Relaxation functions, memory functions, and random forces in the one-dimensional spin-1/2XYand transverse Ising models

We investigate the dynamics of the one-dimensional S=(1/2) isotropic XY model and transverse Ising model in the high-temperature limit by using the method of recurrence relations. We obtain the relaxation functions as well as some Brownian analogs of the generalized Langevin equation for a tagged spin $S_j^x$ in these models, namely, the memory functions and the random forces. We find that the realized dynamical Hilbert spaces of the two models have the same structure, which leads to similar dynamical behavior apart from a time scale.