Superfluid Stiffness Bounds in Time-Reversal Symmetric Superconductors

Quantum geometry has been shown to make an important contribution to the superfluid stiffness of flatband superconductors including moiré materials. In this Letter we use mean-field theory to derive an expression for the superfluid stiffness of time-reversal symmetric superconductors at zero temperature by computing the energy of the mean-field ground state as a function of pairing momentum. We show that the quantum geometric contribution to superfluid stiffness is a consequence of broken Galilean invariance in the interaction Hamiltonian, arising from momentum-dependent form factors related to the momentum dependence of Bloch states. The effects of broken Galilean invariance are not fully parametrized by the quantum metric considered in previous work. We obtain general lower and upper bounds that apply to both continuum and lattice models and present numerical calculations of the precise value in several important cases. The lower bound that we obtain is saturated by systems with Landau-level form factors and closely approximates the precise mean-field stiffness in all examples that we consider. Our results clarify the physical origin of the geometric contribution to superfluid stiffness and provide a new perspective on quantum geometry effects in interacting electron systems.