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

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 (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(t)=〈〈σᵢ(0)σᵢ(t)〉〉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 $t=0$, then [∂m(H,t)∂H]H=0=(1/kT)[1-q(t)] holds exactly in any number of dimensions, where m is the magnetization per spin. A time-dependent susceptibility χ(H,t) is defined and shown to vanish (as H→0) for low enough temperatures and finite t. The MC results for χ 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.