Time-Dependent Order Parameters in Spin-Glasses

Abstract

It is proposed that the spin-glass phase is described by two order parameters $q\left(x\right)={\left[〈S_i\mathrm{}\right(0\left)\mathrm{}{S}_i\right(t_x\left)〉\right]}_{\mathrm{a}\mathrm{v}}$ and $\chi \mathrm{}\left(x\right)=\int 0^{t_x}\chi \mathrm{}\left(t\right)\mathrm{}\mathrm{dt}$, that measure the relaxations of the average autocorrelation and susceptibility along macroscopic time scales, $t_x$, which are parametrized in a decreasing order by $x\in \mathrm{}[0,1]\mathrm{}$. The function $q$ decays from a finite value at $t_1$ to zero at $t_0$, while $\chi$ increases from $\chi \mathrm{}\left(1\right)\mathrm{}$ to a value $\chi \mathrm{}\left(0\right)\mathrm{}$ which is independent of temperature in the mean-field limit. The equilibrium results agree with Parisi's replica solution.