Changing the universality class of the three-dimensional Edwards-Anderson spin-glass model by selective bond dilution
aa r X i v : . [ c ond - m a t . d i s - nn ] S e p Changing the universality class of the three-dimensional Edwards-Andersonspin-glass model by selective bond dilution
F. Rom´a Departamento de F´ısica, Universidad Nacional de San Luis, Instituto de F´ısica Aplicada (INFAP),Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas (CONICET), Chacabuco 917, D5700BWS San Luis, Argentina (Dated: September 29, 2020)The three-dimensional Edwards-Anderson spin-glass model present strong spatial heterogeneities well char-acterized by the called backbone , a magnetic structure that arises as a consequence of the properties of theground state and the low-excitation levels of such a frustrated Ising system. Using extensive Monte Carlo sim-ulations and finite size scaling, we study how these heterogeneities affect the phase transition of the model.Although, we do not detect any significant difference between the critical behavior displayed by the wholesystem and that observed inside and outside the backbone, surprisingly, a selective bond dilution of the comple-ment of this magnetic structure induces a change of the universality class, whereas no change is noted when thebackbone is fully diluted. This finding suggests that the region surrounding the backbone plays a more relevantrole in determining the physical properties of the Edwards-Anderson spin-glass model than previously thought.Furthermore, we show that when a selective bond dilution changes the universality class of the phase transition,the ground state of the model does not undergo any change. The opposite case is also valid, i. e., a dilution thatdoes not change the critical behavior significantly affects the fundamental level.
PACS numbers: 75.10.Nr, 75.40.Mg
Both, the singular phenomenology that glassy materials dis-play and the enormous technical difficulties that must be over-come in order to study them, have promoted them as one ofthe central topics in modern physics. The most serious at-tempts to elucidate the physical origin of this behavior haveled to the development of new highly sophisticated experi-mental, theoretical and simulation techniques. Although overthe years this effort has paid off, it is still unclear how, undercertain conditions relatively simple models are able to exhibitthis phenomenon.Typical glassy models have been studied countless timesby performing increasingly powerful simulations that have al-lowed to characterize with greater precision some few aspectsof the problem, but that have not been enough to get to thebottom of this matter. At present, however, studying the het-erogeneities that characterize these complex systems seems tobe a promising way to understand them a little more in depth[1].These ideas apply to the case of the spin glasses, mag-netic systems that have both quenched disorder and frustra-tion [2, 3]. The paradigmatic Edwards-Anderson (EA) spin-glass model [4] present both spatial and nonequilibrium dy-namical heterogeneities which are linked to each other [5–7]. These spatial heterogeneities are well characterized by astructure called backbone [8] which, in general, originates asa consequence of the properties of the ground state and thelow-excitation levels of this system [9, 10]. Extensive simu-lations have shown that the nonequilibrium dynamic behaviordisplayed within the backbone differs qualitatively from whatis observed outside of this structure. Such numerical resultssuggest that the separation of the system in two components(the backbone and its complement) is not trivial, so a suitable backbone picture could be essential to describe the physics ofspin glasses. Unfortunately, there are great difficulties to perform suchcalculations. On one hand, since the computation of groundstate configurations of the three-dimensional (3D) EA modelis a NP-hard problem, considerable numerical effort is re-quired to calculate the backbone of a particular realization ofthe quenched disorder (sample). As a consequence, only alimited number of samples of small sizes can be calculatedefficiently. On the other hand, although in average approxi-mately 57% of bonds belong to the backbone and the rest toits complement, for a considerable number of samples thesestructures can have very different sizes, i. e., their size dis-tributions are very broad [9]. In addition, both the backboneand its complement percolates (simultaneously), and are com-posed by a giant component and several finite clusters. Thesefactors make it extremely difficult to determine which pro-cesses dominate physics within these regions.In this work we focus on studying how these spatial het-erogeneities affect the critical behavior of the 3D EA model.Using Monte Carlo simulations we calculate at equilibriumand for different lattice sizes, the correlation length and thespin-glass susceptibility for the whole system but also for thebackbone and its complement. A finite size scaling study sug-gests that the critical behavior is unaffected by such hetero-geneities. However, surprisingly, the universality class of thephase transition can be changed by a selective bond dilution:While no changes are observed when the backbone is com-pletely diluted, in the opposite case in which the complementof this structure is removed