Analytic ground state wavefunctions of mean-field $p_x + ip_y$ superconductors with vortices and boundaries

TL;DR

This paper derives exact analytic ground state wavefunctions for a special mean-field model of $p_x+ip_y$ superconductors, including vortices and edges, revealing non-Abelian statistics and aiding the construction of number-conserving Hamiltonians.

Contribution

It provides explicit analytic expressions for topologically degenerate ground states and Majorana modes in finite systems, including vortex and boundary effects, at a special solvable point.

Findings

Exact ground state wavefunctions for $p_x+ip_y$ superconductors with vortices and edges.

Explicit calculation of non-Abelian vortex statistics via wavefunction continuation.

Wavefunctions similar to Moore-Read Pfaffian states, exact at a special point.

Abstract

We study Read and Green's mean-field model of the spinless $p_x+ip_y$ superconductor [N.Read and D.Green, Phys. Rev. B 61, 10267 (2000)] at a special set of parameters where we find the analytic expressions for the topologically degenerate ground states and the Majorana modes, including in finite systems with edges and in the presence of an arbitrary number of vortices. The wavefunctions of these ground states are similar (but not always identical) to the Moore-Read Pfaffian states proposed for the $\nu=5/2$ fractional quantum Hall system, which are interpreted as the p-wave superconducting states of composite fermions. The similarity in the long-wavelength universal properties is expected from previous work, but at the special point studied herein the wavefunctions are exact even for short-range, non-universal properties. As an application of these results, we show how to obtain the non-Abelian statistics of the vortex Majorana modes by explicitly calculating the analytic continuation of the ground state wavefunctions when vortices are adiabatically exchanged, an approach different from the previous one based on universal arguments. Our results are also useful for constructing particle number-conserving (and interacting) Hamiltonians with exact projected mean-field states.