Ground state stability in two spin glass models
aa r X i v : . [ c ond - m a t . d i s - nn ] J a n GROUND STATE STABILITY IN TWO SPIN GLASS MODELS
L.-P. ARGUIN, C.M. NEWMAN, AND D.L. STEIN
Abstract.
An important but little-studied property of spin glasses is the stability of theirground states to changes in one or a finite number of couplings. It was shown in earlierwork that, if multiple ground states are assumed to exist, then fluctuations in their energydifferences — and therefore the possibility of multiple ground states — are closely relatedto the stability of their ground states. Here we examine the stability of ground states intwo models, one of which is presumed to have a ground state structure that is qualitativelysimilar to other realistic short-range spin glasses in finite dimensions. Introduction and definitions
Vladas was a remarkable mathematician, collaborator, colleague and friend: often exciting,always interesting, sometimes frustrating but never boring. We will miss him greatly, butare confident that his memory will survive for a very long time.Although he never worked directly on spin glasses himself, Vladas maintained a longstand-ing interest in the problem, and we enjoyed numerous discussions with him about possibleways of proving nonuniqueness of Gibbs states, energy fluctuation bounds, overlap proper-ties, and many other open problems. In this paper we discuss another aspect of spin glasses,namely ground state stability and its consequences, a topic we think Vladas would haveenjoyed.The stability of a spin glass ground state can be defined in different ways; here we will adoptthe notion introduced in [14, 15] and further developed and exploited in [3, 4]. For specificityconsider the Edwards-Anderson (EA) Ising model [8] in a finite volume Λ L = [ − L, L ] d ∩ Z d centered at the origin, with Hamiltonian(1) H Λ ,J ( σ ) = − X h xy i∈ E (Λ) J xy σ x σ y , σ ∈ {− , } Λ , where E (Λ) denotes the set of nearest-neighbor edges h xy i with both endpoints in Λ. Thecouplings J xy are i.i.d. random variables sampled from a continuous distribution ν ( dJ xy ),which for specificity we take to be N (0 , L and accompanying boundary condition, the ground state configuration(or the ground state pair if the boundary condition has spin-flip symmetry) is denoted by α . One may now ask the question, how does the lowest-energy spin configuration α changewhen one selects an arbitrary edge b and varies its associated coupling J from −∞ to + ∞ ?If J is satisfied, increasing its magnitude will only increase the stability of α and so thelowest-energy spin configuration pair is unchanged. However, if its magnitude is decreased, α becomes less stable, and there exists a specific value J c for which a cluster of connected spins (which we shall refer to as the “critical droplet”) will flip, leading to a new groundstate pair α ′ . The same result follows if J is unsatisfied and its magnitude is then increased.More precisely, note that a ground state pair (hereafter GSP) is a spin configuration suchthat the energy E ∂ D of any closed surface ∂ D in the dual lattice satisfies the condition(2) E ∂ D = X h xy i∈ ∂ D J xy σ x σ y > . The critical value J c corresponds to the coupling value at which P h xy i∈ ∂ D J xy σ x σ y = 0 in α for a single closed surface whose boundary passes through b , while all other such closedsurfaces satisfy (2). The cluster of spins enclosed by the zero-energy surface ∂ D c ( b , α ) isdenoted the “critical droplet” of b in the GSP α . Because the couplings are i.i.d., J c dependson α and all coupling values except that associated with b ; that is, the critical value J c isindependent of J . For a fixed coupling realization in which J ( b ) = J , we can thereforedefine the flexibility F b ,α of b in α as(3) F b ,α = | E ∂ D c ( b ,α ) ( J c ) − E ∂ D c ( b ,α ) ( J ) | . Because the couplings are i.i.d. and drawn from a continuous distribution, all flexibilities arestrictly positive with probability one.The presentation just given is informal; a complete discussion requires use of the excitationmetastate [16, 3, 2, 5] which we omit here for the sake of brevity. A precise definition of theabove concepts and quantities can be found in [4].The concepts of critical droplets and flexibilities for a particular GSP in a fixed couplingrealization provide a foundation for quantifying (at least one version of) the stability of agiven ground state. From an energetic standpoint, one can consider, e.g., the distribution offlexibilities over all bonds. One can also approach the problem from a geometric perspective,by considering the sizes and geometries of the critical droplets associated with each of thebonds. This latter approach has recently proved to be useful, in that the distribution ofcritical droplet sizes has been shown [4] to be closely related to the energy fluctuations asso-ciated with collections of incongruent GSP’s, i.e., GSP’s whose mutual interfaces comprisea positive fraction of all edges in the infinite-volume limit [10, 9].The problem with this approach, for now at least, is that there currently exist no tools orinsights into determining ground state stability properties in ordinary EA models. In thispaper we discuss two models, one of which should belong in the same universality class asthe ordinary EA model, in which some information on these properties can be determined.2.
