Extracting quantum-geometric effects from Ginzburg-Landau theory in a multiband Hubbard model

We first apply the functional-integral approach to a multiband Hubbard model near the critical pairing temperature and derive a generic effective action that is quartic in the fluctuations of the pairing order parameter. Then we consider time-reversal-symmetric systems with uniform (i.e., at both low momentum and low frequency) pairing fluctuations in a unit cell and derive the corresponding time-dependent Ginzburg-Landau (TDGL) equation. In addition to the conventional intraband contribution that depends on the derivatives of the Bloch bands, we show that the kinetic coefficients of the TDGL equation have a geometric contribution that is controlled by both the quantum-metric tensor of the underlying Bloch states and their band-resolved quantum-metric tensors. Furthermore, we show that thermodynamic properties such as the London penetration depth, Ginzburg-Landau (GL) coherence length, GL parameter, and upper critical magnetic field have an explicit dependence on quantum geometry.