The Percolation Signature of the Spin Glass Transition
TThe Percolation Signature of the Spin Glass Transition
J. Machta
Physics Department, University of Massachusetts, Amherst, MA 010003 USA ∗ C.M. Newman
Courant Institute of Mathematical Sciences, New York University, New York, NY 100012 USA † D.L. Stein
Physics Department and Courant Institute of Mathematical Sciences,New York University, New York, NY 100012 USA ‡ Magnetic ordering at low temperature for Ising ferromagnets manifests itself within the associ-ated Fortuin-Kasteleyn (FK) random cluster representation as the occurrence of a single positivedensity percolating network. In this paper we investigate the percolation signature for Ising spinglass ordering — both in short-range (EA) and infinite-range (SK) models — within a two-replicaFK representation and also within the different Chayes-Machta-Redner two-replica graphical rep-resentation. Based on numerical studies of the ± J EA model in three dimensions and on rigorousresults for the SK model, we conclude that the spin glass transition corresponds to the appearanceof two percolating clusters of unequal densities.
I. INTRODUCTION
Ising type spin glass models, both of the short-range Edward-Anderson (EA) [1] and the infinite-range Sherrington-Kirkpatrick (SK) [2] varieties, have been studied for decades (for some recent reviews, see [3] and [4]). Nevertheless, toa large extent, they remain a mystery — especially the short-range variety, with competing views as to the nature oftheir ordered phases at low temperature T [3, 4, 5]. Indeed, from the perspective of rigorous results, it is striking thatthere is no proof of broken symmetry (e.g., of a nonzero EA order parameter) for any dimension d or temperature T .Graphical representations such as the Fortuin-Kastelyn (FK) random cluster model [6, 7] are important tools inthe study of spin systems. They relate correlations in spin systems to geometrical properties of associated randomgraphs. Graphical representations are useful in obtaining rigorous results concerning spin systems (e.g., [8, 9]), theyyield geometric insights into the nature of phase transitions and they are the basis for powerful Monte Carlo methodsfor simulating phase transitions [10, 11, 12]. However, graphical representations have, thus far, played a much lessimportant role in the study of spin glasses than they have for ferromagnets.In this paper, we investigate two different graphical representations — the two-replica graphical representationof Chayes, Machta and Redner (CMR) [13, 14] and a two-replica version of the FK representation (see Sec. 4.1of [15]). Our purpose is to understand the “percolation signature” of spin glass ordering within these graphicalrepresentations. For ferromagnets, ordering corresponds to the occurrence of percolating networks or clusters in thesingle replica version of the FK representation. As we shall explain, we believe we have elucidated the somewhat morecomplicated percolation signature for spin glasses.This should help in understanding better the differences between the nature of the phase transition in ferromagnetsand in spin glasses. It is also our hope that for short-range models, this will be a significant step towards developinga rigorous proof for spin glass ordering and eventually lead to a clean analysis of the differences between short- andinfinite-range spin glass ordering.Understanding the percolation signature for spin glasses requires two ingredients beyond what is needed for ferro-magnets. The first is the need to consider percolation within a two-replica representation. As mentioned, we considertwo different such representations — one is the percolation of a certain class of bonds (these are the “blue bonds”introduced and explained in Subsection II B below) in the CMR two-replica graphical representation and the otheris percolation of bonds that are doubly FK occupied — i.e., occupied in both replicas — in the two-replica Fortuin-Kasteleyn (TRFK) representation. The two different types of percolation, which we will often refer to simply as ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] a r X i v : . [ c ond - m a t . s t a t - m ec h ] J un CMR and TRFK percolation, give relatively similar (but not identical) results, with the major qualitative distinctionoccuring within the SK spin glass.The second ingredient, initially unexpected by us but in retrospect rather natural, is that spin glass orderingcorresponds to a more subtle percolation phenomenon than simply the appearance of a percolating cluster — one thatinvolves a pair of percolating clusters. In the case of a ferromagnet, there are general theorems [16] which ensure thatwhen percolation occurs, there is a unique percolating cluster, whether in single or double replica representations. Itis