| Summary: | 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_{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^{*}.
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