diff --git "a/T9E2T4oBgHgl3EQfCga9/content/tmp_files/2301.03615v1.pdf.txt" "b/T9E2T4oBgHgl3EQfCga9/content/tmp_files/2301.03615v1.pdf.txt"
new file mode 100644--- /dev/null
+++ "b/T9E2T4oBgHgl3EQfCga9/content/tmp_files/2301.03615v1.pdf.txt"
@@ -0,0 +1,3726 @@
+Study of the de Almeida-Thouless (AT) line in the one-dimensional diluted power-law
+XY spin glass
+Bharadwaj Vedula,1 M. A. Moore,2 and Auditya Sharma1
+1Department of Physics, Indian Institute of Science Education and Research, Bhopal, Madhya Pradesh 462066, India
+2Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
+(Dated: January 11, 2023)
+We study the AT line in the one-dimensional power-law diluted XY spin glass model, in which
+the probability that two spins separated by a distance r interact with each other, decays as 1/r2σ.
+Tuning the exponent σ is equivalent to changing the space dimension of a short-range model. We
+develop a heat bath algorithm to equilibrate XY spins; using this in conjunction with the standard
+parallel tempering and overrelaxation sweeps, we carry out large scale Monte Carlo simulations.
+For σ = 0.6, which is in the mean-field regime above six dimensions – it is similar to being in 10
+dimensions – we find clear evidence for an AT line. For σ = 0.75 and σ = 0.85, which are in the
+non-mean-field regime and similar to four and three dimensions respectively, our data is like that
+found in previous studies of the Ising and Heisenberg spin glasses when reducing the temperature
+at fixed field. For σ = 0.75, there is evidence from finite size scaling studies for an AT transition
+but for σ = 0.85, the evidence for a transition is non-existent. We have also studied these systems
+at fixed temperature varying the field and discovered that at both σ = 0.75 and at σ = 0.85 there
+is evidence of an AT transition! Confusingly, the correlation length and spin glass susceptibility as
+a function of the field are both entirely consistent with the predictions of the droplet picture and
+hence the non-existence of an AT line. In the usual finite size critical point scaling studies used to
+provide evidence for an AT transition, there is seemingly good evidence for an AT line at σ = 0.75
+for small values of the system size N, which is strengthening as N is increased, but for N > 2048 the
+trend changes and the evidence then weakens as N is further increased. We have also studied with
+fewer bond realizations the system at σ = 0.70, which is the analogue of a system with short-range
+interactions just below six dimensions, and found that it is similar in its behavior to the system at
+σ = 0.75 but with larger finite size corrections. The evidence from our simulations points to the
+complete absence of the AT line in dimensions outside the mean-field region and to the correctness
+of the droplet picture. Previous simulations which suggested there was an AT line can be attributed
+to the consequences of studying systems which are just too small. The collapse of our data to the
+droplet scaling form is poor for σ = 0.75 and to some extent also for σ = 0.85, when the correlation
+length becomes of the order of the length of the system, due to the existence of excitations which
+only cost a free energy of O(1), just as envisaged in the TNT picture of the ordered state of spin
+glasses. However, for the case of σ = 0.85 we can provide evidence that for larger system sizes,
+droplet scaling will prevail even when the correlation length is comparable to the system size.
+I.
+INTRODUCTION
+While the spin glass problem at mean-field level is now
+well-understood [1], questions remain as to the nature of
+the ordered state in three dimensional spin glasses. A
+key question is whether the ordered phase of real spin
+glasses has the broken replica symmetry features found
+in mean-field theory.
+This question is most easily an-
+swered by finding whether on application of a magnetic
+field hr there is a line, the so-called de Almeida Thou-
+less (AT) line [2], below which in the hr − T plane there
+is replica symmetry breaking. This line exists at mean-
+field level (see Fig. 1) and its possible existence in three
+dimensions can be studied experimentally and with sim-
+ulations. Simulational studies of the existence of replica
+symmetry breaking within the zero-field spin glass state
+itself are plagued by finite size effects: it is expected that
+the difference between the predictions of droplet scaling
+and those of replica symmetry breaking will only become
+visible for very large systems (for a review see [3]). A
+recent review of simulations, including studies of the ex-
+istence of the AT line, can be found in Ref. [4].
+Right from the early days of spin glass studies there
+have been doubts raised as to whether the AT line existed
+below six dimensions. For example Bray and Roberts [6]
+attempted to do an expansion in 6 − ϵ dimensions for
+the critical exponents at the AT line but failed to find
+a stable fixed point. They suggested that maybe that
+indicated that there might be no AT line below six di-
+mensions. A renormalization group calculation also gave
+indications that the AT line was going away as d → 6
+from above
+[7]. As it is difficult to do simulations in
+dimensions around 6 to check these speculations, simula-
+tors have had to turn instead to one-dimensional models
+with long-range power-law interactions.
+These models go back to Kotliar, Anderson and Stein
+[8], who in turn were inspired by the long-range ferro-
+magnet that was studied by Dyson [9, 10].
+The long-
+range power-law model has the advantage that by tuning
+the power-law exponent σ, one has access to both the
+mean-field and the regimes with non-mean-field critical
+behavior. However, the full power-law model is expen-
+sive for numerics. Fortunately a clever workaround was
+introduced by Leuzzi et al. [11] where instead of the in-
+arXiv:2301.03615v1  [cond-mat.dis-nn]  9 Jan 2023
+
+2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+T/Tc
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+hr/J
+exact AT
+approximate AT
+SK data
+σ = 0.6 (T∗ data)
+σ = 0.6 (h∗ data)
+Figure 1. The AT line. The solid line is the exact AT line
+for the SK model, calculated as in Ref. [5]. The dashed line
+is the approximation to it of Eq. (8). We have marked on
+the diagram the results of our simulations on the SK model,
+which were done to check our Monte Carlo procedures. The
+points in red and green are the results of our simulations at
+σ = 0.6, which lie in the mean-field region. The data on the
+horizontal axis for σ = 0.6 are normalized to the transition
+temperature Tc in zero field for that value of σ. For the XY
+SK model the AT line goes to infinity as T → 0.
+teractions falling off as a power law, it is the probability
+of there being a bond between two spins that falls off as
+a power law. The fewer bonds in the model means that a
+significantly smaller computational cost is involved, thus
+allowing for the simulation of larger system sizes.
+While the vast literature on spin glasses is mostly fo-
+cussed on Ising spins [11–16], there has been a revival of
+interest in classical m-component vector spin glass mod-
+els [5, 17–26] in the last decade or so. The XY model
+has m = 2 and the Heisenberg model has m = 3. One
+of the triggers for this revival has been the finding that
+the infinite-range vector spin glass exhibits an AT line
+provided a magnetic field that is random in all the com-
+ponent directions is applied [5]. Furthermore analytical
+studies of the AT transition in m-vector models shows
+that the field theory of these AT transitions is that of the
+Ising spin glass [5]. Thus it has become possible to study
+the question of whether or not an Ising AT transition ex-
+ists in various dimensions by studying one-dimensional
+vector spin glasses with long-range interactions [18]!
+In this paper, we study the one-dimensional diluted
+XY spin glass subjected to a random vector magnetic
+field, with the aid of large scale Monte Carlo simula-
+tions. While Monte Carlo simulations are a time-tested
+tool for the study of phase transitions in spin glasses, the
+exorbitant cost of equilibration makes them rather chal-
+lenging in practice. It has been argued that vector spins
+tend to equilibrate faster compared to Ising spins [27], be-
+cause of the soft nature of the spins involved, even though
+the presence of more components adds to the cost. The
+Heisenberg spin glass [5, 17–19, 27–32] has been the pop-
+ular vector spin to have been considered, because of the
+availability of the heatbath algorithm [28], which works
+very efficiently to equilibrate it. The XY spin glass is
+less effectively handled by the heatbath algorithm [33]
+because of the technicalities involved in inverting a prob-
+ability distribution for which a simple closed form ex-
+pression is unavailable in the XY case. In this paper, we
+develop a method, which is outlined in Appendix A to
+perform this inversion numerically with the hope of ben-
+efiting from the vector nature of XY spins, while simul-
+taneously reducing the components to as small a number
+as possible.
+The improved algorithm yields mixed fruits. The gains
+from the reduced number of components seems to be
+largely counterbalanced by the additional resources con-
+sumed by the numerical inversion. However, with the aid
+of extensive computational power, we are able to access
+system sizes comparable to those in the corresponding
+study with Heisenberg spins. Our findings for the XY
+diluted spin glass closely mimic those obtained for the
+Heisenberg version of the same model [5, 17, 18] and the
+Ising spin glass in three dimensions [34] when we investi-
+gate crossing the possible AT line by varying T at fixed
+values of hr. In the mean-field regime, (we studied here
+the case of σ = 0.6, which corresponds to 10 dimen-
+sions, which is above the upper critical dimension of 6),
+there is clear evidence of an Almeida-Thouless line (see
+Sec. V A). There is rather weak evidence for an Almeida-
+Thouless line for σ = 0.85 using the commonly employed
+finite size critical point scaling methods of analysis (see
+Sec. V C). At this value of σ, our system should be simi-
+lar to the Edwards-Anderson model in three dimensions
+with short-range interactions. For the in-between case
+at σ = 0.75 which lies in the non-mean-field regime, but
+closer to the mean-field boundary at σ = 2/3, our data
+do provide stronger evidence for a phase transition in
+the presence of small magnetic fields than at σ = 0.85.
+However, by varying the magnetic field hr at fixed tem-
+perature T we find in Sec. V that at both σ = 0.75 and
+at σ = 0.85 there is quite decent evidence for an AT
+line. Confusingly, the field dependence of the correlation
+length is very well-described by the Imry-Ma prediction
+of the droplet picture, which implies the complete ab-
+sence of the AT transition! In the droplet picture the
+correlation length in a field remains finite and only di-
+verges as hr → 0. However, when this correlation length
+becomes comparable to the system size L the Imry-Ma
+formula needs to be modified and we give in Sec.
+VI
+a scaling form for this modification.
+It is based upon
+the usual finite size scaling approach used in studying
+critical phenomena, and just as for critical phenomena
+we find that there are finite size corrections to this scal-
+ing form. In addition to these scaling corrections there
+
+3
+are corrections which arise when ξSG ∼ L < L∗ which
+are of different origin and are connected to TNT effects
+[35, 36]. TNT effects arise from droplets of free energy
+cost of O(1), which certainly exist in systems whose sizes
+L < L∗ [3]. The length scale L∗ is always large and is
+expected to diverge as d → 6 or as σ → 2/3. It is only
+for the case of σ = 0.85 that we can reach sizes where
+TNT effects seem to be getting small. These matters are
+discussed in Sec. VII.
+Furthermore, we can use the droplet scaling picture to
+explain some of the features of the apparent AT transi-
+tion which arise on performing the usual finite size crit-
+ical scaling analyses, and show that these are the conse-
+quence of not studying large enough systems. Unfortu-
+nately these arguments will only become compelling for
+system sizes which we cannot reach. Our chief evidence
+for the droplet picture is its very successful prediction
+of the correlation length as a function of the field in the
+region when finite size and TNT effects are unimportant.
+Our claim that the evidence favors the absence of the
+AT line for values of σ outside the mean-field region is
+consistent with the attempt [21] to calculate the AT field
+at T = 0 using an expansion in 1/m. This indicated that
+as d → 6 from above in the Edwards-Anderson model, the
+AT field would go to zero, implying the absence of the AT
+line below 6 dimensions (which in the one-dimensional
+long-range model corresponds to 2/3 < σ < 1).
+For
+σ > 1 there is no finite temperature spin glass phase.
+The plan of this paper is as follows. In Sec. II we de-
+scribe the model in detail. In Sec. III we describe the
+quantities which were studied in our Monte Carlo simu-
+lations, the details of which are given in Appendix A. Our
+data is analysed in Sec. V on the assumption that there
+is an AT transition, while in Sec. VI the data is analysed
+according to droplet scaling assumptions. In Sec. VII we
+discuss the effect of TNT behavior on our results. Finally
+in Sec. VIII we summarize our conclusions.
+II.
+MODEL HAMILTONIAN
+The general Hamiltonian for vector spin glasses is:
+H = −
+�
+⟨i,j⟩
+JijSi · Sj −
+�
+i
+hi · Si ,
+(1)
+where Si is the spin on the ith lattice site (i
+=
+1, 2, . . . , N), which is chosen to be a unit vector. m rep-
+resents the number of components of the vector Si. In
+this work we concentrate on XY spins, and set m = 2.
+The Cartesian components hµ
+i (µ = 1, 2) of the on-site
+external magnetic field are i.i.d random variables drawn
+from a Gaussian distribution of zero mean and variance
+h2
+r and satisfy the relation:
+�
+hµ
+i hν
+j
+�
+av = h2
+rδijδµν.
+(2)
+We use the notation ⟨· · · ⟩ for thermal average and [· · · ]av
+for an average over quenched disorder throughout this
+paper.
+The spins are arranged on a circle so the geometric
+distance between a pair of spins (i, j) is given by [16]
+rij = N
+π sin
+� π
+N |i − j|
+�
+,
+(3)
+which is the length of the chord connecting the ith and
+jth spins. The interactions Jij are independent random
+variables such that the probability of having a non-zero
+interaction between a pair of spins (i, j) falls with the
+distance rij between the spins as a power law:
+P(Jij) ∝
+1
+r2σ
+ij
+.
+(4)
+If the spins i and j are linked the magnitude of the inter-
+action between them is drawn from a Gaussian distribu-
+tion whose mean is zero and whose standard deviation is
+unity, i.e:
+[Jij]av = 0
+and
+�
+J2
+ij
+�
+av = 1.
+(5)
+The mean number of non-zero bonds from a site is fixed
+to be ˜z (co-ordination number). So, the total number
+of bonds among all the spins on the lattice is fixed to
+be Nb = N ˜z/2.
+When ˜z = 6 this model mimics the
+3D simple cubic lattice model and we use this value for
+˜z for all the σ values studied.
+(For σ = 0 and ˜z =
+N −1, the model becomes the infinite-range Sherrington-
+Kirkpatrick (SK) model [37]).
+To generate the set of interaction pairs [11, 17] (i, j)
+with the desired probability we pick a site i randomly and
+uniformly and then choose a second site j with probabil-
+ity given by:
+pij =
+r−2σ
+ij
+�
+j̸=i
+r−2σ
+ij
+.
