Evidence of many thermodynamic states of the three-dimensional Ising spin glass
EEvidence of Many Thermodynamic States of the Three-dimensional Ising Spin Glass
Wenlong Wang,
1, 2, ∗ Mats Wallin, and Jack Lidmar Department of Physics, Royal Institute of Technology, Stockholm, SE-106 91, Sweden College of Physics, Sichuan University, Chengdu 610065, China
We present a large-scale simulation of the three-dimensional Ising spin glass with Gaussian disorder to lowtemperatures and large sizes using optimized population annealing Monte Carlo. Our primary focus is inves-tigating the number of pure states regarding a controversial statistic, characterizing the fraction of centrallypeaked disorder instances, of the overlap function order parameter. We observe that this statistic is subtly andsensitively influenced by the slight fluctuations of the integrated central weight of the disorder-averaged over-lap function, making the asymptotic growth behaviour very difficult to identify. Modified statistics effectivelyreducing this correlation are studied and essentially monotonic growth trends are obtained. The effect of tem-perature is also studied, finding a larger growth rate at a higher temperature. Our detailed examination andstate-of-the-art simulation provide a coherent picture of many pure states, explain the previous findings, and thecontroversy is solved. The pertinent status of the number of pure states beyond this statistic is also discussed,and we find the spin glass balance is overall tilting towards many pure states studied by simulations.
Introduction–
Spin glasses are fascinating disordered andfrustrated magnets with a wide array of applications in diversefields such as biology, computer science, and optimizationproblems [1, 2]. The mean-field Sherrington-Kirkpatrick (SK)model [3] has an unusual low-temperature phase of many purestates described by the replica symmetry breaking (RSB) [4–6]. Here, a pure state refers to a self-sustained thermody-namic state characterized by a time-averaged spin orienta-tional pattern. Despite several decades of efforts, it is stillan outstanding problem whether the more realistic Edwards-Anderson (EA) spin glass [7] in three dimensions has a singlepair or many pairs of pure states. The RSB picture [8, 9] as-sumes that the mean-field theory is qualitatively correct forthe EA model. On the other hand, the droplet picture [10–14]based on the domain-wall renormalization group method pre-dicts only a single pair of pure states much like a ferromagnet.The two pictures also differ on the geometrical aspect of ex-citations (fractals or space-filling) [15], and the existence of aspin-glass phase in a weak external magnetic field [16]. Thereare other pictures as well [2]. The applicability of RSB is ofbroad interest and is related to, e.g., the Gardner transition instructural glasses [17, 18].Despite much research aiming at discriminating which pic-ture is suitable in three (and also four) dimensions, the prob-lem has not been definitely settled. Our approach is to fo-cus on individual spin glass properties, and a solid answer onone property can put stringent constraint on possible theories.Therefore, in this work we solely discuss computationally thenumber of pure states. There is mounting evidence that thedisorder-averaged overlap function is nontrivial (correspond-ing to many pure states) for the sizes available, which havebeen steadily growing over time. One exception is the worksfocusing on the ground states at zero temperature [19, 20].However, we conjecture that a single pair of ground state isa strong support for neither a two state nor many state pic-ture. It seems likely what is going on in this case is that thereare nonzero (finite if there are many pure states) energy gaps ∗ Electronic address: [email protected] between the lowest pure states. In this way, even O (1) large-scale droplet excitations are forbidden at T = 0 . This conjec-ture is motivated by the observation that the disorder-averagedcentral weight [see Eq. (2)] decreases approximately linearlywith decreasing temperature [21]. Next, we turn to the finitetemperatures, which is the focus of this work.We find the computational controversies regarding the num-ber of pure states are essentially from investigating new statis-tics, new boundary conditions, or a combination of the two.As mentioned above, the overlap function is found to be non-trivial for the sizes available at finite temperatures. It is helpfulto rephrase this important statement more precisely as: “Thedisorder-averaged overlap distribution function is nontrivialunder periodic boundary conditions at a typical low temper-ature”. To our best knowledge, there is no numerical workthat contradicts this result. Therefore, to argue against thisstatement, it is necessary that one or several of the conditionshave to be altered. At finite temperatures, these efforts comebroadly in three types: a new statistic, a new boundary condi-tion, or a combination of the two. It should be mentioned thatnot all of these pioneering works concluded strongly againstmany pure states, some only argued their results may haveviolated many pure states, and many of these