Approach to Solving Quasiclassical Equations with Gauge Invariance

TL;DR

This paper develops a gauge-invariant formulation of quasiclassical equations for unconventional superconductors, enabling accurate analysis of magnetic field effects in high-temperature superconductors.

Contribution

It introduces a gauge-invariant Eilenberger-like equation and Ricatti parametrisation for quasiclassical propagators, specifically addressing deviations caused by magnetic fields.

Findings

Derived a first-order gauge-invariant correction to the quasiclassical propagator.

Formulated a new gauge-invariant Eilenberger-like equation.

Applicable to homogeneous d-wave superconductors in magnetic fields.

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 Eilenberger-like equation with a normalisation condition. A Ricatti parametrisation with manifest gauge invariance is proposed. This theory is directly applicable to homogenous d-wave order parameters in the presence of magnetic fields, such as in high-temperature superconductors.