Phase crystallization near surfaces of unconventional superconductors

We demonstrate that a pairbreaking surface leads to an instability in the phase of the superconducting order parameter. Initially uniform U(1) phase acquires spatial variation, and thus breaks the time-reversal symmetry, below a transition temperature T^* that is only a few times smaller than the bulk T_c. The phase modulation appears through a second-order phase transition from the trivial superconducting phase. The most favorable phase distortion occurs at a finite wave vector, thus spontaneously breaking continuous translational symmetry along the surface, and generating periodic superflow and current patterns localized near the surface, which have been previously seen in numerical simulations. We analytically derive the non-local Ginzburg-Landau coefficient that is responsible for this transition, and clarify conditions under which this instability occurs, the reason for the periodic modulation, and the geometry of emerging superflow and current patterns.