Monte Carlo simulation of a canonical spin-glass

Abstract

We perform a Monte Carlo simulation of systems of $N$ spins randomly located on an fcc lattice with an atomic concentration of 0.5% and interacting via a Ruderman-Kittel-Kasuya-Yoshida exchange. Our interest lies on the character of the paramagnetic—to—spin-glass transition. We evaluate the time-dependent equilibrium correlation function ${\chi }_{\mathrm{SG}}\mathrm{}\left(t\right)=N^{-1\mathrm{}}{\Sigma }_{\mathrm{ij}}〈\left[{\overrightarrow{\mathrm{S}}}_i\mathrm{}\right(0\left)\mathrm{}·{\overrightarrow{\mathrm{S}}}_j\mathrm{}\right(0\left)\right]\left[{\overrightarrow{\mathrm{S}}}_i\mathrm{}\right(t\left)\mathrm{}·{\overrightarrow{\mathrm{S}}}_j\mathrm{}\right(t\left)\right]〉$, which has the following limiting behavior: (a) it becomes the generalized susceptibility, $N^{-1\mathrm{}}{\Sigma }_{\mathrm{ij}}{〈{\overrightarrow{\mathrm{S}}}_i·{\overrightarrow{\mathrm{S}}}_j〉}^2$ as $t\to \infty$; (b) $N^{-1\mathrm{}}{\chi }_{\mathrm{SG}}\mathrm{}\left(t\right)\mathrm{}$ becomes an order parameter ($q_2$) as $N\to \infty$ and $t\to \infty$. We study ${\chi }_{\mathrm{SG}}\mathrm{}\left(t\right)\mathrm{}$ as a function of $t$ for systems of $N=22,40,87\mathrm{}$, and 169 spins for various temperatures. Our results indicate that: (a) $q_2$ vanishes for all $T>\mathrm{}0\mathrm{}$; (b) ${\chi }_{\mathrm{SG}}\to \infty$ at some finite temperature ($T_{\mathrm{SG}}$); (c) the corresponding correlation length appears to fulfill $\xi \sim \exp \left[b{\left(\frac{T}{T_0-1\mathrm{}}\right)}^{-x}\right]$ and $x\gtrsim 2$; (d) both ${\chi }_{\mathrm{SG}}\mathrm{}\left(t\right)\mathrm{}$ and the equilibrium time-dependent correlation function $〈\overrightarrow{\mathrm{M}}\mathrm{}\left(0\right)\mathrm{}·\overrightarrow{\mathrm{M}}\mathrm{}\left(t\right)\mathrm{}〉$ relax logarithmically with $t$.