Universality-class dependence of energy distributions in spin glasses (original) (raw)
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Scalings of Domain Wall Energies in Two Dimensional Ising Spin Glasses
Physical Review Letters, 2003
We study domain wall energies of two dimensional spin glasses. The scaling of these energies depends on the model's distribution of quenched random couplings, falling into three different classes. The first class is associated with the exponent θ ≈ −0.28, the other two classes have θ = 0, as can be justified theoretically. In contrast to previous claims we find that θ = 0 does not indicate d = d c l but rather d ≤ d c l , where d c l is the lower critical dimension. PACS numbers: 75.10.Nr, 75.40.Mg, 02.60.Pn
Hierarchical Random Energy Model of a Spin Glass
Physical Review Letters, 2010
We introduce a Random Energy Model on a hierarchical lattice where the interaction strength between variables is a decreasing function of their mutual hierarchical distance, making it a nonmean field model. Through small coupling series expansion and a direct numerical solution of the model, we provide evidence for a spin glass condensation transition similar to the one occurring in the usual mean field Random Energy Model. At variance with mean field, the high temperature branch of the free-energy is non-analytic at the transition point. PACS numbers: 05.10.-a,05.50.+q,75.10.Nr
Finite-Size Scaling in the Energy-Entropy Plane for the 2D ± Ising Spin Glass
Journal of Statistical Physics, 2006
For L × L square lattices with L ≤ 20 the 2D Ising spin glass with +1 and -1 bonds is found to have a strong correlation between the energy and the entropy of its ground states. A fit to the data gives the result that each additional broken bond in the ground state of a particular sample of random bonds increases the ground state degeneracy by approximately a factor of 10/3. For x = 0.5 (where x is the fraction of negative bonds), over this range of L, the characteristic entropy defined by the energy-entropy correlation scales with size as L 1.78(2) . Anomalous scaling is not found for the characteristic energy, which essentially scales as L 2 . When x = 0.25, a crossover to L 2 scaling of the entropy is seen near L = 12. The results found here suggest a natural mechanism for the unusual behavior of the low temperature specific heat of this model, and illustrate the dangers of extrapolating from small L.
Physical Review B, 2013
At the mean-field level, on fully connected lattices, several disordered spin models have been shown to belong to the universality class of "structural glasses" with a "random first-order transition" (RFOT) characterized by a discontinuous jump of the order parameter and no latent heat. However, their behavior in finite dimensions is often drastically different, displaying either no glassiness at all or a conventional spin-glass transition. We clarify the physical reasons for this phenomenon and stress the unusual fragility of the RFOT to short-range fluctuations, associated, e.g., with the mere existence of a finite number of neighbors. Accordingly, the solution of fully connected models is only predictive in very high dimension, whereas despite being also mean-field in character, the Bethe approximation provides valuable information on the behavior of finite-dimensional systems. We suggest that before embarking on a full blown account of fluctuations on all scales through computer simulation or renormalization-group approach, models for structural glasses should first be tested for the effect of short-range fluctuations and we discuss ways to do it. Our results indicate that disordered spin models do not appear to pass the test and are therefore questionable models for investigating the glass transition in three dimensions. This also highlights how nontrivial is the first step of deriving an effective theory for the RFOT phenomenology from a rigorous integration over the short-range fluctuations.
Physical Review B, 2007
The statistics of the ground-state and domain-wall energies for the two-dimensional random-bond Ising model on square lattices with independent, identically distributed bonds of probability p of Jij = −1 and (1 − p) of Jij = +1 are studied. We are able to consider large samples of up to 320 2 spins by using sophisticated matching algorithms. We study L × L systems, but we also consider L×M samples, for different aspect ratios R = L/M . We find that the scaling behavior of the groundstate energy and its sample-to-sample fluctuations inside the spin-glass region (pc ≤ p ≤ 1 − pc) are characterized by simple scaling functions. In particular, the fluctuations exhibit a cusp-like singularity at pc. Inside the spin-glass region the average domain-wall energy converges to a finite nonzero value as the sample size becomes infinite, holding R fixed. Here, large finite-size effects are visible, which can be explained for all p by a single exponent ω ≈ 2/3, provided higher-order corrections to scaling are included. Finally, we confirm the validity of aspect-ratio scaling for R → 0: the distribution of the domain-wall energies converges to a Gaussian for R → 0, although the domain walls of neighboring subsystems of size L × L are not independent.
