A simple relation between frustration and transition points in diluted spin glasses
aa r X i v : . [ c ond - m a t . d i s - nn ] J a n A simple relation between frustration and transition points in diluted spin glasses
Ryoji Miyazaki, ∗ Yuta Kudo, Masayuki Ohzeki,
1, 2 and Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan Institute of Innovative Research, Tokyo Institute of Technology, Kanagawa, 226-8503, Japan (Dated: January 14, 2020)We investigate a possible relation between frustration and phase-transition points in spin glasses.The relation is represented as a condition of the number of frustrated plaquettes in the lattice atphase-transition points at zero temperature and was reported to provide very close points to thephase-transition points for several lattices. Although there has been no proof of the relation, thegood correspondence in several lattices suggests the validity of the relation and some importantrole of frustration in the phase transitions. To examine the relation further, we present a naturalextension of the relation to diluted lattices and verify its effectiveness for bond-diluted square lattices.We then confirm that the resulting points are in good agreement with the phase-transition pointsin a wide range of dilution rate. Our result supports the suggestion from the previous work fornon-diluted lattices on the importance of frustration to the phase transition of spin glasses.
I. INTRODUCTION
Spin glasses have been one of the most attractive sub-jects in statistical mechanics and have been extensivelyinvestigated for decades [1, 2]. Study of spin glasses, inparticular, in infinite dimensions has developed severalelaborate concepts and techniques. For instance, replicasymmetry breaking has had a great influence on subse-quent studies, e.g., structural glasses [3, 4] and informa-tion theory [5, 6], for revealing their complicated energylandscapes. This success motivates us to tackle a nexttask that is to establish theories of more realistic mod-els, namely, finite-dimensional spin glasses. It, however,is a very difficult task. This is partially because tech-niques exploited for the infinite-dimensional models arenot so useful in finite dimensions, while it is still difficultto obtain conclusive proofs with numerical simulations.The gauge transformation [5] has been utilized as atool for analytically investigating finite-dimensional spinglasses [7–15]. A consequence of this approach is theconjecture on the verticality of a phase boundary [16].The conjecture states that the phase boundary betweenthe ferromagnetic and another phases at low tempera-tures does not depend on temperature but is determinedonly by geometrical properties. In other words, the phaseboundary is a vertical line in the p – T plane, where p and T denote the ratio of antiferromagnetic bonds of spinsand temperature, respectively. This conjecture was de-nied by subsequent detailed studies [10, 17–23]. Theestablished phase boundary, however, is almost verti-cal. This fact implies that geometrical properties takea primary role for the phase transition, even though theconjectured relation does not exactly hold. Note thatthe importance of revealing the property of this phasetransition is not limited in the study of spin glasses. Itcan influence on the study of quantum computation. In-deed, this phase boundary can be interpreted to give the ∗ Present address: System Platform Research Laboratories, NECCorporation, Tsukuba, Ibaraki 305-8501, Japan error-correction threshold for topological quantum error-correction codes [24, 25].Another possible relation between a geometrical prop-erty and this phase transition was reported by one of theauthors without utilizing the gauge transformation [26].The relation is represented as a condition on a quantityconcerning frustration [27–29] in the lattice. The condi-tion gives a very close point to the phase-transition pointat zero temperature. The good correspondence is foundin several two-dimensional lattices and hierarchical lat-tices [30]. Moreover, the condition for the Sherrington-Kirkpatrick model [31] exactly gives the replica symme-try solution for its transition point at zero temperature.Unfortunately, we have no proof that this agreement isnot just an accidental one. The above instances, however,allow us to expect some important role of frustration inthe phase transition.Recently, some of the authors extended this argumentto bond-diluted lattices [32]. Their method mainly fol-lows the above one for the non-diluted lattices. Resultingpoints from their method qualitatively agrees with thecorrect phase transition points. However, the methodwas not exactly executed. They instead used an addi-tional ansatz to complete the calculations because of thedifficulty due to the inhomogeneity in the diluted lattices.A natural question is whether the good correspondence isalso found by the canonical extension without the ansatzor is just caused by the ansatz.In this paper, we examine the natural extension of themethod for non-diluted lattices [26] to bond-diluted lat-tices without any extra ansatz. The next section givesthe prescription of the method. We describe two naturalways of extension of the method for non-diluted latticesto the diluted case. The effectiveness of the method isverified for the diluted square lattice in Sec. III. We applythe method with perturbative analysis expanded fromthe non-diluted case and numerical calculations. Theobtained points are compared with the correct phase-transition points. We summarize and discuss our resultsin Sec. IV.
