The Effect of the Non-Abelian Quantum Metric on Superfluidity

Abstract: The quantum geometric tensor, which encodes the full geometric information of quantum states in projective Hilbert space, plays a crucial role in condensed matter physics. In this work, we examine the effect of the non-Abelian quantum metric -- the real part of the non-Abelian quantum geometric tensor -- on the superfluid weight in time-reversal symmetric systems. For conventional $s$-wave pairing, we demonstrate that the superfluid weight includes a contribution proportional to the trace of the non-Abelian quantum metric. Notably, this contribution remains significant even when the total Chern number of a set of degenerate bands is zero and can exceed the conventional contribution, as confirmed using lattice models. Ab initio density functional theory (DFT) calculations for MoS$_2$ and TiSe$_2$ further corroborate these findings, revealing that the non-Abelian quantum metric accounts for up to 20% of the superfluid weight in MoS$_2$ and 50% in TiSe$_2$. Our results provide new insights into the nontrivial relationship between the geometric properties of quantum states and superconductivity, opening avenues for further exploration in topological and superconducting materials.