Leading-Order Auxiliary Field Theory of the Bose-Hubbard Model

TL;DR

This paper introduces the LOAF theory for the Bose-Hubbard model, providing a non-perturbative, conserving approximation that captures phase transitions, critical temperatures, and phase diagrams consistent with experiments and simulations.

Contribution

The paper develops the leading-order auxiliary field (LOAF) theory for the Bose-Hubbard model, offering a novel non-perturbative approach that accurately describes phase transitions and critical phenomena.

Findings

LOAF predicts both first and second-order phase transitions.

The phase diagram includes a line of first-order transitions ending at a critical point.

LOAF results align qualitatively with experimental data and Monte Carlo simulations.

Abstract

We discuss the phase diagram of the Bose-Hubbard (BH) model in the leading-order auxiliary field (LOAF) theory. LOAF is a conserving non-perturbative approximation that treats on equal footing the normal and anomalous density condensates. The mean-field solutions in LOAF correspond to first-order and second-order phase transition solutions with two critical temperatures corresponding to a vanishing Bose-Einstein condensate, $T_c$, and a vanishing diatom condensate, $T^\star$. The \emph{second-order} phase transition solution predicts the correct order of the transition in continuum Bose gases. For either solution, the superfluid state is tied to the presence of the diatom condensate related to the anomalous density in the system. In ultracold Bose atomic gases confined on a three-dimensional lattice, the critical temperature $T_c$ exhibits a quantum phase transition, where $T_c$ goes to zero at a finite coupling. The BH phase diagram in LOAF features a line of first-order transitions ending in a critical point beyond which the transition is second order while approaching the quantum phase transition. We identify a region where a diatom condensate is expected for temperatures higher than $T_c$ and less than $T_0$, the critical temperature of the non-interacting system. The LOAF phase diagram for the BH model compares qualitatively well with existing experimental data and results of \emph{ab initio} Monte Carlo simulations.