Investigation of Finite-Size 2D Ising Model with a Noisy Matrix of Spin-Spin Interactions

Boris Kryzhanovsky,Magomed Malsagov,Iakov Karandashev

doi:10.3390/e20080585

Boris Kryzhanovsky, Magomed Malsagov + Show 1 more

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https://doi.org/10.3390/e20080585

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Abstract

We analyze changes in the thermodynamic properties of a spin system when it passes from the classical two-dimensional Ising model to the spin glass model, where spin-spin interactions are random in their values and signs. Formally, the transition reduces to a gradual change in the amplitude of the multiplicative noise (distributed uniformly with a mean equal to one) superimposed over the initial Ising matrix of interacting spins. Considering the noise, we obtain analytical expressions that are valid for lattices of finite sizes. We compare our results with the results of computer simulations performed for square N = L × L lattices with linear dimensions L = 50 ÷ 1000. We find experimentally the dependencies of the critical values (the critical temperature, the internal energy, entropy and the specific heat) as well as the dependencies of the energy of the ground state and its magnetization on the amplitude of the noise. We show that when the variance of the noise reaches one, there is a jump of the ground state from the fully correlated state to an uncorrelated state and its magnetization jumps from 1 to 0. In the same time, a phase transition that is present at a lower level of the noise disappears.

Highlights

Calculation of the partition function is an essential of statistical physics and informatics.A few conceptual models allow exact solutions [1,2,3,4,5,6]
We introduce adjusting parameters in Functional (1), which allows us to go from the 2D Ising model to Edwards-Anderson model in a smooth manner and investigate the thermodynamic characteristics of the system in the transient state
The disappearance of the phase transition is easy to notice if we look at the curve of the second derivative d2 Ψ/dE2

Summary

Introduction

A few conceptual models allow exact solutions [1,2,3,4,5,6]. Among these a 2D Ising model [7], though simple, deserves special attention because of its importance for investigating critical effects. Having contributed a lot to the development of the spin glass theory, the Edwards-Anderson model [8]. Sherrington-Kirkpatrick model [9] are worth mentioning. There are not many models that permit exact solutions. This is the reason why numerical methods are mostly used for tackling complex systems.

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