The highly disordered model
Definition and properties.
The highly disordered model was introduced in [12, 13](see also [6]). It is an EA-type model defined on the lattice Z d whose Hamiltonian in anyΛ ⊂ Z d is still given by (1); the difference is that now the coupling distribution is volume-dependent even though the coupling values remain i.i.d. for each Λ. The idea is to “stretchout” the coupling distribution so that, with probability one, in sufficiently large volumeseach coupling magnitude occurs on its own scale. More precisely, each coupling magnitudeis at least twice as large as the next smaller one and no more than half as large as the nextlarger one. ROUND STATE STABILITY IN TWO SPIN GLASS MODELS 3
While there are many possibilities for the volume-dependent distribution of couplings, wehave found it convenient to work with the following choice. First, we associate two new i.i.d.random variables with each edge h xy i : ǫ xy = ± K xy which isuniformly distributed in the closed interval [0 , J ( L ) xy within Λ L as follows:(4) J ( L ) xy = c L ǫ xy e − λ ( L ) K xy , where c L is a scaling factor chosen to ensure a sensible thermodynamic limit (but whichplays no role in ground state selection), and λ ( L ) is a scaling parameter that grows quicklyenough with L to ensure that the condition described at the end of the previous paragraphholds. It was shown in [13] that λ ( L ) ≥ L d +1+ δ for any δ > J ( L ) xy depend on L , the K xy ’s and ǫ xy ’sdo not; hence there is a well-defined infinite-volume notion of ground states for the highlydisordered model on all of Z d . This is the subject of the theorem in the next subsection.When the highly disordered condition is satisfied, the problem of finding ground statesbecomes tractable; in fact, a simple greedy algorithm provides a fast and efficient way to findthe exact ground state in a fixed volume with given boundary conditions [12, 13]. Moreover,the ground state problem can be mapped onto invasion percolation [12, 13] which facilitatesanalytic study. It was further shown in [12, 13] that in the limit of infinite volume the highlydisordered model has a single pair of ground states in low dimension, and uncountably manypairs in high dimension. The crossover dimension was found to be six in [11]. It should benoted that this result, related to the minimal spanning tree, is rigorous only in dimensiontwo (or in quasi-planar lattices [18]).The details of ground state structure in the highly disordered model have been describedat length in [12, 13] (see also [17, 11]) and are not recounted here. In this contribution wepresent a new result, concerning the ground state stability of the highly disordered model,where it turns out that this model is tractable as well. The result we prove below is twofold:first, that with probability one all couplings have finite critical droplets in any ground state,and moreover this result is dimension-independent, and therefore independent of groundstate pair multiplicity. We caution, however, that (as with all other results pertaining tothis model) these results may be confined to the highly disordered model alone and have notbeen shown to carry over to the Edwards-Anderson or other realistic spin glass models. Wewill address this question more in the following section.Before proceeding, we need to introduce some relevant properties and nomenclature per-taining to the highly disordered model. One of its distinguishing features — and the centralone for our purposes — is the separation of all bonds into two distinct classes [12, 13]. Thefirst class, which we denote as S1 bonds are those that are satisfied in any ground state re-gardless of the sign of the coupling, i.e., that of ǫ xy . These are bonds that are always satisfied,in every ground state. The remaining bonds, which we call S2 , are those in which a changeof sign of their ǫ xy value changes their status in any ground state from satisfied to unsatisfiedor vice-versa. (Obviously, any unsatisfied bond in any ground state is automatically S2, buta satisfied bond could a priori be of either type.)To make this distinction formal, we introduce the concept of rank : In a given Λ L , thecoupling with largest magnitude (regardless of sign) has rank one (this is the coupling with L.-P. ARGUIN, C.M. NEWMAN, AND D.L. STEIN highest rank and the smallest value of K xy ); the coupling with the next largest magnitudehas rank two; and so on. We then define an S1 bond as follows: Definition 2.1.
