Lifshitz Transitions and Weyl Semimetals from a Topological Superconductor with Supercurrent Flow

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