Universality of Abelian and non-Abelian Wannier functions in one dimension

TL;DR

This paper demonstrates the universal scaling behavior of Wannier functions in one-dimensional insulators using a Dirac model, revealing fundamental relations between boundary charges, fluctuations, and bulk properties across various lattice models.

Contribution

It introduces a field-theoretical framework for Wannier functions and boundary phenomena, establishing their universal scaling and relations in 1D insulators, including Abelian and non-Abelian gauges.

Findings

Universal scaling functions depend only on gap phase and correlation length.

Boundary charge and fluctuations relate universally to bulk properties.

Wannier functions of maximal localization exhibit universal scaling and are related to Dirac Wannier functions.

Abstract

Within a Dirac model in $1+1$ dimensions, a prototypical model to describe low-energy physics for a wide class of lattice models, we propose a field-theoretical version for the representation of Wannier functions, the Zak-Berry connection, and the geometric tensor. In two natural Abelian gauges we present universal scaling of the Dirac Wannier functions in terms of four fundamental scaling functions that depend only on the phase $\gamma$ of the gap parameter and the charge correlation length $\xi$ in an insulator. The two gauges allow for a universal low-energy formulation of the surface charge and surface fluctuation theorem, relating the boundary charge and its fluctuations to bulk properties. Our analysis describes the universal aspects of Wannier functions for the wide class of one-dimensional generalized Aubry-Andr\'e-Harper lattice models. In the low-energy regime of small gaps we demonstrate universal scaling of all lattice Wannier functions and their moments in the corresponding Abelian gauges. We also discuss non-Abelian lattice gauges and find that lattice Wannier functions of maximal localization show universal scaling and are uniquely related to the Dirac Wannier function of the lower band. Our results present evidence that universal aspects of Wannier functions and of the boundary charge are uniquely related and can be elegantly described within universal low-energy theories.