A bond h xy i is S1 in Λ L if it has greater rank than at least one coupling inany path (excluding the bond itself ) that connects its two endpoints x and y . In the above definition, we need to specify what is meant by a path if each endpointconnects to a point on the boundary. For fixed boundary conditions of the spins on ∂ Λ L , allpoints on the boundary are considered connected (often called wired boundary conditions),so disjoint paths from x to ∂ Λ L and y to ∂ Λ L are considered as connecting x and y . Itfollows from the definition that for wired boundary conditions an S1 bond in Λ L remains S1in all larger volumes. These bonds completely determine the ground state configurations,while the S2 bonds play no role. For free boundary conditions, a path connects x and y onlyif it stays entirely within Λ L , never touching the boundary; i.e., points on the boundary areno longer considered connected. For periodic or antiperiodic boundary conditions, boundarypoints are considered connected to their image points but to no others. The reasons forthese distinctions are provided in [13], but are not relevant to the present discussion and arepresented only for completeness.It was proved in [12] and [13] that the set of all S1 bonds forms a union of trees, thatevery site belongs to some S1 tree, and that every S1 tree touches the boundary of Λ L . TheS1 bonds in a given Λ L in some fixed dimension form either a single tree or else a unionof disjoint trees. Although not immediately obvious, it was proved in [12, 13] that the treestructure has a natural infinite volume limit, and moreover every tree is infinite. Moreover,a result from Alexander [1], adapted to the current context, states that if the correspondingindependent percolation model has no infinite cluster at p c , then from every site there is asingle path to infinity along S1 edges; i.e., there are no doubly-infinite paths. It is widelybelieved that in independent percolation there is no infinite cluster at p c in any dimension,but this has not yet been proven rigorously for 3 ≤ d ≤ Z d (corresponding to a single pair ofground states), while above six dimensions that are infinitely many trees (corresponding toan uncountable infinity of ground states).2.2. Ground state stability in the highly disordered model.
Unlike in realistic spinglass models, the ground state structure in the highly disordered model can be analyzed andunderstood in great detail. This allows us to solve other, related properties of the model, inparticular some of the critical droplet properties that have so far been inaccessible in mostother spin glass models. In particular we can prove the following result:
Theorem 2.2.
In the highly disordered model on the infinite lattice Z d in any d , if thereis no percolation at p c in the corresponding independent bond percolation model, then fora.e. realization of the couplings, any ground state α , and any bond b , the critical droplet One can define S1 and S2 bonds for the EA model as well, in the sense that the EA model also possessesbonds that are satisfied in every ground state (though the precise definition used above no longer applies).There are of course far fewer of these in the EA model than in the highly disordered model, and there is noevidence that these “always satisfied” bonds play any special role in ground state selection in that model.(One possibly relevant result, that unsatisfied edges don’t percolate in the ground state, was proved in [7].)
ROUND STATE STABILITY IN TWO SPIN GLASS MODELS 5 boundary ∂ D c ( b , α ) is finite. Correspondingly, in finite volumes Λ L with sufficiently large L , the size of the droplet is independent of L .Remark. As noted above, it has been proved that there is no percolation at p c in thecorresponding independent bond percolation model in all dimensions except 3 ≤ d ≤ O ( L − (1+ ǫ ) ), ǫ >
0, for large L . Proof.
