Finite Temperature Time-Dependent Effective Theory for the Phase Field in 2D d-wave and s-wave Superconductor

G. Sordi, P. Sémon

Sergei G. Sharapov, H. Beck, В. М. Локтев

We derive finite-temperature time-dependent effective actions for the phase of the pairing field, which are appropriate for a two-dimensional electron system with both nonretarded d- and s-wave attraction. As for s-wave pairing, the d-wave effective action contains terms with Landau damping, but their structure appears to be different from the s-wave case due to the fact that the Landau damping is determined by the quasiparticle group velocity ${\mathbf{v}}_{g},$ which for d-wave pairing does not have the same direction as the noninteracting Fermi velocity ${\mathbf{v}}_{F}.$ we show that for d-wave pairing the Landau term has a linear low-temperature dependence and in contrast to the s-wave case is important for all finite temperatures. A possible experimental observation of the phase excitations is discussed.

Chien-Hua Pao

Jacek Dziarmaga

Time-dependent Ginzburg-Landau model with noise for neutral s-wave superconductor is derived from real-time finite-temperature quantum field theory by integrating over fermions in the density matrix propagator. Quantum decoherence due to environment of pair breaking fluctuations is identified. Dynamics is described by Langevin equation in the near critical regime and by a nonlinear Schrodinger equation at zero temperature.

H. C. Krahl, CHRISTOF WETTERICH

Pedro D Sacramento, Wing Chi Yu

The short time evolution of a topological system after a sudden quench is considered, calculating the Loschmidt rate and studying in detail the distribution of Fisher zeros. Singular behavior in the Loschmidt rate has been predicted and found in several systems and, in particular, in one dimension (1D) one finds sharp features in the time evolution of the Loschmidt rate. These are associated with critical times that may be seen as the imaginary part of the Fisher zeros, under the condition that their real part vanishes. In the complex plane one finds lines of zeros that cross the imaginary axis at certain points. In a 2D system, in general, the Fisher zeros occupy areas instead of lines, as shown before, with the consequences that a continuous set of critical times results along the imaginary axis and that one does not find a well defined singular behavior in the Loschmidt rate. Here we show that several 2D topological systems also display very narrow regions (effectively lines) of zeros, in addition to finite extent areas of zeros. As a consequence, well defined sharp critical times are found, under appropriate conditions. An example is a $2\text{D}\phantom{\rule{4pt}{0ex}}p$-wave superconductor in the presence of a magnetic field, but other systems share the same property, such as a Haldane insulator, a system with an anomalous Hall effect, and topological insulators. Details on the momentum distribution of the real and imaginary parts of the Fisher zeros are discussed, clarifying the existence of the lines of zeros and taking as an example the $p$-wave superconductor in magnetic field. The time dependence of the Loschmidt rate phase and of the Pancharatnam geometrical phase is considered, for different times after the quench and as a function of momentum. Pairs of vortices and antivortices in the phase associated with the geometric phase and the Loschmidt rate phase, that have been proposed for some 2D systems, are also shown to appear in the regime where we find lines of zeros.