Geometric treatment of electromagnetic phenomena in conducting materials: variational principles

TL;DR

This paper develops a geometric, variational framework for electromagnetic phenomena in conducting materials, naturally deriving superconductivity theories and addressing dissipative behaviors within a unified covariant approach.

Contribution

It introduces a covariant, gauge-invariant variational formulation that naturally derives superconducting responses and models dissipative conduction by breaking covariance in a phenomenological manner.

Findings

Superconducting response emerges from the geometric variational framework.

Ginzburg-Landau theory is derived from London equations within this approach.

Dissipative conduction can be modeled using a spatial variational principle.

Abstract

The dynamical equations of an electromagnetic field coupled with a conducting material are studied. The properties of the interaction are described by a classical field theory with tensorial material laws in space-time geometry. We show that the main features of superconducting response emerge in a natural way within the covariance, gauge invariance and variational formulation requirements. In particular, the Ginzburg-Landau theory follows straightforward from the London equations when fundamental symmetry properties are considered. Unconventional properties, such as the interaction of superconductors with electrostatic fields are naturally introduced in the geometric theory, at a phenomenological level. The BCS background is also suggested by macroscopic fingerprints of the internal symmetries. It is also shown that dissipative conducting behavior may be approximately treated in a variational framework after breaking covariance for adiabatic processes. Thus, nonconservative laws of interaction are formulated by a purely spatial variational principle, in a quasi-stationary time discretized evolution. This theory justifies a class of nonfunctional phenomenological principles, introduced for dealing with exotic conduction properties of matter [Phys. Rev. Lett. 87, 127004 (2001)].