# Extended time-dependent Ginzburg-Landau theory

## TL;DR

This paper develops a Lorentz covariant, gauge-invariant Ginzburg-Landau theory for nonstationary superconducting phenomena, including eigen and forced oscillations, and explores implications for two-band superconductors and electromagnetic penetration.

## Contribution

It introduces an extended, Lorentz covariant Ginzburg-Landau framework that describes nonstationary regimes and internal mode oscillations in superconductors, incorporating the Anderson-Higgs mechanism.

## Key findings

- Goldstone oscillations generate only transverse fields.
- Dissipative processes violate Lorentz covariance.
- Proposed experimental test for electromagnetic penetration.

## 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.