Mean-Field Dynamics of the Bose-Hubbard Model in High Dimension
Shahnaz Farhat, Denis Périce, Sören Petrat
TL;DR
This work rigorously justifies a mean-field dynamical description for the Bose–Hubbard model on a high-dimensional lattice by proving trace-norm convergence of the one-site reduced density to a nonlinear mean-field evolution as the dimension $d$ grows. The authors introduce two complementary approaches—the moment method and an excitation-energy method—to bound the deviation between the many-body Schrödinger dynamics and the mean-field dynamics, obtaining convergence rates and time-growth bounds under different assumptions on the parameters $J,$, and $U$. The analysis hinges on precise control of reduced densities, energy and moment bounds, and Gronwall-type estimates, making crucial use of a truncated, well-posed mean-field problem and detailed decompositions of the hopping and interaction contributions. The results provide rigorous underpinning for dynamical mean-field theory in bosonic lattice systems and illuminate how mean-field behavior emerges in the high-dimensional limit, with implications for understanding real-space phase dynamics such as the Mott–superfluid transition. The methods and bounds developed offer a template for analyzing beyond-mean-field corrections and for extending DMFT-type approximations to more complex lattice models.
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 Schrödinger 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.