+(6)
+If the spins at i and j are already connected we repeat
+this process until we find a pair of sites (i, j) which have
+not been connected. Once we find such a pair of spins,
+we connect them with a bond whose strength Jij is a
+Gaussian random variable with attributes given by Eq.
+(5). We repeat this process exactly Nb times to generate
+Nb pairs of interacting spins.
+The advantage of the diluted model over the fully con-
+nected model is that, in a fully connected model, there
+are N(N − 1)/2 interactions. The ratio of the number of
+interactions of the diluted model to the fully connected
+model is ˜z/N which is a very small value as N becomes
+large. Hence it is possible to go to much larger system
+sizes with a diluted model as compared to a fully con-
+nected model.
+At zero-field, the mean-field spin glass transition tem-
+perature for the m-component vector spin glass is given
+by [17, 18, 38]
+T MF
+c
+= 1
+m
+�
+��
+j
+�
+J2
+ij
+�
+av
+�
+�
+1/2
+=
+√
+˜z
+m J.
+(7)
+
+4
+The approximate location of the AT line for an m-
+component infinite-range spin glass near the zero-field
+transition temperature Tc is [5]
+�hr
+J
+�2
+=
+4
+m(m + 2)
+�
+1 − T
+Tc
+�3
+.
+(8)
+The accuracy of this approximation for the SK model can
+be judged from Fig. 1.
+A one-dimensional chain with power law diluted in-
+teractions for a particular value of σ is equivalent to a
+short-range model [14] of effective dimension deff, where
+deff =
+2
+2σ − 1,
+(9)
+i.e., there is a one-to-one mapping between a long-range
+diluted network with exponent σ and a short-range model
+with space dimension deff, at least when 1/2 < σ < 2/3.
+Thus when σ = 0.60, deff = 10.
+For the interval
+2/3 < σ < 1 other relations are required [15, 39]. For
+example, for Ising spin glasses, it was suggested in Ref.
+[39] that d = 4 corresponded to σ ≈ 0.790, while d = 3
+corresponded to σ ≈ 0.896. Unfortunately, the mapping
+for the XY model has been less studied.
+III.
+CORRELATION LENGTHS AND
+SUSCEPTIBILITIES
+In this section we discuss the quantities which were
+obtained from our Monte Carlo simulations and used to
+extract a correlation length ξSG and the spin glass suscep-
+tibility χSG. The simulations themselves are described in
+detail in Appendix A.
+The thermal average of a quantity is calculated using
+multiple replicas in the following standard way:
+⟨A⟩⟨B⟩⟨C⟩⟨D⟩ = ⟨A(1)B(2)C(3)D(4)⟩
+(10)
+where (1),(2),(3), and (4) are four copies of the system at
+the same temperature. The wave-vector-dependent spin
+glass susceptibility is given by [5]
+χSG(k) = 1
+N
+�
+i,j
+1
+m
+�
+µ,ν
+��
+χµν
+ij
+�2�
+av eik(i−j),
+(11)
+where
+χµν
+ij =
+�
+Sµ
+i Sν
+j
+�
+− ⟨Sµ
+i ⟩
+�
+Sν
+j
+�
+.
+(12)
+The spin glass correlation length is then determined from
+ξSG =
+1
+2 sin(kmin/2)
+�
+χSG(0)
+χSG (kmin) − 1
+�1/(2σ−1)
+(13)
+where kmin = (2π/N).
+IV.
+FINITE-SIZE ANALYSES ASSUMING A
+TRANSITION EXISTS
+In this section we detail the method of finite-size anal-
+ysis when a transition is assumed to exist. When study-
+ing the AT line, which is a line of phase transitions in
+the hr − T plane, it can be crossed on an infinite number
+of trajectories. The most commonly used trajectory is
+the one where hr is kept constant and the temperature
+T is varied. In this work we also consider the trajectory
+in which T is kept constant and hr is varied. We refer
+to the zero-field transition temperature as Tc while we
+denote a generic transition temperature on the AT line
+by TAT(hr). Similarly we denote the field on the AT line
+by hAT(T).
+The spin glass susceptibility χSG ≡ χSG(0) of a finite
+system of N spins has the finite size scaling form (near
+the transition temperature TAT(hr)) [5]:
+χSG
+N 2−η = C
+�
+N 1/ν (T − TAT(hr))
+�
+,
+(2/3 ≤ σ < 1),
+(14a)
+χSG
+N 1/3 = C
+�
+N 1/3 (T − TAT(hr))
+�
+,
+(1/2 < σ ≤ 2/3),
+(14b)
+where η is given by 2 − η = 2σ − 1. These forms are
+examples of finite size scaling expressions which would
+be expected to hold in the critical region when N → ∞,
+(T − TAT(hr)) → 0, with (say) N 1/ν(T − TAT(hr)) finite.
+The scaling function C will depend on the value of σ.
+There are always finite size corrections to these forms.
+For example, the corrections to Eq. (14b) will be of the
+form
+χSG
+N 1/3 = C
+�
+N 1/3 (T − TAT(hr))
+�
++ N −ωG
+�
+N 1/3(T − TAT(hr))
+�
+.
+(15)
+It has been suggested [17, 40] that the correction to scal-
+ing exponent is given at least in the mean-field region
+by
+ω = 1/3 − (2σ − 1).
+(16)
+Curves of χSG/N 2−η
+(χSG/N 1/3
+in the mean-field
+regime) plotted for different system sizes should inter-
+sect at the transition temperature TAT(hr). In reality,
+finite-size corrections to Eq. (14) are always present and
+cause the intersection point between the curves for size N
+and 2N to depend on N. The intersection temperatures
+vary as [40–43]
+T ∗(N, 2N) = TAT(hr) + A
+N λ ,
+(17)
+where A is the amplitude of the leading correction, and
+the exponent λ is
+λ = 1/3 + ω,
+(1/2 < σ ≤ 2/3),
+(18a)
+λ = 1/ν + ω,
+(2/3 < σ < 1),
+(18b)
+
+5
+where ω is the leading correction to the scaling exponent.
+When σ = 0.6, ω = −2σ + 4/3, so λ = 5/3 − 2σ = 0.467
+[40]. In the regime when σ > 2/3 the values of both ν
+and λ are not well-determined, so there we shall treat λ
+as a fitting parameter.
+The spin glass correlation length has a similar finite
+size scaling form in the critical region
+ξSG
+N
+= X
+�
+N 1/ν (T − TAT(hr))
+�
+,
+(2/3 ≤ σ < 1),
+(19a)
+ξSG
+N deff/6 = X
+�
+N 1/3 (T − TAT(hr))
+�
+,
+(1/2 < σ ≤ 2/3).
+(19b)
+ν, the correlation length critical exponent, has to be de-
+termined numerically in the interval 2/3 < σ < 1.
+We have also studied crossing the AT line at fixed T
+and varying hr. Then Eq. (19) takes the form,
+ξSG
+N
+= X
+�
+N 1/ν (hr − hAT(T))
+�
+,
+(2/3 ≤ σ < 1),
+(20a)
+ξSG
+N deff/6 = X
+�
+N 1/3 (hr − hAT(T))
+�
+,
+(1/2 < σ ≤ 2/3),
+(20b)
+where hAT(T) denotes the field at the AT line at tem-
+perature T. Similarly, the spin glass susceptibility χSG
+of the finite system near the AT transition line takes the
+form
+χSG
+N 2−η = C
+�
+N 1/ν (hr − hAT(T))
+�
+,
+(2/3 ≤ σ < 1),
+(21a)
+χSG
+N 1/3 = C
+�
+N 1/3 (hr − hAT(T))
+�
+,
+(1/2 < σ ≤ 2/3).
+(21b)
+In the thermodynamic limit, Eq. (19) is similar to
+Eq. (20); the effect of finite size corrections to the two
+can differ. For example, while the correction to scaling
+exponent λ does not depend on the choice of the trajec-
+tory, the magnitude of the scaling corrections can differ.
+Thus in the intersection formulae when applied to fields
+h∗(N, 2N) = hAT(T) +
+˜A
+N λ ,
+(22)
+the coefficient ˜A will be different from A in Eq. (17). Cor-
+rections to scaling of, say, Eq. (21a), are more generally
+of the form
+χSG
+N 2−η = C
+�
+N 1/ν (hr − hAT(T))
+�
++ N −ωG
+�
+N 1/ν (hr − hAT(T))
+�
+,
+(23)
+where ω is the correction to scaling exponent, and G is
+another scaling function. This type of scaling form holds
+in the limit where N 1/ν(hr −hAT(T)) is fixed as N → ∞,
+which of course can only be realized approximately in
+numerical studies.
+A key feature of the finite size critical point scaling
+analysis is that right on the AT line itself, that is when
+hr = hAT(T), R = χSG/N 2−η (χSG/N 1/3 for σ ≤ 2/3)
+should be finite as N → ∞. We find (see Sec. VI) that R
+is at least not increasing with N, and perhaps finite (see
+Fig. 37), for σ = 0.60 but for σ = 0.70, 0.75 and 0.85 it is
+in fact increasing with N, at the crossing field h∗(N, 2N).
+We deduce from this observation that at these values of
+σ the crossings at h∗(N, 2N) are not associated with a
+true critical point at all but are consequences of droplet
+scaling. At a true critical point R would tend to a finite
+constant as N increases, but we find it increases with N,
+provided N > 1024 (or system sizes 2N > 2048) for the
+case of σ = 0.75 (see Fig. 38).
+In Sec. V we shall present our attempts at analysing
+the data for σ = 0.6, σ = 0.75, σ = 0.85 at fixed values of
+hr but varying T, and also at a fixed value of T and vary-
+ing hr, on the assumption that there is an AT line and
+using the finite-size scaling methods of this subsection.
+We have also obtained data at fixed T and varying hr
+for σ = 0.60, 0.70, 0.75, 0.85 and analysed them using
+finite size generalizations of well-known droplet scaling
+relations. In this case the droplet picture provides a sim-
+ple set of formulae for analysing the data in the assumed
+absence of an AT line.
+V.
+ANALYSES OF THE SIMULATION DATA
+ASSUMING THERE IS AN AT LINE
+We shall study the phase transitions at hr
+= 0,
+and determine the zero-field transition temperature Tc
+(= TAT(hr = 0)), and seek evidence of an AT transition
+at non-zero hr using the standard critical point finite size
+scaling method of determining the “crossings” or inter-
+sections of the curves of, say, χSG/N z (with z = 1/3
+when σ ≤ 2/3, and with z = 2 − η = 2σ − 1 for σ ≥ 2/3)
+at values of N and 2N as we reduce T through the AT
+transition temperature at fixed hr, or the field hr at fixed
+T in the vicinity of the AT field hAT(T) as outlined in
+Sec. IV. There seems no reason to doubt the existence
+of an AT line for any value of σ in the mean-field region
+σ < 2/3, and our results are entirely consistent with the
+existence of an AT transition at σ = 0.60. They serve
+as a useful comparison for the studies in the non-mean
+regime σ > 2/3, where the evidence will be found to favor
+the droplet picture. We have studied N values 128, 256,
+512, 1024, 2048, 4096, 8192 and 16384 for both σ = 0.60
+and σ = 0.75, but went up to N = 32768 for the case
+of σ = 0.75 when the field hr was varied at fixed T.
+When σ = 0.85 the largest N values used was 4096. In
+this case the zero-field transition temperature Tc is quite
+low and as a consequence all the investigations have to
+be done also at low temperatures, where equilibration
+times are long, preventing the study of larger systems.
+We are mainly interested in the question as to whether
+
+6
+outside the mean-field region, that is for σ > 2/3, an AT
+transition actually exists and whether (say) the depen-
+dence of ξSG on the field hr can be understood as will
+be suggested in Sec. VI on the droplet picture without
+invoking an AT transition at all. If it can, this would
+provide support to the argument that the droplet scaling
+picture rather than replica symmetry breaking describes
+spin glasses below 6 dimensions. We have analysed the
+data for σ = 0.70, 0.75 and for σ = 0.85 using the usual
+“crossing” method, (the finite size scaling approach out-
+lined in Sec. IV), which indeed works well for σ = 0.6.
+The evidence for the existence of an AT transition at
+σ = 0.75 and 0.85 will be contrasted with the evidence
+against an AT transition at these values of σ.
+Our main focus was the case σ = 0.75.
+We looked
+briefly at the case σ = 0.70 to find whether or not it
+might be practical to study whether σ = 2/3 is the value
+of σ above which the AT line might disappear. We found
+that it was similar to σ = 0.75, but that the corrections
+to scaling were larger. This means that for a given level
+of accuracy, larger N values are required. We studied
+σ = 0.85 because it should behave similarly to physical
+systems in three dimensions but we could not equilibrate
+systems at the larger N values in this case because the
+temperatures T of interest have to be less than Tc, which
+is rather small.
+A.
+σ = 0.6
+We shall focus on σ = 0.60 in this subsection. It corre-
+sponds according to Eq. (9) to an effective dimension of
+10 dimensions, which is in the mean-field region; it lies
+above the upper critical dimension of spin glasses, which
+is 6 (or in the mean-field region σ < 2/3 in the one-
+dimensional long-range model). It is natural to expect
+that for this value of σ there will be an AT line and this
+is amply confirmed by our simulations. For this value of
+σ, simulations of the corresponding Ising model [12, 15]
+and the Heisenberg model [17, 18] also found an AT line.
+Our results for hr = 0 are given in Figs. 2 and 3. Ac-
+cording to Eq. (14b), the data for χSG/N 1/3 when plot-
+ted for different system sizes should intersect at the tran-
+sition temperature Tc. Similarly, according to Eq. (19b),
+the data of ξSG/N deff/6 with deff = 2/(2σ − 1) should in-
+tersect at the same transition temperature. Fig. 2 shows
+the data for different system sizes. We find the tempera-
+ture T ∗(N, 2N) at which the curves corresponding to the
+system sizes N and 2N intersect. We then fit this data
+with Eq. (17) to find the transition temperature. The
+exponent λ ≡ 5/3 − 2σ is known to equal 0.467 in this
+case [17, 40]. The result is displayed in Fig. 3, where the
+T ∗(N, 2N) data obtained from intersections of χSG are
+fitted against N −λ with a straight line for the largest 6
+pairs of system sizes to give Tc = 0.8873 ± 0.0017. The
+corresponding intersections of the ξSG data (omitting the
+two smallest system sizes) gives Tc = 0.8893 ± 0.0046.
