Optimal decay of Wannier functions in Chern and Quantum Hall insulators

TL;DR

This paper establishes a fundamental link between the decay properties of Wannier functions and the topological nature of Chern insulators and Quantum Hall systems, showing that non-trivial topology prevents exponential localization.

Contribution

It proves a localization dichotomy: systems are either topologically trivial with exponentially localized Wannier functions or topologically non-trivial with delocalized Wannier functions.

Findings

Exponential localization implies trivial topology and zero Hall conductivity.

Topologically non-trivial systems lack exponentially localized Wannier functions.

The decay of Wannier functions correlates with the system's topological phase.

Abstract

We investigate the localization properties of independent electrons in a periodic background, possibly including a periodic magnetic field, as e.g. in Chern insulators and in Quantum Hall systems. Since, generically, the spectrum of the Hamiltonian is absolutely continuous, localization is characterized by the decay, as $|x| \rightarrow \infty$, of the composite (magnetic) Wannier functions associated to the Bloch bands below the Fermi energy, which is supposed to be in a spectral gap. We prove the validity of a localization dichotomy, in the following sense: either there exist exponentially localized composite Wannier functions, and correspondingly the system is in a trivial topological phase with vanishing Hall conductivity, or the decay of any composite Wannier function is such that the expectation value of the squared position operator, or equivalently of the Marzari-Vanderbilt localization functional, is $+ \infty$. In the latter case, the Bloch bundle is topologically non-trivial, and one expects a non-zero Hall conductivity.