Superfluidity and geometry of Bloch bands

In flat Bloch bands the critical temperature for superconductivity is linear in the coupling constant rather than exponentially suppressed as in conventional BCS theory. I will present our ongoing work on flat-band superconductivity and superfluidity. We show that in the flat-band limit the superfluid weight $D_{\rm s}$ is not controlled by the effective mass, but by a geometric invariant, the quantum metric [2], namely the gauge-invariant part of the Marzari-Vanderbilt localization functional for Wannier orbitals. The quantum metric is related to the Berry curvature, as a consequence we obtain the inequality $D_{\rm s} \geq |C|$ between superfluid weight and Chern number $C$ [2]. This effect is important in a number of lattice models relevant for material science and ultracold gases [2-5]. References: 1) N. B. Kopnin, T. T. Heikkil\"a, and G. E. Volovik, Phys. Rev. B 83, 220503(R) (2011); 2) SP and P. T\"orm\"a, Nature Communications 6, 8944 (2015); 3) A. Julku, SP, T. Vanhala, D.-H. Kim, P. T\"orm\"a, Phys. Rev. Lett. 117, 045303 (2016); 4) M. Tovmasyan, SP, P. T\"orm\"a, S. D. Huber, arXiv:1608.00976; 5) L. Liang, T. I. Vanhala, SP, T. Siro, A. Harju, and P. T\"orm\"a, in preparation.