II. PRESCRIPTION
We investigate ± J Ising spin glass [5], defined by H = − X h i,j i J ij σ i σ j , (1)where h i, j i denotes a pair of nearest-neighbor sites ona lattice. The coupling constants J ij for spin i and j are taken from an independent, identical distribution P ( J ij ) = pδ ( J ij + J )+(1 − p ) δ ( J ij − J ) with J >
0, where δ ( · ) is the Dirac delta function. Note that we define p asthe probability that a bond of spins is an antiferromag-netic one. Ising spin σ i takes 1 or −
1. This model hasbeen extensively used as an elementary model of spinglasses in finite dimensions [5]. We mainly consider themodel on two-dimensional lattices.We focus on frustration for plaquettes [27–29]. A pla-quette is an elementary loop of edges on a lattice, whichcannot be divided into multiple sub-loops. For exam-ple, a plaquette on a square lattice is a square composedof four edges. When a plaquette has an odd number ofantiferromagnetic couplings, there is no spin configura-tion that all bonds in the plaquette take the lower energystate. Consequently, there is frustration at the plaque-tte. Such a plaquette is called frustrated plaquette. Theaverage number of frustrated plaquettes on a lattice overthe bond distribution plays a central role in the argumentbelow. That average is calculated as N fra ( p ) = *X c − Y h i,j i∈ c J ij J + af p , (2)where c are indices for plaquettes, and h·i af p denotes theaverage over the antiferromagnetic-bond distribution for p . The coupling constants are independent of each other,and hence we can rewrite the function as N fra ( p ) = X n N ( n )pla f n ( p ) , (3) f n ( p ) = 12 [1 − (1 − p ) n ] , (4)where N ( n )pla is the number of plaquettes composed of n edges on the lattice. The function f n ( p ) gives the proba-bility that a plaquette composed of n edges is frustrated.For the square lattice, the sum reduces to the single termfor n = 4. This expression also concerns lattices withmultiple types of plaquettes, e.g., the Kagom´e lattice.One of the authors focused on the function [26] definedby v ( p ) = dN fra ( p ) dp (cid:18) dN af ( p ) dp (cid:19) − . (5)Here N af ( p ) is the average number of antiferromagneticbonds over the bond distribution for p , calculated as N af ( p ) = pN edg , where N edg is the number of edges inthe lattice. He reported [26] that the condition v ( p ) = 1gives a value of p that is close to the phase transi-tion point for the model at zero temperature. For in-stance, the value p ≃ . p =0 . . ± J Isingspin glass on bond-diluted lattices. Each edge in thelattices is absent with probability q . The probability thata bond is an antiferromagnetic one thus turns to (1 − q ) p .We introduce two ways of extension to this case. The firstone generalizes the function in Eq. (3) as the averagenumber of frustrated plaquettes over diluted lattices aswell as the antiferromagnetic-bond distributions, namely N fra ( q, p ) = X n D N ( n )pla E di q f n ( p ) , (6)where h·i di q denotes the average over the diluted-bond dis-tribution for q . We have utilized the fact that f n ( p ) isnot affected by the bond dilution. Accordingly, we definea generalized function of v ( p ) in Eq. (5) by v ( q, p ) = ∂N fra ( q, p ) ∂p (cid:18) ∂N af ( q, p ) ∂p (cid:19) − . (7)Here N af ( q, p ) is the average number of antiferromagneticbonds over the bond distribution for q and p , calculatedas N af ( q, p ) = (1 − q ) pN edg , where N edg is the numberof edges in the lattice without dilution. The condition v ( p ) = 1 is generalized as v ( q, p ) = 1 for the bond-dilutedlattices. For the other extension we calculate the aver-age number of frustrate plaquettes and antiferromagneticbonds over only the antiferromagnetic-bond distributionon a given bond-diluted lattice. Equation (5) then givesthe function v ( p ) for the lattice. We consider a bond-diluted lattice for this extension, whereas we took theaverage over bond-diluted lattices for q for the first ex-tension. The second extension could give different solu-tions of v ( p ) = 1 for different diluted lattices. However,if the variance of obtained solutions is small for the lat-tices for q , we would find a typical value of p for thecondition v ( p ) = 1 for the diluted lattices. The possi-ble typical one is regarded as the solution obtained fromour method for q . In addition, we can consider a minorchange of this extension in estimation of the typical so-lution; we estimate the average of the function v ( p ) overbond-diluted lattices and obtain the solution of v ( p ) = 1for the averaged function instead of the average of so-lutions themselves over different lattices. Hereafter, weexamine whether the two procedures of extension give p close to the correct phase transition point of the model (a) (b) (c)(d) (e) (f)(g) (h) FIG. 1. Examples of square lattices without/with dilutionstreated in calculations of N fra ( q, p ) for small q . Plaquettes,which are not composed of four edges, generated by removingedges are highlighted with thick red lines. (a) The squarelattice without dilution. Examples of square lattices lacking(b) an edge, (c), (d) two edges, and (e)–(h) three edges. Theexamples are distinguished by the number of lacked edges andthe number of edges for generated plaquettes. on diluted lattices for q observed with varying p at zerotemperature. III. SQUARE LATTICEA. Perturbative calculations