+The values of Tc obtained from χSG data and ξSG data
+0.75
+0.80
+0.85
+0.90
+0.95
+T
+0.1
+1
+χSG/N 1/3
+σ = 0.6
+hr = 0
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+N = 4096
+N = 8192
+N = 16384
+0.7
+0.8
+0.9
+T
+10−3
+10−2
+10−1
+100
+101
+ξSG/N deff /6
+Figure 2.
+The main figure shows the plot of χSG/N 1/3 as
+a function of the temperature T for different system sizes,
+for σ = 0.6 with hr = 0. The inset figure shows the corre-
+sponding data for ξSG/N deff/6, with deff = 2/(2σ − 1) in the
+mean-field regime (Eq. (9)).
+The exponents of N are cho-
+sen according to Eqs. (14b) and (19b). Both the plots show
+that the curves for different system sizes intersect. The data
+for the intersection temperatures T ∗(N, 2N) between pairs of
+adjacent system sizes are presented in Fig. 3.
+are in agreement with each other. The mean-field pre-
+diction of Eq. (7) is much higher, T MF
+c
+=
+√
+6/2 = 1.2247.
+Fluctuation effects not present in the SK limit must be
+responsible for this large difference.
+For hr = 0.1, the data is as shown in Figs. 4 and 5.
+When the T ∗(N, 2N) data obtained from χSG are fitted
+against N −λ with a straight line for the largest 4 pairs
+of system sizes we get TAT(hr = 0.1) = 0.6735 ± 0.0120.
+The corresponding ξSG data (omitting the two smallest
+system sizes) gives TAT(hr = 0.1) = 0.6745 ± 0.0148.
+Thus we have found that the AT line passes through
+the point (T, hr) = (0.674, 0.1). To compare that with
+the predictions from the SK model, we use the zero-field
+transition temperature Tc = 0.887 obtained above. Then
+for hr = 0.1, the predicted value of the AT transition
+temperature ratio of the SK model would be TAT(hr =
+0.1)/Tc = 0.74, while the Monte Carlo determined value
+at σ = 0.6 is 0.7590 ± 0.0113. (For the SK model, the
+Monte Carlo value of the ratio is 0.7641 ± 0.0341). Thus
+while the zero-field transition temperature at σ = 0.6 is
+not close to the mean-field value of Eq. (7), the SK form
+of the AT line is a good approximation provided it is ex-
+pressed in terms of the renormalized zero-field transition
+temperature Tc (see also Fig. 1).
+The AT line can be approached not only by reducing
+the temperature T but also by reducing the field at fixed
+T. In Figs. 6 and 7 we have constructed the crossing plots
+
+7
+0.000
+0.025
+0.050
+0.075
+0.100
+N −λ
+0.70
+0.75
+0.80
+0.85
+T ∗(N, 2N)
+σ = 0.6
+hr = 0
+χSG
+−0.72N −λ + 0.89
+ξSG
+−1.56N −λ + 0.89
+Figure 3. A plot of the intersection temperatures T ∗(N, 2N)
+for χSG/N 1/3 and ξSG/N deff/6 obtained from the data in
+Fig. 2, as a function of N −λ, for σ = 0.6 with hr = 0. The
+value of the exponent λ is fixed to be 0.467 which is known
+exactly [17, 40]. The fits give Tc = 0.8873 ± 0.0017 from χSG
+and Tc = 0.8893 ± 0.0046 from ξSG.
+0.60
+0.65
+0.70
+0.75
+T
+0.2
+0.3
+0.4
+0.5
+0.6
+χSG/N 1/3
+σ = 0.6
+hr = 0.1
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+N = 4096
+N = 8192
+N = 16384
+0.55
+0.60
+0.65
+0.70
+0.75
+T
+0.1
+1
+10
+ξSG/N deff /6
+Figure 4. A finite size scaling plot of χSG (main figure) and
+ξSG (inset figure), for σ = 0.6 in a magnetic field of hr =
+0.1. Both the datasets clearly indicate that a phase transition
+occurs.
+The transition temperature in the thermodynamic
+limit is estimated in Fig. 5.
+as a function of hr for ξSG and χSG respectively. Analysis
+of the crossing plots of h∗(N, 2N) in Fig. 8 shows that
+the behavior is again consistent with the existence of an
+AT line at least at σ = 0.60. The same value of λ was
+used as when plotting T ∗(N, 2N). The h∗(N, 2N) data
+for all the pairs of system sizes are fitted against N −λ
+to give hAT(T = 0.6) = 0.1569 ± 0.0061 from χSG and
+0.000
+0.025
+0.050
+0.075
+0.100
+N −λ
+0.58
+0.60
+0.62
+0.64
+0.66
+0.68
+T ∗(N, 2N)
+σ = 0.6
+hr = 0.1
+χSG
+−0.23N −λ + 0.67
+ξSG
+−0.58N −λ + 0.67
+Figure 5.
+The intersection temperatures T ∗(N, 2N), for
+σ = 0.6 with hr = 0.1 (look at Fig. 4). Both the datasets
+are consistent with a spin glass transition temperature of
+TAT(hr = 0.1) = 0.67.
+10−2
+10−1
+100
+hr
+10−7
+10−5
+10−3
+10−1
+101
+103
+ξSG/N deff/6
+σ = 0.6
+T = 0.6
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+N = 4096
+N = 8192
+N = 16384
+Figure 6. A finite size scaling plot of ξSG as a function of
+magnetic field hr, for σ = 0.6 at a temperature of T = 0.6.
+hAT(T = 0.6) = 0.1571 ± 0.0067 from ξSG. We found
+two points on the AT line, (T, hr) = (0.674, 0.1) from
+T ∗(N, 2N), and (T, hr) = (0.6, 0.157) from h∗(N, 2N)
+data. These points are plotted in Fig. 1 for comparison
+with the exact AT line for the SK model.
+B.
+σ = 0.75
+The case σ = 0.75 corresponds to the non-mean-field
+regime: the long-range diluted model for this value of σ
+is equivalent to a short-range model with d ≈ 4 dimen-
+
+8
+10−2
+10−1
+100
+hr
+10−2
+10−1
+100
+χSG/N 1/3
+σ = 0.6
+T = 0.6
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+N = 4096
+N = 8192
+N = 16384
+Figure 7. A finite size scaling plot of χSG as a function of
+magnetic field hr, for σ = 0.6 at a temperature of T = 0.6.
+0.000
+0.025
+0.050
+0.075
+0.100
+N −λ
+0.08
+0.10
+0.12
+0.14
+0.16
+h∗(N, 2N)
+σ = 0.6
+T = 0.6
+λ = 5
+3 − 2σ = 0.467
+χSG
+−0.36N −λ + 0.16
+ξSG
+−0.61N −λ + 0.16
+Figure 8. The intersection fields h∗(N, 2N), for σ = 0.6 with
+T = 0.6. Both the datasets are consistent with a spin glass
+transition at hAT(T = 0.6) ≈ 0.16.
+sions. In this regime, simulations of the corresponding
+Heisenberg model [17, 18] were thought consistent with
+an AT transition.
+According to Eq. (14a), the data for χSG/N 2−η,where
+2 − η = 2σ − 1, plotted for different system sizes should
+intersect at the transition temperature Tc. Similarly, ac-
+cording to Eq. (19a), the curves of ξSG/N should also
+intersect at the transition temperature. The main plots
+of Figs. 9 and 11 show the finite-size-scaled data of χSG,
+and the corresponding inset plots show the finite-size-
+scaled data of ξSG. The curves for different system sizes
+show a clear tendency to intersect close to the same tem-
+perature. The data for T ∗(N, 2N) are then fitted with
+0.50
+0.55
+0.60
+0.65
+0.70
+T
+0.1
+χSG/N 2−η
+σ = 0.75
+hr = 0
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+N = 4096
+N = 8192
+0.5
+0.6
+0.7
+T
+0.1
+1
+ξSG/N
+Figure 9. The main figure shows data for χSG/N 2−η (with
+2 − η = 2σ − 1) for different system sizes, for σ = 0.75 with
+hr = 0. The inset shows the data for ξSG/N. According to
+Eqs. (14a) and (19a) the data should intersect at Tc, which is
+shown in Fig. 10.
+0.0
+0.1
+0.2
+0.3
+0.4
+N −λ
+0.54
+0.56
+0.58
+0.60
+0.62
+0.64
+T ∗(N, 2N)
+σ = 0.75
+hr = 0
+λ = 0.1583
+χSG
+−0.15N −λ + 0.64
+ξSG
+−0.15N −λ + 0.64
+Figure 10. A plot of the intersection temperatures T ∗(N, 2N)
+obtained from the data in Fig. 9, for σ = 0.75 with hr = 0.
+Using the value of the scaling exponent λ = 0.1583 (ob-
+tained from the h∗(N, 2N) data), the T ∗(N, 2N) data are
+fitted against N −λ using a straight line. The resulting values
+for the transition temperature are Tc = 0.6397 ± 0.0051 from
+χSG and Tc = 0.6440 ± 0.0154 from ξSG.
+Eq. (17) where the value of the exponent λ is not known
+in the non-mean-field regime and hence should be con-
+sidered as a fitting parameter.
+If there were an AT transition there would be a unique
+value of λ, the same for both the ξSG and χSG intersec-
+tions corresponding to both h∗(N, 2N) and T ∗(N, 2N).
+The h∗(N, 2N) data obtained from χSG intersections in
+
+9
+0.30
+0.35
+0.40
+0.45
+0.50
+T
+0.3
+0.4
+χSG/N 2−η
+σ = 0.75
+hr = 0.05
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+N = 4096
+N = 8192
+0.35
+0.40
+0.45
+0.50
+0.55
+T
+1
+ξSG/N
+Figure 11.
+A finite size scaling plot of χSG (main figure)
+and ξSG (inset figure) for σ = 0.75.
+The magnetic field is
+hr = 0.05.
+0.0
+0.1
+0.2
+0.3
+0.4
+N −λ
+0.2
+0.3
+0.4
+0.5
+T ∗(N, 2N)
+σ = 0.75
+hr = 0.05
+λ = 0.1583
+χSG
+−0.35N −λ + 0.48
+ξSG
+0.73N −λ + 0.19
+Figure 12. The intersection temperatures T ∗(N, 2N) for σ =
+0.75 with hr = 0.05. The data from Fig. 11 did not fit well
+with Eq. (17). So we used the value of λ obtained from the
+h∗(N, 2N) data and did a linear fitting which gives TAT(hr =
+0.05) = 0.4832 ± 0.0394 from χSG and TAT(hr = 0.05) =
+0.1894 ± 0.0383 from ξSG, and the values do not agree with
+each other.
+Fig. 14 (which is described later) are fitted with Eq. (22)
+by considering λ, Tc and ˜A as fitting parameters. This
+is a non-linear fitting procedure for which we use effi-
+cient methods like the Trusted Region Reflective (TRF)
+algorithm and the Levenberg-Marquardt(LM) algorithm
+(for which packages are available in python) to determine
+the fitting parameters, and we obtain λ = 0.1583. Since
+the exponent giving the leading correction to scaling λ is
+universal, we use the same value of λ with both intersec-
+10−3
+10−2
+10−1
+100
+hr
+10−4
+10−3
+10−2
+10−1
+100
+ξSG/N
+σ = 0.75
+T = 0.55
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+N = 4096
+N = 8192
+N = 16384
+N = 32768
+10−3
+10−2
+10−1
+100
+hr
+10−3
+10−1
+ξSG/N
+Figure 13. A finite size scaling plot of ξSG as a function of
+magnetic field hr, for σ = 0.75 at a temperature of T = 0.55.
+The inset shows our two largest system sizes.
+10−3
+10−2
+10−1
+100
+hr
+10−3
+10−2
+10−1
+χSG/N 2−η
+σ = 0.75
+T = 0.55
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+N = 4096
+N = 8192
+N = 16384
+N = 32768
+Figure 14. A finite size scaling plot of χSG as a function of
+magnetic field hr, for σ = 0.75 at a temperature of T = 0.55.
+tions h∗(N, 2N) and T ∗(N, 2N) obtained from χSG and
+ξSG data. We substitute the value of λ obtained above
+in Eq. (17) and fit the T ∗(N, 2N) data against N −λ with
+a straight line.
+As shown in Fig. 10, for hr = 0, the
+χSG fit (considering all the pairs of system sizes) gives
+Tc = 0.6397 ± 0.0051. The corresponding ξSG fit (omit-
+ting the smallest system size) gives Tc = 0.6440±0.0154.
+For hr = 0.05, the intersection temperatures data
+are shown in Fig. 12.
+Omitting the smallest system
+
+10
+0.0
+0.1
+0.2
+0.3
+0.4
+N −λ
+0.005
+0.010
+0.015
+0.020
+h∗(N, 2N)
+σ = 0.75
+T = 0.55
+λ = 0.1583
+χSG
+0.03N −λ + 0.00
+ξSG
+−0.01N −λ + 0.02
+Figure 15. The intersection fields h∗(N, 2N) for σ = 0.75 with
+T = 0.55.
+size, the T ∗(N, 2N) data are fitted with Eq.
+(17) to
+give TAT(hr = 0.05) = 0.4832 ± 0.0394 from χSG and
+TAT(hr = 0.05) = 0.1894 ± 0.0383 from ξSG. Compared
+to Fig.
+5 which gives the equivalent plot for the case
+with σ = 0.60, the data in Fig. 12 does not look like
+data which is converging to the same asymptotic limit
+when N is large. If the crossings were actually due to a
+genuine AT transition, then the asymptotic limit should
+be the same for both.
+We have also studied ξSG and χSG at fixed T, but vary-
+ing hr and the finite size scaling plots for these are given
+in Figs.
+13 and 14.
+There appears to be good inter-
+sections in the curves, supporting therefore the possible
+existence of an AT transition at the temperature studied
+T = 0.55. A plot of h∗(N, 2N) versus 1/N λ is in Fig. 15,
+using the same value of λ = 0.1583. In the intersections
+of ξSG there is a clear rising trend of h∗(N, 2N) with in-
+creasing N until N = 1024, followed by decreasing values
+of h∗(N, 2N) for N > 2048. For the case of σ = 0.60,
+where there is almost certainly a genuine AT transition,
+(Fig. 8) only the rising trend is seen. It is as if for the
+smaller systems N < 2048 the system at σ = 0.75 is
+behaving similarly to its mean-field cousin at σ = 0.60.
