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arXiv:cond-mat/9802211 · Physical review. B, Condensed matter

Phase structure of the three-dimensional Edwards-Anderson spin glass

Enzo Marinari, Giorgio Parisi A, Juan Jesus Ruiz-Lorenzo

Published Dec 1, 1998• 108 citations
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Abstract

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.

DOI
10.1103/physrevb.58.14852

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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 present numerical simulations of the four-dimensional Edwards Anderson Ising spin glass with binary couplings. Our results, in the midst of strong finite-size effects, suggest the existence of a spin-glass phase transition. We present a preliminar determination of critical exponents. We discuss spin-glass susceptibilities, cumulants of the overlap and energy overlap probability distributions, finite-size effects, and the behaviour of the disorder-dependent and integrated probability distributions.
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.
Using the decay of the out equilibrium spin-spin correlation function we compute the equilibrium Edward-Anderson order parameter in the three dimensional binary Ising spin glass in the spin glass phase. We have checked that the Edward-Anderson order parameter computed from out of equilibrium numerical simulations follows with good precision the critical law as determined in experiments and in numerical studies at equilibrium. We have also studied the dependence of the order parameter with the lattice size. Finally we present a large time study of the scaling of the off-equilibrium fluctuation-dissipation relations.
We have characterized numerically, using the Janus computer, the Lee–Yang complex singularities related to the overlap in the 3 D Ising spin-glass with binary couplings over a wide range of temperatures (both in the critical and in the spin-glass phase). Studying the behavior of the zeros at the critical point, we have obtained an accurate measurement of the anomalous dimension in very good agreement with the values quoted in the literature. In addition, by studying the density of the zeros we have been able to characterize the phase transition and to investigate the Edwards–Anderson order parameter in the spin-glass phase, finding agreement with the values obtained using more conventional techniques.