Topological strings linking with quasiparticle exchange in superconducting Dirac semimetals

Abstract

We demonstrate a topological classification of vortices in three dimensional time-reversal invariant topological superconductors based on superconducting Dirac semimetals with an s-wave superconducting order parameter by means of a pair of numbers (NΦ, N ), accounting how many units NΦ of magnetic fluxes hc/4e and how many N chiral Majorana modes the vortex carries. From these quantities, we introduce a topological invariant which further classifies the properties of such vortices under linking processes. While such processes are known to be related to instanton processes in a field theoretic description, we demonstrate here that they are, in fact, also equivalent to the fractional Josephson effect on junctions based at the edges of quantum spin Hall systems. This allows one to consider microscopically the effects of interactions in the linking problem. We therefore demonstrate that associated to links between vortices, one has the exchange of quasi-particles, either Majorana zero-modes or e/2 quasi-particles, which allows for a topological classification of vortices in these systems, seen to be Z8 classified. While NΦ and N are shown to be both even or odd in the weakly-interacting limit, in the strongly interacting scenario one loosens this constraint. In this case, one may have further fractionalization possibilities for the vortices, whose excitations are described by SO(3)3-like conformal field theories with quasi-particle exchanges of more exotic types.

From Dirac Semimetals to Topological Phases in Three Dimensions: A Coupled-Wire Construction

“…intra-valley) s-wave pairing [207]. Pairing vortices host gapless chiral Majorana channels [207][208][209]. An array of these chiral vortices can form the basis in modeling superconducting manybody topological phases in three dimensions.…”

Microwave signatures of the Z2 and Z4 fractional Josephson effects

“…Since Eq. ( 3) has been previously solved, [2][3][4]23 here we review the main results rapidly but pause on some intricacies that are rarely discussed in the literature.…”