Abstract
Flat bands of zero-energy states at the edges of quantum materials have a topological origin. However, their presence is energetically unfavorable. If there is a mechanism to shift the band to finite energies, a phase transition can occur. Here we study high-temperature superconductors hosting flat bands of midgap Andreev surface states. In a second-order phase transition at roughly a fifth of the superconducting transition temperature, time-reversal symmetry and continuous translational symmetry along the edge are spontaneously broken. In an external magnetic field, only translational symmetry is broken. We identify the order parameter as the superfluid momentum p(s), that forms a planar vector field with defects, including edge sources and sinks. The critical points of the vector field satisfy a generalized Poincaré-Hopf theorem, relating the sum of Poincaré indices to the Euler characteristic of the system.