Exponentially Localized Wannier Functions in Periodic Zero Flux Magnetic Fields

TL;DR

This paper establishes conditions for the existence of exponentially localized Wannier functions in periodic systems with zero flux magnetic fields, extending previous results and introducing magnetic symmetry considerations.

Contribution

It extends prior work to include zero flux magnetic fields and introduces magnetic symmetry, providing new criteria for Wannier function localization in various dimensions.

Findings

Magnetic time-reversal symmetry ensures trivial Bloch bundle in 1-3D.

Existence of localized Wannier basis in 4D depends on trace conditions.

In higher dimensions, Chern classes are insufficient for guaranteeing localization.

Abstract

In this work, we investigate conditions which ensure the existence of an exponentially localized Wannier basis for a given periodic hamiltonian. We extend previous results in [Pan07] to include periodic zero flux magnetic fields which is the setting also investigated in [Kuc09]. The new notion of magnetic symmetry plays a crucial role; to a large class of symmetries for a non-magnetic system, one can associate "magnetic" symmetries of the related magnetic system. Observing that the existence of an exponentially localized Wannier basis is equivalent to the triviality of the so-called Bloch bundle, a rank m hermitian vector bundle over the Brillouin zone, we prove that magnetic time-reversal symmetry is sufficient to ensure the triviality of the Bloch bundle in spatial dimension d=1,2,3. For d=4, an exponentially localized Wannier basis exists provided that the trace per unit volume of a suitable function of the Fermi projection vanishes. For d>4 and d \leq 2m (stable rank regime) only the exponential localization of a subset of Wannier functions is shown; this improves part of the analysis of [Kuc09]. Finally, for d>4 and d>2m (unstable rank regime) we show that the mere analysis of Chern classes does not suffice in order to prove trivility and thus exponential localization.