Kinetic Lifshitz invariants and dynamics of nonreciprocal fluctuations in superconductors

Abstract

We derive the generalized time-dependent Ginzburg-Landau theory of a disordered noncentrosymmetric superconductor from the Keldysh nonlinear sigma model, using a two-dimensional electron gas with Rashba spin-orbit coupling and an in-plane Zeeman field as a minimal model. On the thermodynamic side we construct the Lifshitz invariants of the free energy, the linear and cubic gradient terms and the momentum-odd part of the quartic vertex, and trace their dependence on disorder. These couplings are governed by a single closed-form kernel controlled by the ratio of the Dyakonov-Perel spin-relaxation rate to temperature, interpolating between the weak-relaxation regime, where the invariants are suppressed, and the relaxation-dominated regime, where the helical modulation of the order parameter saturates at a universal, disorder-independent value. This crossover reconciles conflicting results for the magnetoelectric couplings of dirty Rashba superconductors. Because the theory is formulated on the Keldysh contour, it also determines the dissipative dynamics: the relaxation rate of a fluctuation with pair momentum $\mathbf{q}$, and hence, by the fluctuation-dissipation theorem, the Langevin noise power, acquires a term odd in $\mathbf{q}$ and odd in the magnetic field. The structure of this kinetic Lifshitz invariant is dictated by Onsager reciprocity: friction and noise renormalize in lockstep, so equal-time fluctuations remain Gibbsian while the dynamics are nonreciprocal. As applications we compute the superconducting diode efficiency near $T_c$, where the cubic invariant competes with the quartic vertex and with even higher-gradient terms rendered odd by the helical shift, reversing the sign of the diode coefficient, and the fluctuation-induced magnetochiral anisotropy above $T_c$, where the current-resolved nonreciprocal resistance forms a plateau across the Gaussian regime.