# Topology-Bounded Superfluid Weight In Twisted Bilayer Graphene

## TL;DR

This paper demonstrates that the nontrivial topology of flat bands in twisted bilayer graphene enhances the superfluid weight, which is crucial for superconductivity, through a topological lower bound related to the Fubini-Study metric.

## Contribution

It derives the superfluid weight for TBLG flat bands as an integral of the Fubini-Study metric and links it to topological invariants, revealing a topological lower bound.

## Key findings

- Superfluid weight is expressed as an integral of the Fubini-Study metric.
- Topological lower bound of superfluid weight is proportional to the pairing gap.
- Trivial flat bands have zero superfluid weight.

## Abstract

While regular flat bands are good for enhancing the density of states and hence the gap, they are detrimental to the superfluid weight. We show that the predicted nontrivial topology of the two lowest flat bands of twisted bilayer graphene plays an important role in the enhancement of the superfluid weight and hence of superconductivity. We derive the superfluid weight (phase stiffness) of the TBLG superconducting flat bands with a uniform pairing, and show that it can be expressed as an integral of the Fubini-Study metric of the flat bands. This mirrors results already obtained for nonzero Chern number bands even though the TBLG flat bands have zero Chern number. We further show the metric integral is lower bounded by the topological $C_{2z}T$ Wilson loop winding number of the TBLG flat bands, which renders the superfluid weight has a topological lower bound proportional to the pairing gap. In contrast, trivial flat bands have a zero superfluid weight. The superfluid weight is crucial in determining the BKT transition temperature of the superconductor. Based on the transition temperature measured in TBLG experiments, we estimate the topological contribution of the superfluid weight in TBLG.