Magnetic field dependent thermodynamics of spontaneous translational symmetry breaking at edges of d-wave superconductors

We present results for the magnetic field dependent thermodynamics of a low-temperature phase with spontaneous loop currents at edges of layered d-wave superconductors. As we showed in Ref. [1] for zero external magnetic field, time-reversal symmetry and translational symmetry along the edge are broken spontaneously in a second order phase transition at a temperature T^*≈0.18 T_c with a jump in the specific heat, where T_c is the superconducting transition temperature. The phase with broken symmetry is characterized by a gauge invariant superfluid momentum p_S that forms a non-trivial vector field with a chain of sources and sinks along the edges with a period of approximately 12ξ₀, where ξ₀ is the superconducting coherence length. At finite external magnetic field, explicitly breaking time-reversal symmetry, the phase transition is still of second order and is associated with spontaneous breaking of translational symmetry along the edge and formation of the non-trivial p_S vector field. Due to a competing paramagnetic response at the edges, the phase transition temperature T^* is slowly suppressed with increasing magnetic field strength, but survives into the mixed state.

[1] M. Håkansson, T. Löfwander, and M. Fogelström, Nat. Phys. 11, 755 (2015).