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arXiv · 2403.05291

A discrete formulation for three-dimensional winding number

Ken Shiozaki
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Abstract

For a smooth map $g: X \to U(N)$, where $X$ is a three-dimensional, oriented, and closed manifold, the winding number is defined as $W_3 = \frac{1}{24\pi^2} \int_{X} \mathrm{Tr}\left[(g^{-1}dg)^3\right]$. We present a discrete formulation to compute $W_3$ based on the concept of $\theta$-gaps. Our approach provides a robust scheme that is directly applicable even to systems with accidental or symmetry-enforced degeneracies. Furthermore, we define two versions of the discrete flux: a simple unmodified flux that is highly practical and almost always quantized for fine grids, and a modified flux that strictly ensures integer quantization.

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Ken Shiozaki. 2024-03-08. A discrete formulation for three-dimensional winding number. https://doi.org/10.21468/scipostphyscore.9.2.026

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