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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2412.18796 |
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| _version_ | 1866913626927923200 |
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| author | Shiozaki, Ken Chen, Jing-Yuan |
| author_facet | Shiozaki, Ken Chen, Jing-Yuan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18796 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Discrete Formulation of Second Stiefel-Whitney Class for Band Theory Shiozaki, Ken Chen, Jing-Yuan Mesoscale and Nanoscale Physics High Energy Physics - Lattice 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. |
| title | A Discrete Formulation of Second Stiefel-Whitney Class for Band Theory |
| topic | Mesoscale and Nanoscale Physics High Energy Physics - Lattice |
| url | https://arxiv.org/abs/2412.18796 |