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Physical Review Letters

Exact critical behavior of a random bond two-dimensional Ising model

R. Shankar

Published Jun 8, 1987• 160 citations
Publisher

Abstract

A 2D Ising model in which the bonds K fluctuate randomly about ${\mathrm{K}}_{\mathrm{c}}$, the critical value of the pure system, is considered. The ensemble average of the square of the two-point function, ${\mathrm{s}}_{0}$${\mathrm{s}}_{\mathrm{R}}$${\mathrm{〉}}^{2}$${\mathrm{〉}}_{\mathrm{Av}}$, is shown to decay as (lnR${)}^{1/4}$${\mathrm{R}}^{\mathrm{\ensuremath{-}}1/2}$ aat the critical point. This implies that ${\mathrm{s}}_{0}$${\mathrm{s}}_{\mathrm{R}}$${}_{\mathrm{Av}}$ is bounded above by (lnR${)}^{1/8}$${\mathrm{R}}^{\mathrm{\ensuremath{-}}1/4}$ in disagreement with the exp [-(ln lnR${)}^{2}$] decay law found by Dotsenko and Dotsenko by a different method. On the other hand, the present calculation reproduces their specific-heat singularity C\ensuremath{\sim}ln\ensuremath{\Vert}ln\ensuremath{\tau}\ensuremath{\Vert} (\ensuremath{\tau}=K-${\mathrm{K}}_{\mathrm{c}}$).