have alreadybeen solved. The free boundary condition was thought to po-tentially support a two state picture as I (0 . [see Eq. (2)]is found to rapidly decrease for small sizes and remarkablyagrees with the /L θ DW scaling, where θ DW is the interfacefree energy exponent [22]. However, this was later found tobe a finite-size effect from the surfaces [23]. The scaling nolonger holds for larger sizes and I (0 . of the free boundarycondition and periodic boundary condition run together forlarger sizes, supporting many pure states. By contrast, vari-ous statistics have been proposed other than I (0 . , but mostof these are not very successful; see, e.g., [24] and the refer-ences therein for a collection of examples. One controversialbut stimulating statistic is the fraction of centrally peaked in-stances [25–27], which we discuss in detail below. For newstatistics and boundary conditions, a work on sample stiff-ness in thermal boundary conditions argued against many purestates [28]. This is also partially addressed [29], and to fullyresolve this problem a contrast experiment of the SK model a r X i v : . [ c ond - m a t . d i s - nn ] M a y should be conducted, which shall be discussed elsewhere.In this work we focus on the controversial statistic ∆ of thefraction of centrally peaked instances [25–27], and find againthat there is no violation of many pure states. This is signifi-cant since ∆ appears to do the best job among the new statis-tics supporting the two state picture [24]. In [25], it was foundthat ∆(0 . , [see Eq. (3)] at T = 0 . grows with systemsize up to about L = 10 , then it levels off or stops growing ap-preciably. By contrast, ∆ of the SK model at a similar T /T C grows prominently for the similar range of sizes. A criticismin [26] suggested that comparing T = 0 . T C for differentmodels has no theoretical basis, and the difference is from theeffective lower temperature of the EA model, i.e., a smallercentral weight I (0 . . Increasing the temperature of the EAmodel such that it has a similar I (0 . as the SK model, itwas found that ∆ also grows prominently in the EA model.However, the problem was not fully addressed in spite of theprofound insight. An explanation of the irregular low tem-perature data is missing, and slightly different models wereinvestigated. The former group used Gaussian disorder and alow temperature [25], while the latter group used ± J disorderand a relatively high temperature [26].The main purpose of this Letter is to resolve this problemsystematically by carrying out a large-scale simulation of thethree-dimensional EA model at the same parameters using thesame model as the original work [25] but including largersystem sizes. In light of [26], data at a higher temperatureare also collected for comparison. Using massively parallelpopulation annealing Monte Carlo [30–34] and MPI parallelcomputing, and taking further advantage of the recent opti-mizations of the algorithm [34–36], we have managed withconsiderable efforts to increase the largest size from spins[25] to spins. We refer to [35] for a discussion of hownotoriously the spin glass computational complexity grows atlow temperatures with the number of spins. Our larger rangeof sizes crucially enables us to identify a subtle correlationbetween ∆( q , κ ) and I ( q ) , showing that even small I ( q ) fluctuations can significantly influence the behaviour of ∆ .Motivated by our observation, we define a slightly modified ∆ compensating effectively for this correlation effect. The mod-ified ∆ essentially grows monotonically, providing a coherentmany state picture. Our results also reveal that the notablydifferent results of earlier works originate from the use of dif-ferent effective temperatures or central weights, as suggestedby [26]. Model, method, and observables–
We study the three-dimensional Edwards-Anderson Ising spin glass [7] definedby the Hamiltonian H = − (cid:80) (cid:104) ij (cid:105) J ij S i S j , where S i = ± are Ising spins and the sum is over nearest neighbours on asimple cubic lattice under periodic boundary conditions. For alinear size L , there are N = L spins. The random couplings J ij are chosen independently from the standard Gaussian dis-tribution n (0 , . A set of quenched couplings J = { J ij } defines a disorder realization or an instance. The model has aspin-glass phase transition at T C ≈ [37].Population annealing cools gradually a population of R ran-dom replicas starting from β = 0 with alternating resam-pling and Metropolis sweeps until reaching the lowest tem- TABLE I: Preliminary parameters of the population annealing sim-ulation. L is the linear system size, R is the number of replicas, T is the lowest temperature simulated, N T is the number of tempera-tures used in the annealing schedule, N S is the number of sweepsapplied to each replica after resampling, and M is the number of in-stances studied. Note that the unequilibrated instances were rerunwith (much) larger simulation parameters; see the text for details. L R T N T N S M × .
42 101 10 50006 1 . × .
42 101 10 50008 4 × .
42 201 10 500010 9 . × .
42 301 10 500012 9 . × .
42 301 10 500014 9 . × .
42 401 20 350016 9 . × .
