Bounds on<i> </i>low energy spectral weight and superfluid stiffness in strongly correlated systems

We have previously obtained rigorous upper bounds [1] on the superfluid stiffness of multi-band systems with arbitrary interactions in any dimension. These results were then used to obtain upper bounds for the superconducting Tc of 2D systems and led to insightful results for strongly correlated systems like Twisted Bilayer Graphene, Monolayer FeSe/STO, and cold atoms across the BCS-BEC crossover. These bounds were expressed in terms of the optical spectral weight of all the bands; we now improve upon these results by projecting down to the active bands crossing the chemical potential. As a nontrivial example, we focus on the case of an isolated flat band near E_F for which one cannot use the usual Peierls’ substitution and the projected interaction terms couple to the external vector potential. We show how the resulting low-energy optical spectral weight depends on the geometry of the band eigenstates (without making any mean-field approximations). We further derive exact upper bounds on the low-energy spectral weight and show that it is related to the Marzari-Vanderbilt localization function, and in some special cases, to the quantum metric.

[1] T. Hazra, N. Verma and M. Randeria, PRX 9, 031049 (2019).