Breaking time-reversal and translational symmetry at edges of $d$-wave superconductors: microscopic theory and comparison with quasiclassical theory

TL;DR

This paper presents a microscopic calculation of a phase transition in $d$-wave superconductors that breaks time-reversal and translational symmetry at edges, confirming previous quasiclassical results and analyzing the effects of realistic band structures.

Contribution

It extends prior quasiclassical studies by performing a microscopic Bogoliubov-de Gennes analysis with realistic band structures, showing the transition's robustness and quantifying differences.

Findings

Transition occurs below 10-20% of $T_c$

Current loops of a few coherence lengths form at edges

Quantitative differences in $T^*$ explained by spectral weights of Andreev bound states

Abstract

We report results of a microscopic calculation of a second-order phase transition into a state breaking time-reversal and translational invariance along pair-breaking edges of $d$-wave superconductors. By solving a tight-binding model through exact diagonalization with the Bogoliubov-de~Gennes method, we find that such a state with current loops having a diameter of a few coherence lengths is energetically favorable below $T^*$ between 10%-20% of $T_{\mathrm{c}}$ of bulk superconductivity, depending on model parameters. This extends our previous studies of such a phase crystal within the quasiclassical theory of superconductivity, and shows that the instability is not qualitatively different when including a more realistic band structure and the fast oscillations on the scale of the Fermi wavelength. Effects of size quantization and Friedel oscillations are not detrimental. We also report on a comparison with quasiclassical theory with the Fermi surfaces extracted from the tight-binding models used in the microscopic calculation. There are quantitative differences in for instance the value of $T^*$ between the different models, but we can explain the predicted transition temperature within each model as due to the different spectral weights of zero-energy Andreev bound states and the resulting gain in free energy by breaking time-reversal and translational invariance below $T^*$.