Quantum action for time-dependent Ginzburg-Landau equations

Adriaan M. J. Schakel

David Reginald Tilley

M. Iskin

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.

S. M. Alamoudi, D. Boyanovsky, Shang-Yung Wang

We present a field-theoretical method to obtain consistently the equations of motion for small amplitude fluctuations of the order parameter directly in real time for a homogeneous, neutral BCS superconductor. This method allows to study the nonequilibrium relaxation of the order parameter as an initial value problem. We obtain the Ward identities and the effective actions for small phase the amplitude fluctuations to one-loop order. Focusing on the long wavelength, low-frequency limit near the critical point, we obtain the time-dependent Ginzburg-Landau effective action to one-loop order, which is nonlocal as a consequence of Landau damping. The nonequilibrium relaxation of the phase and amplitude fluctuations is studied directly in real time. The long-wavelength phase fluctuation (Bogoliubov-Anderson-Goldstone mode) is overdamped by Landau damping and the relaxation time scale diverges at the critical point, revealing critical slowing down.

SC Althorpe, Pavel Soldán, Gabriel G. Balint‐Kurti

Giancarlo Calvanese Strinati

This chapter derives the time-dependent Ginzburg–Landau equation from first principles, by relying on the same formal approach that has led to the time-dependent Gross–Pitaevskii equation of Chapter 28. Specifically, the time-dependent Ginzburg–Landau equation holds close to the critical temperature of the initial equilibrium preparation and in the (extreme) BCS limit of the BCS–BEC crossover, when the Cooper pairs are largely overlapping with each other. Care has to be exerted when dealing with the analytic properties in the wave-vector and frequency space of the normal and anomalous particle–particle bubbles.