also possible (by averaging over disorder realizations) to show (see, e.g., [17] and [15]) that the same conclusionsare valid in the single replica FK representation of spin glasses. However for blue bond percolation in the (two-replica) CMR representation of spin glasses, both our numerical evidence for the d = 3 EA model and our rigorousresults for the SK model (in the CMR representation) show that at temperatures well above the spin glass transition,there already is percolation, but that there are two percolating networks which are equal in density (and presumablyotherwise macroscopically indistinguishable). The SG transition corresponds to the breaking of indistiguishability between the two percolating networks — in particular by having a nonzero difference in densities. The latter featurealso occurs for doubly occupied bonds in the TRFK representation of the SK model, except that in that representationthere are no percolating networks at all above the transition temperature.From a numerical perspective, spin glasses also pose major challenges. Some of the numerical techniques, e.g., thatof Swendsen and Wang (SW) [11, 18, 19] based on graphical representations, such as that of Fortuin and Kasteleyn,which have proven so useful for ferromagnets [6, 7], are in principle applicable to spin glasses. However, they are veryinefficient in practice for values of d and T where ordering is believed to occur. The CMR graphical representationis related to a two-replica algorithm originally introduced by Swendsen and Wang and developed by these and otherauthors [20, 21, 22, 23]. These authors have shown that algorithms incorporating two-replica cluster moves aresomewhat useful in simulating spin glasses. In particular, J¨org [22, 23] has shown that an algorithm based on a two-replica representation performs reasonably efficiently for diluted spin glass models in three dimensions. Two-replicacluster methods have also been successfully applied to Ising systems in a staggered field [14] and to the random fieldIsing model [24]. The Monte Carlo method that we use takes advantage of the full set of moves allowed by the CMRgraphical representation. These moves are a superset of the moves used in [11, 18, 19, 21, 22, 23].The paper is organized as follows. In Sec. II we introduce the idea of graphical representations, describe the CMRand TRFK two-replica graphical representations and present properties of these representations. In Sec. III we analyzeboth two-replica representations on the complete graph — i.e., for the SK spin glass. In Sec. IV we present numericalresults for the three-dimensional EA model. The paper concludes with a discussion. II. GRAPHICAL REPRESENTATIONS FOR SPIN GLASSESA. Fortuin-Kastelyn Graphical Representation
Graphical representations for the Ising model originated with the work of Fortuin and Kastelyn [6, 7]. Theywere re-discovered and given a physical interpretation by Coniglio and Klein [25], applied as the basis of a powerfulalgorithm for simulating the Ising model by Swendsen and Wang [11, 18, 19] and then reformulated as a joint spin-bond distribution by Edwards and Sokal [20]. Edwards and Sokal introduced a joint distribution of spin variables { σ x } and bond variables { ω xy } . Here { x } represents the set of sites (vertices) of an arbitrary lattice (graph) and { xy } the set of bonds (edges). The Ising spin variables take values ± W for the Edwards-Sokal distribution is W ( σ, ω ; p ) = p | ω | (1 − p ) N b −| ω | ∆( σ, ω ) (1)Here | ω | = (cid:80) { xy } ω xy is the number of occupied bonds and N b is the total number of bonds on the lattice. The factor∆( σ, ω ) is defined by, ∆( σ, ω ) = (cid:26) xy : ω xy = 1 → σ x σ y = 10 otherwise (2)The ∆ factor requires that every occupied bond is satisfied. Without the ∆ factor we would have independent Bernoullipercolation. Given the choice, p = P FK ( β ) = 1 − exp( − βJ ), it is easy to verify that the spin and bond marginals ofthe Edwards-Sokal distribution are the ferromagnetic Ising model with coupling strength J and the Fortuin-Kastelynrandom cluster model, respectively.Bond and spin configurations in the ferromagnet contain essentially the same information. For example, the spin-spin correlation function (cid:104) σ x σ y (cid:105) is equal to the probability that sites x and y are connected by occupied bonds in thebond representation, (cid:104) σ x σ y (cid:105) = Prob { x and y connected } . (3)This relationship implies that the phase transition in the spin system is accompanied by a percolation transition inthe bond system.Given a typical equilibrium bond configuration one can construct a typical equilibrium spin configuration by iden-tifying connected components or clusters and independently populating every spin in each cluster with one randomlychosen spin