A key prediction of the Bardeen-Cooper-Schrieffer theory of superconductivity is that the critical temperature increases with the density of states. Therefore a promising way to increase the critical temperature and eventually achieve room temperature superconductivity is to engineer electronic bands with very small bandwidth. In the limit of vanishing bandwidth, called the flat band limit, the density of states is diverging and the critical temperature of the superconducting transition is linearly proportional to the interaction strength, and thus much higher than in the case of a dispersive band.
In the first part of this talk, I will first review the theory of superconducting transport in the flat band limit, more specifically the relation between superfluid weight and quantum metric [1]. The quantum metric is a band structure invariant closely related to the Berry curvature and topological invariants such as the Chern and winding numbers. As consequence, the superfluid weight of a flat band with nonzero Chern number is always nonzero. Recent experimental results obtained with twisted bilayer graphene provide evidence that the quantum metric indeed plays an important rôle in superfluid transport [2, 3].
In the second part of the talk, I will discuss two recent works [4, 5] addressing an important conceptual problem: while it can be shown that the superfluid weight is a geometry-independent quantity, that is an observable depending only the hopping matrix elements but not on the spatial arrangements of the orbitals, the quantum metric actually does depend on the lattice geometry. This discrepancy leads to paradoxical results, which I illustrate using the Su-Schrieffer-Heeger model as an example. The paradox is resolved by using the generalize random phase approximation to show that the superfluid weight is in fact proportional to the minimal quantum metric, the integral over the Brillouin zone of the quantum metric minimized over the orbital positions. The minimal quantum metric is a novel band structure invariant with potentially interesting applications in condensed matter physics, in particular regarding the classification of topological states of matter. For instance, I will discuss how the minimal quantum metric can be used to provide a geometry independent formulation of the winding number.
References
[1] S. Peotta and P. Törmä, Nature Communications 6, 8944 (2015)
[2] Tian et al., Nature 614, 440 (2023)
[3] P. Törmä, S. Peotta and B. A. Bernevig, Nature Reviews Physics 4, 528 (2022)
[4] K. E. Huhtinen, J. Herzog-Arbeitman, A. Chew, B. A. Bernevig, P. Törmä, Physical Review B
106, 014518 (2022)
[5] M. Tam, S. Peotta, Phys. Rev. Research 6, 013256 (2024).
