Extended Time-Dependent Ginzburg–Landau Theory

Abstract

We formulate the gauge invariant Lorentz covariant Ginzburg–Landau theory which describes nonstationary regimes: relaxation of a superconducting system accompanied by eigen oscillations of internal degrees of freedom (Higgs mode and Goldstone mode) and also forced oscillations under the action of an external gauge field. The theory describes Lorentz covariant electrodynamics of superconductors where Anderson–Higgs mechanism occurs, at the same time the dynamics of conduction electrons remains non-relativistic. It is demonstrated that Goldstone oscillations cannot be accompanied by oscillations of charge density and they generate the transverse field only. In addition, we consider Goldstone modes and features of Anderson–Higgs mechanism in two-band superconductors. We study dissipative processes, which are caused by movement of the normal component of electron liquid and violate the Lorentz covariance, on the examples of the damped oscillations of the order parameter and the skin-effect for electromagnetic waves. An experimental consequence of the extended time-dependent Ginzburg–Landau theory regarding the penetration of the electromagnetic field into a superconductor is proposed.

Appendix A: “Derivation” of the Second London Equation from the First London Equation and Vice Versa

Following [46], let us write the Newton equation for SC electrons (they do not experience friction) in the electric field ${\mathbf {E}}$: $m\frac{\mathrm{d}{\mathbf {v}}}{\mathrm{d}t}=e{\mathbf {E}}$, which can be given a form:

$$\begin{aligned} \frac{\mathrm{d}{\mathbf {j}}_{s}}{\mathrm{d}t}=\frac{n_{\mathrm {s}}e^{2}}{m}{\mathbf {E}}, \end{aligned}$$

(A1)

where ${\mathbf {j}}_{s}=n_{\mathrm {s}}e{\mathbf {v}}$ is supercurrent, $n_{\mathrm {s}}$ is density of SC electrons. Equation (A1) is the first London equation (7). Making the operation $\mathrm {curl}$ for both sides of the equation and taking into account the Maxwell equation $\mathrm {curl}{\mathbf {E}}=-\frac{1}{c}\frac{\partial {\mathbf {H}}}{\partial t}$, we obtain:

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\left( \mathrm {curl}{\mathbf {j}}_{s}\right) =-\frac{n_{\mathrm {s}}e^{2}}{mc}\frac{\partial {\mathbf {H}}}{\partial t}. \end{aligned}$$

(A2)

Integrating over time, we obtain:

$$\begin{aligned} \text {curl}{\mathbf {j}}_{s}=-\frac{n_{\mathrm {s}}e^{2}}{mc}\left( {\mathbf {H}}-{\mathbf {H}}_{0}\right) , \end{aligned}$$

(A3)

where ${\mathbf {H}}_{0}$—a constant of integration which does not depend on time, but it can be function of coordinates. Thus, by setting the field ${\mathbf {H}}_{0}$ we set the initial condition, i.e., configuration of the field ${\mathbf {H}}$ is determined by both response of the medium and the initial field. Supposing ${\mathbf {H}}_{0}=0$ (a sample is introduced into magnetic field), we obtain the second London equation (8). However, supposing ${\mathbf {H}}_{0}\ne 0$ (a sample in the normal state is in the magnetic field ${\mathbf {H}}_{0}$, then it is cooled below the transition temperature $T_{c}$), we obtain the freezing of the magnetic field inside the sample. Thus, the field ${\mathbf {H}}_{0}$ is a constant of motion, that characterizes an ideal conductor: $\mathrm {curl}{\mathbf {E}}=-\frac{1}{c}\frac{\partial {\mathbf {B}}}{\partial t}\Rightarrow {\mathbf {B}}=\mathrm {const}$ since we have inside the sample ${\mathbf {E}}=\rho {\mathbf {j}}=0$ due to ideal conductivity $\rho =0$. Therefore, Eq. (A3) does not describe thermodynamically steady state, this equation describes the ideal conductor which pushes out or freezes magnetic field due to the electromagnetic induction and Lenz’s rule.

From other hand, following [5] we can differentiate the second London equation in a form ${\mathbf {j}}_{s}=-\frac{c}{4\pi \lambda ^{2}}{\mathbf {A}}$, where $\mathrm {div}{\mathbf {A}}=0$ (i.e., ${\mathbf {A}}={\mathbf {A}}_{\perp }$), with respect to time:

$$\begin{aligned} \frac{\partial {\mathbf {j}}_{s}}{\partial t}=-\frac{c}{4\pi \lambda ^{2}}\frac{\partial {\mathbf {A}}}{\partial t}= \frac{c^{2}}{4\pi \lambda ^{2}}{\mathbf {E}}=\frac{n_{s}e^{2}}{m}{\mathbf {E}}. \end{aligned}$$

(A4)

Thus, we obtain the first London equation (A1) which is the second Newton law for SC electrons, i.e., it is equation for an ideal conductor. At the same time, the second London equation is result of minimization of the free energy functional, i.e., it is equation for a superconductor. This contradiction is resolved in Sect. 3.2 within the Anderson–Higgs mechanism. It should be noticed that in Eq. (A4) the field ${\mathbf {E}}$ is transverse field ${\mathbf {E}}={\mathbf {E}}_{\perp }=-\frac{1}{c}\frac{\partial {\mathbf {A}}_{\perp }}{\partial t}$ only. At the same time in Eq. (A1), the electric field can be longitudinal $-\nabla \varphi -\frac{1}{c}\frac{\partial {\mathbf {A}}_{||}}{\partial t}$ also (here $\mathrm {div}{\mathbf {A}}_{||}\ne 0$). Thus, the first London equation (A1) cannot be obtain from the second London equation ${\mathbf {j}}_{s}=-\frac{c}{4\pi \lambda ^{2}}{\mathbf {A}}_{\perp }$ completely.