Localization in disordered superconducting wires with broken spin-rotation symmetry

TL;DR

This paper investigates how certain symmetry-breaking terms influence localization in disordered superconducting wires, revealing that these terms cause localization at long distances and unify the universality class of phase transitions across multiple symmetry classes.

Contribution

It demonstrates that symmetry-breaking terms, often neglected in continuum limits, are crucial for understanding localization and critical behavior in superconducting wires with broken spin-rotation symmetry.

Findings

Symmetry-breaking terms lead to localization in long wires.

Additional terms are irrelevant at short distances but relevant at critical fixed points.

Transitions in symmetry classes BD, DIII, and chiral classes share the same universality class.

Abstract

Localization and delocalization of non-interacting quasiparticle states in a superconducting wire are reconsidered, for the cases in which spin-rotation symmetry is absent, and time-reversal symmetry is either broken or unbroken; these are referred to as symmetry classes BD and DIII, respectively. We show that, if a continuum limit is taken to obtain a Fokker-Planck (FP) equation for the transfer matrix, as in some previous work, then when there are more than two scattering channels, all terms that break a certain symmetry are lost. It was already known that the resulting FP equation exhibits critical behavior. The additional symmetry is not required by the definition of the symmetry classes; terms that break it arise from non-Gaussian probability distributions, and may be kept in a generalized FP equation. We show that they lead to localization in a long wire. When the wire has more than two scattering channels, these terms are irrelevant at the short distance (diffusive or ballistic) fixed point, but as they are relevant at the long-distance critical fixed point, they are termed dangerously irrelevant. We confirm the results in a supersymmetry approach for class BD, where the additional terms correspond to jumps between the two components of the sigma model target space. We consider the effect of random $\pi$ fluxes, which prevent the system localizing. We show that in one dimension the transitions in these two symmetry classes, and also those in the three chiral symmetry classes, all lie in the same universality class.