSR J0726−2612 (Rigoselli et al. 2019) and PSR J2251−3711
+(Morello et al. 2020), support the hypothesis that XDINS are in-
+deed not intrinsically radio quiet, but have a radio beam pointed
+away from us. These shared properties could reflect a potential
+link between the radio and X-ray emitting pulsars with XDINSs
+and RRATs. A firm low-frequency radio detection of INSs would
+thus tie together these observationally distinct populations of
+neutron stars.
+In this work we present LOFAR observations of four INSs
+that brightly pulsate at X-ray or γ-ray energies, but have not been
+detected in radio. These sources are listed in Section 2, and their
+parameters are presented in Table 1.
+2. Targeted sources
+2.1. J1412+7922
+The INS 1RXS J141256.0+792204, dubbed "Calvera" and here-
+after J1412+7922, was first detected with ROSAT (Voges et al.
+1999) as an X-ray point source, and subsequently with Swift and
+Chandra (Rutledge et al. 2008; Shevchuk et al. 2009). X-ray ob-
+servations confirmed its neutron star nature through the detection
+of P ≃ 59 ms pulsations by Zane et al. (2011), and allowed for
+the determination of its spin-down luminosity ˙E ∼ 6 × 1035 erg
+s−1, characteristic age τc ≡ P/2 ˙P ∼ 3 × 105 years, and surface
+dipole magnetic field strength Bs = 4.4×1011 G by Halpern et al.
+(2013). Although these values are not unusual for a rotationally-
+powered pulsar, the source is not detected in radio (Hessels et al.
+2007; Zane et al. 2011) or γ-rays (Mereghetti et al. 2021). The
+X-ray emission can be modelled with a two-temperature black
+body spectrum (Zane et al. 2011), similar to other XDINS (Pires
+et al. 2014). However, J1412+7922 shows a spin period much
+faster than typically observed in XDINS. Since the source is lo-
+cated at high galactic latitudes and its inferred distance is rel-
+atively low (∼3.3 kpc; Mereghetti et al. 2021) the path through
+the interstellar medium is not long enough to explain the radio
+non-detections by high dispersion measure (DM) or scattering
+values.
+2.2. J1958+2846
+Discovered by Abdo et al. (2009) through a blind frequency
+search of Fermi-LAT γ-ray data, INS PSR J1958+2846, here-
+after J1958+2846, has shown no X-ray or radio continuum emis-
+sion counterpart so far (Ray et al. 2011; Frail et al. 2016).
+Arecibo observations have put very constraining upper limits of
+0.005 mJy at 1510 MHz (Ray et al. 2011). Searches for pulsa-
+tions from the source using the single international LOFAR sta-
+tion FR606 by Grießmeier et al. (2021) also found no periodic
+signal.
+The double-peaked pulse profile of J1958+2846 can be inter-
+preted as a broad γ-ray beam. The earlier higher-frequency radio
+non-detections could be due to a narrower radio beam and to an
+unfavourable rotation geometry with respect to the line of sight.
+If the radio beam is indeed wider at lower frequencies, LOFAR
+would have higher chances of detecting it. In that case, a setup
+more sensitive than the Grießmeier et al. (2021) single-station
+search is required.
+Modeling by Pierbattista et al. (2015) indicates that the γ-
+ray pulse profile of J1958+2846 can be well fitted by One Pole
+Caustic emission (OPC, Romani & Watters 2010, Watters et al.
+2009) or an Outer Gap model (OG, Cheng et al. 2000). In both
+cases, the γ-rays are generated at high altitudes above the NS
+surface. Each model constrains the geometry of the pulsar. For
+the OPC model, the angle between the rotation and magnetic
+axes α = 49◦, while the angle between the observer line-of-sight
+and the rotational axis ζ = 85◦. The OG model reports similarly
+large angles, with the NS equator rotating in the plane that also
+contains Earth, and an oblique dipole: α = 64◦, ζ = 90◦. If this
+model is correct, the low-frequency radio beam would thus need
+to be wider than ∼30◦ to encompass the telescope. That is un-
+commonly wide; only 8 out of the 600 pulsars in the ATNF cat-
+alogue that are not recycled and have a published 400 MHz flux,
+have a duty cycle suggestive of a beam wider than 30% (Manch-
+ester et al. 2005). As such a width is unlikely, a total-intensity
+detection would thus suggest to first order a geometry where α
+and ζ are closer than follows from Pierbattista et al. (2015), even
+if that suggestion would only be qualitative. Subsequent follow-
+up measurements of polarisation properties throughout the pulse,
+and fitting these to the rotating vector model (RVM; Radhakrish-
+nan & Cooke 1969), can quantify allowed geometries to within
+a relatively precise combinations of α and ζ. As a matter of fact,
+in a similar study on radio-loud γ-ray pulsars, Rookyard et al.
+(2015) already find that RVM fits suggest that the magnetic in-
+clination angles α are much lower than predicted by the γ-ray
+light curve models. This, in turn, affirms that deep radio searches
+can lead to detections even when the γ-ray light curves suggest
+the geometry is unfavorable.
+2.3. J1932+1916
+The INS PSR J1932+1916, hereafter J1932+1916, was dis-
+covered in Fermi-LAT data through blind searches with the
+Einstein@Home volunteer computing system (Clark et al.
+2017). J1932+1916 is the youngest and γ-ray brightest among
+the four γ-ray pulsars presented from that effort in (Pletsch et al.
+2013). The period is 0.21 s, the characteristic age is 35 kyr. Frail
+et al. (2016) find no continuum 150 MHz source at this position
+with GMRT at a flux density upper limit of 27 mJy beam−1, with
+1σ errors. If the flux density they find at the position of the pulsar
+is in fact the pulsed emission from J1932+1916, then a LOFAR
+periodicity search as described here should detect the source at
+a S/N of 15 if the duty cycle is 10%. Karpova et al. (2017) re-
+port on a potential pulsar wind nebula (PWN) association from
+Swift and Suzaku observations. However, no X-ray periodicity
+searches have been carried out before.
+Article number, page 2 of 7
+
+I. Pastor-Marazuela et al.: Upper limits on radio emission from INSs with LOFAR
+2.4. J1907+0919
+The Soft Gamma Repeater J1907+0919, also known as SGR
+1900+14, was detected through its bursting nature by Mazets
+et al. (1979). Later outbursts were detected in 1992 (Kouveliotou
+et al. 1993), 1998 (Hurley et al. 1999) and 2006 (Mereghetti
+et al. 2006). The August 1998 outburst allowed the detection of
+an X-ray period of ∼ 5.16 s, and thus confirmed the nature of
+the source as a magnetar (Hurley et al. 1999; Kouveliotou et al.
+1999). Frail et al. (1999) detected a transient radio counterpart
+that appeared simultaneous to the 1998 outburst, and they
+identified the radio source as a synchrotron emitting nebula.
+Shitov et al. (2000) claimed to have found radio pulsations at
+111 MHz from four to nine months after the 1998 burst, but the
+number of trials involved in the search, the small bandwidth
+of the system, and the low S/N of the presented plots, lead us
+to conclude the confidence level for these detections is low.
+No other periodic emission has been found at higher radio
+frequencies (Lorimer & Xilouris 2000; Fox et al. 2001; Lazarus
+et al. 2012).
+This paper is organised as follows: in Section 3 we explain
+how we used LOFAR (van Haarlem et al. 2013) to observe the
+sources mentioned above; in Section 4 we detail the data reduc-
+tion procedure, including the periodicity and the single pulse
+searches that we carried; in Section 5 we present our results,
+including the upper limit that we set on the pulsed emission; in
+Section 6 we discuss the consequences of these non-detections
+for the radio-quiet pulsar population, and in Section 7 we give
+our conclusions on this work.
+3. Observations
+We observed the four sources with the largest possible set of
+High Band Antennas (HBAs) that LOFAR can coherently beam
+form. Each observation thus added 22 HBA Core Stations, cov-
+ering 78.125 MHz bandwidth in the 110 MHz to 190 MHz
+frequency range (centered on 148.92 MHz), with 400 channels
+of 195 kHz wide. The LOFAR beam-forming abilities allow
+us to simultaneously observe different regions of the sky (van
+Leeuwen & Stappers 2010; Stappers et al. 2011; Coenen et al.
+2014). For our point-source searches of INSs, we used three
+beams per observation; one beam pointed to the source of in-
+terest, one on a nearby known pulsar, and one as a calibrator
+blank-sky beam to cross-check potential candidates as possibly
+arising from Radio Frequency Interference (RFI). We carried out
+observations between 16 January 2015 and 15 February 2015
+under project ID LC3_0361. We integrated for 3 hours on each
+of our sources. The data was taken in Stokes I mode. Since the
+periods of the γ-ray pulsars are known, the time resolution of
+each observation was chosen such to provide good coverage of
+the pulse period, at a sampling time between 0.16−1.3 ms. The
+observation setup is detailed in Table 1.
+4. Data reduction
+The data was pre-processed by the LOFAR pulsar pipeline after
+each observation (Alexov et al. 2010; Stappers et al. 2011) and
+stored on the LOFAR Long Term Archive2 in PSRFITS format
+1 After we completed the current manuscript as Pastor-Marazuela
+(2022, PhD Thesis, Ch. 2), Arias et al. (2022) posted a pre-print pre-
+senting partly the same data.
+2 LTA: https://lta.lofar.eu/
+(Hotan et al. 2004). The 1.5 TB of data was then transferred
+to one of the nodes of the Apertif real-time FRB search cluster
+ARTS (van Leeuwen 2014; van Leeuwen et al. 2022).
+We performed a periodicity search as well as a single-pulse
+search using Presto3 (Ransom 2001). The data was cleaned of
+RFI using first rfifind, and then removing impulsive and peri-
+odic signals at DM=0 pc cm−3. Next we searched the clean data
+for periodic signals and single pulses. We searched for counter-
+parts around the known P and ˙P of each pulsar. Additionally, we
+performed a full blind search in order to look for potential pul-
+sars in the same field of view, since many new pulsars are found
+at low frequencies (Sanidas et al. 2019) and chance discover-
+ies happen regularly (e.g., Oostrum et al. 2020). Since the DM
+of our sources is unknown, we searched over a range of DMs
+going from 4 pc cm−3 to 400 pc cm−3. The DM-distance rela-
+tion is not precise enough to warrant a much smaller DM range,
+even for sources for which a distance estimate exists; and a wider
+DM range allows for discovery of other pulsars contained in our
+field of view. The highest DM pulsar detected with LOFAR has
+a DM = 217 pc cm−3 (Sanidas et al. 2019). We thus searched
+up to roughly twice this value to make sure that any detectable
+sources were covered. We determined the optimal de-dispersion
+parameters with DDplan from Presto. The sampling time varia-
+tion between some of the four observations had a slight impact
+on the exact transitions of the step size but generally the data was
+de-dispersed in steps of 0.01 pc cm−3 up to DM = 100 pc cm−3;
+then by 0.03 pc cm−3 steps up to 300 pc cm−3 and finally using
+0.05 pc cm−3 steps.
