Low-energy optical absorption in correlated insulators: Projected sum rules and the role of quantum geometry
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arXiv
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| Format: | Preprint |
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2024
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| author | Mao, Dan Mendez-Valderrama, Juan Felipe Chowdhury, Debanjan |
| author_facet | Mao, Dan Mendez-Valderrama, Juan Felipe Chowdhury, Debanjan |
| contents | Inspired by the discovery of a variety of correlated insulators in the moiré universe, controlled by interactions projected to a set of isolated bands with a narrow bandwidth, we examine here a partial sum-rule associated with the inverse frequency-weighted optical conductivity restricted to low-energies. Unlike standard sum-rules that extend out to $infinite$ frequencies, which include contributions from $all$ inter-band transitions, we focus here on transitions associated $only$ with the $projected$ degrees of freedom. We analyze the partial sum-rule in a non-perturbative but "solvable" limit for a variety of correlation-induced insulators. This includes (i) magic-angle twisted bilayer graphene at integer-filling with projected Coulomb interactions, starting from the chiral flat-band limit and including realistic perturbations, (ii) fractional fillings of Chern-bands which support generalized Laughlin-like states, starting from a Landau-level and including a periodic potential and magnetic-field, respectively, drawing connections to twisted MoTe$_2$, and (iii) integer filling in toy-models of non-topological flat-bands with a tunable quantum geometry in the presence of repulsive interactions. The partial sum-rule in all of these examples is implicitly constrained by the form of the band quantum geometry via the low-lying excitation spectrum, but is not related to it explicitly. For interacting Slater-determinant insulators, the partial sum-rule is related to a new quantity -- "many-body projected quantum geometry" -- obtained from the interaction-renormalized electronic bands. We also point out an intriguing connection between the partial sum-rule and the quantum Fisher information associated with the projected many-body position operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16352 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Low-energy optical absorption in correlated insulators: Projected sum rules and the role of quantum geometry Mao, Dan Mendez-Valderrama, Juan Felipe Chowdhury, Debanjan Strongly Correlated Electrons Inspired by the discovery of a variety of correlated insulators in the moiré universe, controlled by interactions projected to a set of isolated bands with a narrow bandwidth, we examine here a partial sum-rule associated with the inverse frequency-weighted optical conductivity restricted to low-energies. Unlike standard sum-rules that extend out to $infinite$ frequencies, which include contributions from $all$ inter-band transitions, we focus here on transitions associated $only$ with the $projected$ degrees of freedom. We analyze the partial sum-rule in a non-perturbative but "solvable" limit for a variety of correlation-induced insulators. This includes (i) magic-angle twisted bilayer graphene at integer-filling with projected Coulomb interactions, starting from the chiral flat-band limit and including realistic perturbations, (ii) fractional fillings of Chern-bands which support generalized Laughlin-like states, starting from a Landau-level and including a periodic potential and magnetic-field, respectively, drawing connections to twisted MoTe$_2$, and (iii) integer filling in toy-models of non-topological flat-bands with a tunable quantum geometry in the presence of repulsive interactions. The partial sum-rule in all of these examples is implicitly constrained by the form of the band quantum geometry via the low-lying excitation spectrum, but is not related to it explicitly. For interacting Slater-determinant insulators, the partial sum-rule is related to a new quantity -- "many-body projected quantum geometry" -- obtained from the interaction-renormalized electronic bands. We also point out an intriguing connection between the partial sum-rule and the quantum Fisher information associated with the projected many-body position operator. |
| title | Low-energy optical absorption in correlated insulators: Projected sum rules and the role of quantum geometry |
| topic | Strongly Correlated Electrons |
| url | https://arxiv.org/abs/2410.16352 |