42 401 40 3424 perature T = 0 . . Population annealing is used because itis both efficient and massively parallel [33, 38]. Our crite-rion for equilibration is based on the replica family entropy S f ≥ log(100) and we ensure equilibration for each indi-vidual instance [28, 33]. The preliminary simulation param-eters are summarized in Table I, and the unequilibrated in-stances were rerun with larger parameters until equilibrationis reached. It should be noted that the hard instances mayrequire substantially more work than a typical instance. Forexample, our typical top 5% hard instances at L = 16 requireapproximately R ∼ replicas, N T = 501 temperatures,and as large as N S = 200 sweeps on each replica per temper-ature; cf. the preliminary parameters in the table.Our primary observable is the spin overlap distributionfunction P J ( q ) where the spin overlap q is defined as: q = 1 N (cid:88) i S (1) i S (2) i , (1)where spin configurations “(1)” and “(2)” are chosen inde-pendently from the Boltzmann distribution or the population.Measuring the overlap distribution is a costly process in paral-lel computing, and we have collected the distributions at twotypical low temperatures T = 2 / and T = 0 . . Other regu-lar observables like energy, free energy, and the replica familyentropy are collected at all temperatures.We further introduce two statistics of the individual overlapfunction: the central weight and central peakedness [25]: I J ( q ) = (cid:90) q − q P J ( q ) dq, (2) ∆ J ( q , κ ) = max | q |≤ q [( P J ( q ) + P J ( − q )) / > κ. (3)Here, q characterizes the half length of a chosen intervalaround q = 0 and κ is a chosen threshold to determinewhether or not an instance is centrally peaked. The statistic ∆ J takes either or . When the subscript is dropped, werefer to the disorder-averaged quantity. Hence, ∆ refers to thefraction of centrally peaked instances. ∆ should decrease to for a two state picture while it should increase to for a manystate picture in the thermodynamic limit. Results–
The disorder-averaged spin overlap function andthe central weight I (0 . at both T = 2 / and . are shownin Fig. 1. It is clear that the central weight is essentially flatup to fluctuations as a function of the system size, in agree-ment with numerous previous results [25, 33, 39, 40]. Thisresult is well known, except that we here further extend it totwo larger system sizes at low temperatures. The result is inexcellent agreement with RSB. By contrast, this is strikinglydifferent from the /L θ DW ( θ DW ≈ . ) scaling predictedby the droplet picture. To our best knowledge, there is nostraightforward way to explain this as a finite-size effect, be-cause the interface free energy scales well with this exponentfor the same range of sizes. Finally, the weights of T = 0 . are smaller than those of T = 2 / as expected, as the effectivenumber of “active” pure states is suppressed at lower temper-atures.Next, we discuss the statistic ∆ in detail. The results of ∆(0 . , and ∆(0 . , are shown in the top left panel ofFig. 2. At first sight, the data look quite irregular as observedin [25]. There appears to be a growing trend in general, andthe trend is much clearer at T = 2 / than at the lower temper-ature T = 0 . . We shall discuss the effect of the temperaturebelow, and first focus on the origin of the irregular low temper-ature data. We take the ∆(0 . , as an example without lossof generality. After an increasing trend at small sizes, the growfrom L = 10 to L = 12 is very marginal (marginal or slightlynegative depending on disorder realizations, here we call itcollectively as marginal), in agreement with [25]. Then weobserve a noticeable increase from L = 12 to L = 14 , a some-what promising result for many pure states. But the statisticsubsequently grows rather marginally again from L = 14 to L = 16 , resulting a rather confusing situation. This puzzleis finally understood after checking back and forth when werealized a subtle correlation between the ∆ data and the cen-tral weight data. As illustrated in Fig. 1, I (0 . drops slightlyfrom L = 10 to L = 12 , this is where the corresponding ∆ has a marginal increase. Then I (0 . increases slightly from L = 12 to L = 14 and then drops slightly at L = 16 , explain-ing the correlated trend in ∆ . This together with other similarobservations throughout our data collection process stronglysuggest that the two statistics are indeed correlated. Then ∆ is presumably a growing function with increasing system size,but the growth is very sensitively affected by the fluctuationsof the central weights, producing an irregular growth trend.This also partially explains why the data at a higher tempera-ture have a clearer growth trend, as the relative fluctuation ofthe central weights with respect to the size-averaged mean issmaller at higher temperatures. An additional reason is that ∆ has a larger growth rate at higher temperatures, note thecrossings of the data at the two temperatures.The correlation between ∆ and I is reasonable from the fol-lowing argument. The central weight is like a supply of peaks,and higher weights supply more peaks, which tend to statis-tically produce more peaked instances. Take two extreme ex-amples, if I (0 .