We first restrict our interest to the systems in which thenumber of lacked edges is small and obtain its expansionin terms of q . Motivated by the fact that p for the condi-tion v ( p ) = 1 for the square lattice is extremely close tothe phase-transition point [26], we analyze the model onthe square lattice. Here, we only attempt the first way ofextension, where we calculate v ( q, p ) in Eq. (7), becauseit is intractable to analytically obtain the solutions withthe second extension. The second one will be examinedwith numerical calculations in Sec. III B. It should benoted that N edg and N pla used below denote the num-bers of edges and plaquettes, respectively, for the latticewithout dilution.We obtain N fra ( q, p ) in Eq. (6) for terms up to q n by considering lattices in which the number of lackededges is smaller than n + 1. This is because the prob-ability that a lattice lacks n edges is q n (1 − q ) N edg − n , and because h N ( n )pla i di q in N fra ( q, p ) is the average num-ber of n -edge plaquettes over those lattices. Figure 1shows examples of square lattices removed 1, 2, or 3edges. Note that there can exist edges which do notbelong to any plaquette and thus do not contribute tofrustration. For instance, the edge in the 8-edge squareon the lattice shown in Fig. 1 (h) does not belong to anyloop of edges. As an example of computing N fra ( q, p )let us consider a lattice lacking an edge as shown inFig. 1 (b). The probability that such a lattice is real-ized is q (1 − q ) N edg − . The number of positions at whichan edge is absent is N edg . By removing an edge fromthe primary square lattice, the number of four-edge pla-quettes reduces to N pla −
2, while a six-edge plaquette isgenerated. The contribution of such lattices to N fra ( q, p )is thus q (1 − q ) N edg − N edg [( N pla − f ( p ) + f ( p )]. Tak-ing into account the lattices removed 1, 2, or 3 edges, weobtain N fra ( q, p )=(1 − q ) N edg N pla f ( p )+ q (1 − q ) N edg − N edg [( N pla − f ( p ) + f ( p )]+ q (1 − q ) N edg − N edg × (cid:26) N edg −
72 [( N pla − f ( p ) + 2 f ( p )]+ 3[( N pla − f ( p ) + f ( p )] (cid:27) + q (1 − q ) N edg − N edg × (cid:26) [2( N edg −
12) + 14( N edg − N edg − N edg − ×
13! [( N pla − f ( p ) + 3 f ( p )]+ 3( N edg − N pla − f ( p ) + f ( p ) + f ( p )]+ 9[( N pla − f ( p ) + f ( p )]+ 2[( N pla − f ( p ) + f ( p )] (cid:27) + O ( q )= N pla f ( p ) + N edg [ − f ( p ) + f ( p )] q + 3 N edg [ f ( p ) − f ( p ) + f ( p )] q + N edg [ − f ( p ) + 15 f ( p ) − f ( p ) + 9 f ( p )] q + O ( q ) . (8)Substituting this into Eq. (7), we then have v ( q, p ) =2 r (cid:2) − (cid:0) − r (cid:1) q + 3 (cid:0) − r + 4 r (cid:1) q + (cid:0) − r − r + 45 r (cid:1) q (cid:3) + O ( q )(9)where r = 1 − p . We have used a relation N pla = N edg / p v ( q ) of v ( q, p ) = 1 Numerical simulations3rd order2nd order1st orderWithout perturbation
FIG. 2. Solutions p v ( q ) obtained with the perturbative calcu-lations of the first extension and with the numerical calcula-tions of the second one. The n th-order perturbative solutionis a function of q taking into account lower order terms than q n +1 given in Eq. (10) for n = 0 (without perturbation), 1,2, and 3. The numerical simulations are done for the latticesgenerated by removing edges from the square lattice of L × L units, where L for the result shown here is 128. The numericalsolutions are estimated by averaging 10 instances. for q is expanded in terms of q as p v ( q ) = 12 − − / − h − / − − i q − (cid:20) − / − (cid:21) q − (cid:20) × − / − × − / − (cid:21) q + O ( q ) . (10)The solutions containing the terms up to q n for n = 0, 1,2, and 3 are drawn in Fig. 2, where a result of the secondextension given in Sec. III B is also shown for comparison. B. Numerical calculations