+Note that this change of trend cannot be attributed to the
+correction to scaling terms of Eq. (23). These only apply
+in the limit N → ∞ with N 1/ν(hr −hAT(T)) fixed. For a
+genuine AT transition the intersections h∗(N, 2N) from
+both ξSG and χSG should both extrapolate as N → ∞ to
+the same field hAT(T). It is hard to argue that Fig. 15
+provides good evidence for this. On the other hand, on
+the droplet picture, it would be expected that h∗(N, 2N)
+should extrapolate to zero. The evidence that is happen-
+ing is also weak.
+0.25
+0.30
+0.35
+0.40
+T
+0.1
+0.2
+0.3
+χSG/N 2−η
+σ = 0.85
+hr = 0
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+N = 4096
+0.3
+0.325
+0.35
+0.375
+0.4
+T
+1
+ξSG/N
+Figure 16. A finite size scaling plot of χSG (main figure) and
+ξSG (inset figure), for σ = 0.85 in a magnetic field of hr = 0
+(with 2 − η = 2σ − 1). Both the datasets clearly indicate that
+a phase transition occurs. The transition temperature in the
+thermodynamic limit is estimated in Fig. 17.
+0.000
+0.002
+0.004
+0.006
+N −λ
+0.32
+0.34
+0.36
+0.38
+T ∗(N, 2N)
+σ = 0.85
+hr = 0
+λ = 1.0315
+χSG
+−2.93N −λ + 0.33
+ξSG
+9.78N −λ + 0.33
+Figure 17. The intersection temperatures T ∗(N, 2N), for σ =
+0.85 with hr = 0. A non-linear fit of the χSG data from Fig. 16
+with the Eq. (17) using the Levenberg-Marquadt algorithm
+gives λ = 1.0315. A linear fit of the data using this value of λ
+gives Tc = 0.3336±0.0013 from χSG and Tc = 0.3297±0.0036
+from ξSG.
+C.
+σ = 0.85
+For σ = 0.85 we are further into the non-mean-field
+region.
+According to Eq. (9), σ = 0.85 corresponds
+to a short-range model close to three dimensions.
+In
+this regime, simulations of the corresponding Heisenberg
+model [17, 18] did not find an AT line.
+
+11
+0.1
+0.2
+0.3
+0.4
+T
+0.1
+χSG/N 2−η
+σ = 0.85
+hr = 0.05
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+Figure 18. A finite size scaling plot of χSG for σ = 0.85. The
+magnetic field is hr = 0.05.
+0.1
+0.2
+0.3
+0.4
+T
+0.1
+0.2
+χSG/N 2−η
+σ = 0.85
+hr = 0.02
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+Figure 19.
+A finite size scaling plot of χSG, for σ = 0.85
+with hr = 0.02.
+The data do not intersect even at very
+low temperatures (much lower than the mean field value of
+TAT(hr = 0.02) = 0.2997 obtained using Eq. (8)) indicating
+that there is no phase transition in this regime.
+For hr = 0, Fig. 16 clearly shows that the curves for
+different system sizes are intersecting. The data for in-
+tersection temperatures are shown in Fig. 17. Similar to
+the case of σ = 0.75, the T ∗(N, 2N) data obtained from
+χSG are fitted with Eq. (17) by considering λ, Tc, and A
+as fitting parameters. We obtain λ = 1.0315 from both
+TRF and LM methods. The fit using the χSG data for
+0.1
+0.2
+0.3
+0.4
+T
+1
+ξSG/N
+σ = 0.85
+hr = 0.05
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+Figure 20.
+A finite size scaling plot of ξSG, for σ = 0.85
+with hr = 0.05.
+The data show merging behavior at low
+temperatures.
+0.1
+0.2
+0.3
+0.4
+T
+100
+101
+ξSG/N
+σ = 0.85
+hr = 0.02
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+Figure 21.
+A finite size scaling plot of ξSG, for σ = 0.85
+with hr = 0.02.
+The data show merging behavior at low
+temperatures.
+all the pairs of system sizes gives Tc = 0.3336 ± 0.0013.
+The corresponding ξSG fit (omitting the smallest system
+size) gives Tc = 0.3297 ± 0.0036. The two values of Tc
+are quite close.
+For hr = 0.05 the χSG/N 2−η data do not intersect as
+shown in Fig. 18. Such a field could conceivably be above
+the largest AT field even at T = 0 so we also studied a
+smaller field: hr = 0.02 shown in Fig. 19. There is no
+
+12
+10−3
+10−2
+10−1
+100
+hr
+10−2
+10−1
+100
+ξSG/N
+σ = 0.85
+T = 0.3
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+N = 4096
+10−3
+10−2
+10−1
+100
+hr
+10−2
+10−1
+100
+ξSG/N
+Figure 22. A finite size scaling plot of ξSG as a function of
+magnetic field hr, for σ = 0.85 at a temperature of T = 0.3.
+The inset shows our two largest system sizes.
+10−3
+10−2
+10−1
+100
+hr
+10−3
+10−2
+10−1
+χSG/N 2−η
+σ = 0.85
+T = 0.3
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+N = 4096
+Figure 23. A finite size scaling plot of χSG as a function of
+magnetic field hr, for σ = 0.85 at a temperature of T = 0.3.
+sign of any crossing at this field either!. The ξSG data is
+less clearcut. Fig. 20 shows there are no intersections at
+a field of hr = 0.05 while a merging behavior is seen for
+the larger systems at hr = 0.02, as shown in Fig. 21. In
+our simulations we went to very low temperatures such as
+T = 0.1, which is small in comparison with the mean-field
+values of TAT for hr = 0.02 and hr = 0.05 using Eq. (8),
+but we still could not find any clear intersections in the
+χSG or ξSG data. This suggests that there is no phase
+0.000
+0.002
+0.004
+0.006
+N −λ
+0.01
+0.02
+0.03
+h∗(N, 2N)
+σ = 0.85
+T = 0.3
+λ = 1.0315
+χSG
+0.36N −λ + 0.00
+ξSG
+3.96N −λ + 0.00
+Figure 24. The intersection fields h∗(N, 2N) for σ = 0.85 with
+T = 0.3. We substitue the value of the exponent λ = 1.0315
+obtained from the T ∗(N, 2N) data at hr = 0 in Eq. (22)
+and fit h∗(N, 2N) data agianst N −λ to get hAT(T = 0.3) =
+0.0046±0.0006 from χSG and hAT(T = 0.3) = 0.0047±0.0016
+from ξSG.
+transition in this regime in the presence of a magnetic
+field. Our data are consistent with the scenario where
+the external magnetic field destroys the phase transition,
+just as happens for a ferromagnet when a uniform field
+is turned on.
+Very similar features were seen for the
+Heisenberg version of this model [17, 18] and in the three
+dimensional Ising model [34].
+Confusingly, intersections are seen at fixed T = 0.3
+as hr is varied in the plots of ξSG/N in Fig. 22 and of
+χSG/N 2−η in Fig. 23. The usual analysis of h∗(N, 2N)
+is given in Fig. 24. Thus in crossing the AT line along a
+trajectory of fixed T we have seen intersections, suggest-
+ing there might be an AT transition. However, the large
+N limit of h∗(N, 2N) in Fig. 24 in the case of σ = 0.85,
+suggests that hAT(T) might actually be zero, consistent
+with the droplet scaling picture. In the next section the
+dependence of ξSG and χSG on hr will be explained using
+the droplet scaling approach.
+VI.
+DATA ANALYSES ON THE DROPLET
+PICTURE
+In this section we give the field dependence of ξSG and
+χSG according to the droplet picture [44–46], including
+also their finite size modifications, and compare these
+with our simulation data.
+In the droplet picture one uses an Imry-Ma argument
+[47] for the correlation length ξ and identifies it with
+the size of the region or domain within which the spins
+become re-oriented in the presence of the random field.
+The free energy gained from such a reorientation by the
+the random field is of order
+�
+q(T)hrξd/2. The size of
+such domains ξ is determined by equating this free energy
+
+13
+10−3
+10−2
+10−1
+100
+101
+hr
+10−1
+101
+103
+105
+107
+ξSG
+σ = 0.75
+T = 0.55
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+N = 4096
+N = 8192
+N = 16384
+N = 32768
+fit for N = 32768
+(∼ h−2.71
+r
+)
+Figure 25. A plot of ξSG as a function of magnetic field hr,
+for σ = 0.75 at a temperature of T = 0.55.
+10−3
+10−2
+10−1
+100
+101
+hr
+100
+102
+104
+106
+ξSG
+σ = 0.85
+T = 0.3
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+N = 4096
+fit for N = 4096
+(∼ h−2.20
+r
+)
+Figure 26. A plot of ξSG as a function of magnetic field hr,
+for σ = 0.85 at a temperature of T = 0.3.
+to the free energy cost of the interface of this domain of
+re-ordered spins with the rest of the system, which is of
+the form Υ(T)ξθ [48]. Equating these two free energies
+gives
+ξ ∼
+�
+Υ(T)
+�
+q(T)hr
+�1/(d/2−θ)
+.
+(24)
+While there is a considerable literature on the depen-
+dence of the interface exponent θ on σ for the case of
+10−2
+10−1
+100
+101
+hr
+10−6
+10−3
+100
+103
+106
+109
+1012
+ξSG
+σ = 0.6
+T = 0.6
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+N = 4096
+N = 8192
+N = 16384
+fit for N = 16384
+(∼ h−6.88
+r
+)
+Figure 27. A plot of ξSG as a function of magnetic field hr,
+for σ = 0.6 at a temperature of T = 0.6.
+10−1
+101
+N 1/xhr
+10−6
+10−4
+10−2
+100
+ξSG/N
+σ = 0.75
+T = 0.55
+x = 2.7077
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+N = 4096
+N = 8192
+N = 16384
+N = 32768
+Figure 28.
+A complete finite size scaling plot of ξSG as a
+function of magnetic field hr, plotted on a log-log scale, for
+σ = 0.75 at a temperature of T = 0.55.
+Ising spin glasses [49], we know of no equivalent studies
+for the case of the XY spin glass. (Our data suggests
+that its θ might be close to that of the Ising spin glass).
+Eq. (24) shows that as hr → 0, the length scale be-
+comes infinite; ξ diverges as ξ ∼ 1/hx
+r, where
+x =
+1
+d/2 − θ.
+(25)
+The exponent x is the analogue of ν at the AT transition;
+
+14
+10−1
+101
+N 1/xhr
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+ξSG/N
+σ = 0.85
+T = 0.3
+x = 2.2019
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+N = 4096
+Figure 29.
+A complete finite size scaling plot of ξSG as a
+function of magnetic field hr, plotted on a log-log scale, for
+σ = 0.85 at a temperature of T = 0.3.
+10−1
+100
+101
+102
+N 1/xhr
+10−6
+10−4
+10−2
+100
+ξSG/N
+σ = 0.75
+T = 0.55
+x = 2.7077
+N = 16384
+N = 32768
+Figure 30.
+A complete finite size scaling plot of ξSG as a
+function of magnetic field hr, for σ = 0.75 at a temperature
+of T = 0.55, showing our two largest system sizes.
+it is as if the AT transition hAT(T) = 0.
+We would
+expect this formula to apply until finite size effects limit
+its growth, which will occur when ξ is of O(L) (or O(N)
+in our one-dimensional system). Identifying ξSG with ξ,
+Figs. 25 and 26 show that the Imry-Ma fit indeed works
+well at the larger fields; the data for the larger hr collapse
+nicely onto a power law form as predicted by Eq. (24)
+for all sizes N. It only departs from this formula when
+10−1
+100
+101
+102
+N 1/xhr
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+ξSG/N
+σ = 0.85
+T = 0.3
+x = 2.2019
+N = 2048
+N = 4096
+Figure 31.
+A complete finite size scaling plot of ξSG as a
+function of magnetic field hr, for σ = 0.85 at a temperature
+of T = 0.3, showing our two largest system sizes.
+10−1
+101
+N 1/xhr
+10−5
+10−4
+10−3
+10−2
+10−1
+χSG/N z
+σ = 0.75
+T = 0.55
+x = 2.7077
+xχ = 1.6114
+z = 0.5951
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+N = 4096
+N = 8192
+N = 16384
+N = 32768
+Figure 32.
+A complete finite size scaling plot of χSG as a
+function of magnetic field hr, for σ = 0.75 at a temperature
+of T = 0.55.
+ξSG becomes of order N, when finite size corrections to
+the Imry-Ma formula are needed. Also TNT effects (see
+Sec. VII) produce corrections to the Imry-Ma formula
+when ξSG is of O(N) unless N = L > L∗. The crossover
+scale L∗ is thought to be large, especially as σ approaches
+2/3 (or d → 6) [3].
+To allow for finite size effects on the Imry-Ma formula
+we use the analogue of Eq. (21a) with hAT = 0 and ν = x
+
+15
+10−1
+100
+101
+102
+N 1/xhr
+10−5
+10−4
+10−3
+10−2
+10−1
+χSG/N z
+σ = 0.75
+T = 0.55
+x = 2.7077
+xχ = 1.6114
+z = 0.5951
+N = 16384
+N = 32768
+Figure 33.
+A complete finite size scaling plot of χSG as a
+function of magnetic field hr, for σ = 0.75 at a temperature
+of T = 0.55, for our two largest system sizes.
+10−1
+101
+N 1/xhr
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+χSG/N z
+σ = 0.85
+T = 0.3
+x = 2.2019
+xχ = 1.8919
+z = 0.8592
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+N = 4096
+Figure 34.
+A complete finite size scaling plot of χSG as a
+function of magnetic field hr, for σ = 0.85 at a temperature
+of T = 0.3.
+to write:
+ξSG/N = X(N 1/xhr).
+(26)
+Our results for σ = 0.75 are shown in Fig. 28 and for σ =
+0.85 are shown in Fig.
+29. There are clearly finite size
+corrections to this formula. It is a formula which formally
+would be expected to hold in the scaling limit of N → ∞
+with N 1/xhr fixed. The crossover function X(y) ∼ 1/yx
+10−1
+100
+101
+102
+N 1/xhr
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+χSG/N z
+σ = 0.85
+T = 0.3
+x = 2.2019
+xχ = 1.8919
+z = 0.8592
+N = 2048
+N = 4096
+Figure 35.
+A complete finite size scaling plot of χSG as a
+function of magnetic field hr, for σ = 0.85 at a temperature
+of T = 0.3, for our two largest system sizes.
