Papers › A discrete formulation for three-dimensional winding number
A discrete formulation for three-dimensional winding number
Ken Shiozaki
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For a smooth map g: X →U(N), where X is a three-dimensional, oriented, and closed manifold, the winding number is defined as W₃ = 1/24π² ∫_X Tr[(g⁻¹dg)³]. We present a discrete formulation to compute W₃ based on the concept of θ-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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