Constructing optimal Wannier functions via potential theory: isolated single band for matrix models

TL;DR

This paper introduces a fast, convergent method for computing globally optimal Wannier functions in two-dimensional matrix models, effectively handling topological obstructions and symmetry considerations.

Contribution

It develops a new scheme for constructing optimal Wannier functions directly from parallel transport, including corrections, and addresses topological and symmetry effects.

Findings

Scheme converges rapidly in numerical tests

Constructs real Wannier functions under time-reversal symmetry

Transforms non-optimal Wannier functions into global optima using gauge transformations

Abstract

We present a rapidly convergent scheme for computing globally optimal Wannier functions of isolated single bands for matrix models in two dimensions. The scheme proceeds first by constructing provably exponentially localized Wannier functions directly from parallel transport (with simple analytically computable corrections) when topological obstructions are absent. We prove that the corresponding Wannier functions are real when the matrix model possesses time-reversal symmetry. When a band has a nonzero Berry curvature, the resulting Wannier function is not optimal, but it is transformed into the global optimum by a single gauge transformation that eliminates the divergence of the Berry connection. Complete analysis of the construction is presented, paving the way for further improvements and generalizations. The performance of the scheme is illustrated with several numerical examples.