DOI
10.1103/physrevlett.58.2466

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A 2D Ising model in which the bonds K fluctuate randomly about Kc{\mathrm{K}}_{\mathrm{c}}Kc​, the critical value of the pure system, is considered. The ensemble average of the square of the two-point function, {\mathrm{s}}_{0}$${\mathrm{s}}_{\mathrm{R}}$${\mathrm{〉}}^{2}$${\mathrm{〉}}_{\mathrm{Av}}, is shown to decay as (lnR{)}^{1/4}$${\mathrm{R}}^{\mathrm{\ensuremath{-}}1/2} aat the critical point. This implies that {\mathrm{s}}_{0}$${\mathrm{s}}_{\mathrm{R}}$${}_{\mathrm{Av}} is bounded above by (lnR{)}^{1/8}$${\mathrm{R}}^{\mathrm{\ensuremath{-}}1/4} in disagreement with the exp [-(ln lnR)2{)}^{2})2] decay law found by Dotsenko and Dotsenko by a different method. On the other hand, the present calculation reproduces their specific-heat singularity C\ensuremath{\sim}ln\ensuremath{\Vert}ln\ensuremath{\tau}\ensuremath{\Vert} (\ensuremath{\tau}=K-Kc{\mathrm{K}}_{\mathrm{c}}Kc​).
The random bond Ising model in two dimensions is considered in terms of a random matrix representation. A generalization of this approach enables us to study an N → ∞ limit and the corresponding 1/N expansion. With g as the strength of the random bond fluctuations, the critical temperature T c of the pure system is split into two critical temperatures T ± = T c (1 ± c 0 exp[− c 1 / g ]) with a new phase for T − < T < T + . The specific heat, for instance, grows logarithmically (or double logarithmically in a crossover region near T ± ) with|1− T / T c | up to a bound of the order 1/ g .
PACS. 05.50 – Lattice theory and statistics; Ising problems PACS. 75.10 – General theory and models of magnetic ordering We describe results of the cluster algorithm Special Purpose Processor simulations of the 2D Ising model with impurity bonds. Use of large lattices, with the number of spins up to 10 6, permitted to define critical region of temperatures, where both finite size corrections and corrections to scaling are small. High accuracy data unambiguously show increase of magnetization and magnetic susceptibility effective exponents β and γ, caused by impurities. The M and χ singularities became more sharp, while the specific heat singularity is smoothed. The specific heat is found to be in a good agreement with Dotsenko-Dotsenko theoretical predictions in the The problem of inhomogenities influence on phase transitions has a long history. The first model with some kind of impurities to be studied, is, due to its simplicity, the Ising model. Effect of randomness on the critical behaviour of different Ising
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The energy-energy correlation function of the two-dimensional Ising model with weakly fluctuating random bonds is evaluated in the large scale limit. Two correlation lengths exist in contrast to one correlation length in the pure 2D Ising model: one is finite whereas the other is divergent at the critical points. The corresponding exponent of the divergent correlation length is νe=1/2ν_e=1/2νe​=1/2 in contrast to the pure system where νe=1ν_e=1νe​=1. The calculation is based on a previously developed effective field theory for the energy density fluctuations.
The two-dimensional (2D) random-bond Ising model has a novel multicritical point on the ferromagnetic to paramagnetic phase boundary. This random phase transition is one of the simplest examples of a 2D critical point occurring at both finite temperatures and disorder strength. We study the associated critical properties, by mapping the random 2D Ising model onto a network model. The model closely resembles network models of quantum Hall plateau transitions, but has different symmetries. Numerical transfer matrix calculations enable us to obtain estimates for the critical exponents at the random Ising phase transition. The values are consistent with recent estimates obtained from high-temperature series.
The purpose of this paper is to show how some interesting numerically exact thermal averages can be computed for finite two-dimensional Ising models by a transfer-matrix method, and to apply these methods to the Ising version of the Edwards-Anderson (EA) model of a spin-glass to ascertain whether there is a phase transition. We do not study the spatial dependence of $〈{\ensuremath{\sigma}}_{0}{\ensuremath{\sigma}}_{l}〉$ for, as we show with an example, its behavior for finite systems can be misleading. It is first shown how to obtain ${\ensuremath{\chi}}_{\mathrm{EA}}^{\ensuremath{'}}$, defined by ${\ensuremath{\chi}}_{\mathrm{EA}}^{\ensuremath{'}}={N}^{\ensuremath{-}1}\ensuremath{\Sigma}{〈{〈{\ensuremath{\sigma}}_{i}{\ensuremath{\sigma}}_{j}〉}_{T}^{2}〉}_{J}$. We compute ${\ensuremath{\chi}}_{\mathrm{EA}}^{\ensuremath{'}}$ and study the quantity $\ensuremath{\Lambda}=\frac{\ensuremath{-}\ensuremath{\partial}\mathrm{ln}({\ensuremath{\chi}}_{\mathrm{EA}}^{\ensuremath{'}})}{\ensuremath{\partial}T}$, both as a function of temperature (TTT) and of the number of spins (NNN) in the system. The results obtained for square systems of up to 100 spins in the case where $J=\ifmmode\pm\else\textpm\fi{}1$ with equal probability and for square systems of up to 121 spins in the case where each JJJ is normally distributed about J=0J=0J=0 are in accord with the existence of a critical point at ${T}_{0}\ensuremath{\simeq}1.0$ and at ${T}_{0}\ensuremath{\simeq}0.6$, respectively. In addition the value $\ensuremath{\nu}\ensuremath{\approx}1$ is obtained. The value q=0q=0q=0 for T>0T>0T>0 is consistent with the results obtained. The low-temperature entropy per spin (SSS) is computed for long strips of different widths. Extrapolation to an infinite width yields $\frac{S}{k}\ensuremath{\simeq}0.07$. It is also shown how to calculate the probability, $P(\ensuremath{\eta})$, that the quantity, $\ensuremath{\eta}={N}^{\ensuremath{-}1}\ensuremath{\Sigma}{\ensuremath{\tau}}_{i}{\ensuremath{\sigma}}_{i}$, where each ${\ensuremath{\tau}}_{i}=0,\ifmmode\pm\else\textpm\fi{}1$ take any value in the range $1<~\ensuremath{\eta}<~1$. The probability, $P(\ensuremath{\eta})$, obtained for the EA model at low temperatures often has several maxima separated by regions of improbable values of $\ensuremath{\eta}$, as is to be expected of a system with metastable states.