Tight-binding models for ultracold atoms in optical lattices: general formulation and applications

TL;DR

This paper develops a comprehensive framework for constructing tight-binding models for ultracold atoms in optical lattices using maximally localized Wannier functions, covering various lattice geometries and applications.

Contribution

It introduces a general formulation for tight-binding models based on Wannier functions and explores multiple applications including honeycomb lattices and Dirac equations.

Findings

Two methods for computing tight-binding coefficients are compared.

The phase diagram of the Haldane model is analyzed.

Conditions for the breakdown of the Peierls substitution are discussed.

Abstract

Tight-binding models for ultracold atoms in optical lattices can be properly defined by using the concept of maximally localized Wannier functions for composite bands. The basic principles of this approach are reviewed here, along with different applications to lattice potentials with two minima per unit cell, in one and two spatial dimensions. Two independent methods for computing the tight-binding coefficients - one ab initio, based on the maximally localized Wannier functions, the other through analytic expressions in terms of the energy spectrum - are considered. In the one dimensional case, where the tight-binding coefficients can be obtained by designing a specific gauge transformation, we consider both the case of quasi resonance between the two lowest bands, and that between s and p orbitals. In the latter case, the role of the Wannier functions in the derivation of an effective Dirac equation is also reviewed. Then, we consider the case of a two dimensional honeycomb potential, with particular emphasis on the Haldane model, its phase diagram, and the breakdown of the Peierls substitution. Tunable honeycomb lattices, characterized by movable Dirac points, are also considered. Finally, general considerations for dealing with the interaction terms are presented.