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Wannier decay and the Thouless conjecture

Simon Becker, Zhongkai Tao, Mengxuan Yang

TL;DR

The paperaddresses how topology, encoded by Chern classes, constrains the localization of Wannier functions in periodic systems. By decomposing Bloch bundles and analyzing singular sections, it constructs explicit Wannier bases with optimal nonexponential decay rates: in 2D, a full asymptotic expansion yields decay $O(|x|^{-2})$ in agreement with Thouless, while in 3D the method delivers a uniform decay $O(|x|^{-7/3})$ for the last Wannier function. The results tie decay to the regularity of Bloch frames, showing exponential localization when $c_1$ vanishes and precise asymptotics when it does not. This advances understanding of topological obstruction to localization and sharpens predictions for decay rates in two- and three-dimensional topological insulators.

Abstract

Non-trivial Chern classes pose an obstruction to the existence of exponentially decaying Wannier functions which provide natural bases for spectral subspaces. For non-trivial Bloch bundles, we obtain decay rates of Wannier functions in dimensions $d=2,3$. For $d=2$, we construct Wannier functions with full asymptotics and optimal decay rate $\mathcal{O}(|x|^{-2})$ as conjectured by Thouless; for $d=3$, we construct Wannier functions with the uniform decay rate $\mathcal{O}(|x|^{-7/3})$.

Wannier decay and the Thouless conjecture

TL;DR

The paperaddresses how topology, encoded by Chern classes, constrains the localization of Wannier functions in periodic systems. By decomposing Bloch bundles and analyzing singular sections, it constructs explicit Wannier bases with optimal nonexponential decay rates: in 2D, a full asymptotic expansion yields decay in agreement with Thouless, while in 3D the method delivers a uniform decay for the last Wannier function. The results tie decay to the regularity of Bloch frames, showing exponential localization when vanishes and precise asymptotics when it does not. This advances understanding of topological obstruction to localization and sharpens predictions for decay rates in two- and three-dimensional topological insulators.

Abstract

Non-trivial Chern classes pose an obstruction to the existence of exponentially decaying Wannier functions which provide natural bases for spectral subspaces. For non-trivial Bloch bundles, we obtain decay rates of Wannier functions in dimensions . For , we construct Wannier functions with full asymptotics and optimal decay rate as conjectured by Thouless; for , we construct Wannier functions with the uniform decay rate .
Paper Structure (8 sections, 7 theorems, 66 equations)

This paper contains 8 sections, 7 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Backgrounds and main results
  3. Wannier decay and regularities of Bloch frames
  4. Complex vector bundles and Wannier functions
  5. Classification of complex vector bundles
  6. Chern numbers and localization dichotomy
  7. Asymptotics of Wannier functions on ${\mathbb T}^2$
  8. Decay of Wannier function on ${\mathbb T}^3$

Key Result

Theorem 1

Let $\mathcal{E}$ be a Bloch bundle of rank $r$ with Chern number $c_1(\mathcal{E}) = m \neq 0$ over $\mathbb T^2$, then there exists a Wannier basis $\mathcal{W} = \{\psi_a(x-\gamma): \gamma\in\Gamma, a\in\{1,\cdots, r\}\}$ such that $\psi_a(x)$ decays exponentially for $a \in\{1,\cdots, r-1\}$ and with the leading coefficient $c_{m,0}(\theta_x)=2\pi (-1)^m me^{-im \theta_x}$, $\sin\theta_x=x_1/|

Theorems & Definitions (21)

  • Theorem 1
  • Theorem 2
  • Definition 1.1
  • Remark 1
  • Definition 1.2
  • Definition 1.3
  • Definition 2.1
  • Remark 2
  • Proposition 2.2
  • proof
  • ...and 11 more