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arXiv:cond-mat/0103534 · Physical review. B, Condensed matter

Equilibrium valleys in spin glasses at low temperature

Enzo Marinari, Olivier Martin, Francesco Zuliani

Published Oct 18, 2001• 6 citations
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Abstract

We investigate the three-dimensional Edwards-Anderson spin glass model at low temperature on simple cubic lattices of sizes up to $L=12.$ Our findings show a strong continuity among $T>0$ physical features and those found previously at $T=0,$ leading to a scenario with emerging mean field like characteristics that are enhanced in the large volume limit. For instance, the picture of space filling sponges seems to survive in the large volume limit at $T>0,$ while entropic effects play a crucial role in determining the free-energy degeneracy of our finite volume states. All of our analysis is applied to equilibrium configurations obtained by a parallel tempering on $512$ different disorder realizations. First, we consider the spatial properties of the sites where pairs of independent spin configurations differ and we introduce a modified spin overlap distribution which exhibits a nontrivial limit for large L. Second, after removing the ${Z}_{2}$ $(\ifmmode\pm\else\textpm\fi{}1)$ symmetry, we cluster spin configurations into valleys. On average these valleys have free-energy differences of $O(1),$ but a difference in the (extensive) internal energy that grows significantly with L; there is thus a large interplay between energy and entropy fluctuations. We also find that valleys typically differ by spongelike space filling clusters, just as found previously for low-energy system-size excitations above the ground state.

DOI
10.1103/physrevb.64.184413

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We investigate the three-dimensional Edwards-Anderson spin glass model at low temperature on simple cubic lattices of sizes up to L=12.L=12.L=12. Our findings show a strong continuity among T>0T>0T>0 physical features and those found previously at T=0,T=0,T=0, leading to a scenario with emerging mean field like characteristics that are enhanced in the large volume limit. For instance, the picture of space filling sponges seems to survive in the large volume limit at T>0,T>0,T>0, while entropic effects play a crucial role in determining the free-energy degeneracy of our finite volume states. All of our analysis is applied to equilibrium configurations obtained by a parallel tempering on 512512512 different disorder realizations. First, we consider the spatial properties of the sites where pairs of independent spin configurations differ and we introduce a modified spin overlap distribution which exhibits a nontrivial limit for large L. Second, after removing the Z2{Z}_{2}Z2​ (\ifmmode±\else\textpm\fi1)(\ifmmode\pm\else\textpm\fi{}1)(\ifmmode±\else\textpm\fi1) symmetry, we cluster spin configurations into valleys. On average these valleys have free-energy differences of O(1),O(1),O(1), but a difference in the (extensive) internal energy that grows significantly with L; there is thus a large interplay between energy and entropy fluctuations. We also find that valleys typically differ by spongelike space filling clusters, just as found previously for low-energy system-size excitations above the ground state.
We numerically extract large-scale excitations above the ground state in the 3-dimensional Edwards-Anderson spin glass with Gaussian couplings. We find that associated energies are O(1), in agreement with the mean field picture. Of further interest are the position-space properties of these excitations. First, our study of their topological properties show that the majority of the large-scale excitations are sponge-like. Second, when probing their geometrical properties, we find that the excitations coarsen when the system size is increased. We conclude that either finite size effects are very large even when the spin overlap q is close to zero, or the mean field picture of homogeneous excitations has to be modified.
The question of the number of thermodynamic states present in the low-temperature phase of the three-dimensional Edwards-Anderson Ising spin glass is addressed by studying spin- and link-overlap distributions using population annealing Monte Carlo simulations. We consider overlaps between systems with the same boundary condition, which are the usual quantities measured, and also overlaps between systems with different boundary conditions, both for the full systems and also within a smaller window within the system. Our results appear to be fully compatible with a single pair of pure states such as in the droplet-scaling picture. However, our results for whether or not domain walls induced by changing boundary conditions are space filling or not are also compatible with scenarios having many thermodynamic states, such as the chaotic-pairs picture and the replica-symmetry-breaking picture. The differing results for spin overlaps in the same and different boundary conditions suggest that finite-size effects are very large for the system sizes currently accessible in low-temperature simulations.
The three-dimensional Edwards-Anderson and mean-field Sherrington-Kirkpatrick Ising spin glasses are studied via large-scale Monte Carlo simulations at low temperatures, deep within the spin-glass phase. Performing a careful statistical analysis of several thousand independent disorder realizations and using an observable that detects peaks in the overlap distribution, we show that the Sherrington-Kirkpatrick and Edwards-Anderson models have a distinctly different low-temperature behavior. The structure of the spin-glass overlap distribution for the Edwards-Anderson model suggests that its low-temperature phase has only a single pair of pure states.
We investigate the nature of the low-energy, large-scale excitations in the three-dimensional Edwards-Anderson Ising spin glass with Gaussian couplings and free boundary conditions, by studying the response of the ground state to a coupling-dependent perturbation introduced previously. The ground states are determined exactly for system sizes up to 123{12}^{3}123 spins using a branch-and-cut algorithm. The data are consistent with a picture where the surface of the excitations is not space filling, such as the droplet or the trivial-non-trivial (TNT) picture, with only minimal corrections to scaling. When allowing for very large corrections to scaling, the data are also consistent with a picture with space-filling surfaces, such as replica symmetry breaking. The energy of the excitations scales with their size with a small exponent ${\ensuremath{\theta}}^{\ensuremath{'}},$ which is compatible with zero if we allow moderate corrections to scaling. We compare the results with data for periodic boundary conditions obtained with a genetic algorithm, and discuss the effects of different boundary conditions on corrections to scaling. Finally, we analyze the performance of our branch-and-cut algorithm, finding that it is correlated with the existence of large-scale, low-energy excitations.
We study zero-temperature properties of the three-dimensional Edwards-Anderson Ising spin glass on finite lattices up to size 123{12}^{3}123. Using multicanonical sampling we generate large numbers of ground-state configurations in thermal equilibrium. Finite-size scaling fits of the data are carried out for two hypothetical scenarios: Parisi mean-field theory versus a droplet scaling ansatz. With a zero-temperature scaling exponent y=0.72\ifmmode\pm\else\textpm\fi{}0.12 the data are well described by the droplet scaling ansatz. Alternatively, a description in terms of the Parisi mean-field behavior is still possible. The two scenarios give significantly different predictions on lattices of size \ensuremath{\ge}123{12}^{3}123.
We characterize numerically the properties of the phase transition of the three-dimensional Ising spin glass with Gaussian couplings and of the low-temperature phase. We compute critical exponents on large lattices. We study in detail the overlap probability distribution and the equilibrium overlap-overlap correlation functions. We find a clear agreement with off-equilibrium results from previous work. These results strongly support the existence of a continuous spontaneous replica symmetry breaking in three-dimensional spin glasses.