Computing the $\mathbb{Z}_2$ Invariant in Two-Dimensional Strongly-Correlated Systems
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| Format: | Preprint |
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2024
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| author | Sinha, Sounak Pan, Derek Y. Bradlyn, Barry |
| author_facet | Sinha, Sounak Pan, Derek Y. Bradlyn, Barry |
| contents | We show that the two-dimensional $\mathbb{Z}_2$ invariant for time-reversal invariant insulators can be formulated in terms of the boundary-condition dependence of the ground state wavefunction for both non-interacting and strongly-correlated insulators. By introducing a family of quasi-single particle states associated to the many-body ground state of an insulator, we show that the $\mathbb{Z}_2$ invariant can be expressed as the integral of a certain Berry connection over half the space of boundary conditions, providing an alternative expression to the formulations that appear in [Lee et al., Phys. Rev. Lett. $\textbf{100}$, 186807 (2008)]. We show the equivalence of the different many-body formulations of the invariant, and show how they reduce to known band-theoretic results for Slater determinant ground states. Finally, we apply our results to analytically calculate the invariant for the Kane-Mele model with nonlocal (orbital) Hatsugai-Kohmoto (HK) interactions. This rigorously establishes the topological nontriviality of the Kane-Mele model with HK interactions, and represents one of the few exact calculations of the $\mathbb{Z}_2$ invariant for a strongly-interacting system. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_12120 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computing the $\mathbb{Z}_2$ Invariant in Two-Dimensional Strongly-Correlated Systems Sinha, Sounak Pan, Derek Y. Bradlyn, Barry Strongly Correlated Electrons Mesoscale and Nanoscale Physics We show that the two-dimensional $\mathbb{Z}_2$ invariant for time-reversal invariant insulators can be formulated in terms of the boundary-condition dependence of the ground state wavefunction for both non-interacting and strongly-correlated insulators. By introducing a family of quasi-single particle states associated to the many-body ground state of an insulator, we show that the $\mathbb{Z}_2$ invariant can be expressed as the integral of a certain Berry connection over half the space of boundary conditions, providing an alternative expression to the formulations that appear in [Lee et al., Phys. Rev. Lett. $\textbf{100}$, 186807 (2008)]. We show the equivalence of the different many-body formulations of the invariant, and show how they reduce to known band-theoretic results for Slater determinant ground states. Finally, we apply our results to analytically calculate the invariant for the Kane-Mele model with nonlocal (orbital) Hatsugai-Kohmoto (HK) interactions. This rigorously establishes the topological nontriviality of the Kane-Mele model with HK interactions, and represents one of the few exact calculations of the $\mathbb{Z}_2$ invariant for a strongly-interacting system. |
| title | Computing the $\mathbb{Z}_2$ Invariant in Two-Dimensional Strongly-Correlated Systems |
| topic | Strongly Correlated Electrons Mesoscale and Nanoscale Physics |
| url | https://arxiv.org/abs/2409.12120 |