+0.3
+0.4
+0.5
+0.6
+0.7
+1/N z−(2σ−1)
+0.2
+0.4
+0.6
+0.8
+h∗(N, 2N) N 1/x
+χSG(σ = 0.7)
+ξSG(σ = 0.7)
+χSG(σ = 0.75)
+ξSG(σ = 0.75)
+χSG(σ = 0.85)
+ξSG(σ = 0.85)
+Figure 36. A plot of h∗(N, 2N)N 1/x versus 1/N ω, for σ =
+0.70, 0.75 and 0.85. The values of x and ω, which is obtained
+from Eq. (36) were taken from Table I.
+when y is large, in order to recover Eq. (24). It goes to
+a constant when y → 0. However, a closer look at our
+two largest system sizes N = 16384 and N = 32768 at
+σ = 0.75 (Fig. 30) and our two largest system sizes at
+σ = 0.85, N = 2048 and N = 4096 (Fig. 31) shows that
+the finite size corrections are becoming small, and are
+smaller the further the system is away from the mean-
+field region. If one moves in the other direction, towards
+
+16
+103
+104
+N
+0.24
+0.26
+0.28
+0.30
+0.32
+0.34
+0.36
+R
+σ = 0.6, T = 0.6
+Figure 37. A plot of R = χSG(h∗(N, 2N), N)/N 1/3 versus N,
+for σ = 0.6 at a temperature of T = 0.6.
+103
+104
+N
+0.28
+0.29
+0.30
+0.31
+0.32
+0.33
+0.34
+R
+σ = 0.75, T = 0.55
+Figure 38. A plot of R = χSG(h∗(N, 2N), N)/N 2σ−1 versus
+N, for σ = 0.75 at a temperature of T = 0.55.
+the start of the mean-field region σ = 2/3, the finite size
+corrections are larger, as seen in Fig. 40 for σ = 0.70.
+The finite size scaling form for these corrections to the
+scaling of Eq. (26) will be of the form
+ξSG
+N
+= X(N 1/xhr) + N −ωH(N 1/xhr),
+(27)
+where ω is the correction to scaling exponent. However,
+TNT effects (see Sec. VII) produce large further correc-
+tions to these asymptotic forms when L < L∗. Since in
+103
+N
+0.2200
+0.2225
+0.2250
+0.2275
+0.2300
+0.2325
+0.2350
+0.2375
+R
+σ = 0.85, T = 0.3
+Figure 39. A plot of R = χSG(h∗(N, 2N), N)/N 2σ−1 versus
+N, for σ = 0.85 at a temperature of T = 0.3.
+our studies L∗ is probably larger than the length N of
+our system, at least for σ = 0.75, the scaling form of
+Eq. (27) does not work in the region where ξSG is of or-
+der N (see Fig. 42). For σ = 0.85 where L∗ is expected
+to be smaller, Fig. 43 hints that Eq. (27) might apply as
+the plots at adjacent sizes for the larger N values seem
+to be getting closer together as N is increased, which is
+a feature predicted by Eq. (27).
+In Fig.
+27 we show a similar plot to those in Figs.
+25 and 26 but for the case of σ = 0.60. Notice however
+that because of the AT transition at this value of σ, at
+which ξSG would diverge to infinity as N → ∞ at some
+finite field hr = hAT(T), a shoulder above the dashed
+line has started to appear which is the beginning of this
+divergence. Such a feature is absent in the figures for
+both σ = 0.75 and at σ = 0.85.
+The spin glass susceptibility according to the droplet
+picture is a similar generalization of the finite size scaling
+form of Eq. (14a):
+χSG
+N z = C(N 1/xhr),
+(2/3 ≤ σ < 1).
+(28)
+The crossover function C(y) ∼ 1/yxχ when y is large, so
+that then χSG ∼ ξz and becomes independent of N. Its
+form is then
+χSG ∼ 1/hxχ
+r ,
+(29)
+which implies that xχ = xz.
+In the opposite limit as
+y → 0, C(y) goes to a finite constant. The exponent z
+depends upon whether we are dealing with short-range
+interactions, (such as nearest-neighbor interactions) or
+with the long-range interactions employed in this paper.
+For short-range interactions, the average value of χ2
+ij falls
+
+17
+Table I. Values of the exponents xχ, x, and z obtained from
+our simulations for different values of σ and T.
+σ
+T
+xχ
+x
+z
+0.7
+0.6
+1.5747 ± 0.0009
+3.3220 ± 0.0413
+0.4740 ± 0.0062
+0.75
+0.55
+1.6114 ± 0.0005
+2.7077 ± 0.0531
+0.5951 ± 0.0119
+0.85
+0.3
+1.8919 ± 0.0014
+2.2019 ± 0.0152
+0.8592 ± 0.0066
+off with spin separation rij as
+χ2
+ij ∼ q(T)2T
+Υ(T)rθ
+ij
+,
+(30)
+[45, 46]. This result applies in the zero-field spin glass
+state. Then as,
+χSG = 1
+N
+N
+�
+i,j=1
+χ2
+ij,
+(31)
+so in d dimensions for the zero-field spin glass χSG ∼
+Ld−θ. Hence
+z = d − θ
+d
+,
+(32)
+in order to recover the result χSG → N z as N 1/xhr goes
+to zero. We caution that this formula for z will only hold
+for short-range interactions.
+With long-range interactions a “droplet” is not a single
+connected region but a set of isolated islands of flipped
+spins [49] and this will make the decay of χ2
+ij with rij
+faster than in Eq. (30). This is an effect which has not
+been studied before, and so in our problem the exponent
+z has to be determined by fitting the data. The results
+of our determinations of the droplet exponents x, xχ and
+z for the different values of σ which we have studied are
+summarised in Table 1.
+The resulting excellent data collapse (at least when
+ξSG < N), is shown in Figs. 32, 33,
+34, and
+35. The
+value of z was determined from the observation that when
+N 1/xhr is large, χSG should be independent of N. It is
+remarkable that z determined at large values of N 1/xhr
+results in a decent collapse of the data in the opposite
+limit where N 1/xhr → 0. Nevertheless corrections to the
+Imry-Ma scaling form are visible in the figures (and are
+sizeable in the region where N 1/xhr is small when viewed
+in a linear plot rather than a log scale plot, (just as in the
+ξSG plots Figs. 28 and 29). In the limit when N 1/xhr is
+held fixed with N → ∞ the leading correction to scaling
+will be
+χSG
+N z = C(N 1/xhr) + N −ωG(N 1/xhr),
+(33)
+where G(y) is an unknown scaling function and the cor-
+rection to scaling exponent ω is not known with any cer-
+tainty (but see Eq. (36)).
+Let us suppose that the droplet picture is correct and
+that (say) the spin glass susceptibility χSG is described
+by Eq. (28). This equation predicts that there will be a
+crossing in the plots of χSG/N 2σ−1 used in AT line criti-
+cal scaling studies. (Note we are setting 2 − η = 2σ − 1).
+The correction to scaling term of Eq. (33) is not needed
+for this, but this correction does strongly influence where
+the crossings take place for the N values which are
+reached in our simulations. The crossing arises as fol-
+lows. At small values of hrN 1/x, the function C goes to a
+constant. It turns out that z > (2σ − 1), so χSG/N 2σ−1
+diverges as N is increased as N z−(2σ−1) as hr → 0. On
+the other hand when hrN 1/x is large, χSG → 1/hz
+r, so
+χSG/N 2σ−1 → 1/N 2σ−1hz
+r → 0 as N goes to infinity.
+Because at small fields, χSG/N 2σ−1 is larger for large N,
+but at bigger hr fields it is smaller at the larger N val-
+ues, so there must be a crossing point. We shall denote
+the crossing value between the lines at N and 2N by
+h∗(N, 2N) = H. Then H is determined by the solution
+of the following
+χSG(H, N)
+N 2σ−1
+= C(N 1/xH)N z−(2σ−1) =
+χSG(H, 2N)
+(2N)2σ−1
+= C((2N)1/xH)(2N)z−(2σ−1).
+(34)
+Assuming C(y) → a − by, when y → 0, it is easy to
+show then that the N dependence of h∗(N, 2N) at very
+large N will be as 1/N 1/x. In reality we have no data
+in this region of very large N where the corrections to
+scaling term in Eq. (33) can be ignored. The corrections
+to scaling are numerically small but are very important
+in determining the values of h∗(N, 2N).
+There is a similar crossing predicted in the plots of
+ξSG/N as a function of hr when Eq. (27) holds, using the
+analogue of Eq. (34). In this case it is the scaling cor-
+rection which causes the curves to cross, (which requires
+H(0) to be negative), and for these curves the crossings
+h∗(N, 2N) at very large N will decrease as 1/N 1/x+ω,
+(compare with Eq. (18b)) on taking χ(y) = c − dy and
+H(y) → constant as y → 0. Once again we have no data
+in this very large N regime. In Fig. 36 we have plotted
+h∗(N, 2N)N 1/x versus 1/N ω, assuming that ω is given by
+Eq. (36). Note that the size of the corrections to scaling
+∼ 1/N ω is simply not small for the values of N which we
+can study, contrary to what was assumed in the above.
+h∗(N, 2N)N 1/x should go to a constant as N goes to in-
+finity and it is only for the case of σ = 0.85, where the
+corrections to scaling are the smallest of the three cases
+studied, does that look remotely possible. For the case
+of σ = 0.70 the corrections look to be very large. We
+conclude that for the values of σ = 0.70 and σ = 0.75,
+the crossing data on h∗(N, 2N) is not close to the large
+N asymptotic form predicted by the droplet picture. But
+the droplet picture does predict that the existence of such
+intersections.
+If we only had information on the values of the cross-
+ing fields h∗(N, 2N) it would be difficult to really be sure
+whether the droplet picture or the RSB picture best de-
+scribed the data. The results on h∗(N, 2N) alone are in-
+conclusive as regards both the AT transition line picture
+
+18
+100
+101
+102
+N 1/xhr
+10−7
+10−5
+10−3
+10−1
+101
+ξSG/N
+σ = 0.7
+T = 0.6
+x = 3.3220
+N = 8192
+N = 16384
+Figure 40.
+A complete finite size scaling plot of ξSG as a
+function of magnetic field hr, for σ = 0.70 at a temperature
+of T = 0.6 for our two largest system sizes.
+and the droplet picture. While on the droplet picture
+h∗(N, 2N) are predicted to go to zero as N → ∞, the
+values of h∗(N, 2N) are not convincingly going to zero as
+N is increased (see Fig. 36). Fortunately, there is another
+way of distinguishing the two approaches, which does
+not require us to reach the N values at which h∗(N, 2N)
+starts to approach zero. We define
+R ≡ χSG(h∗(N, 2N), N)/N 2σ−1.
+(35)
+(Because we only determine χSG(hr, N) at a finite num-
+ber of values of hr, we use linear interpolation to calcu-
+late χSG(h∗(N, 2N), N) using the χSG(hr, N) values at
+the two determined values of hr which lie on either side
+of h∗(N, 2N)). On the phase transition picture, R should
+approach a finite constant as N → ∞. On the droplet
+picture R should increase as N z−(2σ−1) as N → ∞. For
+σ = 0.60 where an AT line is expected R should go to
+a constant but at the N values studied it actually still
+appears to be decreasing (see Fig. 37) and has yet to be-
+come constant, presumably due to finite size effects. This
+indicates that trying to determine whether σ = 2/3 is the
+exact value at which the crossover to droplet scaling be-
+havior will also be challenging from the side below 2/3.
+However, for σ = 0.75, Fig. 38 shows that R is clearly
+increasing with N for large N values. But if we had had
+only data for system sizes < 2048 we might have indeed
+concluded that there was good evidence for an AT transi-
+tion in that R seemed to be an N independent constant.
+While at the sizes we can reach R is clearly increasing
+with N it has yet to reach its asymptotic form of in-
+crease as N z−(2σ−1). The quantity R also increases with
+N for σ = 0.70 and σ = 0.85, (see for example Fig. 39).
+In order for χSG to match as σ → 2/3 from either the
+100
+101
+102
+N 1/xhr
+10−4
+10−3
+10−2
+10−1
+χSG/N z
+σ = 0.7
+T = 0.6
+x = 3.3220
+xχ = 1.5747
+z = 0.4740
+N = 8192
+N = 16384
+Figure 41. A complete finite size scaling plot of χSG a function
+of magnetic field hr, for σ = 0.70 at a temperature of T = 0.6
+for our two largest system sizes.
+mean-field side, (where z = 1/3) with its value in the
+non-mean field region, we would expect that z should
+approach 1/3 as σ → 2/3 from above.
+At σ = 0.85,
+z ≈ 0.8409, at σ = 0.75, z ≈ 0.6065, while at σ = 0.70,
+we find z ≈ 0.4737. Thus it seems quite plausible that
+z could approach 1/3 as σ → 2/3 from above. Then the
+combination z − (2σ − 1) would approach zero in this
+limit, which means that the divergence of R with N will
+become harder and harder to see as σ approaches 2/3.
+We conclude that it will be challenging to do numerical
+work which shows that the AT line disappears at precisely
+σ = 2/3. On the mean-field side of 2/3 the correction to
+scaling exponent ω = 1/3 − (2σ − 1). It therefore seems
+natural to expect that on the non-mean field regime
+ω = z − (2σ − 1).
+(36)
+If valid, this would imply that corrections to scaling
+should be larger at σ = 0.70 than at σ = 0.75, and this
+is what we observed in Figs. 40 and 41, in comparison
+with Figs. 30 and 33.
+In the presence of a genuine AT transition, as hr is
+reduced one would pass through three regions: first the
+paramagnetic state at larger values of hr, then the criti-
+cal region, then the low-temperature phase with RSB at
+smaller values of hr. The good data collapse for all val-
+ues of hr using Eq. (26), and Eq. (28) shows that at any
+finite value of hr there is just one region, the paramag-
+netic region. Studying “intersections��� as in Sec. V is an
+attempt to find the critical region. But the intersections
+at finite values of hr for σ = 0.75 and σ = 0.85 are not
+signs of a genuine phase transition, but at least in the
+case of χSG these crossings are also just a consequence of
+droplet scaling. The behavior of h∗(N, 2N) as a function
+
+19
+10−2
+10−1
+100
+N 1/xhr
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+ξSG/N
+σ = 0.75
+T = 0.55
+x = 2.7077
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+N = 4096
+N = 8192
+N = 16384
+N = 32768
+Figure 42. A finite size scaling plot of ξSG as a function of
+magnetic field hr, for σ = 0.75 at a temperature of T = 0.55.
+of N is greatly complicated by finite size effects and will
+only become clear at much larger N values than those
+which we have been able to study.
+Because on the droplet picture there is no AT line and
+so one is always in the paramagnetic phase at any non-
+zero field (just as in a ferromagnet).