we obtain a different set of criticalexponents. This finding suggests that the region surround-ing the backbone plays a more relevant role than previouslythought and therefore we will call it the glass region. Further-more, we show that when a selective bond dilution changesthe universality class of the phase transition, the ground stateof the system does not undergo any change. The opposite caseis also valid, i. e., a dilution that does not change the criticalbehavior significantly affects the fundamental level.In the 3D EA spin-glass model [4], a set of N Ising spins σ i = ± L ( N = L ). Its Hamiltonian is H = − ∑ ( i , j ) J i j σ i σ j , (1)where ( i , j ) indicates a sum over the six nearest neighbors.The coupling constants or bonds, J i j ’s, are independent ran-dom variables drawn from a distribution with mean value zeroand variance one. Here, we use a bimodal distribution, i.e., J i j = ± rigid bonds which do not change their state (satisfied or frustrated)in all its ground-state configurations [8]. Those bonds formthe backbone while the complementary set, the flexible bonds,composes the glass region. Using the algorithm proposed inRef. [9], we have calculated both structures for 10 samplesfor each size L = L =
6, 10 for L =
8, and 320 for L =
10. In addition, to calculate different observables at equi-librium we use a parallel tempering algorithm [13]. Details ofthe simulations are given in the Supplemental Material (SM).At low temperatures spatial heterogeneities affect almostany observable. For example, Fig. 1 (a) shows the averageenergies per bond u , u B , and u G , as function of tempera-ture T for, respectively, the whole system, the backbone, andthe glass, for samples of L =
10. Note that the curve of u G display a minimum at approximately the critical temperature T c = . ( ) [14] and u G > u > u B for finite T , whichevidence that the glass region concentrates the greatest frus-tration of the system. We indicate with arrows two particularpoints, a and b , to show that it is possible to have the samevalue of u G at temperatures, respectively, below and above T c ,one of the reasons it was assumed (wrongly) that this regionis in a paramagnetic phase for T > Ω B ( Ω G ) to the set of spins connected toalmost a rigid (flexible) bond, where the superscript B ( G ) in-dicates that such region is dominated by the backbone (glass).Since some spins are connected to both rigid and flexible f B f G f F (b)1/L T u u B u G T C (a) a b FIG. 1: (Color online) (a) Average energies per bond u , u B , and u G ,as function of T . Arrows indicate the critical temperature and twoparticular points on the u G curve, a and b , located below and above T c . (b) Average fractions of spins f B , f G , and f F , as function of 1 / L . bonds these two sets intersect, Ω F = Ω B T Ω G , where now thesuperscript F denotes the frontier between these structures.In Fig. 1 (b) we can see the average fractions of spins thatbelong to the sets Ω B ( f B ), Ω G ( f G ), and Ω F ( f F ), as functionof 1 / L . In the thermodynamic limit these quantities tend ap-proximately to f B ∼ . f G ∼ .
73, and f F ∼ .
51, whichshows that the mean numbers of spins in the backbone andglass regions are very similar, and the frontier has half of thespins of the system, i. e., both structures interpenetrates eachother sharing a region whose size is proportional to N .Having separated the spins in different sets as describedabove, it is possible to calculate other observables within eachof these regions. In particular, to study the critical behavior ofthe 3D EA model, we calculate the correlation length ξ x de-fined as [15] (in cases where a given quantity is not calculatedover the whole system, we use a superscript x to indicate theregion over which it is evaluated) ξ x =
12 sin ( | k min | / ) " χ x ( ) χ x ( k min ) − / , (2)where k min = ( π / L , , ) is the smaller nonzero wave vectorand χ x ( k ) is the wave vector dependent spin-glass suscepti-bility, χ x ( k ) = N Ω x ∑ i , j ∈ Ω x [ h q i q j i T ] av e i k · ( r i − r j ) . (3)Here, q i = σ α i σ β i is the single spin overlap between two repli-cas of the system α and β , N Ω x is the number of spins of re-gion Ω x , and r i is the vector of position of the i -th spin. h· · · i T and [ · · · ] av represent, respectively, the thermal and disorder av-erages. The correlation length divided by L is a dimensionlessquantity which scales as [16] ξ x L = ˜ S x [ L / ν x ( T − T xc ) / T xc ] , (4)where ˜ S x is a universal scaling function, T xc is the critical tem-perature, and ν x is a critical exponent. If the system expe-riences a phase transition, according to Eq. (4) the curves of ξ x / L for different lattice sizes should intersect at T xc .Figure 2(a) shows the correlation length calculated for thewhole system, ξ , for different lattice sizes as indicated. Thecurves intersect at approximately the true critical tempera-ture and, using precise values of T c = . ( ) and ν = . ( ) taken from recent literature [14], we obtain a gooddata collapse (see inset). This example shows that, despite thelimitations of our calculations (performed for few samples ofvery small sizes), it is still possible to study the critical behav-ior of the model with a certain degree of accuracy.Now, we focus on the backbone and glass regions. Figure2(b) shows that the curves of ξ B