Optical Lattice Emulators: Bose and Fermi Hubbard Models

TL;DR

This paper reviews the Hubbard model for bosons and fermions, comparing mean field theory and quantum Monte Carlo methods to understand quantum phase transitions and experimental results.

Contribution

It provides a pedagogical comparison of two key computational methods for Hubbard models and discusses their implications for quantum phase transitions and experiments.

Findings

Quantum Monte Carlo offers more accurate results than mean field theory.

Fluctuations significantly influence quantum phase transitions.

Comparison with experiments validates the theoretical approaches.

Abstract

This chapter is a pedagogical review of the Hubbard model for bosons with repulsion and for fermions with attraction and repulsion primarily using two methods, one chosen for its simplicity and insights (mean field theory) and the other chosen for its accuracy and reliability (quantum Monte Marlo methods). From a comparison of the two methods we glean valuable information into the effects of fluctuations that dominate quantum phase transitions. The chapter includes an in-depth comparison with experiments. We conclude with a discussion of future developments where the technical methods expounded on here, mean field theory and quantum Monte Carlo, could be useful.