+However, length
+scales like ξSG become very large as hr → 0 for temper-
+atures T < Tc(hr = 0). Once they become comparable
+to the system dimensions L and one is in the regime
+hr < h∗(N, 2N), the system will have many of the fea-
+tures which might be associated with being in the bro-
+ken replica symmetric phase which is envisaged to exist
+below the AT line. For physical systems in three dimen-
+sions the relevant length scale is not the linear dimension
+of the system L, but the linear dimension of a fully equi-
+librated region. This may explain why both simulations
+and experiments have failed for many years to resolve the
+debate.
+Might it be possible to find by simulations whether the
+borderline between RSB ordering and droplet ordering is
+at σ = 2/3, which is the equivalent of d = 6 with short-
+range interactions? To this end we looked at the case
+of σ = 0.70. We found from studying the crossings of
+ξSG and χSG for the zero field case that the zero field
+transition temperature is ≈ 0.724. Figs. 40 and 41 show
+our attempt to collapse the data with the droplet scaling
+forms. Clearly the effects of corrections to scaling are
+larger than was the case at σ = 0.75 in Figs. 30 and
+33. This is in accord with Eq. (36) which predicts that
+the correction to scaling exponent ω will go to zero as
+σ → 2/3 if also z → 1/3 as expected. We conclude that
+it will be difficult to provide good numerical evidence
+that σ = 2/3 is the lower critical dimension of the AT
+transition.
+10−2
+10−1
+100
+101
+N 1/xhr
+0.5
+1.0
+1.5
+2.0
+ξSG/N
+σ = 0.85
+T = 0.3
+x = 2.2019
+N = 128
+N = 256
+N = 512
+N = 1024
+N = 2048
+N = 4096
+Figure 43. A finite size scaling plot of ξSG as a function of
+magnetic field hr, for σ = 0.85 at a temperature of T = 0.3.
+VII.
+TNT VERSUS THE DROPLET SCALING
+PICTURE
+Newman and Stein [50] (see also the recent review
+[51]), have suggested that the ordered phase of spin
+glasses in finite dimensions will fall into one of 4 cate-
+gories, (and which one might depend on the dimension-
+ality d of the system): The RSB state is one of these,
+and is somewhat similar to that envisaged by Parisi for
+the SK model, but there is also the chaotic pairs state
+picture of Newman and Stein. In both of these pictures
+there is an AT transition. The other two pictures are
+the so-called TNT picture of Krzakala and Martin [35]
+and Palassini and Young [36] and the droplet scaling pic-
+ture [44–46]. In neither the TNT picture nor the droplet
+scaling picture is there an AT transition. In the droplet
+picture the Parisi overlap function P(q) is trivial, consist-
+ing of two delta functions at ±qEA in zero field, whereas
+in the TNT picture the form of P(q) is quite similar to
+the non-trivial (NT) form which Parisi found for the SK
+model.
+The TNT picture accounts for the non-trivial
+form of the Parisi overlap function by postulating that
+there exist droplets of the linear size L of the system,
+which contain O(Ld) spins, and which do not have a free
+energy of order Lθ (as they would in the droplet scaling
+picture), but which have instead a free energy of O(1). It
+is the presence of such droplets which makes P(q) non-
+trivial, which is a feature observed in all simulations of
+it to date.
+In a recent paper [3] one of us argued that once the
+linear dimension of the system became larger than a
+crossover length L∗ the non-trivial behavior observed in
+P(q) will change to the trivial form predicted by droplet
+scaling. Estimates of L∗ in d = 3 suggest it might be
+
+20
+large, of the order of several hundred lattice spacings
+and it is probably the case that to date the regime where
+L > L∗ has not been reached. Furthermore it was sug-
+gested that as d → 6, L∗ would grow towards infinity,
+as the droplets of O(1) evolve to the O(1) excitations in
+the Parisi RSB solution, where the pure states have free
+energies which differ from each other by O(1). In our
+one dimensional proxy system we would therefore expect
+to find that L∗ is much larger when σ = 0.75 than it is
+when σ = 0.85.
+In this paper there are TNT-like effects visible in the
+behavior of ξSG/N in the region where ξSG is of O(N)
+(see Figs.
+28 and 29).
+When ξSG is of order N the
+droplets which are important are those of size N and if
+L < L∗ some of these will have free energy of O(1) rather
+than Lθ. As a consequence the good scaling collapse of
+the data visible when ξSG/N ≪ 1 will be lost. In Figs.
+42 and 43 we have plotted ξSG/N on a linear scale versus
+N 1/xhr focussing only on the region where ξSG/N is of
+O(1). If the droplet scaling collapse had been good and
+of the form of Eq. (27) then as N is increased the collapse
+should get better and better. In fact due to TNT effects
+the data in Fig. 42 for σ = 0.75 show the opposite trend,
+and the lines get further apart with increasing N in the
+region where ξSG is of O(N).
+However, for σ = 0.85
+Fig. 43 shows the lines seem to be getting closer with
+increasing N.
+It suggests that for this value of σ we
+are getting into the region where L > L∗ when droplet
+scaling applies even when ξSG is of O(N). Data at larger
+values of N than 4096 would be nice to confirm this trend
+but because these simulations have to be done at quite
+low temperatures compared to those for σ = 0.75 it will
+be challenging to do this. Despite this limitation on the
+size of N which can be reached for σ = 0.85, there is
+evidence that for it, TNT and finite size scaling effects
+are less troublesome than for σ = 0.75, despite the fact
+that much larger values of N can be studied at this σ
+value.
+VIII.
+SUMMARY AND CONCLUSIONS
+In this paper, we have studied the phase transitions
+in the one-dimensional power-law diluted XY spin glass,
+both in the zero-field limit, and in the presence of a mag-
+netic field random in the component directions. Whether
+or not an AT line exists for various values of the param-
+eter σ is a question of fundamental interest. To address
+this, we have performed large scale Monte-Carlo sim-
+ulations using a new heatbath algorithm, described in
+Appendix A. This algorithm hopefully speeds up equi-
+libration, so cutting computational costs. We certainly
+do gain some advantage in terms of computational time
+due to the smaller number of components of XY spins
+compared to those of the Heisenberg model. Alas, the
+heatbath algorithm for XY spins suffers from an intrin-
+sic disadvantage. Because our algorithm has to generate
+two random numbers during each Monte Carlo step, the
+benefits of the smaller number of components are largely
+counterbalanced by the additional labor involved in the
+heatbath step. We were unable to go to larger system
+sizes than in the corresponding work with Heisenberg
+spins [18].
+The largest system sizes that we are able
+to simulate are: N = 16384 for σ = 0.6, N = 32768 for
+σ = 0.75, while the largest N for σ = 0.85 was 4096. The
+total CPU time spent in generating all the data that we
+presented at fixed hr and varying T was 1183636.2 hrs,
+which is 135.12 years. The total CPU time consumed in
+generating the data at fixed T and varying hr was 96101.6
+days which is 263.29 years. Thus despite the algorithm
+not producing significant dividends, we are able to study
+fairly large system sizes owing to the expenditure of a
+large amount of computer time.
+The results from our work are broadly in accord with
+those for the corresponding Heisenberg spin glass model.
+For σ = 0.6, which is in the mean-field regime, we find
+a phase transition in the absence of an external mag-
+netic field, and in the presence of a magnetic field, which
+indicates the existence of an AT line. The location of
+the AT line is close to the mean-field predictions. For
+σ = 0.75, which is in the non-mean-field regime, the con-
+ventional data collapse suggests the existence of an AT
+line, but the behavior of the intersections as a function
+of N indicate that the data is not close to its large N
+asymptotic form. The estimated location of the AT field
+based upon intersections that we get from our data at
+σ = 0.75 is strikingly smaller than estimates based on
+the mean-field theory formulas. For σ = 0.85, which is
+deep in the non-mean-field regime and corresponds to a
+space dimension of about 3, our data are consistent with
+the absence of an AT line. In this case there is no crossing
+of the curves of χSG/N 2−η versus T at various N values.
+But confusingly intersections h∗(N, 2N), as a function of
+hr, seem to exist, whereas intersections T ∗(N, 2N) are
+absent at least for σ = 0.85.
+However, for σ = 0.75 and for σ = 0.85 we found that
+the droplet picture provided a much better description
+of our data from that obtained assuming the existence
+of an AT transition line. The Imry-Ma formula for the
+field dependence of ξSG works well until ξSG becomes
+comparable to the system size. A similar behavior was
+reported for the Ising spin glass at σ = 0.75 in Ref. [48].
+A finite-size scaling formulation was developed to treat
+the data at small fields when ξSG is comparable to the
+system size N, and with it an excellent collapse of all
+our data on ξSG and χSG was obtained. We showed that
+droplet scaling predicts the existence of the intersections
+h∗(N, 2N). Our data unfortunately does not extend to
+values of N large enough to be in the asymptotic region
+where the N-dependence of h∗(N, 2N) is simple. Fortu-
+nately there exists a way of testing whether the intersec-
+tions are due to an AT transition or are just those pre-
+dicted by droplet scaling, which is to study the N depen-
+dence of R = χSG/N 2σ−1, calculated at h∗(N, 2N), and
+this test supports the droplet picture provided N > 1024
+at σ = 0.75. Thus it is only for large systems that one
+
+21
+can obtain good evidence for the droplet picture.
+We now summarize our main results. The strongest
+evidence for droplet scaling is the success of the Imry-Ma
+formula for the field dependence of ξSG for σ = 0.75 and
+σ = 0.85 (see Figs. 30 and 31). If droplet scaling works,
+then no AT line is to be expected. When ξSG ∼ N there
+are visible sizeable corrections to the Imry-Ma formula
+which are related to TNT effects. However for σ = 0.85
+there is tentative evidence in Fig. 43 that if even larger
+systems could be studied then the TNT effects might be
+absent, and so there could exist a length scale L∗ above
+which TNT effects become unimportant (see Ref. [3]).
+If instead of droplet scaling one assumes that there is
+an AT phase transition then the usual finite size scaling
+plots used to determine hAT as in Fig. 15 for σ = 0.75
+are unsatisfactory: for example the values of hAT which
+would be derived from the crossings of ξSG and χSG as
+N becomes large look to be significantly different. In the
+equivalent data plot for σ = 0.60 (see Fig.
+8) they are
+in good agreement. Furthermore the quantity R of Eq.
+(35) should approach a constant as N → ∞ if there is a
+genuine AT transition, but instead for the cases σ = 0.75
+(Fig. 38) and 0.85 (Fig. 39), it is increasing with N once
+N becomes large enough.
+The simulations of this paper provide numerical evi-
+dence that the AT line and hence RSB is absent in spin
+glasses below six dimensions. What is now needed is an
+explanation of why this might be the case. Better still
+would be a rigorous proof that the lower critical dimen-
+sion for replica symmetry breaking is six. Our work indi-
+cates that showing that σ = 2/3 is the precise value of the
+critical value of σ will be challenging using simulations
+as finite size effects are large in its vicinity.
+ACKNOWLEDGMENTS
+We are grateful to the High Performance Comput-
+ing (HPC) facility at IISER Bhopal,
+where large-
+scale
+calculations
+in
+this
+project
+were
+run.
+We
+thank Peter Young and Dan Stein for helpful discus-
+sions.
+B.V is grateful to the Council of Scientific
+and Industrial Research (CSIR), India, for his PhD
+fellowship.
+A.S acknowledges financial support from
+SERB via the grant (File Number: CRG/2019/003447),
+and from DST via the DST-INSPIRE Faculty Award
+[DST/INSPIRE/04/2014/002461].
+Appendix A: The simulation method
+We now give some technical aspects of how the simula-
+tions are run. In the simulations we start with a random
+initial configuration and allow it to evolve according to
+the prescription given in this section. To incorporate par-
+allel tempering, we simultaneously simulate NT copies of
+the system over NT different temperatures ranging from
+Tmin ≡ T1 to Tmax ≡ TNT .
+In order to facilitate the
+computation of the observables outlined in this section,
+it is convenient to simulate 4 sets of NT copies (2 for
+hr = 0), which we label (1),(2),(3), and (4). We perform
+overrelaxation, heatbath and parallel tempering sweeps
+over all these copies keeping track of the labels appropri-
+ately.
+For every 10 overrelaxation sweeps we perform
+1 heatbath and 1 parallel tempering sweep, since the
+overrelaxation sweep involves a significantly lower com-
+putational cost, and is known to speed up equilibration.
+The parameters of the simulations are shown in Tables II
+and III. Once the system reaches equilibrium, we perform
+the same number of sweeps in the measurement phase,
+so Nsweep is the total number of sweeps over which the
+simulation is run, inclusive of both the equilibration and
+measurement phases. The last column in the table shows
+the amount of computer time expended to generate the
+data corresponding to the parameters in that row.
+In
+the measurement phase, we perform one measurement
+on the system for every 4 sweeps. The following sections
+contain the details of our Monte Carlo simulation pro-
+cedures.
+In order to equilibrate the system as quickly
+as possible, we perform three kinds of sweeps: overre-
+laxation or microcanonical sweeps, heatbath sweeps, and
+parallel tempering sweeps.
+1.
+Overrelaxation sweep
+We sweep sequentially through all the lattice sites and
+compute the local field Hi = �
+j JijSj+hi at a particular
+lattice site. The new spin direction S′
+i at the ith lattice
+site is taken to be the mirror image of the vector Si about
+Hi, i.e.,
+S′
+i = −Si + 2Si · Hi
+H2
+i
+Hi.
+(A1)
+Since S′
+i · Hi = Si · Hi, the energy of the system does
+not change due to these sweeps. Hence these sweeps are
+also called microcanonical sweeps. These sweeps help us
+in sampling out the microstates with the same energy.
+The process of equilibration speeds up when we include
+overrelaxation sweeps along with the other sweeps [33,
+52].
+2.
+Heatbath sweep
+The overrelaxation sweeps generate states with the
+same energy and hence they cannot directly equilibrate
+the system. Therefore, we also perform a heatbath sweep
+for every 10 microcanonical sweeps. Similar to the micro-
+canonical case, we sweep sequentially through the lattice.
