Exact expression for the Berry connection in the projection gauge

TL;DR

This paper derives an exact, local expression for the Berry connection in the projection gauge, improving the accuracy and stability of geometric property calculations in topological insulators and related materials.

Contribution

It introduces a closed-form, local formula for the non-Abelian Berry connection in the projection gauge, enhancing computational stability and accuracy.

Findings

Validated in 1D and 3D models for Berry phase and axion angle

Provides a stable, local framework for Wannier-based geometric calculations

Reduces errors from discretization and gauge misalignment in numerical evaluations

Abstract

The Berry connection encodes the momentum-space geometry of occupied Bloch states in gapped insulators and plays a central role in topological materials. While gauge-invariant quantities can be evaluated from overlap matrices between neighboring $k$ points, accessing the Berry connection itself as a smooth field requires specifying a continuous gauge over the Brillouin zone. Wannier-based workflows achieve this through projection onto localized orbitals, enabling stable evaluation of geometric quantities and response functions. In this setting, the Berry connection enters directly in Wannier-interpolated calculations of polarization, Berry curvature, optical conductivity, and related response functions. In practical implementations, however, the projection-gauge Berry connection is typically constructed from finite-difference overlaps between neighboring $k$ points, discretizing momentum derivatives and introducing errors tied to $k$-mesh spacing and gauge alignment. These effects can become numerically delicate in systems with small band gaps or when evaluating higher-order responses such as the Chern-Simons axion angle. Here, we derive an exact expression for the non-Abelian Berry connection in the projection gauge that is local in crystal momentum. Starting from projected and orthonormalized Bloch-like states, we obtain a closed-form equation expressed entirely in terms of $k$-local quantities. We validate the formulation in one and three dimensions by computing the Berry phase and Chern-Simons axion angle in tight-binding models. The resulting framework provides a stable route to evaluating geometric properties within Wannier interpolation schemes and future first-principles implementations.