Canonical-ensemble results for the Ising model with random bonds in two dimensions

Abstract

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 ($T$) and of the number of spins ($N$) 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 $J$ is normally distributed about $J=0$ 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 $q=0$ for $T>0$ is consistent with the results obtained. The low-temperature entropy per spin ($S$) 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.