Finite-Size Scaling of the Domain Wall Entropy Distributions for the 2D ± J Ising Spin Glass
Journal of Statistical Physics, 2006
The statistics of domain walls for ground states of the 2D Ising spin glass with +1 and -1 bonds are studied for L × L square lattices with L ≤ 48, and p = 0.5, where p is the fraction of negative bonds, using periodic and/or antiperiodic boundary conditions. When L is even, almost all domain walls have energy E dw = 0 or 4. When L is odd, most domain walls have E dw = 2. The probability distribution of the entropy, S dw , is found to depend strongly on E dw . When E dw = 0, the probability distribution of |S dw | is approximately exponential. The variance of this distribution is proportional to L, in agreement with the results of Saul and Kardar. For E dw = k > 0 the distribution of S dw is not symmetric about zero. In these cases the variance still appears to be linear in L, but the average of S dw grows faster than √ L. This suggests a one-parameter scaling form for the L-dependence of the distributions of S dw for k > 0.
Frustration and ground-state degeneracy in spin glasses
Physical Review B, 1977
The problem of an Ising model with random nearest-neighbor interactions is reformulated to make manifest Toulouse's recent suggestion that a broken "lattice gauge symmetry" is responsible for the unusual properties of spin glasses. Exact upper and lower bounds on the ground-state energy for models in which the interactions are of constant magnitude but fluctuating sign are obtained, and used to place restrictions on possible geometries of the unsatisfied interactions which must be present in the ground state. Proposed analogies between the ferromagnetspin-glass phase boundary at zero temperature and a percolation threshold for the "strings" of unsatisfied bonds are reviewed in the light of this analysis. Monte Carlo simulations show that the upper bound resulting from a "one-dimensional approximation" to the spin-glass ground-state energy is reasonably close to the actual result. The transition between spin glass and ferromagnet at 0 K appears to be weakly first order in these models. The entropy of the ground state is obtained from the temperature dependence of the internal energy, and compared with the density of free spins at very low temperatures. For a two-dimensional spin glass in which half the bonds are antiferromagnetic, S(0)-0.099 k~; for the analogous three-dimensional spin glass the result is S(0)-0.062 k~. Monte Carlo kinetic simulations are reported which demonstrate the existence and stability of a fieldcooled moment in the spin-glass ground state.
Critical behavior of three-dimensional Ising spin glass models
Physical Review B, 2008
We perform high-statistics Monte Carlo simulations of three-dimensional Ising spin-glass models on cubic lattices of size L: the ±J (Edwards-Anderson) Ising model for two values of the disorder parameter p, p = 0.5 and p = 0.7 (up to L = 28 and L = 20, respectively), and the bond-diluted bimodal model for bond-occupation probability p b = 0.45 (up to L = 16). The finite-size behavior of the quartic cumulants at the critical point allows us to check very accurately that these models belong to the same universality class. Moreover, it allows us to estimate the scaling-correction exponent ω related to the leading irrelevant operator: ω = 1.0(1). Shorter Monte Carlo simulations of the bond-diluted bimodal models at p b = 0.7 and p b = 0.35 (up to L = 10) and of the Ising spinglass model with Gaussian bond distribution (up to L = 8) also support the existence of a unique Ising spin-glass universality class. A careful finite-size analysis of the Monte Carlo data which takes into account the analytic and the nonanalytic corrections to scaling allows us to obtain precise and reliable estimates of the critical exponents ν and η: we obtain ν = 2.45(15) and η = −0.375(10).
Finite-size scaling of the Domain Wall Entropy for the 2D \pm J Ising Spin Glass
2005
The statistics of domain walls for ground states of the 2D Ising spin glass with +1 and -1 bonds are studied for LtimesLL \times LLtimesL square lattices with Lle20L \le 20Lle20, and xxx = 0.25 and 0.5, where xxx is the fraction of negative bonds, using periodic and/or antiperiodic boundary conditions. Under these conditions, almost all domain walls have an energy EdwE_{dw}Edw equal to 0 or 4. The probability distribution of the entropy, SdwS_{dw}Sdw, is found to depend strongly on EdwE_{dw}Edw. The results for SdwS_{dw}Sdw when Edw=4E_{dw} = 4Edw=4 agree with the prediction of the droplet model. Our results for SdwS_{dw}Sdw when Edw=0E_{dw} = 0Edw=0 agree with those of Saul and Kardar. In addition, we find that the distributions do not appear to be Gaussian in that case. The special role of Edw=0E_{dw} = 0Edw=0 domain walls is discussed, and the discrepancy between the prediction of Amoruso, Hartmann, Hastings and Moore and the result of Saul and Kardar is explained.
Distribution of metastable states of Ising spin glasses
Physical Review B
Local minima, also known as inherent structures, are expected to play an essential role in the behavior of spin glasses. Here, we propose techniques to efficiently sample these configurations in Monte Carlo simulations. For the Sherrington-Kirkpatrick and the three-dimensional Edwards-Anderson model their spectra are determined and compared to analytical results.