+We manually inspected all candidates down to σ = 4, result-
+ing in ∼1400 candidates per beam. To verify our observational
+setup, we performed the same blind search technique to our test
+pulsars B1322+83 and B1933+16, which we detected. The test
+pulsar B1953+29 was not detected because the sampling time of
+the observation of J1958+2846 was not adapted to its ∼ 6 ms pe-
+riod. However, we were able to detect B1952+29 (Hewish et al.
+1968) in this same pointing. Even though it is located at >1◦
+from the targeted coordinates, it is bright enough to be visible as
+a side-lobe detection.
+The candidates from Presto’s single pulse search were fur-
+ther classified using the deep learning classification algorithm
+developed by Connor & van Leeuwen (2018), which has been
+verified and successful in the Apertif surveys (e.g. Connor et al.
+2020; Pastor-Marazuela et al. 2021). This reduced the number
+of candidates significantly by sifting out the remaining RFI. The
+remaining candidates were visually inspected.
+5. Results
+In our targeted observations we were unable to detect any
+plausible astronomical radio pulsations or single pulses. We
+determine new 150 MHz flux upper limits by computing
+the sensitivity limits of our observations. To establish these
+sensitivity limits, we apply the radiometer equation adapted to
+pulsars, detailed below. We determine the telescope parameters
+that are input to this equation by following the procedure4
+described in Kondratiev et al. (2016) and Mikhailov & van
+Leeuwen (2016). That approach takes into account the system
+temperature (including the sky temperature), the projection
+effects governing the effective area of the fixed tiles, and the
+amount of time and bandwidth removed due to RFI, to produce
+3 Presto: https://www.cv.nrao.edu/~sransom/presto/
+4 https://github.com/vkond/LOFAR-BF-pulsar-scripts/
+blob/master/fluxcal/lofar_fluxcal.py
+Article number, page 3 of 7
+
+A&A proofs: manuscript no. aanda
+Table 1. Parameters of the observed pulsars and observational setup of the observations in the LC3_036 proposal. The beam of each observation
+was centered in the reported pulsar coordinates. Listed in the bottom rows are the earlier periodicity and single pulse search limits. The upper
+limits from Frail et al. (2016) described in the main text are period-averaged flux densities and are not listed here. The last row lists the limits from
+the current work, for S/N=5, with errors of 50% (Bilous et al. 2016).
+J1412+7922
+J1958+2846
+J1932+1916
+J1907+0919
+Right ascension, α (J2000). . . . . . . . . . . . . .
+14 12 56
+19 58 40
+19 32 20
+19 07 14.33
+Declination, δ (J2000) . . . . . . . . . . . . . . . . .
++79 22 04
++28 45 54
++19 16 39
++09 19 20.1
+Period, P (s) . . . . . . . . . . . . . . . . . . . . . . . . . .
+0.05919907107
+0.29038924475
+0.208214903876
+5.198346
+Period derivative, ˙P (s s−1) . . . . . . . . . . . . . .
+3.29134×10−15
+2.12038×10−13
+9.31735×10−14
+9.2×10−11
+Epoch (MJD) . . . . . . . . . . . . . . . . . . . . . . . . .
+58150a
+54800b
+55214c
+53628d
+LOFAR ObsID . . . . . . . . . . . . . . . . . . . . . . . .
+L257877
+L258545
+L259173
+L216886
+Obs. date (MJD) . . . . . . . . . . . . . . . . . . . . . .
+57038
+57046
+57068
+56755
+Sample time (ms) . . . . . . . . . . . . . . . . . . . . .
+0.16384
+1.31072
+1.31072
+0.65536
+Test pulsar detected . . . . . . . . . . . . . . . . . . . .
+B1322+83
+B1952+29
+B1933+16
+B1907+10
+Periodic flux density (mJy @ GHz) . . . . . .
+<4 @ 0.385e
+<2.0 @ 0.15g
+<2.9 @ 0.15g
+50 @ 0.111h
+<0.05 @ 1.36f
+<0.005 @ 1.51b
+<0.075 @ 1.4c
+<0.4 @ 0.43i
+<0.3 @ 1.38e
+<0.3 @ 1.41i
+<0.012 @ 1.95 j
+LOFAR periodic sensitivity S lim,p (mJy). .
+0.26 ± 0.13
+0.53 ± 0.26
+0.73 ± 0.36
+1.39 ± 0.69
+LOFAR single pulse sensitivity S lim,sp (Jy)
+1.47 ± 0.73
+1.35 ± 0.68
+2.20 ± 1.10
+0.84 ± 0.82
+Notes. aBogdanov et al. (2019), bRay et al. (2011), cPletsch et al. (2013), dMereghetti et al. (2006), eHessels et al. (2007), f Zane et al. (2011),
+gGrießmeier et al. (2021), hShitov et al. (2000), iLorimer & Xilouris (2000), jLazarus et al. (2012)
+the overall observation system-equivalent flux density (SEFD).
+For the sensitivity limit on the periodic emission we use the
+following equation (see., e.g., Dewey et al. 1985):
+S lim,p = β
+Tsys
+G �np ∆ν tobs
+× S/Nmin ×
+�
+W
+P − W ,
+(1)
+where β ≲ 1 is a digitisation factor, Tsys (K) is the system temper-
+ature, G (K Jy−1) is the telescope gain, ∆ν (Hz) is the observing
+bandwidth, and tobs (s) is the observation time. P (s) represents
+the spin period, while W (s) gives the pulsed width assuming
+a pulsar duty cycle of 10%. To facilitate direct comparison of
+the periodic emission limits to values reported in e.g., Ray et al.
+(2011) and Grießmeier et al. (2021), we use a minimum signal-
+to-noise ration S/Nmin = 5. A more conservative option, given
+the high number of candidates per beam, would arguably be to
+use a limit of S/N=8. We did, however, review by eye all can-
+didates with S/N>4; and the reader can easily scale the reported
+sensitivity limits to a different S/N value.
+The sensitivity limit on the single pulse emission, S lim,sp, is
+computed as follows:
+S lim,sp = β
+Tsys
+G �np ∆ν tobs
+× S/Nmin ×
+�
+tobs
+W ,
+(2)
+where all variables are the same as in Equation 2. We searched
+for single pulses down to a signal-to-noise ratio S/Nmin = 7.
+We report these periodic and single pulse sensitivity limits,
+computed at the coordinates of the central beam of each obser-
+vation, in Table 1. Even though all observations are equally long,
+the estimated S lim,p values are different. That is mostly due to the
+strong dependence of the LOFAR effective area, and hence the
+sensitivity, on the elevation.
+In Fig. 1, we compare our upper limits to those established in
+previous searches, mostly using the same techniques. Our upper
+limit on the flux of J1907+0919 is ∼50× deeper than the claimed
+1998-1999 detections, at the same 3-m wavelength, with BSA
+(Shitov et al. 2000). Other searches were generally undertaken
+at higher frequencies (Hessels et al. 2007; Zane et al. 2011; Ray
+et al. 2011; Pletsch et al. 2013; Grießmeier et al. 2021). If we
+assume that these four pulsars have radio spectra described by
+a single power-law S ν ∝ να with a spectral index of α = −1.4
+(Bates et al. 2013; Bilous et al. 2016), the upper limits we present
+here for J1412+7922 and J1932+1916 are the most stringent so-
+far for any search. The upper limits on J1958+2846 (Arecibo;
+Ray et al. 2011) and J1907+0919 (GBT; Lazarus et al. 2012)
+are a factor of 2–3 more sensitive than ours. However, pulsars
+present a broad range of spectral indices. If we take the mean
+±2σ measured by Jankowski et al. (2018), spectral indices can
+vary from −2.7 to −0.5. The flux upper limits we measure would
+be the deepest assuming a −2.7 spectral index, but the shallowest
+at −0.5.
+6. Discussion
+6.1. Comparison to previous limits
+For J1958+2846 and J1932+1916, we can make a straightfor-
+ward relative comparisons between our results presented here
+and the existing limit at 150 MHz, from the single-station LO-
+FAR campaign by Grießmeier et al. (2021). Our 22 Core Sta-
+tions are each 1/4th of the area of the FR606 station and are co-
+herently combined, leading to a factor
+Acore
+AFR606 = 22
+4 difference in
+area A for the radiometer equation and S lim. The integration time
+t of 3 h is shorter than the FR606 total of 8.3 h (J1958+2846) and
+4.1 h (J1932+1916), leading to a factor
+�
+tcore
+tFR606 =
+�
+3
+8.3 in the ra-
+diometer equation. Other factors such as the sky background and
+the influence of zenith angle on the sensitivity should be mostly
+the same for both campaigns. Our S lim is thus 22
+4
+�
+3
+8.3 = 3.3
+times deeper than the Grießmeier et al. (2021) upper limit for
+Article number, page 4 of 7
+
+I. Pastor-Marazuela et al.: Upper limits on radio emission from INSs with LOFAR
+0.1
+0.15
+0.4
+1.0
+1.4
+2.0
+Frequency (GHz)
+10−3
+10−2
+10−1
+100
+101
+102
+Flux density (mJy)
+J1412+7922
+J1958+2846
+J1932+1916
+J1907+0919
+Fig. 1. Flux density upper limits of this work at 150 MHz (filled sym-
+bols) with S/N = 5 for comparison to earlier searches of the same
+sources (empty symbols). Solid lines going through our upper limit es-
+timates with spectral index α = −1.4 are overlaid to show the scaling of
+our sensitivity limits. Our limits are plotted slightly offset from the 150
+MHz observing frequency (dashed line) for better visibility. The faded
+green marker for SGR J1907+0919 represents the claimed detection
+from Shitov et al. (2000).
+J1958+2846, and 4.7 times for J1932+1916. Those factors are
+in good agreement with the actual limits listed in Table 1.