are very similar to those cal-culated for the whole system, and a good data collapse canbe obtained using the same critical parameters as before (seeinset). For the glass region, however, we do not obtain a re-sult as robust as the previous one, see Fig. 2(c). Each pairof curves of ξ G calculated for two consecutive sizes inter-sect at a temperature slightly higher than T c , and their crossingpoint slowly moves towards lower temperatures as the systemsize increases. The scaling plot shows in the inset, performedagain using T c = . ( ) and ν = . ( ) , does not al-low to achieve a good data collapse. This suggests that theresults obtained for the glass region are probably affected byvery strong finite-size effects.In order to determine with certainty the universality classof a phase transition, it is necessary to analyze the scaling of asecond observable that depends on an independent critical ex-ponent. Therefore, we consider the spin-glass susceptibilityfor a given region, χ x ≡ χ x [ k = ( , , )] , and the correspond-ing critical exponent η x . Although for the whole system andthe backbone we have obtained good data collapses of the cor-relation length using a conventional scaling Eq. (4), for thesusceptibility we choose an extended scaling scheme [17] χ x = ( LT ) − η x ˜ C x [( LT ) / ν x ( − ( T xc / T x ) )] , (5)which is more appropriate for dealing with samples of smallsizes. Using the critical parameters T c = . ( ) , ν = . ( ) , and η = − . ( ) [14], we obtain excellentdata collapses for all regions, and in particular for the glassone (from now on and for reasons of space, the differentcurves of susceptibility and the corresponding scaling plotsare shown in the SM). We conclude then that the critical be-havior is the same in each of the regions in which we havedivided the system.The previous results seem to suggest that the spatial het-erogeneities we are considering are not closely related to thecritical behavior of this system. This conclusion, however, isnot entirely accurate. As we shall see below, a selective bonddilution allows us to unveil surprising features of the back-bone and glass regions, otherwise impossible to detect in asimulation that does not take into account such structures.First, for comparison purposes, we calculate the correla-tion length and the spin-glass susceptibility for the 3D ran- L = 4 L = 6 L = 8 L = 10 / L T (a) -2 0 20.21.0 / L L (T-T c )/T c L = 4 L = 6 L = 8 L = 10 B / L T (b) -2 0 20.21.0 L (T-T c )/T c B / L L = 4 L = 6 L = 8 L = 10 G / L T (c) -2 0 20.21.0 L (T-T c )/T c G / L FIG. 2: (Color online) Correlation length function divided by L asfunction of T for (a) the whole system, and for (b) the backboneand (c) glass regions. Insets show the corresponding data collapsesperformed according to Eq. (4) using the critical parameters T c = . ( ) and ν = . ( ) [14]. dom bond-diluted EA spin-glass model, ξ ∗ and χ ∗ , respec-tively. We use the same lattice sizes and number of samplesas before, with 50% of dilution. In Fig. 3(a) we can observethat the curves of ξ ∗ / L cross at T ∗ c = . ( ) and, using thecritical exponents ν = . ( ) and η = − . ( ) , gooddata collapses are obtained for this quantity (see inset) and forthe spin-glass susceptibility (see SM). This numerical exper-iment corroborates something that is well known, that a ran-dom bond dilution does not change the universality class ofthe 3D EA spin-glass model [18].Surprisingly, a selective bond dilution is capable of induc-ing a change of universality. Figure 3(b) shows the correla-tion length curves calculated for the backbone region, ξ B ∗ ,obtained after diluting the entire glass region. On one hand,we observe a cross at T B ∗ c = . ( ) , a critical temperaturehigher than that of the undiluted system. This is an expectedresult, since by removing the region with the greatest frustra-tion of the system, the phase should be more stable in termsof energy and the critical temperature should rise. But, onthe other hand, even more important is that we obtain twocritical exponents, ν B ∗ = . ( ) and η B ∗ = . ( ) , that areobviously very different from those of the 3D EA spin-glassmodel. These critical parameters were calculated by perform-ing a careful statistical analysis of the data which is describedin the SM. Using them, good data collapses are obtained forthe correlation length [see inset in Fig. 3(b)] and for the cor-responding spin-glass susceptibility, χ B ∗ (see SM).This phase transition probably belongs to the universal-ity class of the 3D ferromagnetic Ising model. There are atleast two reasons to suppose that this is so. Firstly, the back-bone has very little frustration, i. e., approximately only 10%of its bonds are frustrated in the ground state [9]. In com-parison, the ferromagnetic phase transition in the 3D Isingmodel persists even if a fraction of bonds up to 22% are ran-domly replaced by antiferromagnetic bonds, most of whichcontribute directly to the frustration of the fundamental level[19]. Therefore, these energetic considerations and also pre-vious studies of the nonequilibrium dynamic of the 3D EAmodel [20], suggest that the backbone would be able of sus-taining a ferromagnetic-like order.Secondly, simulating the 3D ferromagnetic Ising model forsmall lattice sizes up to L =
10, we obtain a set of effective crit-ical exponents, ν I eff ≈ . η I eff ≈ .
