Non-Markovian dynamics in a spin star system: the failure of thermalisation

In most cases, a small system weakly interacting with a thermal bath will eventually reach a thermal state with the temperature of the bath. We show that this intuitive picture is not always true, by using a spin star model where the non-Markov effect dominates the dynamical process. The spin star system consists of a central spin homogeneously interacting with an ensemble of identical noninteracting spins. We find that the correlation time of the bath is infinite, which implies that the bath has a perfect memory, and that the dynamical evolution of the central spin must be non-Markovian. A direct consequence of this is that the final state of the central spin is not the thermal state in equilibrium with the bath, but a steady state which depends on the initial state of the spin.