We run numerical simulations of the second extension,where the solution of v ( p ) = 1 for each bond-diluted lat-tice is estimated. We first generate a bond-diluted lat-tice under the periodic boundary condition in which anedge is lacked with probability q and then count plaque-ttes and edges. We do not consider whether bonds inthe lattice are ferromagnetic ones or antiferromagneticones, since this matter concerns only the p dependenceof N fra ( p ) that is already determined by f n ( p ) given inEq. (4). We then obtain the function v ( p ) and the solu-tion of v ( p ) = 1 for the lattice. Sampling solutions for anumber of diluted lattices for q by this way, we estimatethe average and variance of the solutions. The average isalso denoted by p v ( q ) for simplicity.Figure 2 shows the plot of the estimated solutionsas a function of q . The perturbative solutions basedon the first extension are also displayed for comparison.The square lattice before the dilution has L × L units(squares), where L for the result shown in Fig. 2 is 128.Solutions of v ( p ) = 1 are sampled from 10 lattices gener-ated from the distribution for q . The variance of the solu- L = 163264128 = (a) (b) (L = 32) FIG. 3. Solutions p v ( q ) of v ( p ) = 1 estimated from 10 in-stances of numerical simulations based on the second exten-sion. (a) The solutions for L = 16, 32, 64, and 128. (b)The solutions for the averaged function over n ( v )lat lattices for n ( v )lat = 1, 10, and 100 for L = 32. tions over different lattices is small. The averaged valueis thus regarded as the probable solution for q obtainedwith the second extension of our method. In addition, theobtained numerical solution of the second extension is ingood agreement with the perturbative solutions of thefirst extension for small q . In particular, the numericalsolution and the third-order perturbative solution showgood correspondence for q ≤ .
4. This result demon-strates that both the ways of extension lead to almostidentical solutions. The perturbative solutions, however,do not exhibit the non-monotonic behavior found in thenumerical ones at q > .
4, where the perturbative analy-sis expanded from q = 0 would be unreliable. We shouldremark that the curve of the numerical solutions con-verges to 0 with q approaching 0.5, which agrees withthe exact phase-transition point at q = 0 . L dependence of the solutions is shown inFig. 3 (a). We find no definite difference in the average ofthe solutions between investigated L except for q ≥ . L .We therefore expect that the finite-size effect of our so-lutions is small. On the other hand, the variance of thesolutions clearly decreases as L increases. As mentionedabove, the average of the solutions over bond-diluted lat-tices agrees well with the solutions of the first extension,where we obtained the solutions with the average num-ber of frustrated plaquettes. This finding and the de-crease of the variance with increasing L suggest that thesmall variance of the numerical solution originates fromthe typicality of the number of frustrated plaquettes thatcould be involved in the self-averaging property [5] of thesystem.Using a rather small lattice ( L = 32), we also executethe other procedure of estimation of the typical solutionmentioned in the end of Sec. II, where we compute the so-lutions of the averaged v ( p ) over n ( v )lat lattices. To observethe variance of the resulting solutions, they are sampled10 times. This estimation for n ( v )lat = 1, hence, corre-sponds to the above method the result of which is shownin Fig. 3 (a) ( L = 32). Figure 3 (b) displays the average MWPM
FIG. 4. Solutions p v ( q ) of v ( p ) = 1 for L = 128 (the same oneas shown in Fig. 2) and the phase-transition points estimatedwith MWPM [33]. of the obtained solutions with error bars over 10 sam-ples for n ( v )lat = 1 , , n ( v )lat does not makeany definite differences in the average of the solutions butjust suppresses the fluctuation of the solutions. There-fore, we use the result for n ( v )lat = 1 as the solution of ourmethod for the bond-diluted lattices. C. Comparison with the minimum-weightperfect-matching algorithm
We compare the obtained solution of v ( p ) = 1 withthe correct phase-transition point. The latter has beenalready estimated in the context of the quantum er-ror correction for the surface code with loss by usingthe minimum-weight perfect-matching (MWPM) algo-rithm [33]. We, however, performed the similar calcula-tions in a number of points of q , because we need detailedillustration of the q dependence of the critical point. Wefollowed the treatment of the diluted lattices as well asthe system size, L = 16, 24, 32, and the number of in-stances of diluted lattices, 5 × , in Ref. [33] and thefinite-size scaling ansatz in Ref. [20].The average p v ( q ) of solutions of v ( p ) = 1 is in goodagreement with the obtained critical values p c in thewhole range of q (0 ≤ q ≤ .