5, that differ from thevalues of this universality class, ν I ≈ .
684 and η I ≈ . ν B ∗ = . ( ) and η B ∗ = . ( ) . In addition, dilut-ing 50% of the bonds of the 3D ferromagnetic Ising systemat random, we determine that ν I ∗ eff ≈ . η I ∗ eff ≈ .
52. Weconclude then that, having removed the glass region of the 3DEA model, the backbone undergoes a phase transition whoseuniversality class probably belongs to the 3D Ising model but,since it is only possible to analyze lattices of small sizes, wecannot confirm that this is the case. Further studies should beperformed to clarify this issue.In the opposite case, when we dilute the backbone butkeep the glass region, we observe in Fig. 3(c) that the cor-relation length curves, ξ G ∗ , intersect at T G ∗ c = . ( ) . Un-like the previous case, this critical temperature is lower than L = 4 L = 6 L = 8 L = 10 * / L T (a) -1 0 10.21.0 * / L L (T-T *c )/T *c L = 4 L = 6 L = 8 L = 10 B * / L T (b) -4 0 40.01.0 B * / L L B* (T-T cB* )/T cB* L = 4 L = 6 L = 8 L = 10 G * / L T (c) -2.5 0.0 2.50.01.0 G * / L L (T-T cG* )/T cG* FIG. 3: (Color online) Correlation length function divided by L asfunction of T for a random bond dilution (a) of 50%, and for a selec-tive bond dilution of (b) the glass and (c) backbone regions. Insetsshow the corresponding data collapses (see text). T c since the most energetically stable region (backbone) hasbeen removed. Using ν = . ( ) , the inset shows a muchbetter data collapse than that obtained in the undiluted case (b) u x -1 00510152025 u x0j P x (a) W B G B* G*
FIG. 4: (Color online) Probability distribution functions of ground-state energies per bond for samples of L =
10, and for the differentregions as indicated. (b) Disorder average of these energies as func-tion of 1 / L . [Fig. 2(c)]. To confirm that this phase transition belong to theuniversality class of the 3D EA spin-glass model, we makea scaling plot of the susceptibility, χ G ∗ , using the exponent η = − . ( ) , which leads to a very good data collapse(see SM). In this way, it is justified that we have named thispart of the system the glass region.Finally, we observe that a selective bond dilution also af-fects other properties of the system, in particular its funda-mental level. We calculate the probability distribution func-tion, P x , of ground-state energies per bond, u x j , where as be-fore the superscript x indicates the region over which this en-ergy is evaluated and the conditions under which the calcula-tions are performed (the undiluted and diluted cases). Figure4(a) presents the distributions that were obtained for samplesof L =
10, while the panel (b) shows the disorder average ofeach energy as function of 1 / L . Here, we can see importantdifferences between the main regions of the system. In fact, aselective bond dilution of the glass region does not change theground state of the backbone (the distributions of u B j and u B ∗ j are equals), while in the opposite case an appreciable effect isobserved: the probability distributions of u G j and u G ∗ j are verydifferent, and the latter overlaps appreciably with the corre-sponding one for the backbone. Therefore, a selective bonddilution can change the universality class of the phase transi-tion of a given region leaving its ground state unchanged, andvice versa.Summarizing, through an extensive analysis of the groundstate of the 3D EA spin-glass model we have separated thelattice of the system in two components, the backbone andthe glass. We show that the phase transitions observed withineach of these regions have the same class of universality thanthat has the whole system, i. e., in the first instance the spa-tial heterogeneities seem not to affect the critical behavior ofthe model. However, diluting the glass we observe