Kavli Affiliate: Debanjan Chowdhury
| First 5 Authors: Dan Mao, Juan Felipe Mendez-Valderrama, Debanjan Chowdhury, ,
| Summary:
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.
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