+In Bilous et al. (2016), they measured the mean flux den-
+sity S mean of 158 pulsars detected with LOFAR, where S lim,p =
+S mean × √W/(P − W) = S mean/3. Compared to those LOFAR
+detections, our upper limit on J1412+7922 is deeper than all 158
+sources (100%), J1958+2846 is deeper than 156 sources (99%),
+J1932+1916 is deeper than 144 sources (93%), and J1907+0919
+is deeper than 109 sources (69%). The flux upper limits we have
+set on each of the sources in our sample are some of the deep-
+est compared to other LOFAR radio pulsar detections. Longer
+observing times are thus unlikely to result in a detection or im-
+prove our flux upper limits. Additional follow up would only be
+constraining with more sensitive radio telescopes.
+6.2. Emission angles and intensity
+Different pulsar emission mechanism models exist that predict
+radio and γ-ray emission to be simultaneously formed in the
+pulsar magnetosphere. The emission sites are not necessarily co-
+located, though. The periodic radio emission is generally thought
+to be formed just above the polar cap. The high-energy polar cap
+(PC) model next assumes that the γ-ray emission is also pro-
+duced near the surface of the NS, and near the magnetic polar
+caps. In the outer magnetosphere emission models, such as the
+Outer Gap (OG) or the One Pole Caustic (OPC) models, on the
+other hand, the γ-ray emission is produced high up in the mag-
+netosphere of the NS, within the extent of the light cylinder.
+For the sources in our sample, specific high-energy geome-
+try models have only been proposed for J1958+2846 (Pierbat-
+tista et al. 2015). A detection could have confirmed one of these
+(Sect. 2.2). But also for our sample in general, conclusions can
+be drawn from the non detections. The two general high-energy
+model classes mentioned above predict different, testable beam
+widths. Our radio non-detections, when attributed to radio beams
+that are not wide enough to encompass Earth, favor outer mag-
+netospheric models (see, e.g., Romani & Watters 2010). That is
+because in the OG/OPC models, the γ-ray beam (which is de-
+tected for our sources) is much broader than the radio beam. The
+radio beam, being much narrower, is unlikely cut through our
+line of sight. Such a model class is thus more applicable than
+one where the radio and high-energy beam are of similar angu-
+lar size, such as the PC model (or, to a lesser extent, the slot
+gap model; Muslimov & Harding 2003; Pierbattista et al. 2015).
+In that case, detections in both radio and high-energy would be
+more often expected. Our results thus favor OG and OPC models
+over PC models for high-energy emission.
+Note that while it is instructive to discuss the coverage of the
+radio pulsar beam in binary terms – it either hits or misses Earth
+– this visibility is not that unambiguous in practice. The beam
+edge is not sharp. In a beam mapping experiment enabled by the
+geometric precession in PSR J1906+0745 (van Leeuwen et al.
+2015), the flux at the edge of the beam is over 100× dimmer
+than the peak, but it is still present and detectable (Desvignes
+et al. 2019). Deeper searches thus continue to have value, even
+if non-detections at the same frequency already exist.
+That said, the detection of PSR J1732−3131 only at 327 and
+potentially even 34 MHz (Maan & Aswathappa 2014) shows that
+emission beam widening (or, possibly equivalently, a steep spec-
+tral index) at low frequencies is a real effect, also for γ-ray pul-
+sars.
+6.3. Emission mechanism and evolution
+Most models explain the radio quietness of an NS through a
+chance beam misalignment, as above. It could, of course, also
+be a more intrinsic property. There are at least two regions in the
+P- ˙P diagram where radio emission may be increasingly hard to
+generate.
+The first parameter space of interest is for sources close to the
+radio death line (Chen & Ruderman 1993). XDINSs are prefer-
+ably found there, which suggests these sources are approach-
+ing, in their evolution, a state in which radio emission gener-
+ally ceases. From what we see in normal pulsars, the death line
+represents the transition into a state in which electron-positron
+pair formation over the polar cap completely ceases. Once the
+pulsar rotates too slowly to generate a large enough potential
+drop over the polar cap, required for this formation, the radio
+emission turns off (Ruderman & Sutherland 1975). The high-
+energy emission also requires pair formation, but these could
+occur farther out. We note that polar cap pair formation can con-
+tinue at longer periods, if the NS surface magnetic field is not
+a pure dipole. With such a decreased curvature radius, the NS
+may keep on shining. Evidence for such higher-order fields is
+present in a number of pulsars, e.g., PSR J0815+0939 (Szary
+& van Leeuwen 2017) and PSR B1839−04 (Szary et al. 2020).
+This would also influence the interpretation of any polarization
+information, as the RVM generally assumes a dipole field.
+None of the sources in our sample are close to this death
+line (See Fig. 2), but SGR J1907+0919 is beyond a different,
+purported boundary: the photon splitting line (Baring & Hard-
+ing 2001). In pulsars in that second parameter space of inter-
+est, where magnetic fields are stronger than the quantum critical
+field, of 4.4 × 1013 G (Fig. 2), pair formation cannot compete
+with magnetic photon splitting. Such high-field sources could
+then be radio quiet but X-ray or γ-ray bright. We mark the criti-
+cal field line for a dipole in Fig. 2, but note, as Baring & Harding
+(2001) do, that higher multipoles and general relativistic effects
+can subtly change the quiescence limit on a per-source basis.
+That said, given its spindown dipole magnetic field strength of
+7 × 1014 G, our non-detection of SGR J1907+0919 supports the
+existence of this limit.
+Article number, page 5 of 7
+
+A&A proofs: manuscript no. aanda
+Death line
+1010 G
+1011 G
+1012 G
+1013 G
+Photon splitting line
+10−3
+10−2
+10−1
+100
+101
+Period (s)
+10−23
+10−21
+10−19
+10−17
+10−15
+10−13
+10−11
+10−9
+Period Derivative
+Magnetars
+Binary
+Radio-IR Emission
+”Radio-Quiet”
+RRAT
+XDINS
+Fig. 2. P − ˙P diagram showing the location of the sources presented in
+this work. All pulsars from the ATNF Pulsar Catalogue (Manchester
+et al. 2005) are shown as grey dots, with different pulsar classifica-
+tions encircled by different symbols. The sources discussed in this work
+are shown as black stars, from left to right: J1412+7922, J1932+1916,
+J1932+1916, and J1907+0919. The orange shaded region is delimited
+by the death line, while the green shaded region is delimited by the
+photon splitting line. Plot generated with psrqpy (Pitkin 2018).
+6.4. Propagation effects
+While the emission beam widening and the negative spectral
+index provide potential advantages when searching for pulsars
+at low frequencies, some propagation effects such as disper-
+sion and scattering intensify there, impeding detection of cer-
+tain sources. The largest pulsar DM detected with LOFAR is
+217 pc cm−3, while many galactic pulsars are known to have
+DM>1000 pc cm−3. Although the sources studied in this work
+do not have radio detections and thus no known DM, we can
+estimate this DM if a hydrogen column density NH was mea-
+sured from soft X-ray detections. He et al. (2013) find a cor-
+relation between NH and DM as follows: NH (1020 cm−2) =
+0.30+0.13
+−0.09 DM (pc cm−3).
+While J1958+2846 and J1932+1916 have only been de-
+tected in γ-rays, J1412+7922 and J1907+0919 have soft X-
+ray detections where NH has been measured. For J1907+0919,
+Kouveliotou et al. (1999) measured a large NH value of
+3.4 − 5.5 × 1022 cm−2. The correlation suggests a DM of
+1100−1800 pc cm−3. At such a large DM the detection limit of
+LOFAR is severly impacted. Because J1907+0919 is a very slow
+rotator, the intra channel dispersion delay still only becomes or
+order 10% of the period, which means peridiocity searches could
+in principle still detect it; but the flux density per bin is of course
+much decreased when the pulse is smeared out over 100s of time
+bins.
+In contrast, Shevchuk et al. (2009) reported a measured NH =
+3.1 ± 0.9 × 1020 cm−2 for J1412+7922. We thus estimate its DM
+to be in the range 5–15 pc cm−3. This low DM would have easily
+been detected with LOFAR.
+7. Conclusion
+We have conducted deep LOFAR searches of periodic and
+single-pulse radio emission from four isolated neutron stars. Al-
+though we validated the observational setup with the detection of
+the test pulsars, we did not detect any of the four targeted pulsars.
+This can be explained with an intrinsic radio-quietness of these
+sources, as was previously proposed. It could also be caused by
+a chance misalignment between the radio beam and the line of
+sight.
+With the new upper limits, we can rule out the hypothesis
+that INSs had not been previously detected at radio frequencies
+around 1 GHz, because of a steeper spectrum than that of regu-
+lar radio pulsars. Since radio emission from magnetars has been
+detected after high energy outbursts (e.g. Maan et al. 2022b),
+additional radio observations of J1907+0919 if the source reac-
+tivates might be successful at detecting single pulse or periodic
+emission in the future.
+Acknowledgements. This research was supported by the Netherlands Research
+School for Astronomy (‘NOVA5-NW3-10.3.5.14’), the European Research
+Council under the European Union’s Seventh Framework Programme (FP/2007-
+2013)/ERC Grant Agreement No. 617199 (‘ALERT’), and by Vici research pro-
+gramme ‘ARGO’ with project number 639.043.815, financed by the Dutch Re-
+search Council (NWO). We further acknowledge funding from National Aero-
+nautics and Space Administration (NASA) grant number NNX17AL74G issued
+through the NNH16ZDA001N Astrophysics Data Analysis Program (ADAP) to
+SMS. This paper is based (in part) on data obtained with the International LO-
+FAR Telescope (ILT) under project code LC3_036 (PI: van Leeuwen). LOFAR
+(van Haarlem et al. 2013) is the low frequency array designed and constructed
+by ASTRON. It has observing, data processing, and data storage facilities in
+several countries, that are owned by various parties (each with their own fund-
+ing sources), and that are collectively operated by the ILT foundation under a
+joint scientific policy. The ILT resources have benefitted from the following re-
+cent major funding sources: CNRS-INSU, Observatoire de Paris and Université
+d’Orléans, France; BMBF, MIWF-NRW, MPG, Germany; Science Foundation
+Ireland (SFI), Department of Business, Enterprise and Innovation (DBEI), Ire-
+land; NWO, The Netherlands; The Science and Technology Facilities Council,
+UK; Ministry of Science and Higher Education, Poland.
