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arXiv:1109.6366 · Journal of Statistical Mechanics Theory and Experiment

Ensemble inequivalence and the spin-glass transition

Zsolt Bertalan, Kazutaka Takahashi

Published Nov 25, 2011• 12 citations
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Abstract

We report on the ensemble inequivalence in a many-body spin-glass model with integer spin. The spin-glass phase transition is of first order for certain values of the crystal-field strength and is dependent on whether it was derived in the microcanonical or the canonical ensemble. In the limit of infinitely-many-body interactions, the model is the integer-spin equivalent of the random-energy model, and is solved exactly. We also derive the integer-spin equivalent of the de Almeida–Thouless line.

DOI
10.1088/1742-5468/2011/11/p11022

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We report on the ensemble inequivalence in a many-body spin-glass model with integer spin. The spin-glass phase transition is of first order for certain values of the crystal-field strength and is dependent on whether it was derived in the microcanonical or the canonical ensemble. In the limit of infinitely-many-body interactions, the model is the integer-spin equivalent of the random-energy model, and is solved exactly. We also derive the integer-spin equivalent of the de Almeida–Thouless line.
We study in Ising spin glasses the finite-size effects near the spin-glass transition in zero field and at the de Almeida-Thouless transition in a field by Monte Carlo methods and by analytical approximations. In zero field, the finite-size scaling function associated with the spin-glass susceptibility of the Sherrington-Kirkpatrick mean-field spin-glass model is of the same form as that of one-dimensional spin-glass models with power-law long-range interactions in the regime where they can be a proxy for the Edwards-Anderson short-range spin-glass model above the upper critical dimension. We also calculate a simple analytical approximation for the spin-glass susceptibility crossover function. The behavior of the spin-glass susceptibility near the de Almeida-Thouless transition line has also been studied, but here we have only been able to obtain analytically its behavior in the asymptotic limit above and below the transition. We have also simulated the one-dimensional system in a field in the non-mean-field regime to illustrate that when the Imry-Ma droplet length scale exceeds the system size one can then be erroneously lead to conclude that there is a de Almeida-Thouless transition even though it is absent.
We study the problem of ensemble equivalence in spin systems with short-range interactions under the existence of a first-order phase transition. The spherical model with nonlinear nearest-neighbour interactions is solved exactly for both canonical and microcanonical ensembles. The result reveals apparent ensemble inequivalence at the first-order transition point in the sense that the microcanonical entropy is non-concave as a function of the energy and consequently the specific heat is negative. In order to resolve the paradox, we show that an unconventional saddle point should be chosen in the microcanonical calculation that represents a phase separation. The XY model with nonlinear interactions is also studied by microcanonical Monte Carlo simulations in two dimensions to see how this model behaves in comparison with the spherical model.
Ensemble inequivalence, where a system's thermodynamic properties depend on the statistical ensemble used to describe it, is a well-established phenomenon in classical systems with long-range interactions. Here, we present a detailed analysis of a quantum ferromagnet spin model that exhibits this same behavior. We find that the microcanonical and canonical phase diagrams are identical at zero temperature ( T = 0 ), but differ significantly at finite temperatures. Our results highlight a breakdown of ensemble equivalence in this quantum long-range model, contrasting with what is observed for short range. These findings have important implications for the study and manipulation of long-range interacting quantum platforms, such as those used in atomic, molecular, and optical physics.