Remanence and non-exponential relaxation in an Ising chain with random bonds

Abstract

We study the time-dependent properties of an Ising chain with random bonds Gaussianly distributed. We obtain some exact as well as Monte Carlo (MC) results. Remanence, as well as a seemingly logarithmic long-time decay of the magnetization and of the energy towards their equilibrium values is observed at low temperatures by means of MC simulation. The remanent values of the magnetization ($\frac{1}{3}$, starting with all spins up) and of the energy are derived. A simple and explicit physical picture of the mechanism behind remanence and the logarithmic relaxation emerges. The equilibrium value of $q\left(t\right)={〈〈{\sigma }_i\mathrm{}\left(0\right)\mathrm{}{\sigma }_i\mathrm{}\left(t\right)\mathrm{}〉〉}_J$ is obtained via the MC technique; it also seems to relax logarithmically for low temperatures. In contrast with the two-and three-dimensional cases, it is shown how any MC calculation can start immediately from an "equilibrium state" in this model, a very convenient feature for very-low-temperature MC computations. We show that if $H$ is switched on at $t=0\mathrm{}$, then ${\left[\frac{\partial m(H,t)\mathrm{}}{\partial H}\right]}_{H=0\mathrm{}}=\left(\frac{1}{\mathrm{kT}}\right)[1-q(t\left)\right]$ holds exactly in any number of dimensions, where $m$ is the magnetization per spin. A time-dependent susceptibility $\chi \mathrm{}(H,t)\mathrm{}$ is defined and shown to vanish (as $H\to 0$) for low enough temperatures and finite $t$. The MC results for $\chi$ are in accord with this result and, if graphed versus $T$, show a hump at a time-dependent temperature. Finally, we compare the exact equilibrium specific heat, $C$, for this model with the results obtained by MC simulation of calorimetric measurements. Thus, a simple explicit case is exhibited of the difficulties which may arise in measurements of $C$ in spin-glasses due to long-time effects.