type. Similarly, given an equilibrium spin configuration, an equilibrium bond configuration can be con-structed by occupying satisfied bonds with probability P FK ( β ). The equivalence between spin and bond configurationsis the basis of the Swendsen-Wang algorithm, which proceeds by successively creating spin configurations from bondconfigurations and then bond configurations from spin configurations. It is easy to verify that this algorithm is ergodicand satisfies detailed balance with respect to the Edwards-Sokal distribution. Power law decay of spin correlationsat criticality imply via Eq. (3) that the connected components of bond configurations at criticality have a powerlaw distribution of sizes. The efficiency of the Swendsen-Wang algorithm is due to the fact that the spin system ismodified on all length scales in a single step.The FK representation is easily adapted to the ± J Ising spin glass. (With minor modifications, it can also beadapted to Gaussian and other distributions for the couplings, but we will generally not consider those in this paper.)The corresponding Edwards-Sokal weight is the same as given in Eq. (1). The ∆ factor must still enforce the rulethat all occupied bonds are satisfied,∆( σ, ω ; J ) = (cid:26) xy : ω xy = 1 → J xy σ x σ y = 10 otherwise. (4)The spin marginal of the corresponding Edwards-Sokal distribution is the Ising spin glass with couplings { J xy } .Unfortunately, the relationship between spin-spin correlations and bond connectivity is complicated by the presenceof antiferromagnetic bonds. Specifically, one has (cid:104) σ x σ y (cid:105) = (5)Prob { x and y connected by even number of antiferromagnetic bonds }− Prob { x and y connected by odd number of antiferromagnetic bonds } . It is no longer the case that the percolation of FK bonds implies long range order [26]. Two spins separated bya large distance may usually be connected by occupied bonds but still be uncorrelated because half the time theconnection has an even number of antiferromagnetic bonds and half the time an odd number of antiferromagneticbonds. Indeed, FK bonds percolate at a temperature that is well above the spin glass transition temperature. Forthe three-dimensional Ising spin glass on the cubic lattice Fortuin-Kastelyn bonds percolate at β FK ,p ≈ .
26 [27]while the inverse critical temperature is β c = 0 . ± .
03 [28]. Near the spin glass critical temperature, the giant FKcluster includes most of the sites of the system. For this reason, the Swendsen-Wang algorithm, though valid, is quiteinefficient for simulating spin glasses.
B. The CMR Two-Replica Graphical Representation
A conceptual difficulty of using the Fortuin-Kastelyn representation to understand spin glass ordering is that FKclusters identify magnetization correlations but the spin glass order parameter is not the magnetization. Spin glassorder is manifest in the Edwards-Anderson order parameter, which can be defined with respect to two independentreplicas of the system, each with the same couplings { J xy } . The spins in the two replicas are { σ x } and { τ x } ,respectively, each taking values ±
1. The Edwards-Anderson order parameter, q EA , is defined in terms of the overlap, Q = N s − (cid:88) { x } σ x τ x , (6)in the limit as the number of sites N s → ∞ . In general, Q is a random variable whose maximum possible value is q EA , but in the case where (in the limit N s → ∞ ) { σ x } and { τ x } are drawn from a single pure state, Q takes on onlythe single value q EA .The two-replica graphical representation, introduced in [13, 14], explicitly relates spin glass order to geometry. Theassociated Edwards-Sokal joint distribution has, in addition to the spin variables, { σ x } and { τ x } , two types of bondvariables ω xy and η xy each taking values in { , } .The Edwards-Sokal weight is W ( σ, τ, ω, η ; J ) = B blue ( ω ) B red ( η )∆( σ, τ, ω ; J )Γ( σ, τ, η ; J ) (7)where the B ’s are Bernoulli factors for the two types of bonds, B blue ( ω ) = P | ω | blue (1 − P blue ) N b −| ω | (8) B red ( ω ) = P | η | red (1 − P red ) N b −| η | (9)and the bond occupation probabilities are P blue = 1 − exp( − β | J | ) (10) P red = 1 − exp( − β | J | ) . (11)The ∆ and Γ factors constrain where the two types of occupied bonds are allowed,∆( σ, τ, ω ; J ) = (cid:26) xy : ω xy = 1 → J xy σ x σ y > J xy τ x τ y >
00 otherwise (12)Γ( σ, τ, η ) = (cid:26) xy : η xy = 1 → σ x σ y τ x τ y <
00 otherwise (13)We refer to the ω occupied bonds as “blue ” and the η occupied bonds as “red ”. The ∆ constraint says that bluebonds are allowed only if the bond is satisfied in both replicas. The Γ constraint says that red bonds are allowed onlyif the bond is satisfied in exactly one replica.It is straightforward to verify that the spin marginal of the CMR Edwards-Sokal weight is the weight for twoindependent Ising spin glasses with the same couplings, (cid:88) { ω }{ η } W ( σ, τ, ω, η ; J ) = const × exp β (cid:88) { xy } J xy ( σ x σ y + τ x τ y ) (14) C. Properties of Graphical Representations for Spin Glasses