+To equilibrate the system, the angle θ between Hi and
+S
+′
+i should be sampled out from the Boltzmann distribu-
+tion given by
+fΘ(θ) = e−βEi
+Z
+= eβHiSi cos θ
+Z
+= ew cos θ
+Z
+,
+(A2)
+
+22
+where w = βHiSi and
+Z =
+π
+�
+−π
+eβHiSi cos θ dθ
+(A3)
+is the normalizing constant. The simplest way to do this
+is to equate the cumulative density function (CDF) of θ,
+FΘ(θ), to that of a uniform distribution:
+FΘ(θ) =
+θ
+�
+−π
+fΘ(θ′) dθ′ = Π(r1) = r1,
+(A4)
+where r1 is a random variable sampled from a uniform
+distribution in the interval (0, 1). The value of θ can be
+obtained by simply inverting this function to get
+θ = F −1
+Θ (r1).
+(A5)
+This method works well with the Heisenberg spins
+as fΘ(θ) is integrable, which gives an invertible CDF
+FΘ(θ) [18, 28]. Since the probabililty density function
+(PDF) fΘ(θ) for the XY spin glasses given by Eq. (A2)
+is not exactly integrable, this method cannot be used.
+To overcome this problem and to sample out θ from the
+Boltzmann distribution (Eq. (A2)) in as few a number of
+sweeps as possible, we develop a heatbath sweep based
+on the rejection method [53]. We generate two random
+numbers r1 ∈ uniform(−π, π) and r2 ∈ uniform(0, fmax).
+If r2 < fθ(r1), we accept the move, i.e., take θ = r1. Else,
+we reject the move and generate another pair of random
+numbers (r1, r2). This process is repeated until we find
+an acceptable value of r1. A graphical representation for
+this method is shown in Fig. 44. The new spin direction
+S′
+i in Cartesian co-ordinates is given by:
+S′
+x = cos(θ + θH),
+(A6a)
+S′
+y = sin(θ + θH),
+(A6b)
+where θH is the angle made by the Hi vector with the X-
+axis. Since the generation of random numbers is involved,
+this sweep is computationally costlier than others. Hence
+we perform more microcanonical sweeps than heatbath
+sweeps.
+3.
+Parallel tempering sweep
+Spin glasses have a complex free energy landscape due
+to which, at low temperatures, they tend to get stuck in-
+side metastable valleys, and true equilibration consumes
+a lot of time. At high temperatures, the system can eas-
+ily escape the valley due to thermal fluctuations, and
+so equilibration is quick. To equilibrate the system in
+as small a number of moves as possible, we perform
+one parallel tempering sweep for every 10 overrelaxation
+sweeps [28, 33]. To benefit from the parallel tempering
+algorithm [54, 55], we simultaneously run the simulation
+−2
+0
+2
+θ
+0.0
+0.1
+0.2
+0.3
+0.4
+fΘ(θ)
+fmax
+(r1, r2)
+(r1, fΘ(r1))
+(r1, r2)
+(r1, fΘ(r1))
+w = βHS = 1.5
+Rejected
+Accepted
+Figure 44. Graphical representation of the rejection method.
+We randomly pick a point (r1, r2) within the rectangle from a
+uniform distribution. If the point lies under the fΘ(θ) curve
+given by Eq. (A2), then the point is accepted, and θ is taken
+to be r1. Otherwise, the point is rejected.
+for NT copies of the system at NT different temperatures
+T1 < T2 < T3 < · · · < TNT . The minimum temperature
+T1 is the low temperature at which we are interested in
+studying the behavior of the system, and the maximum
+temperature TNT is high enough that the system equi-
+librates very fast. We perform overrelaxation and heat-
+bath sweeps separately on each of the NT copies of the
+system. In the parallel tempering sweep, we compare the
+energies of two spin configurations at adjacent tempera-
+tures, Ti and Ti+1, starting from the smallest tempera-
+ture T1. We swap these two spin configurations such that
+the detailed balance condition is satisfied. The Metropo-
+lis probability for such a swap is
+P(T swap) = min{1, exp(∆β∆E)}
+(A7)
+=
+�
+exp(∆β∆E)
+(if ∆β∆E < 0),
+1
+(otherwise),
+(A8)
+where ∆β
+=
+1/Ti − 1/Ti+1 and ∆E
+=
+Ei(Ti) −
+Ei+1(Ti+1). In this way, a given set of spins performs
+a random walk in temperature space.
+4.
+Checks for equilibration
+In order to check whether the system has reached equi-
+librium, we have used a convenient test [56] which is pos-
+sible because of the Gaussian nature of the interactions
+and the onsite external magnetic field. The relation
+U = zJ2
+2T (ql − qs) + h2
+r
+T
+�
+q − |S|2�
+,
+(A9)
+
+23
+Table II. Parameters of the simulations. Nsamp is the number of disorder samples, Nsweep is the number of over-relaxation
+Monte Carlo sweeps for a single disorder sample. The system is equilibrated over the first half of the sweeps, and measurements
+are done over the last half of the sweeps with a measurement performed every four over-relaxation sweeps. Tmin and Tmax are
+the lowest and highest temperatures simulated, and NT is the number of temperatures used for parallel tempering.
+σ
+hr
+N
+Nsamp
+Nsweep
+Tmin
+Tmax
+NT
+ttot(hrs)
+0.6
+0
+128
+10000
+512
+0.6
+1
+18
+0.49
+0.6
+0
+256
+8000
+1024
+0.6
+1
+22
+2.23
+0.6
+0
+512
+6400
+2048
+0.6
+1
+22
+6.46
+0.6
+0
+1024
+8000
+4096
+0.6
+1
+26
+40.74
+0.6
+0
+2048
+3840
+8192
+0.6
+1
+24
+105.41
+0.6
+0
+4096
+3200
+16384
+0.6
+1
+27
+571.49
+0.6
+0
+8192
+3200
+32768
+0.6
+1
+30
+3776.85
+0.6
+0
+16384
+2600
+65536
+0.64
+0.98
+32
+18225.5
+0.6
+0.1
+128
+9600
+2048
+0.5
+0.8
+21
+7.75
+0.6
+0.1
+256
+9600
+2048
+0.5
+0.8
+21
+15.89
+0.6
+0.1
+512
+9600
+8192
+0.5
+0.8
+22
+85.67
+0.6
+0.1
+1024
+8000
+16384
+0.5
+0.8
+22
+414.47
+0.6
+0.1
+2048
+7200
+32768
+0.5
+0.8
+26
+2029.23
+0.6
+0.1
+4096
+7200
+65536
+0.5
+0.8
+24
+10014.8
+0.6
+0.1
+8192
+4380
+131072
+0.55
+0.8
+25
+34810.6
+0.6
+0.1
+16384
+7128
+262144
+0.55
+0.8
+28
+224425
+0.75
+0
+128
+12800
+1024
+0.35
+0.85
+21
+1.6
+0.75
+0
+256
+12800
+2048
+0.35
+0.85
+24
+7.21
+0.75
+0
+512
+8000
+8192
+0.35
+0.85
+24
+35.22
+0.75
+0
+1024
+8000
+16384
+0.35
+0.85
+24
+196.9
+0.75
+0
+2048
+6400
+32768
+0.35
+0.85
+25
+774.55
+0.75
+0
+4096
+4880
+65536
+0.35
+0.85
+27
+3405.2
+0.75
+0
+8192
+3000
+131072
+0.38
+0.82
+30
+14290.9
+0.75
+0.05
+128
+19200
+8192
+0.28
+0.6
+21
+45.27
+0.75
+0.05
+256
+16000
+16384
+0.28
+0.6
+20
+133.35
+0.75
+0.05
+512
+13600
+32768
+0.28
+0.6
+20
+464.77
+0.75
+0.05
+1024
+11000
+65536
+0.28
+0.6
+21
+2075.57
+0.75
+0.05
+2048
+10920
+262144
+0.28
+0.6
+24
+21314.3
+0.75
+0.05
+4096
+10800
+524288
+0.3
+0.58
+26
+123093
+0.75
+0.05
+8192
+5320
+1048576
+0.32
+0.54
+32
+364358
+0.85
+0
+128
+12800
+8192
+0.2
+0.5
+30
+17.97
+0.85
+0
+256
+12800
+16384
+0.2
+0.5
+32
+72.15
+0.85
+0
+512
+12800
+65536
+0.2
+0.5
+30
+752.36
+0.85
+0
+1024
+12800
+131072
+0.2
+0.5
+30
+3219.93
+0.85
+0
+2048
+8000
+262144
+0.2
+0.5
+30
+9504.05
+0.85
+0
+4096
+6480
+524288
+0.24
+0.48
+30
+40322.4
+0.85
+0.02
+128
+8000
+65536
+0.1
+0.4
+30
+194.33
+0.85
+0.02
+256
+4000
+131072
+0.1
+0.4
+32
+470.39
+0.85
+0.02
+512
+4400
+524288
+0.1
+0.4
+34
+4780.6
+0.85
+0.02
+1024
+3000
+2097152
+0.1
+0.4
+35
+30356.9
+0.85
+0.02
+2048
+1800
+4194304
+0.16
+0.4
+36
+84056
+0.85
+0.05
+128
+2000
+65536
+0.1
+0.4
+30
+67.07
+0.85
+0.05
+256
+4000
+131072
+0.1
+0.4
+32
+604.8
+0.85
+0.05
+512
+3500
+524288
+0.1
+0.4
+36
+4958.17
+0.85
+0.05
+1024
+3120
+2097152
+0.1
+0.4
+36
+28028.1
+0.85
+0.05
+2048
+3240
+4194304
+0.16
+0.4
+36
+151503
+is valid in equilibrium. Here
+U = 1
+N [⟨H⟩]av
+= − 1
+N
+�
+��
+⟨i,j⟩
+ϵijJij ⟨Si · Sj⟩ +
+�
+i,µ
+hµ
+i ⟨Sµ
+i ⟩
+�
+�
+av
+(A10)
+is the average energy per spin, q =
+1
+N
+�
+i
+[⟨Si⟩ · ⟨Si⟩]av
+is
+the
+Edwards-Anderson
+order
+parameter,
+ql
+=
+1
+Nb
+�
+⟨i,j⟩
+�
+ϵij ⟨Si · Sj⟩2�
+av is the “link overlap”, and qs =
+1
+Nb
+�
+⟨i,j⟩
+�
+ϵij
+�
+(Si · Sj)2��
+av is the “spin overlap”, where
+
+24
+Table III. Parameters of the simulations done at fixed temperature T and varying field hr. N(hr) is the number of values of
+field taken in the range hr(min,max). The equilibration times are different for different values of the field hr, which lie in the
+range Nsweep(min,max). The number of disorder samples for different fields lie in the range Nsamp(min,max). ttot is the total
+CPU time consumed in hours to generate data for a particular system size.
+σ
+T
+N
+hr(min,max)
+N(hr)
+Nsweep(min,max)
+Nsamp(min,max)
+ttot(hrs)
+0.6
+0.6
+128
+(0.010, 9.000)
+32
+(2048, 2048)
+(2000, 64000)
+11.44
+0.6
+0.6
+256
+(0.010, 9.000)
+32
+(4096, 4096)
+(2000, 48000)
+55.5
+0.6
+0.6
+512
+(0.010, 9.000)
+32
+(8192, 16384)
+(2000, 80000)
+163.91
+0.6
+0.6
+1024
+(0.010, 9.000)
+32
+(4096, 65536)
+(2000, 80000)
+2375
+0.6
+0.6
+2048
+(0.010, 9.000)
+32
+(16384, 131072)
+(1200, 60000)
+10259.8
+0.6
+0.6
+4096
+(0.010, 9.000)
+32
+(65536, 2097152)
+(960, 28600)
+92052.1
+0.6
+0.6
+8192
+(0.010, 9.000)
+32
+(65536, 4194304)
+(400, 19428)
+310948
+0.6
+0.6
+16384
+(0.010, 9.000)
+32
+(131072, 4194304)
+(488, 10689)
+1021481
+0.7
+0.6
+128
+(0.010, 9.000)
+31
+(1024, 1024)
+(4000, 4000)
+1.86
+0.7
+0.6
+256
+(0.010, 9.000)
+31
+(2048, 2048)
+(1000, 8000)
+8.53
+0.7
+0.6
+512
+(0.010, 9.000)
+31
+(4096, 4096)
+(1000, 8000)
+39.46
+0.7
+0.6
+1024
+(0.010, 9.000)
+31
+(8192, 8192)
+(1500, 8000)
+288.41
+0.7
+0.6
+2048
+(0.010, 9.000)
+31
+(16384, 16384)
+(500, 8000)
+559.99
+0.7
+0.6
+4096
+(0.010, 9.000)
+31
+(32768, 32768)
+(400, 8000)
+3020.62
+0.7
+0.6
+8192
+(0.010, 9.000)
+31
+(16384, 131072)
+(2000, 6800)
+17500.2
+0.7
+0.6
+16384
+(0.010, 9.000)
+31
+(32768, 262144)
+(640, 6400)
+77734.3
+0.75
+0.55
+128
+(0.001, 9.000)
+42
+(512, 1024)
+(2000, 40000)
+6.59
+0.75
+0.55
+256
+(0.001, 9.000)
+42
+(1024, 2048)
+(2000, 40000)
+56.67
+0.75
+0.55
+512
+(0.001, 9.000)
+42
+(4096, 4096)
+(1000, 24000)
+165.51
+0.75
+0.55
+1024
+(0.001, 9.000)
+43
+(8192, 8192)
+(1000, 12000)
+255.75
+0.75
+0.55
+2048
+(0.001, 9.000)
+43
+(16384, 16384)
+(1000, 12000)
+1000.52
+0.75
+0.55
+4096
+(0.001, 9.000)
+43
+(32768, 32768)
+(800, 12000)
+5269.11
+0.75
+0.55
+8192
+(0.001, 9.000)
+42
+(65536, 65536)
+(800, 8000)
+21410.3
+0.75
+0.55
+16384
+(0.001, 9.000)
+42
+(131072, 131072)
+(760, 6280)
+62062
+0.75
+0.55
+32768
+(0.001, 9.000)
+42
+(131072, 262144)
+(512, 4995)
+146627
+0.85
+0.3
+128
+(0.001, 9.000)
+36
+(32768, 131072)
+(2000, 30000)
+264.97
+0.85
+0.3
+256
+(0.001, 9.000)
+36
+(65536, 262144)
+(1000, 25000)
+1052.21
+0.85
+0.3
+512
+(0.001, 9.000)
+36
+(131072, 524288)
+(1000, 34800)
+6161.52
+0.85
+0.3
+1024
+(0.001, 9.000)
+36
+(262144, 1048576)
+(1000, 20000)
+17668
+0.85
+0.3
+2048
+(0.001, 9.000)
+36
+(524288, 8388608)
+(320, 3372)
+68933.6
+0.85
+0.3
+4096
+(0.001, 9.000)
+36
+(262144, 16777216)
+(312, 5760)
+439004
+Nb = Nz/2, and ϵij = 1 if the ith and jth spins are in-
+teracting and is zero otherwise. The [· · · ]av in Eq. (A10)
+is analytically evaluated by performing integration over
+Jij and hµ
+i [57] since they have Gaussian distributions.
