Breaking time-reversal and translational symmetry at edges of d -wave superconductors: Microscopic theory and comparison with quasiclassical theory
Journal article, 2020

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 Tc 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∗.


Niclas Wennerdal

Chalmers, Microtechnology and Nanoscience (MC2), Applied Quantum Physics

Andreas Josefsson Ask

Chalmers, Microtechnology and Nanoscience (MC2), Applied Quantum Physics

Patric Holmvall

Chalmers, Microtechnology and Nanoscience (MC2), Applied Quantum Physics

Tomas Löfwander

Chalmers, Microtechnology and Nanoscience (MC2), Applied Quantum Physics

Mikael Fogelström

Chalmers, Microtechnology and Nanoscience (MC2)

Physical Review Research

Vol. 2 043198

Areas of Advance

Nanoscience and Nanotechnology (SO 2010-2017, EI 2018-)

Materials Science


Basic sciences


C3SE (Chalmers Centre for Computational Science and Engineering)

Subject Categories

Condensed Matter Physics



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