Connectivity by occupied bonds in the CMR representation is related to correlations of the local spin glass orderparameter, Q x = σ x τ x . (15)It is straightforward to verify that (cid:104) Q x Q y (cid:105) = Prob { x and y connected by even number of red bonds } (16) − Prob { x and y connected by odd number of red bonds } . As in the case of the FK representation, a minus sign complicates the relationship between correlations and connectivitybut in a conceptually different way. The second term in Eq. (16) is independent of the underlying coupling in themodel and is present for both spin glasses and ferromagnetic models.In the case of a ferromagnet, having a percolating cluster (or clusters) in the (single replica) FK representationeasily shows that there is broken symmetry with respect to global spin flips. For example, one can impose plus orminus boundary conditions on those boundary spins belonging to FK percolating networks in the Edwards-Sokal jointspin-bond represenatation and these two choices of boundary conditions give two different Gibbs states for the spinsystem in the infinite volume limit. More simply, in the ferromagnetic case, the magnetization order parameter equalsthe total density of the percolating network(s), since finite FK clusters do not contribute.We remark, as noted in Section I, that for ferromagnets (in the absence of boundary conditions that force interfaces),the signature of ordering is a single percolating cluster. For spin glasses, the situation is analogous, but morecomplicated. If there is in the CMR graphical representation a percolating blue cluster of strictly larger densitythan any other blue clusters, one can similarly show broken symmetry. Here one can impose “agree” or “disagree”boundary conditions between those σ x and τ x boundary spins belonging to the maximum density blue network.[32]In the infinite volume limit, these two boundary conditions give different Gibbs states for the σ -spin system (for fixed τ ) related to each other by a global spin flip (of σ ). However, in this case, it is not so easy to rigorously relate (ingeneral) the overlap Q to the densities of percolating blue networks, even if one assumes that there are exactly twosuch networks with densities D and D . This is because in a two-replica situation, it is not immediate that there is nocontribution from finite (non-percolating) clusters, which would be enough to imply that Q = D − D . Nevertheless,this identity seems likely to be the case, and indeed is valid for the SK model, as we discuss in the next section of thepaper.For the TRFK representation, similar reasoning shows that the occurrence of exactly two doubly-occupied perco-lating FK clusters with different densities implies broken symmetry for the spin system [15] and that Q should equal(and does equal in the SK model) the density difference. In Sec. IV C we will present peliminary numerical evidencethat there is such a nonzero density difference below the spin glass transition temperature for the d = 3 EA ± J spinglass (for both the TRFK and CMR representations). III. THE SPIN GLASS ON THE COMPLETE GRAPH
The spin glass on the complete graph was introduced by Sherrington and Kirkpatrick (SK) [2]. The ± J versionof the model has couplings given by ± N − / where N is the number of vertices on the graph. This scaling for thecoupling strength insures that the free energy is extensive. In this section, we study the percolation properties of boththe Fortuin-Kastelyn and CMR representations for the SK model, In the high temperature phase, β < β c = 1, boththe magnetization and the EA order parameter, q EA , vanish. The SK solution, valid for the high temperature phase,yields the energy per spin, u = − β/
2. The number of unsatisfied edges minus the number of satisfied edges is equalto uN / . Thus, letting f s be the fraction of satisfied edges, we have that f s ∼ − uN − / . (17)In the FK representation a fraction P FK = 1 − exp( − βN − / ) ≈ βN − / of satisfied edges are occupied. Whenwill the occupied edges first form a giant cluster and how many giant clusters will coexist? The theory of randomgraphs (see [29]) can be used to answer this question. It is known [30] that a giant cluster forms in a random graphof N vertices when a fraction x/N of edges is occupied with x >
1, and that there is then a single giant cluster. Thissuggests that (single replica) FK giant clusters should form with β = xN − / when x >