+On evaluating this integral using integration by parts,
+we get Eq. (A9). As the system reaches equilibrium, the
+two sides of Eq. (A9) approach their common equilibrium
+value from opposite directions.
+In simulations, we evaluate both sides of Eq. (A9) for
+different number of Monte-Carlo sweeps (MCSs), which
+increase in an exponential manner, each value being twice
+the previous one. The averaging is done over the last half
+of the sweeps. We initially start with a random spin con-
+figuration, so the LHS of Eq. (A9) is small and the RHS
+is very large. As the system gets closer to equilibrium,
+these two values come closer to each other from opposite
+directions.
+When we notice that the averaged quanti-
+ties satisfy Eq. (A9) within error bars, consistently for
+at least the last two points, we declare that our system
+has reached equilibrium. Once the system reaches equi-
+librium, we perform the same number of sweeps in the
+measurement phase, where we evaluate different quanti-
+ties (given below) used to study the possible phase tran-
+sitions of the system.
+[1] Marc M´ezard, Giorgio Parisi,
+and Miguel Angel Vira-
+soro, Spin glass theory and beyond: An Introduction to
+the Replica Method and Its Applications, Vol. 9 (World
+Scientific Publishing Company, 1986).
+[2] J R L de Almeida and D J Thouless, “Stability of the
+sherrington-kirkpatrick solution of a spin glass model,”
+Journal of Physics A: Mathematical and General 11,
+983–990 (1978).
+
+25
+[3] M. A. Moore, “Droplet-scaling versus replica symmetry
+breaking debate in spin glasses revisited,” Phys. Rev. E
+103, 062111 (2021).
+[4] V. Martin-Mayor, J. J. Ruiz-Lorenzo, B. Seoane,
+and
+A. P. Young, “Numerical simulations and replica sym-
+metry breaking,” (2022).
+[5] Auditya Sharma and A. P. Young, “de Almeida–Thouless
+line in vector spin glasses,” Phys. Rev. E 81, 061115
+(2010).
+[6] A J Bray and S A Roberts, “Renormalisation-group ap-
+proach to the spin glass transition in finite magnetic
+fields,” Journal of Physics C: Solid State Physics 13,
+5405–5412 (1980).
+[7] M. A. Moore and A. J. Bray, “Disappearance of the de
+Almeida-Thouless line in six dimensions,” Phys. Rev. B
+83, 224408 (2011).
+[8] G. Kotliar, P. W. Anderson,
+and D. L. Stein, “One-
+dimensional spin-glass model with long-range random in-
+teractions,” Phys. Rev. B 27, 602–605 (1983).
+[9] Freeman J. Dyson, “Non-existence of spontaneous mag-
+netization in a one-dimensional Ising ferromagnet,”
+Communications in Mathematical Physics 12, 212–215
+(1969).
+[10] Freeman J. Dyson, “An Ising ferromagnet with discontin-
+uous long-range order,” Communications in Mathemati-
+cal Physics 21, 269–283 (1971).
+[11] L. Leuzzi, G. Parisi, F. Ricci-Tersenghi, and J. J. Ruiz-
+Lorenzo, “Dilute One-Dimensional Spin Glasses with
+Power Law Decaying Interactions,” Phys. Rev. Lett. 101,
+107203 (2008).
+[12] L. Leuzzi, G. Parisi, F. Ricci-Tersenghi, and J. J. Ruiz-
+Lorenzo, “Ising Spin-Glass Transition in a Magnetic Field
+Outside the Limit of Validity of Mean-Field Theory,”
+Phys. Rev. Lett. 103, 267201 (2009).
+[13] A. P. Young and Helmut G. Katzgraber, “Absence of
+an Almeida-Thouless Line in Three-Dimensional Spin
+Glasses,” Phys. Rev. Lett. 93, 207203 (2004).
+[14] Helmut G. Katzgraber and A. P. Young, “Probing the
+Almeida-Thouless line away from the mean-field model,”
+Phys. Rev. B 72, 184416 (2005).
+[15] Helmut G. Katzgraber, Derek Larson, and A. P. Young,
+“Study of the de Almeida–Thouless Line Using Power-
+Law Diluted One-Dimensional Ising Spin Glasses,” Phys.
+Rev. Lett. 102, 177205 (2009).
+[16] Helmut G. Katzgraber and A. P. Young, “Monte Carlo
+studies of the one-dimensional Ising spin glass with
+power-law interactions,” Phys. Rev. B 67, 134410 (2003).
+[17] Auditya Sharma and A. P. Young, “Phase transitions in
+the one-dimensional long-range diluted Heisenberg spin
+glass,” Phys. Rev. B 83, 214405 (2011).
+[18] Auditya Sharma and A. P. Young, “de Almeida–Thouless
+line studied using one-dimensional power-law diluted
+Heisenberg spin glasses,” Phys. Rev. B 84, 014428 (2011).
+[19] Auditya Sharma, Joonhyun Yeo,
+and M. A. Moore,
+“Metastable minima of the Heisenberg spin glass in a
+random magnetic field,” Phys. Rev. E 94, 052143 (2016).
+[20] Frank Beyer, Martin Weigel,
+and M. A. Moore, “One-
+dimensional infinite-component vector spin glass with
+long-range interactions,” Phys. Rev. B 86, 014431 (2012).
+[21] M. A. Moore, “1/m expansion in spin glasses and the de
+Almeida-Thouless line,” Phys. Rev. E 86, 031114 (2012).
+[22] Auditya Sharma, Alexei Andreanov, and Markus M¨uller,
+“Avalanches and hysteresis in frustrated superconductors
+and XY spin glasses,” Phys. Rev. E 90, 042103 (2014).
+[23] Silvio Franz, Flavio Nicoletti, Giorgio Parisi,
+and Fed-
+erico Ricci-Tersenghi, “Delocalization transition in low
+energy excitation modes of vector spin glasses,” SciPost
+Phys. 12, 16 (2022).
+[24] Cosimo Lupo and Federico Ricci-Tersenghi, “Compari-
+son of Gabay–Toulouse and de Almeida–Thouless insta-
+bilities for the spin-glass XY model in a field on sparse
+random graphs,” Phys. Rev. B 97, 014414 (2018).
+[25] Cosimo Lupo,
+Giorgio Parisi,
+and Federico Ricci-
+Tersenghi, “The random field XY model on sparse
+random graphs shows replica symmetry breaking and
+marginally stable ferromagnetism,” Journal of Physics A:
+Mathematical and Theoretical 52, 284001 (2019).
+[26] M. Baity-Jesi, V. Mart´ın-Mayor, G. Parisi, and S. Perez-
+Gaviro, “Soft Modes, Localization, and Two-Level Sys-
+tems in Spin Glasses,” Phys. Rev. Lett. 115, 267205
+(2015).
+[27] L. A. Fernandez, V. Martin-Mayor, S. Perez-Gaviro,
+A. Tarancon,
+and A. P. Young, “Phase transition in
+the three dimensional Heisenberg spin glass: Finite-size
+scaling analysis,” Phys. Rev. B 80, 024422 (2009).
+[28] L. W. Lee and A. P. Young, “Large-scale Monte Carlo
+simulations of the isotropic three-dimensional Heisenberg
+spin glass,” Phys. Rev. B 76, 024405 (2007).
+[29] I. Campos, M. Cotallo-Aban, V. Martin-Mayor, S. Perez-
+Gaviro, and A. Tarancon, “Spin-Glass Transition of the
+Three-Dimensional Heisenberg Spin Glass,” Phys. Rev.
+Lett. 97, 217204 (2006).
+[30] J. A. Olive, A. P. Young, and D. Sherrington, “Computer
+simulation of the three-dimensional short-range Heisen-
+berg spin glass,” Phys. Rev. B 34, 6341–6346 (1986).
+[31] K. Hukushima and H. Kawamura, “Chiral-glass tran-
+sition
+and
+replica
+symmetry
+breaking
+of
+a
+three-
+dimensional Heisenberg spin glass,” Phys. Rev. E 61,
+R1008–R1011 (2000).
+[32] Dao Xuan Viet and Hikaru Kawamura, “Numerical
+Evidence of Spin-Chirality Decoupling in the Three-
+Dimensional Heisenberg Spin Glass Model,” Phys. Rev.
+Lett. 102, 027202 (2009).
+[33] J. H. Pixley and A. P. Young, “Large-scale Monte Carlo
+simulations of the three-dimensional XY
+spin glass,”
+Phys. Rev. B 78, 014419 (2008).
+[34] M Baity-Jesi,
+R A Ba˜nos,
+A Cruz,
+L A Fernan-
+dez, J M Gil-Narvion, A Gordillo-Guerrero, D I˜niguez,
+A Maiorano, F Mantovani, E Marinari, V Martin-
+Mayor, J Monforte-Garcia, A Mu˜noz Sudupe, D Navarro,
+G Parisi, S Perez-Gaviro, M Pivanti, F Ricci-Tersenghi,
+J J Ruiz-Lorenzo, S F Schifano, B Seoane, A Tarancon,
+R Tripiccione,
+and D Yllanes, “The three-dimensional
+Ising spin glass in an external magnetic field: the role
+of the silent majority,” Journal of Statistical Mechanics:
+Theory and Experiment 2014, P05014 (2014).
+[35] F. Krzakala and O. C. Martin, “Spin and Link Overlaps
+in Three-Dimensional Spin Glasses,” Phys. Rev. Lett. 85,
+3013–3016 (2000).
+[36] Matteo Palassini and A. P. Young, “Nature of the Spin
+Glass State,” Phys. Rev. Lett. 85, 3017–3020 (2000).
+[37] David Sherrington and Scott Kirkpatrick, “Solvable
+Model of a Spin-Glass,” Phys. Rev. Lett. 35, 1792–1796
+(1975).
+[38] J R L de Almeida, R C Jones, J M Kosterlitz,
+and
+D J Thouless, “The infinite-ranged spin glass with m-
+component spins,” Journal of Physics C: Solid State
+Physics 11, L871–L875 (1978).
+
+26
+[39] R. A. Ba˜nos, L. A. Fernandez, V. Martin-Mayor,
+and
+A. P. Young, “Correspondence between long-range and
+short-range spin glasses,” Phys. Rev. B 86, 134416
+(2012).
+[40] Derek Larson, Helmut G. Katzgraber, M. A. Moore, and
+A. P. Young, “Numerical studies of a one-dimensional
+three-spin spin-glass model with long-range interac-
+tions,” Phys. Rev. B 81, 064415 (2010).
+[41] K. Binder, “Finite size scaling analysis of Ising model
+block distribution functions,” Zeitschrift f¨ur Physik B
+Condensed Matter 43, 119–140 (1981).
+[42] H.G. Ballesteros, L.A. Fern´andez, V. Mart´ın-Mayor, and
+A. Mu˜noz Sudupe, “Finite size effects on measures of
+critical exponents in d = 3 O(N) models,” Physics Letters
+B 387, 125–131 (1996).
+[43] Martin Hasenbusch, Andrea Pelissetto,
+and Ettore Vi-
+cari, “Critical behavior of three-dimensional Ising spin
+glass models,” Phys. Rev. B 78, 214205 (2008).
+[44] W L McMillan, “Scaling theory of Ising spin glasses,”
+Journal of Physics C: Solid State Physics 17, 3179–3187
+(1984).
+[45] A. J. Bray and M. A. Moore, “Scaling Theory of the
+ordered phase of spin glasses,” in Heidelberg Collo-
+quium on Glassy Dynamics and Optimization, edited
+by L. Van Hemmen and I. Morgenstern (Springer, New
+York, 1986) p. 121.
+[46] Daniel S. Fisher and David A. Huse, “Equilibrium be-
+havior of the spin-glass ordered phase,” Phys. Rev. B
+38, 386–411 (1988).
+[47] Yoseph Imry and Shang-keng Ma, “Random-Field Insta-
+bility of the Ordered State of Continuous Symmetry,”
+Phys. Rev. Lett. 35, 1399–1401 (1975).
+[48] T. Aspelmeier, Helmut G. Katzgraber, Derek Larson,
+M. A. Moore, Matthew Wittmann, and Joonhyun Yeo,
+“Finite-size critical scaling in Ising spin glasses in the
+mean-field regime,” Phys. Rev. E 93, 032123 (2016).
+[49] T. Aspelmeier, Wenlong Wang, M. A. Moore, and Hel-
+mut G. Katzgraber, “Interface free-energy exponent in
+the one-dimensional Ising spin glass with long-range in-
+teractions in both the droplet and broken replica sym-
+metry regions,” Phys. Rev. E 94, 022116 (2016).
+[50] C. M. Newman and D. L. Stein, “Finite-Dimensional Spin
+Glasses: States, Excitations, and Interfaces,” Ann. Henri
+Poincar´e 4, 497 (2003).
+[51] C. M. Newman, N. Read, and D. L. Stein, “Metastates
+and Replica Symmetry Breaking,” (2022).
+[52] J. L. Alonso, A. Taranc´on, H. G. Ballesteros, L. A.
+Fern´andez, V. Mart´ın-Mayor,
+and A. Mu˜noz Sudupe,
+“Monte Carlo study of O(3) antiferromagnetic models in
+three dimensions,” Phys. Rev. B 53, 2537–2545 (1996).
+[53] Richard C Larson and Amedeo R Odoni, Urban opera-
+tions research, Monograph (1981).
+[54] Koji Hukushima and Koji Nemoto, “Exchange monte
+carlo method and application to spin glass simulations,”
+Journal of the Physical Society of Japan 65, 1604–1608
+(1996).
+[55] J. Machta, “Strengths and weaknesses of parallel tem-
+pering,” Phys. Rev. E 80, 056706 (2009).
+[56] Helmut G. Katzgraber, Matteo Palassini,
+and A. P.
+Young, “Monte Carlo simulations of spin glasses at low
+temperatures,” Phys. Rev. B 63, 184422 (2001).
+[57] A J Bray and M A Moore, “Some observations on the
+mean-field theory of spin glasses,” Journal of Physics C:
+Solid State Physics 13, 419–434 (1980).
+