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arXiv:cond-mat/9804017 · Journal of Physics A Mathematical and General

Small window overlaps are effective probes of replica symmetry breaking in three-dimensional spin glasses

Enzo Marinari, Giorgio Parisi A, Federico Ricci‐Tersenghi, Juan Jesus Ruiz-Lorenzo

Published Jul 3, 1998• 34 citations
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Abstract

We numerically compute small window overlaps in the three-dimensional Edwards - Anderson spin glass. We show that they behave in the way implied by the replica symmetry breaking ansatz, that they do not qualitatively differ from the full volume overlap and do not tend to a trivial function when increasing the lattice volume. In contrast we show that they are affected by small finite-volume effects, and are interesting tools for the study of the features of the spin-glass phase.

DOI
10.1088/0305-4470/31/26/001

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We numerically compute small window overlaps in the three-dimensional Edwards - Anderson spin glass. We show that they behave in the way implied by the replica symmetry breaking ansatz, that they do not qualitatively differ from the full volume overlap and do not tend to a trivial function when increasing the lattice volume. In contrast we show that they are affected by small finite-volume effects, and are interesting tools for the study of the features of the spin-glass phase.
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.
We use a sample-dependent analysis, based on medians and quantiles, to analyze the behavior of the overlap probability distribution of the Sherrington-Kirkpatrick and 3D3D3D Edwards-Anderson models of Ising spin glasses. We find that this approach is an effective tool to distinguish between replica symmetry breaking--like and droplet-like behavior of the spin-glass phase. Our results are in agreement with a replica symmetry breaking--like behavior for the 3D3D3D Edwards-Anderson model.
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.
We have argued in recent papers that the Monte Carlo results for the equilibrium properties of the Edwards-Anderson spin glass in three dimensions, which had been interpreted earlier as providing evidence for replica symmetry breaking, can be explained quite simply within the droplet model once finite-size effects and proximity to the critical point are taken into account. In this paper we show that similar considerations are sufficient to explain the Monte Carlo data in four dimensions. In particular, we study the Parisi overlap and the link overlap for the four-dimensional Ising spin glass in the Migdal-Kadanoff approximation. Similar to what is seen in three dimensions, we find that temperatures well below those studied in the Monte Carlo simulations have to be reached before the droplet model predictions become apparent. We also show that the double-peak structure of the link overlap distribution function is related to the difference between domain-wall excitations that cross the entire system and droplet excitations that are confined to a smaller region.