Finite size effects in global quantum quenches: examples from free bosons in an harmonic trap and the one-dimensional Bose-Hubbard model

Abstract

We investigate finite size effects in quantum quenches on the basis of simple energetic arguments. Distinguishing between the low-energy part of the excitation spectrum, below a microscopic energy-scale, and the high-energy regime enables one to define a crossover number of particles that is shown to diverge in the small quench limit. Another crossover number is proposed based on the fidelity between the initial and final ground-states. Both criteria can be computed using ground-state techniques that work for larger system sizes than full spectrum diagonalization. As examples, two models are studied: one with free bosons in an harmonic trap which frequency is quenched, and the one-dimensional Bose-Hubbard model, that is known to be non-integrable and for which recent studies have uncovered remarkable non-equilibrium behaviors. The diagonal weights of the time-averaged density-matrix are computed and observables obtained from this diagonal ensemble are compared with the ones from statistical ensembles. It is argued that the "thermalized" regime of the Bose-Hubbard model, previously observed in the small quench regime, experiences strong finite size effects that render difficult a thorough comparison with statistical ensembles. In addition, we show that the non-thermalized regime, emerging on finite size systems and for large interaction quenches, is not related to the existence of an equilibrium quantum critical point but to the high energy structure of the energy spectrum in the atomic limit. Its features are reminiscent of the quench from the non-interacting limit to the atomic limit.