A discrete formulation for three-dimensional winding number
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911647190220800 |
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| author | Shiozaki, Ken |
| author_facet | Shiozaki, Ken |
| contents | 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 $W_3 = \frac{1}{24π^2} \int_{X} \mathrm{Tr}\left[(g^{-1}dg)^3\right]$. We present a discrete formulation to compute $W_3$ based on the concept of $θ$-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_05291 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A discrete formulation for three-dimensional winding number Shiozaki, Ken Mesoscale and Nanoscale Physics High Energy Physics - Lattice Mathematical Physics 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 $W_3 = \frac{1}{24π^2} \int_{X} \mathrm{Tr}\left[(g^{-1}dg)^3\right]$. We present a discrete formulation to compute $W_3$ based on the concept of $θ$-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. |
| title | A discrete formulation for three-dimensional winding number |
| topic | Mesoscale and Nanoscale Physics High Energy Physics - Lattice Mathematical Physics |
| url | https://arxiv.org/abs/2403.05291 |