Transfer matrix approach to the persistent current in hybrid normal-superconducting rings

TL;DR

This paper introduces an exact transfer matrix method to compute the persistent current in hybrid normal-superconducting rings, capturing effects like topological phase transitions and size-dependent periodicity changes.

Contribution

It develops a novel transfer matrix technique for exact calculation of persistent currents in hybrid rings, including finite temperature extensions and large-size analytic formulas.

Findings

Exact formulas for persistent current in hybrid rings.

Identification of topological phase transition signatures.

Size-dependent periodicity crossover in the current.

Abstract

Using the properties of the transfer matrix of one-dimensional quantum mechanical systems, we derive a technique to exactly compute the persistent current across a hybrid normal-superconducting-mesoscopic ring pierced by a magnetic flux Phi as a single integral of a known function of the system's parameters. Our approach provides exact results at zero temperature, which can be readily extended to finite temperatures T much below the superconducting gap. We apply our technique to derive the persistent current through p-wave and s-wave superconducting-normal hybrid rings, recovering at once a number of effects such as the crossover in the current periodicity on increasing the size of the ring and the signature of the topological phase transition in the p-wave case. In the limit of a large ring size, resorting to a systematic expansion in inverse powers of the ring length, we derive exact analytic closed-form formulas, applicable to a number of cases of physical interest.