1, i.e., that the single replicaFK percolation threshold is at β FK ,p = N − / . (18)It also suggests that above this threshold, there should be a single giant FK cluster.Although the arguments just given are incomplete in that the satisfied edges were treated (without justification)as though they were chosen independently of each other, nevertheless the conclusions can be proved rigorously as wenow explain. Indeed, our rigorous analysis of the much more interesting cases with two replicas will use very similararguments. The idea is to obtain upper and lower bounds for the conditional probability that an edge { x y } issatisfied, given the satisfaction status of all the other edges. If these bounds are close to each other (for large N ) thentreating the satisfied edges as though chosen independently can be justified.A key point is that because of frustration, such approximate independence is impossible if one knows too muchabout the signs of the couplings. Thus, we will not condition on the sign of the single coupling J x y — in fact wewill consider precisely the conditional probability of that sign given the configuration of all other couplings J xy andall spins σ x . For the ± J model that we are considering, it is quite elementary to see first that the ratio Z + /Z − forthe partition functions with J x y = + N − / and J x y = − N − / satisfiesexp( − βN − / ) ≤ | Z + /Z − | ≤ exp(2 βN − / ) , (19)and then that the conditional probabilities P ± that J x y = ± N − / given any configuration of the other J xy ’s andall σ x ’s satisfy e − β/ √ N ≤ e − β/ √ N | Z − /Z + | ≤ P + /P − ≤ e β/ √ N | Z − /Z + | ≤ e β/ √ N . (20)It then follows that the conditional probabilities P s or P u for any edge { x y } to be satisfied or unsatisfied giventhe satisfaction status of all other edges satisfy e − β/ √ N ≤ P s /P u ≤ e β/ √ N , (21)so that 12 − O( β/ √ N ) = ( e β/ √ N + 1) − ≤ P s ≤ ( e − β/ √ N + 1) − = 12 + O( β/ √ N ) . (22)One now obtains rigorously the same conclusions as before — i.e., (18) is valid with a single giant FK cluster for β = β N ≥ xN − / with any x >
1. Before proceeding to our detailed analysis of the situation with two replicas, westate our main conclusions:
The threshold for TRFK percolation is β TRFK ,p = 1 . (23) For β ≤ , there is no giant cluster (containing a strictly positive density, i.e., fraction of sites). For β > there areexactly two giant clusters with unequal densities. (Strictly speaking, we do not rigorously rule out the possibility thatfor some choices of β > , there might be only a single giant cluster, but we explain why that should not be so andalso prove that a nonzero spin-spin overlap rules out the possiblity of two clusters of exactly equal density.)The threshold for percolation of blue bonds in the CMR two-replica graphical representation is β CMR ,p = N − / . (24) Only above that threshold are there one or more giant clusters. The number and density of the giant clusters isdetermined by a second threshold which is exactly the SK spin glass critical value β c = 1 . For xN − / ≤ β N ≤ with x > , there are exactly two giant clusters, which have equal densities; if N / β N → ∞ , then the two densitiesare both exactly / . For β N ≥ x with any x > , there are two giant clusters of unequal densities, whose sum isone. (Strictly speaking, as in the case of TRFK percolation, we do not rigorously rule out the possibility that for some β > , there might be a single blue giant cluster, which would necessarily have density one.) Now we explain our analysis when there are two spin replicas σ and τ . For both TRFK percolation and forblue percolation in the CMR graphical representation, we focus on doubly satisfied edges. For the Fortuin-Kastelynrepresentation, doubly satisfied edges are occupied with probability P TRFK = [1 − exp( − βN − / )] ∼ β /N . For theCMR representation, doubly satisfied edges are occupied with probability P CMR = 1 − exp( − βN − / ) ∼ βN − / .The crucial new ingredient in two-replica situations is that an edge { xy } can be doubly satisfied only if σ x σ y τ x τ y = +1or equivalently if σ x τ x = σ y τ y (and then will be satisifed for exactly one of the two signs of J xy ). Thus, beforeproceeding as in the single replica situation, we first divide all ( σ, τ ) configurations into two groups or sectors — the agree (where σ x = τ x ) and the disagree sectors (where σ x = − τ x ). We also denote by N a and N d the numbers of sitesin the sectors and denote by D a = N a /N and D d = N d /N the sector densities (so that D a + D d = 1). We note thatthe spin overlap Q is just Q = 1 N (cid:88) x σ x τ x = N a − N d N = D a − D d (25)and that for β ≤ β c = 1, Q → N → ∞ while for β > β c = 1, Q is nonzero, e.g., in the sense that Av ( < Q > ) > N → ∞ , where Av denotes the average over couplings.We now proceed similarly to the single replica case, but separately within the agree and disagree sectors. Letting¯ P ± denote the conditional probabilities that J x y = ± N − / given the other J xy ’s and all σ x ’s and τ x ’s, we havewithin either of the two sectors that e − β/ √ N ≤ e − β/ √ N | Z − /Z + | ≤ ¯ P + / ¯ P − ≤ e β/ √ N | Z − /Z + | ≤ e β/ √ N (26)so that the conditional probability within a single sector P ds for x y to be doubly satisfied is (1 /