that theground state of the backbone remain unchanged but, more im-portantly, we detect that the universality class of the phase transition changes. In the opposite case, when the backboneis removed, we observe that the glass undergoes dramaticchanges at its fundamental level, while the critical behaviorremains the same as the undiluted system.These results indicate that the critical behavior of the 3DEA spin-glass model originates from the interaction betweentwo subsystems of very different nature, one of which domi-nates the other. The dilution process further reveals that thereis no direct but subtle connection between the fundamentallevel of the system and this phenomenon. In fact, it is thenon-trivial separation in backbone and glass, two structuresdefined in the ground state, that allows to unveil this phe-nomenon. Taking further advantage of these new elementscould help improve current understanding of spin-glass sys-tems.I acknowledge financial support from CONICET (Ar-gentina) under Project No. PIP 112-201301-00049-CO andUniversidad Nacional de San Luis (Argentina) under ProjectNo. PROICO P-31216. [1] L. Berthier, G. Biroli, J.-P. Bouchaud, L. Cipelletti, and W. vanSaarloos, Dynamical Heterogeneities in Glasses, Colloids, andGranular Media (Oxford Science Publishers, Oxford, 2011),Vol. 150.[2] K. Binder and A.P. Young, Rev. Mod. Phys. , 801 (1986).[3] K. H. Fischer and J. A. Hertz, Spin Glasses (Cambridge Uni-versity Press, Cambridge, 1991).[4] S.F. Edwards and P.W. Anderson, J. Phys. F , 965 (1975).[5] F. Rom´a, S. Bustingorry, and P. M. Gleiser, Phys. Rev. Lett. ,167205 (2006).[6] F. Rom´a, S. Bustingorry, P. M. Gleiser, and D. Dom´ınguez,Phys. Rev. Lett. , 097203 (2007).[7] F. Rom´a, S. Bustingorry, and P. M. Gleiser, Eur. Phys. J. B ,259 (2016).[8] F. Barahona, R. Maynard, and R. Rammal, J. P. Uhry, J. Phys.A: Math. Gen. , 673 (1982).[9] F. Rom´a, S. Risau-Gusman, A. J. Ramirez-Pastor, F. Nieto, andE. E. Vogel, Phys. Rev. B , 214401 (2010).[10] F. Rom´a and S. Risau-Gusman, Phys. Rev. E , 042105 (2013).[11] S. Kirkpatrick, Phys. Rev. B ,4630 (1977).[12] A. K. Hartmann, Phys. Rev. E , 016106 (2000).[13] K. Hukushima and K. Nemoto, J. Phys. Soc. Jpn. , 1604(1996).[14] M. Baity-Jesi et al. , Phys. Rev. B , 224416 (2013).[15] M. Palassini and S. Caracciolo, Phys. Rev. Lett. , 5128(1999).[16] H. G. Katzgraber, M. K¨orner, and A. P. Young, Phys. Rev. B ,224432 (2006).[17] I. A. Campbell, K. Hukushima, and H. Takayama, Phys. Rev.Lett. , 117202 (2006).[18] M. Hasenbusch, A. Pelissetto, and E. Vicari, Phys. Rev. B ,214205 (2008).[19] A. K. Hartmann, Phys. Rev. B , 3617 (1999).[20] F. Rom´a, S. Bustingorry, and P. M. Gleiser, Phys. Rev. B ,104412 (2010).[21] H. G. Ballesteros et al. , Phys. Rev. B , 2740 (1998). r X i v : . [ c ond - m a t . d i s - nn ] S e p Supplemental Material for:Changing the universality class of the three-dimensional Edwards-Andersonspin-glass model by selective bond dilution
F. Rom´a Departamento de F´ısica, Universidad Nacional de San Luis, Instituto de F´ısica Aplicada (INFAP),Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas (CONICET), Chacabuco 917, D5700BWS San Luis, Argentina (Dated: September 29, 2020)
PACS numbers: 75.10.Nr, 75.40.Mg
NUMERICAL SIMULATIONS
Monte Carlo simulations are performed using a paralleltempering algorithm [1, 2]. We use this technique to calculateboth ground-state configurations [3, 4] and average values ofdifferent observables at equilibrium, for 10 samples for eachsize L = L =
6, 10 for L =