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+Under review
+LEVERAGING THE THIRD DIMENSION
+IN CONTRASTIVE LEARNING
+Sumukh K Aithal 1, Anirudh Goyal 1, 4, Alex Lamb 2, Yoshua Bengio 1, Michael Mozer 3
+1 Mila, Universit´e de Montr´eal, 2 Microsoft Research, NYC, 3 Google Research, Brain Team
+4 DeepMind
+ABSTRACT
+Self-Supervised Learning (SSL) methods operate on unlabeled data to learn robust
+representations useful for downstream tasks. Most SSL methods rely on augmen-
+tations obtained by transforming the 2D image pixel map. These augmentations
+ignore the fact that biological vision takes place in an immersive three-dimensional,
+temporally contiguous environment, and that low-level biological vision relies
+heavily on depth cues. Using a signal provided by a pretrained state-of-the-art
+monocular RGB-to-depth model (the Depth Prediction Transformer, Ranftl et
+al., 2021), we explore two distinct approaches to incorporating depth signals into
+the SSL framework. First, we evaluate contrastive learning using an RGB+depth
+input representation. Second, we use the depth signal to generate novel views
+from slightly different camera positions, thereby producing a 3D augmentation
+for contrastive learning. We evaluate these two approaches on three different SSL
+methods—BYOL, SimSiam, and SwAV—using ImageNette (10 class subset of
+ImageNet), ImageNet-100 and ImageNet-1k datasets. We find that both approaches
+to incorporating depth signals improve the robustness and generalization of the
+baseline SSL methods, though the first approach (with depth-channel concatena-
+tion) is superior. For instance, BYOL with the additional depth channel leads
+to an increase in downstream classification accuracy from 85.3% to 88.0% on
+ImageNette and 84.1% to 87.0% on ImageNet-C.
+1
+INTRODUCTION
+Biological vision systems evolved in and interact with a three-dimensional world. As an individual
+moves through the environment, the relative distance of objects is indicated by rich signals extracted
+by the visual system, from motion parallax to binocular disparity to occlusion cues. These signals play
+a role in early development to bootstrap an infant’s ability to perceive objects in visual scenes (Spelke,
+1990; Spelke & Kinzler, 2007) and to reason about physical interactions between objects (Baillargeon,
+2004). In the mature visual system, features predictive of occlusion and three-dimensional structure
+are extracted early and in parallel in the visual processing stream (Enns & Rensink, 1990; 1991), and
+early vision uses monocular cues to rapidly complete partially-occluded objects (Rensink & Enns,
+1998) and binocular cues to guide attention (Nakayama & Silverman, 1986). In short, biological
+vision systems are designed to leverage the three-dimensional structure of the environment.
+In contrast, machine vision systems typically consider a 2D RGB image or a sequence of 2D RGB
+frames to be the relevant signal. Depth is considered as the end product of vision, not a signal that
+can be exploited to improve visual information processing. Given the bias in favor of end-to-end
+models, researchers might suppose that if depth were a useful signal, an end-to-end computer vision
+system would infer depth. Indeed, it’s easy to imagine the advantages of depth processing integrated
+into the visual information processing stream. For example, if foreground objects are segmented from
+the background scene, neural networks would not make the errors they often do by using short-cut
+features to classify (e.g., misclassifying a cow at the beach as a whale) (Geirhos et al., 2020).
+In this work, we take seriously the insight from biological vision that depth signals are extracted
+early in the processing stream, and we explore how depth signals might support computer vision. We
+assume the availability of a depth signal by using an existing state-of-the-art monocular RGB-to-depth
+extraction model, the Dense Prediction Transformer (DPT) (Ranftl et al., 2021).
+1
+arXiv:2301.11790v1  [cs.CV]  27 Jan 2023
+
+Under review
+Augmentation
+Contrastive 
+Self-Supervised 
+Learning Method
+Depth 
+Estimation
+Depth 
+Estimation
+R
+G
+B
++
+D
+R
+G
+B
++
+D
+Figure 1: Improving Self-Supervised Learning by concatenating an input channel with estimated
+depth to the RGB input. Depth is estimated from both an original image and an augmentation, and
+the resulting 4-channel inputs are used to produce the representation. Incorporating the depth channel
+improves downstream accuracy in a variety of SSL techniques, with the largest improvements on
+challenging corrupted benchmarks. (Teaser results are shown. Complete results in Tables 1, 2, 3)
+We focus on using the additional depth information for self-supervised representation learning. SSL
+aims to learn effective representations from unlabelled data that will be useful for downstream tasks
+(Chen et al., 2020a). We investigate two specific hypotheses. First, we consider directly appending
+the depth channel to the RGB and then use the RGB+D input directly in contrastive learning (Fig. 1).
+Second, we consider synthesizing novel image views from the RGB+D representation using a recent
+method, AdaMPI (Han et al., 2022) and treating these synthetic views as image augmentations for
+contrastive learning (Fig. 2).
+Prior work has explored the benefit of depth signals in supervised learning for specific tasks like
+object detection and semantic segmentation (Cao et al., 2016; Hoyer et al., 2021; Song et al., 2021;
+Seichter et al., 2021). Here, we pursue a similar approach in contrastive learning, where the goal is to
+learn robust, universal representations that support downstream tasks. To the best of our knowledge,
+only one previous paper has explored the use of depth for contrastive learning (Tian et al., 2020). In
+their case, ground truth depth was used and it was considered as one of many distinct “views” of the
+world. We summarize our contributions below:
+• Motivated by biological vision systems, we propose two distinct approaches to improving SSL
+using a (noisy) depth signal extracted from a monocular RGB image. First, we concatenate the
+derived depth map and the image and pass the four-channel RGB+D input to the SSL method.
+Second, we use a single-view view synthesis method that utilizes the depth map as input to generate
+novel 3D views and provides them as augmentations for contrastive learning.
+• We show that both of these approaches improve the performance of three different contrastive
+learning methods (BYOL, SimSiam, and SwAV) on ImageNette, ImageNet-100 and large-scale
+ImageNet-1k datasets. Our approaches can be integrated into any contrastive learning framework
+without incurring any significant computational cost and trained with the same hyperparameters
+as the base contrastive method. We achieve a 2.8% gain in the performance of BYOL with the
+addition of depth channel on ImageNette dataset.
+• Both approaches also yield representations that are more robust to image corruptions than the
+baseline SSL methods, as reflected in performance on ImageNet-C and ImageNet-3DCC. On the
+large-scale ImageNet-100 dataset, SimSiam+Depth outperforms base SimSiam model by 4% in
+terms of corruption robustness.
+2
+RELATED WORK
+Self-Supervised Learning. The goal of self-supervised learning based methods is to learn a universal
+representation that can generalize to various downstream tasks. Earlier work on SSL relied on
+handcrafted pretext tasks like rotation (Gidaris et al., 2018), colorization (Zhang et al., 2016) and
+jigsaw (Noroozi & Favaro, 2016). Recently, most of the state-of-the-art methods in SSL are based on
+2
+
+Results on ImageNette
+100
+BYOL
+BYOL+Depth
+Accuracy (in %)
+90
+80
+70
+60
+Top-1 Acc
+IN-C
+IN-3DCCResults on ImageNet-100
+90
+SimSiam
+80
+SimSiam+Depth
+(%
+Accuracy (in 
+70
+60
+50
+40
+30
+Top-1 Acc
+IN-C
+IN-3DCCUnder review
+Single-View View Synthesis 
+Contrastive 
+Self-Supervised 
+Learning Method
+Augmentation
+Augmentation
+Sample one of K Views
+Figure 2: Novel views can be synthesized from a single image by using the estimated depth channel,
+which can be used as additional augmentations across a variety of contrastive self-supervised learning
+techniques. These improve results, especially on benchmarks with image corruptions. (Result
+highlights are shown. Complete results in Tables 1, 2, 3
+contrastive representation learning. The goal of contrastive representation learning is to make the
+representations between two augmented views of the scene similar and also to make representations
+of views of different scenes dissimilar.
+SimCLR (Chen et al., 2020b) showed that augmentations play a key role in contrastive learning and
+the set of augmentations proposed in the work showed that contrastive learning can perform really
+well on large-scale datasets like ImageNet. BYOL (Grill et al., 2020) is one of the first contrastive
+learning based methods without negative pairs. BYOL is trained with two networks that have the
+same architecture: an online network and a target network. From an image, two augmented views
+are generated; one is routed to the online network, the other to the target network. The model learns
+by predicting the output of the one view from the other view. SwAV (Caron et al., 2020) is an
+online clustering based method that compares cluster assignments from multiple views. The cluster
+assignments (or code) from one augmented view of the image is predicted from the other augmented
+view. SimSiam (Chen & He, 2021) explores the role of Siamese networks in contrastive learning.
+SimSiam is an conceptually simple method as it does not require a BYOL-like momentum encoder or
+a SwAV-like clustering mechanism.
+Contrastive Multiview Coding (CMC) Tian et al. (2020) proposes a framework for multiview con-
+trastive learning that maximizes the mutual information between views of the same scenes. Each
+view can be an additional sensory signal like depth, optical flow, or surface normals. CMC is closely
+related to our work but differs in two primary ways. First, CMC considers depth as a separate view
+and applies a mutual information maximization loss across multiple views; in contrast, we either
+concatenate the estimated depth information to the RGB input or generate 3D realistic views using
+the depth signal. Second, CMC considers only ground truth depth maps whereas we show that depth
+maps estimated from RGB are also quite helpful.
+Monocular Depth Estimation in Computer Vision. Monocular depth estimation is a pixel-level
+task that aims to predict the distance of every pixel from the camera using a single image. Though
+monocular depth estimation is a highly ill-posed problem, deep learning based techniques have been
+shown to perform extremely well on this task. A few works (Eitel et al., 2015; Cao et al., 2016; Hoyer
+et al., 2021; Song et al., 2021; Seichter et al., 2021) have explored the benefits of depth estimation
+for semantic segmentation and object detection. Cao et al. (2016) were one of the first efforts to
+perform a detailed analysis showing that augmenting the RGB input with estimated depth map can
+significantly improve the performance on object detection and segmentation tasks. A multi-task
+training procedure of predicting the depth signal along with the semantic label was also proposed
+in Cao et al. (2016). RGB-D segmentation with ground truth depth maps was shown to be superior
+compared to standard RGB segmentation (Seichter et al., 2021). Hoyer et al. (2021) proposed to
+use self-supervised depth estimation as an auxiliary task for semantic segmentation. Multimodal
+Estimated-Depth Unification with Self-Attention (MEDUSA) Song et al. (2021) incorporated inferred
+depth maps with RGB images in a multimodal transformer for object detection tasks. With limited
+analysis on CIFAR-10, He (2017) showed that estimated depth maps aid image classification.
