A Discrete Formulation of Second Stiefel-Whitney Class for Band Theory

TL;DR

This paper introduces a fully discrete, gauge-independent method to compute the second Stiefel-Whitney class in band theory, enabling topological analysis from discrete Bloch states without continuum assumptions.

Contribution

It develops a novel discrete, gauge-fixing-free formula for the second Stiefel-Whitney class applicable to first-principles band calculations.

Findings

Provides a discrete formula for $w_2$ based on sampled Bloch states

Connects topological invariants to lattice field theory concepts

Enables topological classification from discrete data

Abstract

Topological invariants in band theory are often formulated assuming that Bloch wave functions are smoothly defined over the Brillouin zone (BZ). However, first-principles band calculations typically provide Bloch states only at discrete points in the BZ, rendering standard continuum-based approaches inapplicable. In this work, we focus on the second Stiefel-Whitney class $w_2$, a key $\mathbb{Z}_2$ topological invariant under PT symmetry that characterizes various higher-order topological insulators and nodal-line semimetals. We develop a fully discrete, gauge-fixing-free formula for $w_2$ which depends solely on the Bloch states sampled at discrete BZ points. Furthermore, we clarify how our discrete construction connects to lattice field theory, providing a unifying perspective that benefits both high-energy and condensed matter approaches.