The superconducting properties of materials are inherently rooted in the quantum mechanical wavefunctions of electrons and the interactions that govern their behavior. However, our understanding of superconductivity in the regime of extremely narrow electronic bands remains elusive due to the lack of a comprehensive microscopic theory. I will present a novel theoretical framework for computing a critical fundamental characteristic – the superconducting phase stiffness – without relying on the conventional Bardeen–Cooper–Schrieffer (BCS) approximation, which is not applicable to narrow-band superconductivity. Without prior knowledge of the underlying pairing symmetry, our method can give an upper bound on the superconducting phase stiffness, which is related to the superconducting transition temperature in 2D.
Furthermore, we find a relationship between the superconducting phase stiffness and the optical absorption, which leads to the discovery of the so-called projected optical sum rule, which puts fundamental constraints on how much light the system can absorb. I will discuss two types of projected optical sum rule: the f-sum rule and the Souza-Wilkens-Martin sum rule. While the connection between quantum geometry and optical absorption is well-established in the non-interacting limit, whether an analogous connection can be established for the low-energy projected optical sum rules in strongly interacting systems remains an open question.
From the study of twisted bilayer graphene and fractional Chern insulators, we obtain non-perturbative results on the projected optical sum rules, pointing out the intriguing interplay between the “many-body” quantum geometry, symmetry and topology.