Time-Dependent Order Parameters in Spin-Glasses

Abstract

It is proposed that the spin-glass phase is described by two order parameters $q(x)={[〈{S}_{i}(0){S}_{i}({t}_{x})〉]}_{\mathrm{a}\mathrm{v}}$ and $\ensuremath{\chi}(x)=\ensuremath{\int}{0}^{{t}_{x}}\ensuremath{\chi}(t)\mathrm{dt}$, that measure the relaxations of the average autocorrelation and susceptibility along macroscopic time scales, ${t}_{x}$, which are parametrized in a decreasing order by $x\ensuremath{\in}[0,1]$. The function $q$ decays from a finite value at ${t}_{1}$ to zero at ${t}_{0}$, while $\ensuremath{\chi}$ increases from $\ensuremath{\chi}(1)$ to a value $\ensuremath{\chi}(0)$ which is independent of temperature in the mean-field limit. The equilibrium results agree with Parisi's replica solution.