Nonlinear electrodynamics and its possible connection to relativistic superconductivity: Instance of a time-dependent system

In a recent article, we proposed an example suggesting a possible link between the Heisenberg-Euler (HE) model of nonlinear electrodynamics (NLED) and relativistic type-II superconductivity, drawing parallels with the Ginzburg-Landau (GL) theory. In cylindrical coordinates, we considered a concrete static electromagnetic (EM) four-potential in vacuum with Gaussian-like behavior in the radial coordinate. This work extends our analysis to a time-dependent EM potential with a Gaussian-like radial profile and a time-dependent harmonic sinusoidal angular behavior. By deriving nonlinear Maxwell's equations from the HE Lagrangian, we determine the four-current density. Concurrently, we investigate relativistic type-II superconductivity in analogy with the GL theory. This leads us to propose a specific form for the relativistic vortex's supercurrent density and to establish a connection between this current and the HE current. As a result, we deduce a relation that allows us to obtain the relativistic order parameter that characterizes the macroscopic quantum state of the superconductor. Thereafter, we present a discussion on possible corrections to our results in relation to the extended GL (EGL) formalism. This formalism provides a more sophisticated description of vortex dynamics in type-II superconductors. Additionally, we briefly comment on high-temperature condensates in intense EM fields where relativistic effects may become prominent. This scenario is particularly relevant to neutron stars, where the formation of superconducting protons and superfluid neutrons in neutron-star outer cores can lead to coupled multiband-like effects.