Triviality of Bloch and Bloch-Dirac bundles

TL;DR

This paper investigates the triviality of Bloch and Bloch-Dirac bundles in periodic potentials, demonstrating that smooth, periodic quasi-Bloch functions exist for dimensions less than four by leveraging time-reversal symmetry and bundle theory.

Contribution

It generalizes previous results by proving the triviality of Bloch bundles in dimensions less than four using symmetry and bundle-theoretic methods.

Findings

Positive answer for d<4 dimensions

Utilizes time-reversal symmetry in analysis

Applicable to Dirac equation and piezoelectricity

Abstract

In the framework of the theory of an electron in a periodic potential, we reconsider the longstanding problem of the existence of smooth and periodic quasi-Bloch functions, which is shown to be equivalent to the triviality of the Bloch bundle. By exploiting the time-reversal symmetry of the Hamiltonian and some bundle-theoretic methods, we show that the problem has a positive answer for any d < 4, thus generalizing a previous result by G. Nenciu. We provide a general formulation of the result, aiming at the application to the Dirac equation with a periodic potential and to piezoelectricity.