When a physical system is coupled to a heat bath, one expects to observe thermalization to an equilibrium state whose temperature is determined by the bath properties. For an isolated many-body system, i.e. in the absence of a heat bath, the situation is less clear, although some kind of relaxation to equilibrium may be expected for sufficiently large generic systems and suitable observables. Recent progress in experiments with cold atoms and ions has stimulated intense theoretical interest in equilibration and thermalization behaviour of isolated many-body quantum systems. General mechanisms leading to thermalization have been proposed, rigorous proofs of equilibration have been obtained for generic Hamiltonians, and analytic as well as numeric model studies have been reported. Much less is known about the time scales on which relaxation to equilibrium takes place, though a deeper theoretical understanding would be beneficial also for the interpretation of data from cold atom or ion experiments. Here we study the time evolution of correlation functions in isolated long-range interacting quantum Ising systems. By imposing certain restrictions on the initial conditions and/or parameter values, exact analytic results are obtained in arbitrary lattice dimension, for ferromagnetic or antiferromagnetic coupling, and hence also in the presence of geometric frustration. Gaussian (instead of the commonly expected exponential) relaxation to equilibrium is found. Increasing the long-range character of the interactions beyond a certain threshold, relaxation takes place in two steps on two widely separated time scales, showing pronounced prethermalization plateaus. Specializing to a triangular lattice in two spatial dimensions, we propose to utilize these results for benchmarking a recently developed ion-trap based quantum simulator.
Citation: Nature Physics
Pub Type: Journals
ion trap, Ising model, long-range interactions, quantum simulation, relaxation, spin correlations, thermalization