Exact expression for the Berry connection in the projection gauge

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.

Figures (3)

• Figure 1: Berry connection in the projection gauge as a function of crystal momentum (reduced coordinates) for $v=1$ and $w=1.5$. The solid black curve shows the single-point Berry connection. Dashed curves show the discrete connection obtained from Wilson loops at increasing mesh densities. As $N_k$ increases, the discrete connection converges to the single-point result.
• Figure 2: The axion angle $\theta(\beta)$ over one adiabatic cycle $\beta \in [0, 2\pi]$. We compute $\theta$ using three independent approaches: (i) the gauge-invariant formulation based on integrating the second Chern form (four-curvature), shown as a dotted black line; (ii) the Chern--Simons 3-form from Eq. (\ref{['eq:CS3form-curv']}) using the analytic single-point Berry connection of Eq. (\ref{['eq:tildeA_final']}), plotted in blue; and (iii) the same Chern--Simons expression evaluated using the finite difference connection of Eq. (\ref{['eq:fd_connection']}), shown in red. The vertical green dotted lines mark the region within which we use trial wavefunctions that do not respect $\mathcal{T}$. The nontrivial second Chern number ($C_2 = 1$) produces a net $2\pi$ shift of $\theta$ over one adiabatic cycle.
• Figure 3: The convergence of $\theta$ toward its quantized value of $\pi$, when $\beta=\pi$, with respect to the $k$-mesh density. The horizontal axis denotes the number of $k$-points along each reciprocal lattice vector; e.g., a given $N_{k}$ corresponds to an $N_k \times N_k \times N_k$-sized $k$-mesh. The single-point formulation rapidly approaches the continuum result, whereas the finite-difference construction converges more slowly due to additional discretization error in the Berry connection itself.