+3
+
+Results on ImageNette
+100
+BYOL
+BYOL+3DViewS
+Accuracy (in %)
+90
+80
+70
+60
+Top-1 Acc
+IN-C
+IN-3DCCResults on ImageNet-100
+90
+SimSiam
+80
+SimSiam+3DViews
+(%
+Accuracy (in 
+70
+60
+50
+40
+30
+Top-1 Acc
+IN-C
+IN-3DCCUnder review
+Most prior works that utilize depth information do so with the objective of improving certain tasks
+like object detection or semantic segmentation. To the best of our knowledge, ours is the first work
+that focuses specifically on using an estimated depth signal to enhance contrastive learning. The deep
+encoder obtained from contrastive learning can then be used for various downstream tasks like object
+detection or image classification.
+3
+DEPTH IN CONTRASTIVE LEARNING
+We propose two general methods of incorporating depth information into any SSL framework. Both
+of these methods, which we describe in detail shortly, assume the availability of a depth signal.
+We obtain this signal from an off-the-shelf pretrained Monocular Depth Estimation model. We
+generate depth maps for every RGB image in our data set using the state-of-the-art Dense Prediction
+Transformer (DPT) Ranftl et al. (2021) trained for the monocular depth estimation task. DPT is trained
+on a large training dataset with 1.4 million images and leverages the power of Vision Transformers.
+DPT outperforms other monocular depth estimation methods by a significant margin. It has been
+shown that DPT can accurately predict depth maps for in-the-wild images Han et al. (2022). We treat
+the availability of these depth maps for contrastive learning as being similar to the availability that
+people have to extract depth cues via binocular disparity, motion parallax, or occlusion.
+(a) Original Image
+(b) Estimated Depth Map
+(c) Cropped Image
+(d) Estimated Depth Map
+Figure 3: Despite two images of a church (Imagenette) being quite similar visually, the presence of
+a tree occluding the church is a strong hint that the church is in the background, resulting in a very
+different depth map.
+3.1
+CONCATENATING A DEPTH CHANNEL TO THE INPUT
+We analyze the effect of concatenating a depth channel to the RGB image as a means of providing a
+richer input. This four-channel input is then fed through the model backbone. As we argued earlier,
+ample evidence suggests that cues to the three dimensional structure of the world are critical in the
+course of human development (e.g., learning about objects and their relationships), and these cues
+are available to biological systems early in the visual processing stream and are very likely used
+to segment the world into objects. Consequently, we hypothesize that a depth channel will support
+improved representations in contrastive learning.
+We anticipate that the depth channel might particularly assist the model when an image is corrupted,
+occluded, or viewed from an unusual perspective (Fig. 3). Depth might also be helpful in low-light
+environments where surface features of an object may not be clearly visible. This is quite important
+in safety critical applications like autonomous driving. The conjecture that depth cues will support
+interpretation of corrupted images is far from obvious because when the depth estimation method
+is applied to a corrupted image, the resulting depth maps are less than accurate (see Fig. 6 and
+7). We conduct evaluations using two corruption-robustness benchmarks to determine whether the
+depth signal extracted yields representations that on balance improve accuracy in a downstream
+classification task. Sample visualizations of the images and their depth map can be found in App. C.
+As Figure 1 depicts, our proposed method processes each image and each augmentation of an
+image through the DPT depth extractor. However, in accord with practice in SSL, we sample a new
+augmentation on each training step and the computational cost of running DPT on every augmentation
+in every batch is high. To avoid this high cost of training, we perform a one-time computation of depth
+4
+
+Under review
+maps for every image in the dataset and use this cached map in training for the original image, but we
+also transform it for the augmentation. This transformation works as follows. First, an augmentation
+is chosen from the set of augmentations defined by the base SSL method, and the RGB image is
+transformed according to this augmentation. For the depth map, only the corresponding Random
+Crop and Horizontal Flip transforms (i.e., dilation, translation, and rotations) are applied. The
+resulting depth map for the augmentation is cheap to compute, but it has a stronger correspondence
+to the original image’s depth map than one might expect had the depth map been computed for the
+augmentation by DPT. To address the possibility that the SSL method might come to rely too heavily
+on the depth map, we incorporated the notion of depth dropout.
+With depth dropout, the depth channel of any original image or augmentation is cleared (set to 0)
+with probability p, independently decided for each image or augmentation. When depth dropout is
+integrated with a SSL method, it prevents the SSL method from becoming too dependent on the depth
+signal by reducing the reliability of that signal. Consider a method like BYOL, whose objective is to
+predict the representations of one view from the other. With depth dropout, the objective is much
+more challenging. Since the depth channel is dropped out in some views, the network has to learn to
+predict the representations of a view with a depth signal using a view without depth. This leads to the
+model capturing additional 3D structure about the input without any significant computation cost.
+At evaluation, every image in the evaluation set is processed by DPT; the short cut of remapping the
+depth channel from the original image to the augmentation was used only during training.
+3.2
+3D VIEWS WITH ADAMPI
+We now discuss our second method of incorporating depth information in contrastive SSL methods.
+This method is motivated by the fact that humans have two eyes and binocular vision requires us
+to match up the different views of the world seen by each eye. Because each eye has a subtlely
+different perspective, the images impinging on the retina are slightly different. The brain integrates
+the two images by determining the correspondence between regions from each eye. This stereo
+correspondence helps people in understanding and representing the 3D scene. We introduce this idea
+into Self-Supervised Learning with the help of Single-View View Synthesis methods.
+Single-View View Synthesis (Tucker & Snavely, 2020) is an extreme version of the view synthesis
+problem that takes single image as the input and renders images of the scene from new viewpoints. The
+task of view synthesis requires a deep understanding of the objects, scene geometry and appearance.
+Most of the methods proposed for this task make use of multiplane-image (MPI) representation
+(Tucker & Snavely, 2020; Li et al., 2021; Han et al., 2022). MPI consists of N fronto-parallel RGBα
+planes arranged at increasing depths. MINE (Li et al., 2021) introduced the idea of Neural Radiance
+Fields (Mildenhall et al., 2020) into the MPI to perform novel view synthesis with a single image.
+These single-view view synthesis methods have a wide ranging applications in Augmented and
+Virtual Reality as they allow the viewer to interact with the photos.
+Recently, a lot of single-view view synthesis methods have been using layered depth representations
+(Shih et al., 2020; Jampani et al., 2021). These methods have been shown to generalize well on the
+unseen real world images. As mentioned in Section 3.1, monocular depth estimation models like DPT
+(Ranftl et al., 2021) are used when depth maps are not available. AdaMPI (Han et al., 2022) is one
+such recently proposed method that aims to generate novel views for in-the-wild images. AdaMPI
+introduces two novel modules, a plane adjustment network and a color prediction network to adapt to
+diverse scenes. Results show that AdaMPI outperforms MINE and other single image view synthesis
+methods in terms of quality of the synthesized images. We use AdaMPI for all of the experiments in
+our paper, given the quality of synthesized images generated by AdaMPI.
+At inference, AdaMPI takes an RGB image, depth (estimated from the monocular depth estimation
+model), and the target view to be rendered. The single-view view synthesis model then generates a
+multiplane-image representation of the scene. This representation can then be easily used to transform
+the image in the source view to the target view. More details about AdaMPI is present in App. B.
+In a nutshell, AdaMPI generates a “3D photo” of a given scene given a single input. In a way, it can
+be claimed that an image can be “brought to life” by generating the same image from another camera
+viewpoint (Kopf et al., 2019). We propose to use the views generated by AdaMPI as augmentations
+for SSL methods (Fig. 2). The synthesized views captures the 3D scene and generates realistic
+5
+
+Under review
+Table 1: Results on ImageNette Dataset show consistently improved robustness from explicitly
+leveraging depth estimation. Additionally, the depth channel approach consistently outperforms the
+3D view augmentation approach.
+Method
+kNN
+Top-1 Acc.
+ImageNet-C
+ImageNet-3DCC
+BYOL (Grill et al., 2020)
+85.71
+85.27
+84.13
+83.68
++ Depth (p = 0.5)
+88.56
+88.03
+87.00
+86.68
++ 3D Views
+87.01
+87.42
+85.75
+85.86
+SimSiam (Chen & He, 2021)
+85.10
+85.76
+84.08
+84.16
++ Depth (p = 0.5)
+86.52
+87.41
+85.13
+85.08
++ 3D Views
+85.94
+87.62
+83.87
+84.37
+SwAV (Caron et al., 2020)
+89.63
+91.08
+75.31
+82.05
++ Depth (p = 0.5)
+89.20
+90.85
+83.80
+85.02
+augmentations that help the model learn better representations. These augmentations are meant to
+reflect the type of subtle shifts in perspective obtained from the two eyes or from minor head or body
+movements.
+Augmentations are a key ingredient in contrastive learning methods (Chen et al., 2020a). Modifying
+the strength of augmentations or removing certain augmentations leads to significant drop in the
+performance of contrastive methods (Chen et al., 2020a; Grill et al., 2020; Zhang & Ma, 2022). Most
+of these augmentations can be considered as ”2D” as they make changes in the image either by
+cropping the image or applying color jitter. On the other hand, the generated 3D views are quite
+diverse as they bring in another dimension to the contrastive setup. Moreover, they can be combined
+with the existing set of augmentations to achieve the best performance.
+The synthesized views as augmentations allow the model to virtually interact with the 3D world.
+For every training sample, we generate k views synthesized from the camera in the range of x-axis
+range, y-axis range and z-axis range. The x-axis range essentially refers to the shift in the x-axis
+from the position of the original camera. The synthesis of the 3D Views is computed only once for
+the training dataset in an offline manner. Out of the total k views per sample, we sample one view at
+every training step and use it for training. We tried two techniques to augment the synthesized views.
+First, we applied the augmentations of the base SSL method on top of the synthesized view. Second,
+we applied the base SSL augmentations with a probability of q or we used the synthesized view (with
+Random Crop and Flip) with a probability of 1-q. Full details can be found in the Appendix.
