Shaping Maximally Localized Wannier Functions via Discrete Adiabatic Transport

TL;DR

This paper introduces a deterministic, non-variational algorithm for constructing maximally localized Wannier functions by integrating gauge smoothing and eigenvalue problems, with applications to understanding geometric effects in graphene.

Contribution

A novel constructive method that unifies gauge smoothing and eigenvalue problems for Wannier functions, avoiding traditional spread minimization.

Findings

Benchmark calculations show good agreement with standard methods.

The approach isolates the physical origin of mesh-dependent spread scaling in graphene.

Discrete adiabatic transport naturally emerges in the solution process.

Abstract

Maximally localized Wannier functions (MLWFs) are conventionally constructed by iteratively minimizing a spread functional over a high-dimensional gauge landscape. In this work, we present a non-variational constructive algorithm that unifies gauge smoothing and the eigenvalue problem of the projected position operator into a single deterministic framework. We demonstrate that discrete adiabatic transport across band degeneracies emerges naturally as an integral part of the solution procedure for the position eigenvectors. In this transport-aligned gauge, the Bloch overlaps exhibit an approximately linear phase dependence, allowing the Wannier centers to be extracted via deterministic fixed-point iterations and self-consistent updates rather than spread-functional minimization. Benchmark calculations for one- and two-dimensional systems yield spreads and orbital shapes in good agreement with standard minimization schemes. Furthermore, this analytical approach transparently isolates the physical origin of the $\mathcal{O}(L)$ mesh-dependent spread scaling ($L$ being the boundary seam resolution) observed in graphene, demonstrating that it is an intrinsic geometric manifestation of non-commuting projected position operators forcing finite gauge defects to accumulate along a one-dimensional boundary seam.