What do Monte Carlo simulations tell us about compressible Ising models?

Abstract

Monte Carlo simulations have begun to illuminate the nature of phase transitions and universality classes for compressible Ising models. A comprehensive analysis of a Landau-Ginsburg Wilson hamiltonian for systems with elastic degrees of freedom predicts that there should be four cases with different behavior, depending upon symmetries and thermodynamic constraints. We shall describe the results of careful Monte Carlo simulations for a simple compressible Ising model that can be easily modified to correspond to each of the four cases.

Phase transition in compressible Ising systems at fixed volume

“…It is worth noting that the transition depends on the sample shape in hydrogen-metal systems at constant pressure 1,22 , where the proton concentration is linearly coupled to the dilation 13 . (ii) We mention Monte Carlo simulations on a binary alloy by Landau's group 10,11,12 . They assumed that a mixture undergoing unmixing corresponds to ferromagnets and that forming a superstructure to antiferromagnets.…”

“…Such elastic effects strongly depend on the nature of the coupling and their understanding is crucial in technology. In the present work, we will focus on the phase transition behavior of compressible ferromagnets or antiferromagnets, which has long been studied theoretically in the physics community 2,3,4,5,6,7,8,9,10,11,12 . In real materials, the short-range spin interactions depend on the distances among the spins, so the spin fluctuations are coupled to the elastic dilation strain.…”

“…This renormalization effect should trigger a first order phase transition sufficiently close to the critical point. Simulations have been prerformed on compressible Ising systems and a number of numerical results still remain not well understood 10,11,12 . These theories and simulations show that the phase transition depends on whether the pressure or the volume is fixed.…”

Compressible Sherrington-Kirkpatrick spin-glass model

“…Although elastic spin models with uniform interactions have been much investigated, disordered compressible spin models are less explored. Nowadays it has been feasible to carry out detailed computer simulations for compressible models [15,16,17], including disordered compressible models, as in the work of Marshall for a compressible Edwards-Anderson Ising spin-glass on a two-dimensional lattice [18,19]. These works have provided further motivation to study the phase diagram of the simple compressible SK spin-glass model.…”

Modelos estatísticos de campo médio para vidros de spins e fluidos complexos

“…Chamamos a atenção para o fato de que simulações numéricas com alta precisão de modelos compressíveis tornaram-se possíveis apenas recentemente, com o aumento da capacidade de processamento dos computadores modernos e o desenvolvimento de algoritmos mais eficazes (Mitchell, Pereira, e Landau, 2008). Aqui também servem de paradigma os modelos de Ising compressíveis com interações ferro (Landau, 2006) (Marshall, Chacraborty, e Nagel, 2006;Marshall, 2007), não há relatos na literatura sobre estudos de modelos compressíveis a partir de abordagens analíticas 12 .…”