Choose an arbitrary S1 bond and a volume sufficiently large so that the tree itbelongs to has the following property: The branch emanating from one of its endpoints (callit x ) touches the boundary (on which we apply fixed boundary conditions) and the branchemanating from the other endpoint ( x ) does not. This remains the case as the boundarymoves out to infinity: for any S1 bond and a sufficiently large volume, this is guaranteed tobe the case by the result of Alexander mentioned above [1].We use the fact, noted in Sect. 1, that as the coupling value of any bond varies from −∞ to + ∞ while all other couplings are held fixed, there is a single, well-defined critical pointat which a unique cluster of spins, i.e., the critical droplet, flips, changing the ground state.(This is true regardless of whether one is considering a finite volume with specified boundarycondition or the infinite system.) Now keep the magnitude of the S1 bond fixed but changeits sign. Because the S1 bond must still be satisfied, this must cause a droplet flip, which asnoted above must be the critical droplet.Now consider the state of the spins at either endpoint of the bond. Suppose that originallythe bond was ferromagnetic, and the spins at x and x were both +1. After changing thesign of the coupling, the spin at x , remains +1 (because it is connected to the boundary,as explained in the first paragraph of this proof) while the spin at x is now −
1. This mustsimultaneously flip all the spins on the branch of the tree connected to x . This is a finitedroplet and as the chosen S S K xy value sufficiently small) so that it becomes S1. (Thiswill cause a rearrangement of one or more trees, but it can be seen that any correspondingdroplet flip must also be finite.) Now change the sign of the coupling. The same argumentas before shows that the corresponding droplet flip is again finite. But given that the criticaldroplet corresponding to a given bond is unique, this was also the critical droplet of theoriginal S2 bond. ⋄ The strongly disordered model
Although the highly disordered model is useful because of its tractability, it is clearly anunrealistic model for laboratory spin glasses. This leads us to propose a related model that,while retaining some of the simplifying features of the highly disordered model, can shedlight on the ground state properties of realistic spin glass models. We will refer to this newmodel as the strongly disordered model of spin glasses.
L.-P. ARGUIN, C.M. NEWMAN, AND D.L. STEIN
The main difference between the two models is that in the strongly disordered model thecouplings have the same distribution for all volumes. This is implemented by removing thevolume dependence of the parameter λ : Definition 3.1.
The strongly disordered model is identical to the highly disordered model butwith Eq. (4) replaced by (5) J xy = ǫ xy e − λK xy with the constant λ ≫ independent of L . In the strongly disordered model, the condition that every coupling value is no more thanhalf the next larger one and no less than twice the next smaller one breaks down in sufficientlylarge volumes. This can be quantified: let g ( λ ) = Prob(1 / ≤ e − λK xy /e − λK x ′ y ′ ≤ g ( λ ) is the probability that any two arbitrarily chosen bonds have coupling values that do not satisfy the highly disordered condition. A straightforward calculation gives g ( λ ) = 2 ln 2 /λ .The strongly disordered model carries two advantages. On the one hand, its critical dropletproperties are analytically somewhat tractable given its similarity to the highly disorderedmodel. On the other hand, since its coupling distribution is i.i.d. with mean zero andfinite variance, and not varying with L , we expect global properties such as ground statemultiplicity to be the same as in other versions of the EA spin glass with more conventionalcoupling distributions. Theorem 3.2.
If there is no percolation at p c in the corresponding independent bond per-colation model, then in the strongly disordered model, the critical droplet of an arbitrary butfixed bond is finite with probability approaching one as λ → ∞ . Proof.
Consider a fixed, infinite-volume ground state on Z d ; this induces a (coupling-dependent and ground-state-specific) spin configuration on the boundary ∂ Λ L of any finitevolume Λ L ⊂ Z d .Consider an arbitrary edge { x , y } . Let R denote the (random) smallest value in theinvasion/minimal spanning forest model on Z d , defined by the i.i.d. K xy (but with K x y set to zero, for convenience of the argument) such that one of the branches from x or y iscontained within a cube of side length 2 R centered at { x , y } . By the result of Alexander [1]mentioned earlier, R is a finite random variable (depending on the K xy ’s) if there is nopercolation at p c in the corresponding independent bond percolation model.Now choose a deterministic Λ L and consider the two events: (a) A L = { R < L/ } and (b) B L = { the highly disordered condition is valid in the cube of side L centered at { x , y }} .Because R is a finite random variable, Prob( A L ) can be made arbitrarily close to one for L large. Moreover, from the definition of the highly disordered model Prob( B L ) can alsobe made close to one by choosing λ large (for the given L ). Specifically, let P denote theprobability that the critical droplet of { x , y } is finite. Then P ≥ − ǫ if Prob( R >L/
2) + CL d /λ ≤ ǫ for some fixed C >
0. But for any ǫ > L so that Prob( R > L/ ≤ ǫ/
2, and then choose λ such that CL d /λ ≤ ǫ/
2. Theresult then follows. ⋄ Theorem 3.2 sets a strong upper bound O ( λ − ) on the fraction of bonds that might not have a finite critical droplet. We do not yet know whether this gap can be closed in thesense that the strongly disordered model might share the property that all bonds have finite ROUND STATE STABILITY IN TWO SPIN GLASS MODELS 7 critical droplets. It could be that this is not the case, but that if the number of bondswith infinite critical droplets is sufficiently small, theorems analogous to those in [4] can beapplied. Work on these questions is currently in progress.
