Effects of disorder on the dynamics of theXYchain

Abstract

We investigate the effects of disorder on the dynamics of the $s=1/2$ $\mathrm{XY}$ model in one dimension. The energy couplings are randomly drawn independently from a bimodal distribution. We use an extension of the method of recurrence relations, in which an averaging over realizations of disorder is incorporated into the definition of the scalar product of the dynamical Hilbert space of ${\ensuremath{\sigma}}_{j}^{z}(t),$ to determine analytically the first six basis vectors as well as the corresponding recurrants. We then use an ansatz for the higher-order recurrants, based on the behavior of the first ones exactly determined, to obtain the time-dependent correlation functions and spectral densities for several degrees of disorder. We find that the dynamics at long times is governed by the stronger couplings present in the system even if only a very small amount of disorder is present. In the long-time limit, the correlation functions oscillate at the cutoff frequency of the disorderless stronger-coupling case, with the spectral densities displaying tails that end at the stronger-coupling cutoff frequency.