Mean-Field Dynamics of the Bose-Hubbard Model in High Dimension

TL;DR

This paper rigorously justifies a mean-field approximation for the Bose-Hubbard model's dynamics in high dimensions, capturing phase transitions and providing a foundation for dynamical mean-field theory in bosonic systems.

Contribution

It establishes the validity of a mean-field approximation for the Bose-Hubbard model's dynamics in high dimensions, including phase transition phenomena.

Findings

Proves trace norm estimate between true and mean-field dynamics

Derives a non-trivial mean-field equation capturing phase transitions

Supports the validity of dynamical mean-field theory for bosons

Abstract

The Bose-Hubbard model effectively describes bosons on a lattice with on-site interactions and nearest-neighbour hopping, serving as a foundational framework for understanding strong particle interactions and the superfluid to Mott insulator transition. This paper aims to rigorously establish the validity of a mean-field approximation for the dynamics of quantum systems in high dimension, using the Bose-Hubbard model on a square lattice as a case study. We prove a trace norm estimate between the one-lattice-site reduced density of the Schr\"odinger dynamics and the mean-field dynamics in the limit of large dimension. Here, the mean-field approximation is in the hopping amplitude and not in the interaction, leading to a very rich and non-trivial mean-field equation. This mean-field equation does not only describe the condensate, as is the case when the mean-field description comes from a large particle number limit averaging out the interaction, but it allows for a phase transition to a Mott insulator since it contains the full non-trivial interaction. Our work is a rigorous justification of a simple case of the highly successful dynamical mean-field theory (DMFT) for bosons, which somewhat surprisingly yields many qualitatively correct results in three dimensions.