Search for localized Wannier functions of topological band structures via compressed sensing

TL;DR

This paper extends compressed sensing methods to find maximally localized Wannier functions in topological band structures, enabling systematic identification of localized representatives and exploring their existence in topological insulators.

Contribution

It develops a practical, symmetry-preserving compressed sensing framework for Wannier functions, including efficient algorithms and applications to topological superconductors and insulators.

Findings

Efficient $O(N \, log N)$ algorithm for Wannier function localization

Numerical evidence against existence of compact Wannier functions in 2D topological insulators

Benchmark results demonstrating the method's power in topological superconductors

Abstract

We investigate the interplay of band structure topology and localization properties of Wannier functions. To this end, we extend a recently proposed compressed sensing based paradigm for the search for maximally localized Wannier functions [Ozolins et al., PNAS 110, 18368 (2013)]. We develop a practical toolbox that enables the search for maximally localized Wannier functions which exactly obey the underlying physical symmetries of a translationally invariant quantum lattice system under investigation. Most saliently, this allows us to systematically identify the most localized representative of a topological equivalence class of band structures, i.e., the most localized set of Wannier functions that is adiabatically connected to a generic initial representative. We also elaborate on the compressed sensing scheme and find a particularly simple and efficient implementation in which each step of the iteration is an $O(N \log N)$ algorithm in the number of lattice sites $N$. We present benchmark results on one-dimensional topological superconductors demonstrating the power of these tools. Furthermore, we employ our method to address the open question whether compact Wannier functions can exist for symmetry protected topological states like topological insulators in two dimensions. The existence of such functions would imply exact flat band models with strictly finite range hopping. Here, we find numerical evidence for the absence of such functions. We briefly discuss applications in dissipative state preparation and in devising variational sets of states for tensor network methods.