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 wave vector <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>Q</a:mi></a:math> related to the total momentum of a Cooper pair. Here we investigate this phenomenon in a <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>p</b:mi></b:math>-wave topological superconductor, described by a one-dimensional Kitaev model. We demonstrate that, by treating <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>Q</c:mi></c:math> as an extra synthetic dimension, the current-carrying nonequilibrium 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 <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>p</d:mi></d:math>-wave superconductor, where Cooper pairs coexist with unpaired electrons and holes. Such a transition is signaled by the appearance of a sharp cusp in the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>Q</e:mi></e:math>-dependence of the supercurrent, at a critical value <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:msup><f:mi>Q</f:mi><f:mo>*</f:mo></f:msup></f:math> that is robust to variations of the chemical potential <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mi>μ</g:mi></g:math>. We determine the maximal current that the system can sustain in the topological phase, and we discuss possible implementations. Published by the American Physical Society 2024