Approach to Solving Quasiclassical Equations with Gauge Invariance
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Approach to Solving Quasiclassical Equations with Gauge Invariance Priya Sharma1 Received: 1 September 2019 / Accepted: 30 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Quasiclassical equations with manifest gauge invariance are discussed in the context of unconventional singlet superconducting states in the static limit. Deviations of the quasiclassical propagator from its equilibrium solutions in the presence of magnetic fields and Hall terms are analysed in terms of a “small” parameter and a formulation developed to first order in “small”. A modified quasiclassical propagator is defined to this order that is a solution of a new gauge-invariant Eilenbergerlike equation with a normalization condition. A Riccati parametrization with manifest gauge invariance is proposed. Riccati equations are derived to leading order in “small” that are directly applicable to superconducting systems in the presence of magnetic fields. Keywords Quasiclassical theory · Gauge invariance · Superconductivity
1 Introduction The quasiclassical theory [1] is established as a powerful method to describe superconducting systems in the presence of external perturbations that vary slowly on the scale of the Fermi energy and Fermi wavelength. The central object of the theory is the quasiclassical Green’s function for quasiparticles travelling with Fermi velocity, vF along ballistic trajectories defined by their direction v̂ F . This propagator, represented as a Nambu matrix, is given by solutions of the Eilenberger equation [2], along with a normalization condition that picks the physical solution. In the presence of magnetic fields, additional forces on the quasiparticles such as the Lorentz force appear as driving terms in the quasiclassical equation and have been included in a gauge-invariant formulation by Kita [3]. The augmented Eilenberger equations have been applied to study vortex core charging, Hall currents and vortex lattices in superconductors [3–5]. * Priya Sharma [email protected] 1
Department of Physics, Indian Institute of Science, Bangalore 560012, India
13
Vol.:(0123456789)
Journal of Low Temperature Physics
The quasiclassical equation (Eilenberger as well as augmented versions) is a differential equation of the first order and usually, the solutions must be found numerically. In addition, the pair potential or the superconducting order parameter and self-energies (which are functions of the propagator) need to be solved for self-consistently. This is a nonlinear problem that is much simplified by a parametrization of the quasiclassical propagator in terms of coherence amplitudes that are solutions of a differential equation of Riccati form [6]. This Riccati parametrization transforms the Eilenberger equation for the matrix propagator to a scalar differential equation for the Riccati amplitudes. The solution then reduces to that of an initial value problem to a scalar differential equation. It is known that the Riccati parametrization of the quasiclassical propagator le
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