Monte Carlo simulation of a canonical spin-glass

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 ${{χ}}SG(t)={N}^{{-}1}{{Σ}}ᵢⱼ〈[{{{→}}{S}}ᵢ(0)·{}{{{→}}{S}}ⱼ(0)][{{{→}}{S}}ᵢ(t)·{}{{{→}}{S}}ⱼ(t)]〉$, which has the following limiting behavior: (a) it becomes the generalized susceptibility, ${N}^{{-}1}{{Σ}}ᵢⱼ{〈{{{→}}{S}}ᵢ·{}{{{→}}{S}}ⱼ〉}²$ as $t{→}{∞}$; (b) ${N}^{{-}1}{{χ}}SG(t)$ becomes an order parameter (${q}₂$) as $N{→}{∞}$ and $t{→}{∞}$. We study ${{χ}}SG(t)$ as a function of $t$ for systems of $N=22,40,87$, and 169 spins for various temperatures. Our results indicate that: (a) ${q}₂$ vanishes for all $T>0$; (b) ${{χ}}SG{→}{∞}$ at some finite temperature (${T}SG$); (c) the corresponding correlation length appears to fulfill ${ξ}{~}exp[b{({T}{{T}₀{-}1})}^{{-}x}]$ and $x{}2$; (d) both ${{χ}}SG(t)$ and the equilibrium time-dependent correlation function $〈{{→}}{M}(0)·{}{{→}}{M}(t)〉$ relax logarithmically with $t$.