8, and 320 for L = M noninteracting replicas of a system of N spins,each one associated to a different temperature in the interval [ T min , T max ] where, for simplicity, the difference between con-secutive temperatures is chosen constant. A parallel temper-ing algorithm consists of two routines. One of them is a stan-dard Monte Carlo procedure, i. e., an attempt to update a ran-dom selected spin of the ensemble (we randomly choose botha replica and a spin of this replica) with probability given bythe Metropolis rule [5]. The second routine consists of an ex-change of configurations between two replicas at consecutivetemperatures which is attempted with the probability definedin Ref. [2]. The unit time (or step) of a parallel temperingalgorithm consists of N × M elementary steps of the standardMonte Carlo procedure, followed by a single trial of replicaexchange.The total simulation times (number of parallel temperingsteps), t , required to equilibrate the system are chosen as t = × for L = t = × for L = t = × for L =
8, and t = for L =
10. We also use between M = M =
21 replicas of the system in each case. To reach equi-librium under a given condition x (the undiluted and dilutedcases), it is necessary to choose that the highest temperatureis above the critical one, T max > T xc . In addition, once equi-librium is reached the average values of different observablesare calculated over a time interval of the same length t .Previously, it is necessary to find the backbone and glassregions of each realization of the quenched disorder. To deter-mine which are the rigid and flexible bonds of a given sample,we use a very simple strategy [6, 7]:1. A ground-state configuration C is calculated and its en-ergy U is stored (to do so, we use a parallel temperingalgorithm as explained in Ref. [4]).2. Then, a bond J i j of the sample is chosen at random.3. The system being in configuration C , one of the spins joined by the bond J i j , i.e. either σ i or σ j , is flipped.This flip changes the “condition” of the bond from sat-isfied to frustrated, or vice versa.4. The orientations of the spins i and j are frozen and, forthis “constrained” system, a new ground-state configu-ration C ′ of energy U ′ is calculated.5. If U ′ > U , it follows that J i j is a rigid bond (since weverify that there does not exist a ground-state configu-ration of energy U in which this bond appears with itschanged condition).6. If U ′ = U , then J i j is a flexible bond (we find a ground-state configuration of energy U in which this bondappears with its changed condition; this configurationcould have been found by exhaustively exploring thefundamental level of the unconstrained system).7. The bond J i j is added to the list of “checked” bonds,and the restrictions over the spins σ i or σ j are lifted.8. If there are still non-checked bonds, a new bond J i j ischosen among them and the process is repeated fromstep 3.Ground-state configurations were calculated with the sameparallel tempering algorithm as before, using the parametersgiven in Refs. [4, 7]. SPIN-GLASS SUSCEPTIBILITY
In this section we present the spin-glass susceptibilitycurves for the different cases studied.The spin-glass susceptibility is defined as χ x = N Ω x ∑ i , j ∈ Ω x [ h q i q j i T ] av . (1)Figures S1 (a), (b), and (c), show this quantity as functionof T for, respectively, the whole system, i. e., the three-dimensional (3D) Edwards-Anderson (EA) model, and forits backbone and glass regions, for different lattice sizes asindicated. For all cases, we obtain excellent data collapses(see insets) using the critical parameters T c = . ( ) , L = 4 L = 6 L = 8 L = 10
