Berry Phase of Bloch States through Modular Symmetries

TL;DR

This paper derives an expression for the Berry phase in crystalline materials using Gaussian orbitals, linking modular symmetries to the Zak phase and topological properties without requiring inversion symmetry.

Contribution

It introduces a novel approach to compute Berry phases via analytical Bloch states and connects modular symmetries with topological invariants in complex space groups.

Findings

Derived an explicit formula for the Berry phase using Gaussian orbitals.

Linked eigenvalues of modular symmetries to the Zak phase.

Showed modular symmetries can define topological distinctions without inversion symmetry.

Abstract

The theoretical identification of crystalline topological materials has enjoyed sustained success in simplified materials models, often by singling out discrete symmetry operations protecting the topological phase. When band structure calculations of realistic materials are considered, complications often arise owing to the requirement of a consistent gauge in the Brillouin zone, or down to the fineness of its sampling. Yet, the Berry phase, acting as topological label, encodes geometrical properties of the system, and it should be easily accessible. Here, an expression for the Berry phase is obtained, thanks to analytical Bloch states constructed from an infinite series of $s$-type Gaussian orbitals. Two contributions in the Berry phase are identified, with one having an immediate geometric interpretation, being equal to the Zak phase. Eigenvalues of a modular symmetry, considered here for the first time in the context of crystalline solid state systems, are put in correspondence with the Zak phase: modular symmetries allow to define a non-trivial action for the spatial inversion also when the system does not have an inversion centre, as for the considered case of space group no. 22 ($F222$), which is known to host symmetry equivalent Bloch states distinguishable by their Berry phase.