Quantum geometry in superfluidity and superconductivity

TL;DR

This work surveys how quantum geometry, encapsulated by the quantum metric and Berry curvature, governs superfluidity and superconductivity in multiband systems, with flat bands where geometry alone can enable dissipationless transport. It develops a multiband mean-field framework and a two-body analysis to separate conventional and geometric contributions to the superfluid weight, showing that in isolated flat bands $D_s$ is tied to the minimal quantum metric via $[D_s]_{ij} \propto \mathcal{M}^{\min}_{ij}$. A central theme is the Wannier-function interpretation: larger orbital overlaps (and thus larger quantum geometry) stabilize superconductivity, even when single-particle dispersion is flat, with topological aspects (Chern numbers) setting fundamental localization constraints. The discussion extends to the BKT transition in 2D and to Bose–Einstein condensation, highlighting experimental platforms such as moiré materials and twisted bilayer graphene as promising venues to observe quantum-geometry–driven high-temperature superconductivity.

Abstract

We review the theoretical description of the role of quantum geometry in superfluidity and superconductivity of multiband systems, with focus on flat bands where quantum geometry is wholly responsible for supercurrents. This review differs from previous ones in that it is based on the most recent understanding of the theory: the dependence of the self-consistent order parameter on the supercurrent is properly taken into account, and the superfluid weight in a flat band becomes proportional to the minimal quantum metric. We provide a recap of basic quantum geometric quantities and the concept of superfluid density. The geometric contribution of superconductivity is introduced via considering the two-body problem. The superfluid weight of a multiband system is derived within mean-field theory, leading to a topological bound of flat band superconductivity. The physical interpretation of the flat band supercurrent in terms of Wannier function overlaps is discussed.

Figures (1)

• Figure 1: There are two ways for non-interacting particles can be immobile: their Wannier functions can be exponentially localized at the lattice sites, or they can be localized due to destructive interference. In the former case, interacting particles will remain localized too, while in the latter, the interference is distorted by the interactions and the particle movement is facilitated by the finite overlap of the Wannier functions. Wannier function overlaps are determined by the quantum geometry of the band and exponential localization of the Wannier functions is impossible in topologically non-trivial systems. Picture credits Antti Paraoanu.