+The range of novel camera views generated by the single-view view synthesis method can be
+controlled by the user. It is possible to specifically control the x-axis shift, y-axis shift and z-axis
+shift (zoom) during the generation of the novel views. The quality of generated images degrades
+when the novel view to be generated is far from the current position of the camera. This is expected
+because it is not feasible to generate a complete 360-degree view of the scene by using a single image.
+In practice, we observe certain artifacts in the image when views far away from the current position
+of the camera. Additional details can be found in App. A and App. D.
+4
+EXPERIMENTAL RESULTS
+We show results with the addition of depth channel and 3D Views with various SSL methods on
+ImageNette, ImageNet-100 and ImageNet-1k datasets. We also measure the corruption robustness of
+these models by evaluating the performance of these models on ImageNet-C and ImageNet-3DCC.
+4.1
+EXPERIMENTAL SETUP
+ImageNette: is a 10 class subset of ImageNet (Deng et al., 2009) that consists of 9469 images for
+training and 2425 images for testing. We use the 160px version of the dataset for all the experiments
+and train the models with an image size of 128.
+6
+
+Under review
+Table 2: Results on ImageNet-100 Dataset indicates that both addition of the depth channel and 3D
+Views leads to a gain in corruption robustness performance.
+Method
+kNN
+Top-1 Acc.
+ImageNet-C
+ImageNet-3DCC
+BYOL (Grill et al., 2020)
+74.24
+80.74
+47.15
+53.69
++ Depth (p = 0.3)
+74.66
+80.24
+50.17
+55.55
++ 3D Views
+73.42
+80.16
+48.15
+54.88
+SimSiam (Chen & He, 2021)
+67.56
+76.00
+44.39
+50.44
++ Depth (p = 0.2)
+70.90
+76.54
+48.30
+52.93
++ 3D Views
+68.08
+76.40
+45.78
+52.17
+ImageNet-100: is a 100 class subset of ImageNet (Deng et al., 2009) consisting of 126689 training
+images and 5000 validation images. We use the same classes as in (Tian et al., 2020) and train all
+models with image size of 224.
+ImageNet-1k: consists of 1000 classes with 1.2 million training images and 50000 validation images.
+ImageNet-C (IN-C) (Hendrycks & Dietterich, 2019): ImageNet-C dataset is a benchmark to evaluate
+the robustness of the model to common corruptions. It consists of 15 types of algorithmically
+generated corruptions including weather corruptions, noise corruptions and blur corruptions with
+different severity. Refer to Fig. 6 for a visual depiction of the images corrupted with Gaussian Noise.
+ImageNet-3DCC (IN-3DCC) (Kar et al., 2022): ImageNet-3DCC consists of realistic 3D corruptions
+like camera motion, occlusions, weather to name a few. The 3D realistic corruptions are generated
+using the estimated depth map and improves upon the corruptions in ImageNet-C. Some examples of
+these corruptions include XY-Motion Blur, Near Focus, Flash, Fog3D to name a few.
+Experimental Details. We use a ResNet-18 (He et al., 2016) backbone for all our experiments
+except the ImageNet-1k dataset where we use ResNet-50 architecture as used in Chen & He (2021).
+For the pretraining stage, the network is trained using the SGD optimizer with a momentum of 0.9
+and batch size of 256. The ImageNette experiments are trained with a learning rate of 0.06 for
+800 epochs whereas the ImageNet-100 experiments are trained with a learning rate of 0.2 for 200
+epochs. We implement our methods in PyTorch 1.11 (Paszke et al., 2019) and use Weights and Biases
+(Biewald, 2020) to track the experiments. We refer to the lightly (Susmelj et al., 2020) benchmark
+for ImageNette experiments and solo-learn (da Costa et al., 2022) benchmark for ImageNet-100
+experiments. We follow the commonly used linear evaluation protocol to evaluate the representations
+learned by the SSL method. For linear evaluation, we use SGD optimizer with a momentum of 0.9
+and train the network for 100 epochs. For the ImageNette+3D Views experiments, we apply base
+SSL augmentation on top of the synthesized views at every training step. For the ImageNet-100+3D
+Views experiments, we apply the base SSL augmentations with a probability of 0.5. Additional
+experimental details is present in the App. A.
+4.2
+RESULTS ON IMAGENETTE
+Table 1 shows the benefit of incorporating depth with any SSL method on the ImageNette dataset. We
+use the k-nearest neighbor (kNN) classifier and Top-1 Acc from the linear evaluation performance
+to evaluate the learned representation of the SSL method. It can be seen that the addition of depth
+improves the accuracy of BYOL, SimSiam and SwAV. BYOL+Depth indicates that the model is
+trained with depth map with the depth dropout. BYOL+Depth improves upon the Top-1 accuracy
+of BYOL by 2.8% along with a 3% increase in the ImageNet-C and ImageNet-3DCC performance.
+This clearly demonstrates the role of depth information in corrupted images.
+We observe a significant 8.5% increase in the ImageNet-C with SwAV+Depth over the base SwAV.
+On a closer look, it can be seen that the addition of depth channel results in high robustness to
+noise-based perturbations and blur-based perturbations. For instance, the accuracy on the Motion Blur
+corruption increases from 70.32% with SwAV to 86.88% with SwAV+Depth. And the performance
+on Gaussian Noise corruption increases from 69.76% to 84.56% with the addition of depth channel.
+BYOL + 3D Views indicates that the views synthesized by AdaMPI are used as augmentations in the
+contrastive learning setup. We show that proposed 3D Views leads to a gain in accuracy with both
+7
+
+Under review
+Table 3: Results on ImageNet-1k dataset (ResNet-50) illustrates the role of depth channel and 3D
+Views in self-supervised learning methods on large-scale datasets.
+Method
+Epochs
+Top-1 Acc.
+ImageNet-C
+ImageNet-3DCC
+SimSiam (Chen & He, 2021)
+800
+71.70
+36.45
+43.32
++ Depth (p = 0.2)
+800
+71.30
+38.23
+45.11
+SimSiam (Chen & He, 2021)
+100
+68.10
+32.99
+38.94
++ 3D Views
+100
+68.08
+34.43
+40.71
+BYOL and SimSiam. This indicates that the diversity in the augmentations due to the 3D Views helps
+the model capture a better representation of the world. We also observe a decent gain in accuracy on
+IN-C and IN-3DCC with 3D views compared to the baseline BYOL.
+4.3
+RESULTS ON IMAGENET-100
+Table 2 summarizes the results on the large-scale ImageNet-100 with BYOL and SimSiam. We find
+that most of the observations on the ImageNette datasets also hold true in the ImageNet-100 datasets.
+Though the increase in the Top-1 Accuracy with the inclusion of depth is minimal, we observe that
+performance on ImageNet-C and ImageNet-3DCC increases notably. With SimSiam, we notice a
+3.9% increase in ImageNet-C accuracy and a 2.5% increase in ImageNet-3DCC accuracy just by
+the addition of depth channel. These results emphasize the role of the proposed depth channel with
+dropout in contrastive learning.
+We observe that the proposed method of incorporating 3D views outperforms the base SSL method
+on the ImageNet-100 dataset, primarily in the corruption benchmarks. On a detailed look at the
+performance of each corruption, we observe that the 3D Views improves the performance of 3D
+based corruptions by more than 2.5%. (Refer Table 5)
+4.4
+RESULTS ON IMAGENET-1K
+Table 3 shows results on the large scale ImageNet dataset with 1000 classes. We achieve comparable
+Top-1 accuracy with both Depth and 3D Views. Since the training set is large, the additional inductive
+bias is ineffective for the in-distribution test set but useful for out-of-distribution samples. We observe
+significant accuracy boosts in classification of corrupted images: 1.5% for ImageNet-C and 1.8% for
+ImageNet-3DCC. These results indicate that our observations scale up to ImageNet-1k dataset and
+further strengthens the argument about the role of depth channel and 3D Views in SSL methods.
+5
+DISCUSSION
+Depth Dropout. Table 4 shows the ablation of probability of Depth dropout (p) on the ImageNette
+dataset with BYOL. The influence of using the depth dropout can also be understood with these
+results. It can be observed that without depth dropout (p = 0.0), the performance of the model
+is significantly lower than the baseline BYOL, as the network learns to focus solely on the depth
+channel. We find that p = 0.2 leads to the highest Top-1 Accuracy but p = 0.5 achieves the best
+performance on the ImageNet-C and ImageNet-3DCC. As the depth dropout increases (p = 0.8), the
+performance gets closer to the base SSL method as the model completely ignores the depth channel.
+What happens when depth is not available during inference? In this ablation, we examine the
+importance of depth signal at inference. Given a model trained with depth information, we analyze
+what happens when we set the depth to 0 at inference. Table 7 reports these results on ImageNette
+dataset with BYOL. Interestingly, we find that even with the absence of depth information, the
+accuracy of the model is higher than the baseline BYOL. This indicates that the model has implicitly
+learned some depth signal and captured better representations. It can also be seen that the performance
+on IN-3DCC is 1.5% higher than BYOL. Furthermore, we observe that the addition of depth map
+improves the performance on all the benchmarks. This further highlights our message that depth
+signal is a useful signal in learning a robust model.
+8
+
+Under review
+10
+15
+20
+25
+30
+35
+40
+45
+50
+Number of Views
+85.75
+86.00
+86.25
+86.50
+86.75
+87.00
+87.25
+87.50
+Accuracy (in %)
+Number of generated 3D views
+SimSiam (Baseline)
+Figure 4: As the number of 3D
+views increases, the performance of
+the SSL method increases with very
+limited increase in performance.
+Method
+Top-1 Acc.
+IN-C
+IN-3DCC
+BYOL (Grill et al., 2020)
+85.27
+84.13
+83.68
++ Depth (p = 0.0)
+84.38
+72.64
+73.68
++ Depth (p = 0.2)
+89.05
+85.93
+85.33
++ Depth (p = 0.5)
+88.03
+87.00
+86.68
++ Depth (p = 0.8)
+86.57
+85.38
+85.60
+Table 4: Ablation of Depth Dropout hyperparameter (p). A
+large dropout (p = 0.8) leads to the model ignoring the depth
+signal and a low (or zero) depth dropout leads to model
+relying only on depth signal.
+Table 5: Results on ImageNet-100 Corruptions show that while use of 3D view augmentations
+provides a larger improvement on 3D corruptions, the improvements from using depth channel are
+more consistent on a wide range of corruptions. Detailed results in App. E.
