Correlations in the Bose & Fermi Hubbard Model

TL;DR

This paper develops an analytical framework for studying correlations in large-coordination-number Bose-Hubbard and Fermi-Hubbard models, revealing insights into phase coherence growth, local observable relaxation, and tunnelling phenomena with comparisons to numerical simulations.

Contribution

It introduces a 1/Z expansion method to analytically analyze correlations and dynamics in large-coordination Hubbard models, including novel results on phase growth and tunnelling effects.

Findings

Growth of phase coherence after a quench from Mott to superfluid

Local observables approach non-thermal quasi-equilibrium states

Tunnelling probability in tilted lattices resembles Schwinger effect

Abstract

We study the Bose-Hubbard and Fermi-Hubbard model in the limit of large coordination numbers Z (i.e., many tunnelling partners). Via a controlled expansion into powers of 1/Z, we establish a hierarchy of correlations, which facilitates an approximate analytic solution of the quantum evolution. For the Bose-Hubbard model, we derive the growth of phase coherence after a quench from the Mott to the superfluid phase. For a quench within the Mott phase, we find that various local observables approach a quasi-equilibrium state after a finite period of time. However, this state is not thermal, i.e., real thermalisation -- if it occurs -- requires much longer time scales. For a tilted lattice in the Mott state, we calculate the tunnelling probability and find a remarkable analogy to the Sauter-Schwinger effect (i.e., electron-positron pair creation out of the vacuum due to a strong electric field). These analytical results are compared to numerical simulations for finite lattices in one and two dimensions and we find qualitative agreement. Finally, we generalize these studies to the more involved case of the Fermi-Hubbard model.