2) + O( βN − / ).For β ≤ β c , we have D a = 1 / D d = 1 / N → ∞ ) and so in either sector, double FK percolation isapproximately a random graph model with N/ / β N − = β ( N/ − ;thus double FK giant clusters do not occur for β ≤ / βN − / = βN / ( N/ − and so the thresholdfor blue percolation is given by (24). But now there are two giant clusters, one in each of the two sectors, and theyare of equal density for β ≤ β c = 1 since D a = D d . On the other hand, for β > β c , D a (cid:54) = D d and the two giantclusters will be of unequal density. In fact, since βN / → ∞ for β > β c (indeed for any fixed β > D a and D d are also the clusterpercolation densities of the two giant clusters.In the case of two-replica FK percolation for β > β c , let us denote by D max and D min the larger and smaller of D a and D d , so that D max + D min = 1 and D max − D min = Q . Then for β > β c , the bond occupation probabilityin the larger sector is β ( N/ − = 2 β D max ( D max N ) − with 2 β D max > β D min (= β (1 − Q )) >
1. Since Q ≤ q EA , for this to be the case it suffices if for β > β c , q EA < − β . (27)The estimated behavior of q EA both as β →
1+ and as β → ∞ [5] suggests that this is always valid. In any case, wehave rigorously proved that there is a unique maximal density double FK cluster for β > β c . IV. NUMERICAL SIMULATIONS
In this section we describe numerical simulations of the Edwards-Anderson spin glass in three dimensions to testideas about the percolation signature for spin glass ordering. For comparison, we also describe simulations of spinclusters for the 3 D ferromagnetic Ising model. A. Methods
We carried out simulations of the ± J Ising spin glass using a Monte Carlo method that combines CMR clustermoves, Metropolis sweeps and parallel tempering (replica exchange). A similar scheme was used in a study of therandom field Ising model [24]. The cluster moves are closely related to the replica Monte Carlo algorithm introducedby Swendsen and Wang [11, 18, 19] and developed in [21, 22, 23]. The combination of two-replica cluster moves andparallel tempering was first introduced by Houdayer [21]. The new ingredient in the present algorithm is that all ofthe degrees of freedom available in the CMR representation are used. The additional degree of freedom is incorporatedin “grey” moves, described below.The parallel tempering component of the algorithm works with R pairs of replicas at equally spaced inverse temper-atures. Standard temperature exchange moves are carried out between one of the two replicas at one temperature andat one of the neighboring temperatures. The CMR cluster moves begin by identifying all singly and doubly satisfiedbonds and occupying them with probabilities, P blue = 1 − exp( − β ) and P red = 1 − exp( − β ), respectively. Bondsthat are not satisfied in either replica cannot be occupied. The occupied bonds determine blue and grey clusters. Setsof sites connected by blue bonds and singletons are considered to be blue clusters. Sets of sites connected by blue orred bonds are considered to be grey clusters. The cluster moves proceed as follows. For each grey cluster a randombit determines whether to perform a grey move or not. If a grey move is chosen, the sign of Q in each blue cluster inthe grey cluster is reversed. Next, a random bit determines which of the two spin states to put each blue cluster ingiven its Q state. For example, if Q = 1 in a given blue cluster then, with equal probability, the spin state of a givensite in the cluster is either (++) or ( −− ).The two-replica cluster component of the algorithm satisfies detailed balance with respect to the Edwards-Sokalweight for the CMR graphical representation. In addition, the two-replica cluster component of the algorithm is, byitself, ergodic. There is a non-vanishing probability that any given site is a singleton cluster and is flipped to any ofthe four spin states. Thus, there is a non-vanishing probability of a transition from any spin configuration for thepair of replicas to any other spin configuration in a finite number of steps. A single pair of replicas will eventuallyapproach equilibrium under two-replica cluster moves. However, for reasons that will become clear in Sec. IV C, theequilibration by two-replica cluster moves alone is very slow in three dimensions. Thus we supplement these moveswith both temperature exchange moves and Metropolis sweeps.The presence of very long lived metastable states makes