T (a) -1 -2 -1 / ( LT ) - (TL) |1 - ( T c / T ) | L = 4 L = 6 L = 8 L = 10 B T (b) -1 -2 -1 B / ( LT ) - (TL) |1 - ( T c / T ) | L = 4 L = 6 L = 8 L = 10 G T (c) -1 -2 -1 G / ( LT ) - (TL) |1 - ( T c / T ) | FIG. S1: (Color online) Spin-glass susceptibility as function of T for(a) the whole system, and for (b) the backbone and (c) glass regions.Insets show the corresponding data collapses (see text). ν = . ( ) , and η = − . ( ) [8], and an extendedscaling scheme [9] χ x = ( LT ) − η x ˜ C x [( LT ) / ν x ( − ( T xc / T x ) )] . (2) L = 4 L = 6 L = 8 L = 10 * T (a) -1 -1 / ( LT ) - (TL) |1 - ( T *c / T ) | L = 4 L = 6 L = 8 L = 10 B * T (b) -1 -2 -1 B * / ( LT ) - B * (TL) |1 - (T B*c / T ) | L = 4 L = 6 L = 8 L = 10 G * T (c) -1 -2 -1 G * / ( LT ) - (TL) |1 - (T G*c / T ) | FIG. S2: (Color online) Spin-glass susceptibility as function of T for a random bond dilution (a) of 50%, and for a selective bond di-lution of (b) the glass and (c) backbone regions. Insets show thecorresponding data collapses (see text). In Fig. S2 (a) we show the spin-glass susceptibility for the3D random bond-diluted EA spin-glass model for a dilutionof 50%. Instead, Figs. S2 (b) and (c) present, respectively, thesusceptibility calculated for the backbone region after dilutingthe entire glass region, and for the opposite case, when wedilute the backbone but keep the glass region. In the insets ofpanels (a) and (c) we show the data collapses obtained usingthe critical parameters ν = . ( ) and η = − . ( ) [8], and the corresponding critical temperatures. Instead, forthe isolated backbone, panel (b) shows that a good scalingplot is obtained if we use a different set of critical exponents, ν B ∗ = . ( ) and η B ∗ = . ( ) , and T B ∗ c = . ( ) . Theselast critical parameters were calculated as it is detailed in thenext section. STATISTICAL ANALYSIS OF THE DATA
After a selective dilution of the glass region, we observethat the backbone undergoes a phase transition that doesnot belong to the universality class of the 3D EA spin-glassmodel. In fact, a scaling plot of both the correlation length ξ B ∗ and the spin-glass susceptibility χ B ∗ using the critical ex-ponents ν = . ( ) and η = − . ( ) , leads to baddata collapses of these quantities.To determine a suitable set of critical parameters T B ∗ c , ν B ∗ , and η B ∗ , we use a procedure similar to that reportedin Refs. [10, 11]. First, we fit each curve of correlationlength and susceptibility to a fifth-order polynomial and, fromnow on, we work exclusively with these continuous functions.Then, we calculate T B ∗ c and ν B ∗ looking for the values of theseparameters that allow us to achieve the best data collapse ofthe correlation length curves using a conventional scaling ξ B ∗ L = ˜ S B ∗ [ L / ν B ∗ ( T − T B ∗ c ) / T B ∗ c ] . (3) In addition, to determine η B ∗ , we fix T B ∗ c and ν B ∗ to the val-ues obtained previously and we follow a similar procedure forthe spin-glass susceptibility, i. e., we look for the best datacollapse of these curves but now using an extended scaling,Eq. (2). Finally, we calculate error bars on these critical pa-rameters through a bootstrap method as described in Ref. [10]. [1] C. Geyer, Computing Science and Statistics: Proceedings of the23rd Symposium on the Interface (American Statistical Associ-ation, New York, 1991), p. 156.[2] K. Hukushima and K. Nemoto, J. Phys. Soc. Jpn. , 1604(1996).[3] J. L. Moreno, H. G. Katzgraber, and A. K. Hartmann, Int. J.Mod. Phys. C , 285 (2003).[4] F. Rom´a, S. Risau-Gusman, A. J. Ramirez-Pastor, F. Nieto, andE. E. Vogel, Physica A , 2821 (2009).[5] N. Metropolis, A. W. Rosenbluth, N. M. Rosenbluth, A. H.Teller, and E. Teller, J. Chem. Phys. , 1087 (1953).[6] A. J. Ramirez-Pastor, F. Rom´a, F. Nieto, and E. E. Vogel, Phys-ica A , 454 (2004).[7] F. Rom´a, S. Risau-Gusman, A. J. Ramirez-Pastor, F. Nieto, andE. E. Vogel, Phys. Rev. B , 214401 (2010).[8] M. Baity-Jesi et al. , Phys. Rev. B , 224416 (2013).[9] I. A. Campbell, K. Hukushima, and H. Takayama, Phys. Rev.Lett. , 117202 (2006).[10] H. G. Katzgraber, M. K¨orner, and A. P. Young, Phys. Rev. B ,224432 (2006).[11] D. A. Matoz-Fernandez and F. Rom´a, Phys. Rev. B94