2) = 0 , it is clear that ∆(0 . , is bounded to P ( q ) q qT = 2 / T = 0 . L = 4 L = 6 L = 8 L = 10 L = 12 L = 14 L = 16 ( a ) I ( . ) L T = 2 / T = 0 . ( b ) FIG. 1: The disorder-average overlap function [panel ( a ) ] and cen-tral weight I (0 . [panel ( b ) ] for different system sizes L at typicallow temperatures T = 2 / and T = 0 . . The statistic I (0 . is ap-proximately independent of the system size L for both temperatures,in agreement with the many state picture. Note that the maximumsize is L = 16 , compared with L = 12 of [25]. be zero as well by definition. On the other hand, if I (0 . foran instance is large, there is almost certainly a peak present,at least when the size is large. Since the fluctuation of I has aprofound influence on the statistic ∆ , we next look for mod-ified statistics to compensate this correlation effect by a vari-ance reduction method.We define a slightly modified statistic of weighted ∆ as ∆( q , κ )[ (cid:104) I ( q ) (cid:105) /I ( q )] , where the angular bracket is an av-erage over the system size. This definition takes advantage ofthe approximately constant nature of I ( q ) and only slightlymodifies the ∆ data. Before we show that this statistic workswell, we look at another example to first motivate this statis-tic. We look at two extreme windows that have very differentweights. From the overlap distribution functions, it is clearthat at the lower temperature there are wide q ranges wherethe major q EA peaks have little influence. In addition, theweight density is higher at larger q than at the neighbourhoodof q = 0 . Therefore, we select the following two different ∆ ( q , ) LT = 2 / , q = 0 . T = 0 . , q = 0 . T = 2 / , q = 0 . T = 0 . , q = 0 . ( a ) I a nd ∆ L Window A , I
Window B , I
Window A , ∆ Window B , ∆ ( b ) ∆ ( q , ) × (cid:104) I ( q ) (cid:105) / I ( q ) LT = 2 / , q = 0 . T = 0 . , q = 0 . T = 2 / , q = 0 . T = 0 . , q = 0 . ( d ) ∆ / I L Window AWindow B ( c ) FIG. 2: The statistics ∆(0 . , and ∆(0 . , as a function of L at two temperatures T = 2 / and T = 0 . [panel ( a ) ]. While ∆ appears to have a growing trend, it is not fully regular. We find that ∆ is subtly influenced by the central weight; cf. Fig. 1. To illustrate thiscorrelation, two windows of the same size but of very different weights are studied at T = 0 . , and the Window B with a larger weight alsohas consistently larger ∆ [panel ( b ) ]. This correlation can be effectively reduced by looking at the statistic ∆ /I as shown in the panel ( c ) .Similarly, the modified ∆ as shown in panel ( d ) has a much cleaner growth trend and is in fact remarkably monotonic. Here, (cid:104) I (cid:105) is the averageof I over the sizes. See the text for more details. windows and study the behaviours of I and ∆ at T = 0 . :(1) Window A defined as the interval q ∈ [ − . , . with asmall weight and (2) Window B defined as | q | ∈ [0 . , . with the same length but a noticeably larger weight. Here, I and ∆ are measured in the respective supports. The two statis-tics as a function of size for these two windows are shown inthe top right panel of Fig. 2. Since Window B has consistentlylarger weights I , it also has consistently larger ∆ as expected.The ratio ∆ /I is shown in the bottom right panel, and thissimple statistic brings the two ∆ data sets much closer, partic-ularly for the pertinent large sizes. Similar behaviour is alsofound for the higher temperature, despite that Window B isslightly affected by the q EA peaks. These demonstrate that ∆ /I is an effective statistic to reduce the correlation effectfrom the weights.The modified ∆ is shown in the bottom left panel of Fig. 2.It is remarkable that this simple modified ∆ has a clean growthtrend with increasing system size, i.e., the growing trend is not only improved but also the data are monotonic. To bemore quantitative, we have carried out a growing trend test toleading linear order using a linear fit. The computed slopes are . (low T, q = 0 . , . (high T, q = 0 . , . (low T, q = 0 . , . (high T, q = 0 . ,respectively. Note that all of these values, especially the hightemperature data, are significantly larger than , suggesting acollective growth trend of this statistic. We conclude thereforethat the seemingly nonmonotonic growth of ∆ is a result of itssensitivity to the fluctuations of the central weight. From thisperspective, the statistic ∆ is not as good as the central weightin discriminating the number of pure states due to the discretenature of its support.The modified ∆ also shows clearly the effect of temperatureon ∆ . The growth rate at the higher temperature (red sym-bols) is noticeably larger than at the lower temperature (bluesymbols). There is also an interesting crossing in ∆ at eachtemperature despite that there is no crossing in I at the two -0.06-0.04-0.0200.020.040.06 4 6 8 10 12 14 16 ∆ − I Lq = 0 . -0.15-0.1-0.0500.050.1 4 6 8 10 12 14 16 Lq = 0 . T = 2 / T = 0 . T = 2 / T = 0 . FIG. 3: A more traditional statistic ∆ − I to decorrelate ∆ and I ,and the left and right panels are for q = 0 . and . , respectively.The statistic shows a growing behaviour as the modified ∆ , in agree-ment with the many states picture. By contrast, the statistic shouldconverge to for a two state picture. temperatures, showing the complex relations of I and ∆ ingeneral. Nevertheless, the crossings can be qualitatively un-derstood by the two mechanisms of the sharpening of peakseither by increasing the system size or lowering the temper-ature. When the system size is sufficiently large, all peaksregardless of the temperature tend to be sharp and tall. In thiscase I will sufficiently determine the order of ∆ , the ∆ at thehigher temperature will be larger as the weight is larger. Onthe other hand, peaks for small sizes tend to be wide and shortat high temperatures and sharp and tall at low temperatures,with respect to the chosen cutoff κ . This provides a possi-bility that one could register more peaked instances at lowertemperatures due to the sharper and taller peaks, despite thatthe central weight is smaller. This explains the crossings andleads to a conclusion that the growth rate of ∆ is higher athigher temperatures than at lower temperatures.Next we look at an alternative method for variance reduc-tion [41] by considering ∆ − I . The results are shown in Fig. 3.Similar to the modified ∆ , this statistic also has a clean growthtrend. By contrast, this statistic should converge to for a twostate picture. The data are clearly running above and are stillgrowing, and should reach finite limits if there are many purestates. Therefore, the statistics I , modified ∆ , and ∆ − I areall coherently in agreement with the many state picture.Our results are in full qualitative agreement with [25, 26].The former group found a small growth rate of ∆ at a low tem-perature, while the latter group found instead a much larger growth rate by operating at a higher temperature. The size ofthe central weight again has a significant impact on the growthbehaviours of ∆ [25]. Our large range of sizes is crucial inidentifying this subtle correlation. We conclude that the statis-tic ∆ has no evidence of violating the many state picture, andinstead it is consistent with a coherent picture of many purestates [26]. The controversy of ∆ is solved.Finally, we briefly discuss the difficulties of the two statepicture with current numerical results. (1) In the same rangeof sizes, we clearly get a finite domain-wall free energy expo-nent θ DW yet a flat I (0 . . It is unclear what finite-size effectis responsible for a flat central weight, should one predict itwould eventually decay as /L θ DW . (2) The θ DW exponentis a growing function of dimensionality [42] and even the SKmodel [43] has a positive exponent which is clearly describedby RSB [44]. It seems likely that θ DW > is not capable ofexcluding large-scale O (1) droplet excitations; see, e.g., [29]for an interesting possibility of how a positive domain-wallexponent and nontrivial overlap distribution functions can co-exist. These difficulties in our opinion must be addressed fora two state picture to be acceptable and meaningful. Conclusions–
In this work we carried out a state-of-the-artsimulation of the Gaussian Ising spin glass in three dimen-sions and examined the statistic ∆ in detail. Our results revealthat the nonmonotonic growth of the statistic as a function ofthe system size is a result of its sensitivity to the fluctuationsof the central weight I . By looking at a modified ∆ and also ∆ − I compensating for this correlation effect, we find es-sentially monotonic growth of the statistics. Combining withthe relatively flat central weight, we conclude that the statis-tic ∆ is in full agreement with the many state picture but notwith the two state picture. With one more controversy solvedand supporting (again) many pure states and the difficulties ofthe two state picture mentioned above, it appears that a coher-ent picture of many pure states is emerging and that the spinglass balance is significantly tiling towards many pure statesby recent simulations. Further explorations such as the SKspin glass in thermal boundary conditions [28] are currentlyin progress and will be published in the future. Acknowledgments–
We thank J. Machta, M. Weigel, andB. Yucesoy for helpful discussions. W.W. acknowledges sup-port from the Swedish Research Council Grant No. 642-2013-7837, and the Goran Gustafsson Foundation for Research inNatural Sciences and Medicine, and the Fundamental Re-search Funds for the Central Universities, China. M.W. ac-knowledges support from the Swedish Research CouncilGrant No. 621-2012-3984. The computations were performedon resources provided by the Swedish National Infrastructurefor Computing (SNIC) at the National Supercomputer Centre(NSC), and the High Performance Computing Center North(HPC2N), and the Emei cluster at Sichuan university. [1] K. Binder and A. P. Young,
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