Topological nature of the transition between the gap and the gapless superconducting states

TL;DR

This paper explores the topological nature of the transition between gapped and gapless superconducting states, identifying the Euler characteristic as the key invariant and relating it to a Lifshitz-type phase transition with experimental implications.

Contribution

It clarifies the topological invariant involved in the transition, linking it to the Euler characteristic and catastrophe theory, and proposes experimental methods to confirm the topological phase transition.

Findings

The transition is of Lifshitz's type, a 2.5 order phase transition.

The topological invariant changing during the transition is the Euler characteristic.

The transition can be associated with a cuspidal edge in the density of states surface.

Abstract

Recently it was demonstrated that the long-known transition between the gap and gapless superconducting states in the Abrikosov-Gor'kov theory of superconducting alloy with paramagnetic impurities is of the Lifshitz's type, i.e. at zero temperature this is the $2\frac12$ order phase transition. Since transitions of this kind in a normal metal are always associated to certain topological changes, then below we clarify the topological nature of the transition under consideration. Namely, we demonstrate that the topological invariant which in process of the transition undergoes the change is nothing but the Euler characteristic. Alternatively, in terms of the theory of catastrophes one can relate this transition to appearance of the cuspidal edge at the corresponding surface of the density of states as the function of energy and superconducting order parameter. The concept of experiments for the confirmation of $2\frac12$ order topological phase transition is proposed. Obtained theoretical results can be applied for the explanation of recent experiments with lightwave-induced gapless superconductivity, for the interpretation of the disorder induced transition $s_{\pm}$-$s_{++}$ states via gapless phase in two-band superconductors, and the emergence of gapless color superconductivity in quantum chromodynamics.