Study of Oscillations and Correlations in the Relaxation of a Model Spin System

Abstract

A detailed investigation is made of the approach to equilibrium of system of spins with $I=\frac{1}{2}$. The spin-spin interaction ${\ensuremath{\Sigma}}_{k}{\ensuremath{\Sigma}}_{j<k}{B}_{\mathrm{jk}}{{\ensuremath{\sigma}}_{x}}^{j}{{\ensuremath{\sigma}}_{x}}^{k}$ with ${B}_{\mathrm{jk}}$ constant is treated exactly, while the additional interaction ${\ensuremath{\Sigma}}_{k}{\ensuremath{\Sigma}}_{j<k}{L}_{\mathrm{jk}}{{\ensuremath{\sigma}}_{x}}^{j}{{\ensuremath{\sigma}}_{x}}^{k}$ with ${L}_{\mathrm{jk}}$ depending on lattice vibrations is treated by means of the assumption of sufficiently short correlation times for the ${L}_{\mathrm{jk}}$ operators. All correlations between the ${L}_{\mathrm{jk}}$ are included. The effect of the correlations usually neglected is expressed in terms of a sum over states somewhat resembling an Ising-model partition function with the time replacing interaction strength. The oscillatory relaxation via the ${B}_{\mathrm{jk}}$ and the monotonic relaxation via the phonons compete with each other; interference effects between the two relaxation modes also occur; the origin and nature of the irreversibility are very different for the two relaxation modes.