+Method
+IN-C
+Noise
+Blur
+Weather
+Digital
+IN-3DCC
+3D
+Misc
+BYOL (Grill et al., 2020)
+47.15
+36.69
+38.95
+49.57
+59.33
+53.69
+54.53
+51.16
++ Depth (p = 0.3)
+50.17
+42.36
+40.66
+51.88
+62.17
+55.55
+55.85
+54.65
++ 3D Views
+48.15
+34.50
+43.06
+50.16
+60.14
+54.88
+56.56
+49.81
+SimSiam (Chen & He, 2021)
+44.39
+36.20
+36.11
+45.24
+55.86
+50.44
+51.32
+47.83
++ Depth (p = 0.2)
+48.30
+41.90
+38.40
+49.76
+59.84
+52.93
+53.16
+52.25
++ 3D Views
+45.78
+35.00
+40.42
+46.20
+57.14
+52.17
+53.69
+47.63
+Number of Views generated by AdaMPI. Figure 4 investigates the impact of the number of
+generated 3D views on the performance of SimSiam (ImageNette). We observe that as the number of
+views increases, the Top-1 Accuracy increases although the gains are quite minimal. It must be noted
+that even with 10 views, the SimSiam+3D Views outperforms the baseline SimSiam by 1.5%.
+Which corruptions improve due to depth and 3D Views? A detailed analysis of the performance
+of the methods on various type of corruptions is reported in Table 5. We report the average on different
+categories of corruptions to understand the role of various corruptions on the overall performance.
+For ImageNet-C (IN-C), we divide the corruptions into 4 groups: Noise, Blur, Weather and Digital.
+ImageNet-3DCC is split up into two categories based on whether they make use of 3D information.
+We observe that the depth channel leads to a massive 5.7% average gain on the noise corruptions
+and 3.4% increase in digital corruptions over the baseline. The use of 3D Views in SSL results in
+a notable 4.2% improvement on the Blur corruptions over the base SSL method. As expected, the
+performance on 3D Corruptions with the 3D Views is much higher than standard SSL method and
+slightly higher than the method that uses depth channel. More results can be found in App. E.
+Table 6: Ablation on Range of synthesized views generated
+by AdaMPI. Results are shown on ImageNette dataset.
+Method
+Top-1 Acc.
+IN-C
+IN-3DCC
+BYOL (Grill et al., 2020)
+85.27
+84.13
+83.68
++ 3D Views (x = 0.1; y = 0.1)
+86.09
+83.33
+83.63
++ 3D Views (x = 0.4; y = 0.4)
+87.87
+84.78
+85.22
++ 3D Views (x = 0.5; y = 0.5)
+88.08
+85.07
+85.33
++ 3D Views (x = 0.8; y = 0.8)
+87.49
+82.47
+84.35
++ 3D Views (x = 1.0; y = 1.0)
+86.34
+80.81
+83.30
+Range
+of
+Views
+generated
+by
+AdaMPI. The range of 3D Views
+generated by AdaMPI play a huge
+role in the performance of the SSL
+method. Table 6 summarizes the ef-
+fects of moving the target camera on
+the learned representations on Ima-
+geNette dataset. x denotes the amount
+by which the x-axis is moved and y de-
+notes the same for y-axis. We observe
+that a very small change in viewing
+direction (x = 0.1; y = 0.1) does not
+boost the performance very much. As x and y get larger, the quality of generated images also de-
+creases. Thus, a large change in the viewing direction leads to artifacts which hurts the performance.
+9
+
+Under review
+Table 7: These results on ImageNette show that
+the model is robust to the absence of depth signal
+and that estimated depth improves the corruption
+robustness and linear evaluation performance.
+Method
+Top-1 Acc.
+IN-C
+IN-3DCC
+BYOL (Grill et al., 2020)
+85.27
+84.13
+83.68
++ Depth (p = 0.5)
+88.03
+87.00
+86.68
+Depth = 0 at inference
+86.80
+84.95
+85.21
+Table 8: Comparison of two Single-View View
+Synthesis Methods for generating 3D Views on
+ImageNette dataset. Higher quality views leads
+to higher performance.
+Method
+Top-1 Acc.
+IN-C
+IN-3DCC
+BYOL (Grill et al., 2020)
+85.27
+84.13
+83.68
++ 3D Views (MINE)
+87.49
+84.47
+83.93
++ 3D Views (AdaMPI)
+88.08
+85.07
+85.33
+This can be clearly observed in Table 6 where we see a drop in accuracy as the x and y increases
+from 0.5 to 1.0.
+Quality of Synthesized Views. In this ablation, we investigate how the quality of the synthesized
+views affects the representations learnt by Self-Supervised methods. We compare two different
+methods to generate 3D Views of the image namely MINE (Li et al., 2021) and AdaMPI (Han et al.,
+2022). The quantitative and qualitative results shown in Han et al. (2022) indicate that AdaMPI
+generates superior quality images compared to MINE. Table 8 reports the results on ImageNette with
+BYOL comparing the 3D Views synthesized by MINE and AdaMPI methods. We observe that the
+method with 3D Views generated by AdaMPI outperforms the method with 3D Views generated
+by MINE. This is a clear indication that as the quality of 3D view synthesis methods improves, the
+accuracy of the SSL methods with 3D views increases as well.
+6
+CONCLUSION
+In this work, we propose two distinct approaches to improving SSL using a (noisy) depth signal
+extracted from a monocular RGB image. Our results on ImageNette, ImageNet-100 and ImageNet-
+1k datasets with a range of SSL methods (BYOL, SimSiam and SwAV) show that both proposed
+approaches outperform the baseline SSL on test accuracy and corruption robustness. Further, our
+approaches can be integrated into any SSL method to boost performance. We close with several
+critical directions for future research. First, given that our two approaches are complementary
+and compatible, we might evaluate the two approaches in combination. Second, is depth dropout
+necessary when depth extraction with DPT can be run on every augmentation on every training step?
+Third, one might explore the idea of synthesizing views in Single-View View Synthesis methods with
+the goal of maximizing the performance (Ge et al., 2022) or develop better methods to utilize the 3D
+Views.
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+13
+
+Under review
+A
+EXPERIMENTAL DETAILS
+We discuss the detailed experimental setup to allow reproducibility of the results.
+Pretraining:
+BYOL: The architecture of the online and target networks in BYOL consists of three components:
+encoder, projector and predictor. We use ResNet-18 (He et al., 2016) implementation available in
+torchvision as our encoder. The Prediction Network in BYOL is a Multi-Layer Perceptron (MLP) that
+consists of a linear layer with an output dimension of 4096, followed by Batch Normalization (Ioffe
+& Szegedy, 2015), ReLU (Nair & Hinton, 2010) and a final linear layer with a dimension of 256.
+We use the same augmentations as in lightly benchmark which uses a slightly modified version of
+augmentations used in SimCLR (Chen et al., 2020b). The network is trained with an SGD Optimizer
+with a momentum of 0.9 and a weight decay of 0.0005. A batch size of 256 is used and the network
+is trained for a total of 800 epochs with a cosine annealing scheduler.
+For ImageNet-100, we use the ResNet-18 encoder and train the network using an SGD optimizer
+with a momentum of 0.9 and a weight decay of 0.0001. We use the set of augmentations in solo-
+learn benchmark in our experiments. The model is trained for 200 epochs with a batch size of 256.
+The architecture of the prediction head is same as the one used in ImageNette but with the output
+dimension of the linear layer set to 8192.
+SimSiam We follow the same optimization hyperparameters as in BYOL for the ImageNette dataset.
+The architecture of the projection head is a 3-layer MLP with Batch Normalization and ReLU applied
+to each layer. (The output layer does not have ReLU). The prediction head is a 2-layer MLP with a
+hidden dimension of 512. We refer to the official implementation of SimSiam 1 for the ImageNet-1k
+experiments.
+SwAV: For SwAV, we use the Adam optimizer (Kingma & Ba, 2014) with a learning rate of 0.001
+and weight decay of 0.000001. The number of code vectors (or prototypes) is set to 3K with 128
+dimensions. The projection head is a 2-layer MLP with a hidden layer dimension of 2048 and an
+output dimension of 128. SwAV also introduced the idea of multi-crop where a single input image
+is transformed into 2 global views and V local views. 6 local views are used in our ImageNette
+experiments.
+Linear Probing:
+For linear probing, we choose the model with the highest validation kNN accuracy and freeze the
+representations. We then train a linear layer using SGD with momentum optimizer for 100 epochs.
+We do a grid search on {0.2, 0.5, 0.8, 5.0} and report the best accuracy of the best performing
+model. This is commonly followed in the SSL literature (Zhou et al., 2021). We use the standard
+set of augmentations which includes Random Resized Crop and Horizontal Flip for training. For
+ImageNet-100, we observe that a higher learning rate seems to help and we do a grid search on
+{0.5, 0.8, 5.0, 30.0}. In most of the experiments, we observe that using the learning rate of 30.0
+yields the best-performing model.
+Depth Prediction Transformer
+We refer to the official implementation of the DPT 2 to compute the depth maps. The weights of the
+best-performing monocular depth estimation model i.e, DPT-Large, is used for the calculation of the
+depth maps. We use the relative depth maps generated by the DPT model.
+AdaMPI:
+We refer to the official implementation of AdaMPI 3 paper to compute the 3D Views. The depth maps
+generated by DPT are fed as input to the AdaMPI. We generate 50 views per sample. A pretrained
+AdaMPI model with 64 MPI planes is used in our experiments.
+For the ImageNette experiments, we apply base SSL augmentations on top of the generated AdaMPI
+at every training step. We did a grid search on a set of generated views and selected the best
+1https://github.com/facebookresearch/simsiam
+2https://github.com/isl-org/DPT
+3https://github.com/yxuhan/AdaMPI
+14
+
+Under review
+performing model. For both BYOL and SimSiam x = 0.4; y = 0.4 and z = 0.0 was used to generate
+3D Views.
+For ImageNet-100, we apply the base SSL augmentations with a probability of 0.5 and use the
+synthesized views with a probability of 0.5. We use the views synthesized with x = 0.2; y = 0.2 and z
+= 0.2.
+For ImageNet-100 experiments, we use Automatic Mixed Precision training to speed up the training.
+All the ImageNette experiments are run on RTX 8000 GPUs while the ImageNet-100 experiments are
+run on A100 GPUs. We are thankful to the authors of DPT (Ranftl et al., 2021) and AdaMPI (Han
+et al., 2022) for publicly releasing the code and pretrained weights. We will also release the code and
+pretrained weights to enable reproducible research.