Acknowledgments.
The research of L.-P. A. was supported in part by NSF CAREERDMS-1653602. The research of CMN was supported in part by NSF Grant DMS-1507019.
References [1] Alexander, K.S.: Percolation and minimal spanning forests in infinite graphs. Ann. Prob. , 87–104(1995).[2] Arguin, L.-P., Damron, M.: Short-range spin glasses and random overlap structures. J. Stat. Phys. ,226–250 (2011).[3] Arguin, L.-P., Damron, M., Newman, C.M., Stein, D.L.: Uniqueness of ground states for short-rangespin glasses in the half-plane. Commun. Math. Phys. , 641–657 (2010).[4] Arguin, L.-P., Newman, C.M., Stein, D.L.: A relation between disorder chaos and incongruent states inspin glasses on Z d . Commun. Math. Phys. , 1019–1043 (2019).[5] Arguin, L.-P., Newman, C.M., Stein, D.L., Wehr, J.: Fluctuation bounds for interface free energies inspin glasses. J. Stat. Phys. , 221–238 (2014).[6] Banavar, J.R., Cieplak, M., Maritan, A.: Optimal paths and domain walls in the strong disorder limit.Phys. Rev. Lett. , 2320–2323 (1994).[7] Berger, N., Tessler, R. J.: No percolation in low temperature spin glass. Electron. J. Probab. , paperno. 88 (2017).[8] Edwards, S., Anderson, P.W.: Theory of spin glasses. J. Phys. F , 965–974 (1975).[9] Fisher, D.S., Huse, D.A.: Absence of many states in realistic spin glasses. J. Phys. A , L1005–L1010(1987).[10] Huse, D.A., Fisher, D.S.: Pure states in spin glasses. J. Phys. A , L997–L1003 (1987).[11] Jackson, T.S.., Read, N.: Theory of minimum spanning trees. I. Mean-field theory and strongly disor-dered spin-glass model. Phys. Rev. E , 021130 (2010).[12] Newman, C.M., Stein, D.L.: Spin-glass model with dimension-dependent ground state multiplicity.Phys. Rev. Lett. , 2286–2289 (1994).[13] Newman, C.M., Stein, D.L.: Ground state structure in a highly disordered spin glass model.J. Stat. Phys , 1113–1132 (1996).[14] Newman, C.M., Stein, D.L.: Nature of ground state incongruence in two-dimensional spin glasses. Phys.Rev. Lett. , 3966–3969 (2000).[15] Newman, C.M., Stein, D.L.: Are there incongruent ground states in 2 D Edwards-Anderson spin glasses?Commun. Math. Phys. , 205–218 (2001).[16] Newman, C.M., Stein, D.L.: Interfaces and the question of regional congruence in spin glasses.Phys. Rev. Lett. , 077201 (2001).[17] Newman, C.M., Stein, D.L.: Realistic spin glasses below eight dimensions: A highly disordered view.Phys. Rev. E , 16101 (2001).[18] Newman, C.M., Tassion, V., Wu, W.: Critical percolation and the minimal spanning tree in slabs.Commun. Pure Appl. Math. , 2084 – 2120 (2017). L.-P. ARGUIN, C.M. NEWMAN, AND D.L. STEIN
L.-P. Arguin, Department of Mathematics, City University of New York, Baruch Collegeand Graduate Center, New York, NY 10010
E-mail address : [email protected] C.M. Newman, Courant Institute of Mathematical Sciences, New York, NY 10012 USA,and NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan RoadNorth, Shanghai 200062, China
E-mail address : [email protected] D.L. Stein, Department of Physics and Courant Institute of Mathematical Sciences, NewYork University, New York, NY 10003, USA, and NYU-ECNU Institutes of Physics andMathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai, 200062,China, and Santa Fe Institute, 1399 Hyde Park Rd., Santa Fe, NM 87501 USA
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