Low-energy optical absorption in correlated insulators: Projected sum rules and the role of quantum geometry

TL;DR

This paper investigates a partial optical sum-rule in correlated insulators focusing on low-energy transitions within projected bands, revealing connections to quantum geometry and many-body effects in various models including twisted bilayer graphene.

Contribution

It introduces a new partial sum-rule constrained by quantum geometry, applicable to correlated insulators, and links it to many-body projected quantum geometry and quantum Fisher information.

Findings

Partial sum-rule is constrained by quantum geometry.

Connection established between sum-rule and many-body projected quantum geometry.

Relation between sum-rule and quantum Fisher information in correlated insulators.

Abstract

Inspired by the discovery of a variety of correlated insulators in the moir\'e 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.