+B
+ADAMPI
+This section explains about how AdaMPI renders new views. The notation and content of this section
+is heavily derived from Han et al. (2022) and Li et al. (2021).
+Consider a pixel coordinate in a image as [x, y], the camera intrinsic matrix K, camera rotation matrix
+R, camera translation matrix t. A Multiplane image (MPI) is a layered representation that consists of
+N fronto-parallel RGBα planes arranged in the increasing order of depth.
+The first step in rendering a novel view to find the correspondence between the source pixel coordinates
+[xs, ys]T and target pixel coordinates [xt, yt]T . This can be done by using the homography function
+(Hartley & Zisserman, 2004) as shown by the equation below.
+�xs, ys, 1�⊤ ∼ K
+�
+R − tn⊤
+di
+�
+K−1 �xt, yt, 1�⊤ ,
+(1)
+where, n = [0, 0, 1]⊤ is the normal vector of the fronto-parallel plane in the source view. Equation 1
+essentially maps the correspondence between source and target pixel coordinate at a particular MPI
+plane.
+The plane projections at the target plane c′
+di(xt, yt) = c′
+di(xs, ys) and σ′
+di(xt, yt) = σ′
+di(xs, ys).
+Volume rendering (Li et al., 2021; Kajiya & Von Herzen, 1984; Mildenhall et al., 2020) and Alpha
+compositing can then be used to render the image.
+AdaMPI has two major components, a planar adjustment network and color prediction network. In
+previous works Tucker & Snavely (2020), the di was usually fixed. However, in AdaMPI, the planar
+adjustment predicts di and each MPI plane at correct depth. The color prediction network takes this
+adjusted depth planes and predicts the color and density at each plane. For additional details, we refer
+the reader to Han et al. (2022).
+C
+VISUALIZATION OF DEPTH MAPS
+In this section, we show sample visualization of the depth map generated by the DPT model. Figure
+5 shows some sample visualization of the original image and the corresponding depth maps. The
+impact of corrupted images on the estimated depth maps is shown in Fig. 7. It can be seen that high
+severity in Gaussian Noise distorts the estimated depth maps significantly.
+In Figure 3, we show the impact of occlusion on the estimated depth map. Fig 3a contains a tree
+in front of it and thus it looks like the Church building has a low depth (It is far away). When we
+just crop the image and remove the trees (Fig. 3c), it can clearly seen how the estimated depth maps
+changes drastically (Fig. 3d).
+D
+VISUALIZATION OF 3D VIEWS
+We refer the reader to the supplementary zip file for some sample videos and images of synthesized
+views from AdaMPI.
+15
+
+Under review
+(a) Original Image
+(b) Estimated Depth Map
+(c) Original Image
+(d) Estimated Depth Map
+Figure 5: Visualization of Depth Maps of Images from the ImageNette dataset
+(a) Severity = 1
+(b) Severity = 2
+(c) Severity = 3
+(d) Severity = 4
+(e) Severity = 5
+Figure 6: Visualization of Images corrupted by Gaussian Noise (from ImageNet-C dataset)
+(a) Severity = 1
+(b) Severity = 2
+(c) Severity = 3
+(d) Severity = 4
+(e) Severity = 5
+Figure 7: Visualization of Depth Maps of Images corrupted by Gaussian Noise
+16
+
+Under review
+Table 9: Different Augmentations on top of 3D Views.
+Method
+Top-1 Acc.
+IN-C
+IN-3DCC
+BYOL (Grill et al., 2020)
+85.27
+84.13
+83.68
++ 3D Views (Base SSL Aug)
+88.08
+85.07
+85.33
++ 3D Views (Minimal Aug)
+83.54
+68.69
+72.26
+Table 10: Results on ImageNet-100 Noise Corruptions (IN-C). It can be clearly seen that the
+concatenation of the depth channel significantly improves the performance on noise based corruptions
+(by 8% in the case of Impulse noise). On the other hand, the introduction 3D Views hurts the
+performance on noise based corruptions.
+Method
+IN-C
+Gaussian Noise
+Shot Noise
+Impulse Noise
+Speckle Noise
+BYOL (Grill et al., 2020)
+47.15
+37.08
+36.00
+28.31
+45.36
++ Depth (p = 0.3)
+50.17
+41.79
+40.37
+36.98
+50.30
++ 3D Views
+48.15
+34.25
+33.25
+27.04
+43.46
+SimSiam (Chen & He, 2021)
+44.39
+36.51
+34.48
+30.80
+43.00
++ Depth (p = 0.2)
+48.30
+41.36
+39.98
+36.99
+49.29
++ 3D Views
+45.78
+34.85
+33.66
+28.86
+42.61
+E
+ADDITIONAL RESULTS
+What happens when the base SSL augmentations are not applied on 3D Views? Table 9 analyzes
+the role of augmentations applied on top of the synthesized 3D Views. ”Base SSL Aug” refers to
+applying the same augmentations as the base SSL method, whereas ”Minimal Aug” means that only
+Random Resized Crop and Horizontal Flip are used as augmentations. With 3D Views, even without
+the sophisticated augmentations, the model’s linear evaluation performance is close to baseline BYOL
+trained with heavy augmentations.
+Table 10 and 11 summarize the results on Noise Based Corruptions and Blur Corruptions respectively.
+Table 12 and 13 reports the results on Weather based and Digital Corruptions respectively.
+Table 14 and Table 15 report the performance of corruptions in ImageNet-3DCC dataset.
+Table 11: Results on ImageNet-100 Blur Corruptions (IN-C). Both the depth channel and 3D Views
+method improve the accuracy on blur based corruptions. The introduction of the 3D Views helps the
+model capture the 3D structure more easily and thus is highly robust to blur based corruptions.
+Method
+IN-C
+Defocus Blur
+Glass Blur
+Motion Blur
+Zoom Blur
+Gaussian Blur
+BYOL (Grill et al., 2020)
+47.15
+40.77
+33.37
+37.03
+37.76
+46.30
++ Depth (p = 0.3)
+50.17
+40.21
+36.89
+38.50
+41.55
+46.16
++ 3D Views
+48.15
+45.21
+37.32
+39.98
+42.13
+50.70
+SimSiam (Chen & He, 2021)
+44.39
+36.84
+30.92
+34.72
+35.32
+42.76
++ Depth (p = 0.2)
+48.30
+37.34
+34.94
+37.64
+39.17
+42.92
++ 3D Views
+45.78
+40.58
+35.21
+39.19
+41.40
+45.72
+17
+
+Under review
+Table 12: Results on ImageNet-100 Weather Corruptions (IN-C). The proposed method with the
+incorporation of depth channel results in a large increase on the performance of weather-corrupted
+images.
+Method
+IN-C
+Snow
+Frost
+Fog
+Brightness
+BYOL (Grill et al., 2020)
+47.15
+35.93
+41.79
+46.84
+73.71
++ Depth (p = 0.3)
+50.17
+40.15
+46.46
+46.48
+74.42
++ 3D Views
+48.15
+38.43
+42.48
+45.99
+73.72
+SimSiam (Chen & He, 2021)
+44.39
+32.78
+38.62
+40.10
+69.48
++ Depth (p = 0.2)
+48.30
+38.84
+44.11
+45.81
+70.78
++ 3D Views
+45.78
+35.20
+38.86
+41.45
+69.3
+Table 13: Results on ImageNet-100 Digital Corruptions (IN-C). Combining the depth channel with
+the input improves the performance of all kinds of digital corruptions whereas we observe that
+3D Views improves the accuracy on some corruptions and the performance degrades with some
+corruptions.
+Method
+IN-C
+Elastic
+Contrast
+Pixelate
+Saturate
+Spatter
+JPEG
+BYOL (Grill et al., 2020)
+47.15
+53.32
+50.57
+65.94
+71.92
+51.02
+63.22
++ Depth (p = 0.3)
+50.17
+58.50
+51.62
+69.10
+72.55
+54.98
+66.26
++ 3D Views
+48.15
+58.74
+50.32
+66.73
+69.79
+51.26
+63.97
+SimSiam (Chen & He, 2021)
+44.39
+50.32
+49.28
+60.91
+69.44
+47.25
+57.94
++ Depth (p = 0.2)
+48.30
+55.37
+49.95
+66.08
+69.54
+53.33
+64.14
++ 3D Views
+45.78
+54.65
+47.69
+62.50
+68.17
+47.38
+62.48
+Table 14: Results on ImageNet-100 3D Corruptions (Subset of ImageNet-3DCC). Both the proposed
+methods improve upon the base SSL method in terms of the 3D Corruptions with the 3D Views being
+the best of the three.
+Method
+IN-3DCC
+Far Focus
+Flash
+Low Light
+Near Focus
+XY-Motion Blur
+Z Motion Blur
+BYOL (Grill et al., 2020)
+53.69
+59.09
+47.85
+53.98
+64.84
+31.12
+36.22
++ Depth (p = 0.3)
+55.55
+60.42
+50.24
+57.37
+65.18
+34.28
+42.04
++ 3D Views
+54.88
+61.39
+49.36
+53.98
+66.75
+34.73
+41.82
+SimSiam (Chen & He, 2021)
+50.44
+55.31
+44.82
+48.51
+61.67
+28.93
+34.34
++ Depth (p = 0.2)
+52.93
+58.78
+47.16
+52.76
+62.61
+32.62
+39.93
++ 3D Views
+52.17
+57.24
+45.94
+48.88
+63.27
+34.10
+42.29
+Table 15: Results on ImageNet-100 3D Corruptions (Subset of IN-3DCC). Depth Channel improves
+upon the performance of non-3D corruptions like Iso-Noise and Color Quant.
+Method
+IN-3DCC
+Fog3D
+Iso-Noise
+Color Quant
+Bit Error
+BYOL (Grill et al., 2020)
+53.69
+51.68
+33.36
+66.15
+51.78
++ Depth (p = 0.3)
+55.55
+50.64
+39.15
+67.44
+52.09
++ 3D Views
+54.88
+51.55
+29.82
+65.64
+52.30
+SimSiam (Chen & He, 2021)
+50.44
+48.26
+32.56
+62.42
+48.69
++ Depth (p = 0.2)
+52.93
+48.24
+39.53
+64.46
+49.30
++ 3D Views
+52.17
+48.83
+30.87
+63.13
+48.72
+18
+

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