5) except for q ≃ .
34, asshown in Fig. 4. The correspondence at q = 0 previouslyfound [26] is reproduced. Interestingly, the slope of p v ( q )at q = 0 is also very similar to that of p c . Moreover, theyremain almost identical curves for q ≤ .
14. This findingdemonstrates that our method effectively captures the q dependence of the true phase-transition point for small q at least. For larger q , p v ( q ) departures from p c andtakes a little smaller value. The difference between themtakes its maximum around q ≃ .
34, but it is still small.For q ≥ . p c decreases more rapidly than p v ( q ), and they take similar values again at q ≥ .
4. This agree-ment is owed to that the non-monotonic behavior for q ≥ . p v ( q ), mentioned in Sec. III B, appears alsoin p c . Both the curves finally converge to 0 with q ap-proaching 0.5. This good correspondence in the range of q implies a scenario that our simple method could givesome approximate location of the phase-transition pointeven for the diluted lattices, although we have not beenable to directly derive their relationship.The non-monotonic behavior in the curve of p c wasalready reported in the previous work [33]. This was at-tributed to a finite-size effect [33] because of the fact thatthe largest plaquette occupies approximately half of theprimary square lattice in the range for the non-monotonicbehavior. More precisely, the threshold values of q , say q scale ( L ), at which the largest plaquette occupies half ofthe lattice were estimated as a function of L . The non-monotonic behavior indeed appeared for larger q than q scale ( L ) for L used in estimating the critical points [33].Hence, the non-monotonic behavior in our curve for p c would be regarded as a signal of the finite-size effect. Thisargument derives that for the lattice of L = 128, which isused for our numerical analysis of v ( p ) and is much largerthan that for the estimation with MWPM ( L = 16, 24,and 32), the finite-size effect caused by the occupationof the large plaquettes is not supposed to be observedfor q ≤ .
46 at least [33]. p v ( q ), however, exhibits thenon-monotonic behavior in q ≤ .
46. This fact impliesthat the non-monotonic behavior of p v ( q ) is not due tothe occupation of the large plaquettes. If our methodbased on v ( p ) is effective to approximately predict thephase-transition point even for large q , our result sup-ports that the non-monotonic behavior in the estimated p c was accidentally identified to the finite-size effect butis a nontrivial feature of this phase transition. IV. SUMMARY AND DISCUSSION
We presented a possible relation between frustrationand the phase transition of Ising spin glasses on bond-diluted lattices. The relation is represented as the corre-spondence of points obtained by a simple method con-cerning frustration and the phase-transition points atzero temperature observed with varying the ratio of anti-ferromagnetic bonds. The method is based on extensionof a previous one for lattices without dilution [26]. Wecalculate v ( q, p ) defined by Eq. (7) using averaged quan-tities over diluted lattices or v ( p ) defined by Eq. (5) foreach diluted lattice. Both the two functions concern thederivative of the number of frustrated plaquettes withrespect to the number of antiferromagnetic bonds in thelattice. This extension is more natural than another onepreviously proposed with an additional ansatz [32]. Mo-tivated by the work for non-diluted lattices [26], wherethe condition that the obtained function is equal to unityleads to an approximate location of the phase-transitionpoint, we applied the extended method to the dilutedsquare lattice. Consequently, we found that both the twoways of extension typically give almost identical resultand that the obtained curve as a function of q is close tothe correct phase boundary in the range 0 ≤ q ≤ .
5. Aremarkable feature of the curve is non-monotonic behav-ior in q > .
4. Although the similar feature found in thecorrect phase boundary in the same range was attributedto a finite-size effect [33], our case is not simply regardedas the finite-size effect, since we investigated larger lat-tices which are not supposed to exhibit the finite-sizeeffect for given q .Our scheme provides close points to the phase-transition points even for diluted lattices. We, however,never propose it as a method to obtain the phase transi-tion points, since the reason for the good correspondencehas not been revealed. We need further investigation to clarify whether this agreement is reasonable or not. Wewill examine other diluted lattices as the next task. Ifthe good agreement is not an accident, our result sug-gests that geometrical properties can almost fully deter-mine the phase-transition points. Moreover, the non-monotonic behavior in the phase boundary might be agenuine feature of the phase transition. ACKNOWLEDGMENTS
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