Replica Monte Carlo Simulation of Spin-Glasses

Abstract

A new Monte Carlo method is presented for simulations of systems with quenched random interactions. The approach greatly reduces the long correlation times characteristic of standard methods, allowing the investigation of lower temperatures with less computer time than previously necessary.PACS numbers: 05.50. +q, 75.lO.Hk Over the past decade, there have been continuing controversies about the properties of systems with quenched random interactions, such as spin-glasses 1-5 and random-field models. 6 "" 8 Extensive work has been carried out to answer some of the questions by Monte Carlo simulations. While these efforts have been partially successful, they have been greatly hampered by the extremely long relaxation times that are characteristic of systems with frustrated interactions. Similar difficulties are also found in some engineering applications involving optimization subject to conflicting constraints. 9 In this paper, we present a new approach to Monte Carlo computer simulations, which provides rapid relaxation times, making possible the study of equilibrium properties with relatively modest amounts of computer time.In constructing a Markov process for Monte Carlo simulations, two conditions should be met: A sequence of transitions with nonzero probability must connect any two configurations, and the condition of detailed balance must be satisfied. The standard Monte Carlo algorithm satisfies both conditions, ensuring that equilibrium will be achieved eventually (although not necessarily within budget limitations).By retaining standard Monte Carlo methods as part of the new simulation, we satisfy the first condition and provide a fast process for relaxation of local fluctuations. Additional processes satisfying detailed balance are then introduced to reduce relaxation times for large fluctuations. This strategy has been shown to be effective when the form of the important large-scale fluctuations is known. 10 The method described below provides for the automatic recognition of important fluctuations in a spin-glass.To illustrate our method, we will discuss its application to the Ising spin-glass,where s t takes on the values ± 1 and the factor -I/ICBT has been absorbed into the coupling constant K. The Bgj are dimensionless variables, which describe the quenched, random interactions. Instead of simulating different temperatures sequentially, we treat several independent "replicas" of this system at different values of K simultaneously. By including a replica index, /i, we can describe the entire set of replicas with a single Hamiltonian, modifying Eq. (1) to become fR'I,I t B l jK M s l M sj'\ (2) n (ij)Information is transferred by the "mixing" of two neighboring replicas, by use of new variables, t^n\ defined at each site: 5/» + 1 >-*/»ty»>. (3)The pair of replicas can now be represented by the variables {s^} and {t^} and the part of the Hamiltonian describing this pair of replicas becomes //pair -I Bij lK in) + K {n + X V"V W) W"V"^ < 4 > Changing sf n) while holding */ n) fixed is equivalent to ch...

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“…(11). In order to translate them to a temperature variation, the scaling factors should always be positive and either smaller or greater than one, implying a heating and a cooling, respectively, of the involved degrees of freedom.…”

“…The Replica Exchange Method (REM) [11][12][13][14] provides an elegant and simple solution to quasiergodic sampling. In REM, several independent trajectories, called replicas, are simultaneously generated in different thermodynamic conditions.…”

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“…The replica exchange (RE) method [2][3][4][5] , also known as parallel tempering, has emerged as a relatively straightforward but powerful approach for enhanced conformational sampling. The basic idea is to simulate multiple replicas of the system at different temperatures independently using either MC or MD.…”

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