Lifshitz transitions and Weyl semimetals from a topological superconductor with supercurrent flow

A current flowing through a superconductor induces a spatial modulation in its superconducting order parameter, characterized by a wavevector Q related to the total momentum of a Cooper pair. Here we investigate this phenomenon in a p-wave topological superconductor, described by a one-dimensional Kitaev model. We demonstrate that, by treating Q as an extra synthetic dimension, the current carrying non-equilibrium steady state can be mapped into the ground state of a half-filled two-dimensional Weyl semimetal, whose Fermi surface exhibits Lifshitz transitions when varying the model parameters. Specifically, the transition from Type-I to Type-II Weyl phases corresponds to the emergence of a gapless p-wave superconductor, where Cooper pairs coexist with unpaired electrons and holes. Such transition is signaled by the appearance of a sharp cusp in the Q-dependence of the supercurrent, at a critical value Q^* that is robust to variations of the chemical potential μ. We determine the maximal current that the system can sustain in the topological phase, and discuss possible implementations.