Generalized Bloch theorem and topological characterization

TL;DR

This paper generalizes the Bloch theorem to include all crystal symmetries, providing a unified framework for Hamiltonians and Berry curvature calculations in periodic systems.

Contribution

It introduces a symmetry-inclusive generalization of the Bloch theorem, expanding its applicability to anisotropic interactions and clarifying the relation to Berry curvature.

Findings

The generalized Bloch theorem constrains Hamiltonian forms with additional symmetries.

For isotropic interactions, the theorem yields a unique Hamiltonian matching the periodic gauge.

The average Berry curvature over all Hamiltonians in the Bloch gauge equals that in the periodic gauge.

Abstract

The Bloch theorem enables reduction of the eigenvalue problem of the single-particle Hamiltonian that commutes with translational group. Based on a group theory analysis we present generalization of the Bloch theorem that incorporates all additional symmetries of a crystal. The generalized Bloch theorem constrains the form of the Hamiltonian which becomes manifestly invariant under additional symmetries. In the case of isotropic interactions the generalized Bloch theorem gives a unique Hamiltonian. This Hamiltonian coincides with the Hamiltonian in the periodic gauge. In the case of anisotropic interactions the generalized Bloch theorem allows a family of Hamiltonians. Due to the continuity argument we expect that even in this case the Hamiltonian in the periodic gauge defines observables, such as Berry curvature, in the inverse space. For both cases we present examples and demonstrate that the average of the Berry curvatures of all possible Hamiltonians in the Bloch gauge is the Berry curvature in the periodic gauge.