it difficult to gauge whether a spin glass simulation hasreached equilibrium. Here we measure the time it takes for a spin configuration, originally at the highest simulatedtemperature, to diffuse by replica exchange moves, to the lowest temperature. If the entire set of replicas is inequilibrium and if the replica pair at the highest temperature is rapidly equilibrated, then this first passage timeestimates the time it takes to obtain an independent sample at the lowest temperature. We assume that the meanfirst passage time is comparable to the equilibration time for the full set of replicas though it is conceivable thatequilibration time is much longer than the mean first passage time. For the 12 system, the largest system studiedhere, the mean first passage time is of the order of one hundred MC sweeps and does not vary greatly from onerealization to another so we believe that the system is well equilibrated.The two-replica cluster moves complicate the ability to keep track of a single spin configuration as it diffuses intemperature space. If there are two giant clusters and, in one of the replicas, both or neither of the giant clustersare flipped then the identity of the spin configuration is unaffected (though it may have suffered an overall spin flip).On the other hand, if one giant cluster is flipped and the other not then the spin configurations of the replicas areswapped. Below the CMR percolation transition, this same rule is applied to the two largest clusters but the identityof the spin configuration is effectively lost in one two-replica move. B. Improved Estimators
One advantage of cluster algorithms in data collection is the existence of improved estimators [12] in the graphicalrepresentation. For example, in the Swendsen-Wang algorithm, one can obtain the magnetization and magneticsusceptibility from the cluster configurations. The magnetization is the average size of the largest cluster and thesusceptibility is proportional to the sum of the squares of the cluster sizes. Since each occupied bond configurationcorresponds to many spin configurations, the variance of observables measured from the bond configurations is lessthan for the same observables measured in the spin configuration leading to smaller error bars for the same amoutof computational work. Improved estimators exist within the CMR representation for the spin glass order parameterand susceptibility. For example, the order parameter Q can be obtained from the percolating grey cluster if it existsand is unique. In particular, the local order parameter Q x summed over the sites of the percolating grey clustershould equal the Q of the whole system since the contributions of small grey clusters vanishes after averaging overthe possible spin states of these clusters. C. Simulation results
We simulated the three-dimensional ± J Edwards-Anderson model on skew periodic cubic lattices for system sizes6 , 8 , 10 and 12 . For each size we simulated 20 inverse temperatures equally spaced in the range β = 0 .
16 to β = 0 .
92. Since there are two replicas for each temperature, the total number of replicas simulated was 40. A recentestimate of the phase transition temperature of the system is β c = 0 . ± .
03 [28]. For each size we simulated 100realizations of disorder for 50,000 Monte Carlo sweeps of which the first 1 / / C and second largest blue cluster, C and the number of blue CMR wrapping cluster, w CMR , and the number of TRFK “wrapping” clusters, w TRFK . Acluster is said to wrap if it is connected around the system in any of the three directions.Figure 1 shows the average number w CMR of CMR blue wrapping clusters as a function of inverse temperature β . The curves are ordered by system size with largest size on the bottom for the small β and on top for large β .The data suggests that there is a percolation transition at some β CMR ,p . For β > β CMR ,p there are two wrappingclusters while for β < β CMR ,p there are none. Near and above the spin glass transition at β c ≈ .
89 the expectednumber of wrapping clusters falls off but the fall-off diminishes as system size increases. This figure suggests thatin the large size limit there are exactly two spanning clusters near the spin glass transition both above and belowthe transition temperature. Figure 2 is a magnification of Fig. 1 near the CMR percolation transition. The crossingpoints identify the percolation transition as β CMR ,p ≈ . ± J EA model β FK ,p ≈ .
26 reported in [27]. A more careful study would be needed to test the hypothesis that β CMR ,p > β FK ,p .Figure 3 shows the fraction of sites in the largest CMR blue cluster, C , second largest CMR blue cluster, C andthe sum of the two, C + C . The middle set of four curves is C for sizes 6 , 8 , 10 and 12 , ordered from top tothe bottom at β = 0 .
