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

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 $T^{\ast }$ 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 $T^{\ast }$ 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 $T^{\ast }$.

15 pages and 9 figures

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Cited by in corpus (11)

• Spontaneous time-reversal symmetry breaking by disorder in superconductors
• Quasiclassical theory of charge transport across mesoscopic normal metal-superconducting heterostructures with current conservation
• Disorder-robust phase crystal in high-temperature superconductors stabilized by strong correlations
• Robust and tunable coreless vortices and fractional vortices in chiral $d$-wave superconductors
• SuperConga: an open-source framework for mesoscopic superconductivity
• A finite element method for the quasiclassical theory of superconductivity
• Enhanced chiral edge currents and orbital magnetic moment in chiral $d$-wave superconductors from mesoscopic finite-size effects
• Sparse modeling approach for quasiclassical theory of superconductivity
• Impurity strength-temperature phase diagram with phase crystals and competing time-reversal symmetry breaking states in nodal $d$-wave superconductors
• Designing edge currents using mesoscopic patterning in chiral d-wave superconductors
• Strengthened correlations near [110] edges of $d$-wave superconductors in the t-J model with the Gutzwiller approximation