Theory of gauge-invariant response of superconductors to an external electromagnetic field

A consistent method of calculating the linear response to an external magnetic field that allows obtaining the result in a manifestly gauge-invariant form is proposed. Within the diagram technique for nonequilibrium processes, the self-consistency equations for the order parameter in an arbitrary-gauge field allow deriving an equation that determines the phase of the order parameter as a function of the external field. Such a method automatically accounts for the existence of collective excitations in superconductors, which must be taken into consideration in accordance with the continuity equation. The possible types of collective excitations in pure superconductors at different temperatures are considered. The authors present a microscopic theory that explains the possibility of observing collective modes in superconducting tunnel junctions.