5. The bottom set of curves is C with systems sizes ordered from smallest on bottom to largeston top at β = 0 .
5. The difference between the fraction of sites in the two largest clusters, C − C is approximatelythe spin glass order parameter. As the system size increases, this difference decreases below the transition suggestingthat C = C for β < β c in the thermodynamic limit. On the other hand, the sum of the two largest clusters isquite constant independent of system size. Near the transition, approximately 96% of the sites are in the two largestclusters.The large fraction of sites in the two largest clusters makes the CMR cluster moves inefficient. If all sites were inthe two largest clusters then the cluster moves would serve only to flip all spins in one or both clusters or exchangethe identity of the two replicas. Equilibration depends on the small fraction of spins that are not part of the twolargest clusters. One of the reasons that bond diluted spin glasses are more efficiently simulated using two-replicacluster algorithms is the smaller fraction of sites in the two largest clusters. We have carried out simulations on thesame bond diluted Ising spin glass studied by J¨org [22, 23]. This model has 55% of the couplings set to zero and45% set to ±
1. Near the phase transition, we find that only 87% of the sites are contained in the two largest clustersinstead of the 96% found in the undiluted spin glass. Β w C M R FIG. 1: Average number of wrapping CMR clusters, w CMR vs. β for the 3D EA model. Β w C M R FIG. 2: Same as Fig. 1, magnified near the CMR percolation transition
Figure 4 show the average number of wrapping TRFK clusters w TRFK as a function of inverse temperature. Thelargest system size is on the bottom for the small β and on top for the large β . As for the case of CMR clusters,the data suggests a transition at some β TRFK ,p from zero to two wrapping TRFK clusters. Although the number ofTRFK wrapping clusters is significantly less than two for all β and all system sizes, the trend in system size suggeststhat it might approach two for large systems and β > β TRFK ,p . Figure 5 shows a close up of the transitions regionand the crossing points give the inverse percolation temperature as β TRFK ,p ≈ . , 8 , 10 and 12 . The vertical axis is located at the critical temperature.This figure is qualitatively similar to the results for the two largest clusters in the CMR representations. In bothcases, for these system sizes, the phase transition is quite rounded in the sense that a difference in density developswell before the transition and the transition itself cannot be identified by looking at the size of the clusters. The0 Β C L U S TE R S I ZE FIG. 3: C (middle set), C (bottom set) and C + C (top set) vs. β for the CMR graphical representation for the 3D EA model. Β w T R F K FIG. 4: Average number of doubly occupied wrapping Fortuin-Kastelyn clusters, w TRFK vs. β for the 3D EA model. difference in the density of the two clusters is thus not a sharp indicator of the phase transition for small system sizes. V. DISCUSSION
In this paper we have proposed a new percolation-theoretic approach towards understanding the nature of the spinglass phase transition. It is based on the Fortuin-Kasteleyn [6, 7] random cluster method (and some variants), whichhas been enormously useful in analyzing phase transitions in ferromagnets, but had much less impact on systems suchas spin glasses until now (see, however, [18, 22, 23]).There are a number of advantages to our approach. First, and most obviously, it sheds new light on the natureof the spin glass phase transition, particularly its geometric aspects. For example, it provides at least a qualitativeexplanation of why the EA spin glass on a simple planar lattice doesn’t have broken spin flip symmetry at positivetemperature (see below). Second, it indicates a possible new framework towards an eventual rigorous proof of an EAspin glass phase transition (at least in sufficiently high dimensions), while providing a basis for numerical work toexplore, and hopefully resolve, the issue of the lower critical dimension. Finally, it helps to emphasize fundamentaldifferences between phase transitions in spin glasses as opposed to more conventional systems such as ferromagnets.1 Β w T R F K FIG. 5: Same as Fig. 4, magnified near the TRFK percolation transition Β C L U S TE R S I ZE FIG. 6: The size of the largest cluster of satisfied bonds (middle set), the second largest cluster (bottom set) and the sum ofthe two largest clusters (top set) vs. β for the three-dimensional ferromagnetic Ising model. Two different representations, each involving two replicas, were used to apply an FK type formalism — the Chayes-Machta-Redner(CMR) [13, 14] representation and the two-replica FK (TRFK) representation previously consideredby Newman-Stein [15]. It is likely that other representations can also be used, as long as they involve the overlapof independent replicas. While various details will differ, as described in the text, the essential — and somewhatsurprising — feature appears to be that the spin glass transition coincides with the emergence of percolating clustersof unequal densities.Numerical results for the d = 3 EA spin glass seem to indicate that this occurs, as T is lowered below T c , as thebreaking of symmetry in the equal densities of both doubly occupied TRFK and blue CMR clusters that alreadypercolate above the transition temperature. For the SK model, on the other hand, what occurs just above T c isrepresentation-dependent: in the TRFK representation, there is no doubly occupied percolation at all above T c , whilein the CMR representation there are two equal-density blue clusters just above T c , similar to the situation in the d = 3 EA model. This difference in behavior above T c may arise from the peculiarities of the SK model, and a similarrepresentation-dependence may not occur in short-range models.Finally, we speculate about the nature of the lower critical dimension. Our numerical results are consistent withprior studies [28, 31] indicating the appearance of broken spin-flip symmetry in the EA model in three dimensions.2If the percolation signature scenario proposed here is correct for short-range models, it would help explain why thereis no spin glass transition leading to broken spin-flip symmetry on simple planar lattices: two dimensions does notgenerally provide enough “room” for two disjoint infinite clusters to percolate. However, a system that is infinitein extent in two dimensions but finite in the third might be able to support two percolating clusters, with unequaldensities at low temperature. This and other possibilities will be explored in future work. Acknowledgments
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