Majorana braiding in superconductors with fixed total number of particles

TL;DR

This paper explores Majorana modes in superconductors with fixed particle number, demonstrating how braiding operations can be implemented in a many-body context with conserved total charge, using projected wave functions.

Contribution

It extends the mean-field Majorana braiding framework to fixed particle number systems by employing projected wave functions, showing how topological operations can be realized with reduced noise protection.

Findings

Projected wave functions retain key topological features.

Braiding operations can be implemented with fixed particle number.

Protection from noise and disorder is reduced.

Abstract

One-dimensional topological superconductors treated at the mean-field level host zero-energy edge Majorana modes, which encode topological degeneracy of their ground states. Geometric manipulations (braiding) of multiple wires can be used to induce topologically robust transformations within the ground state subspace. The mean-field ground states do not have a definite number of particles and thus cannot describe an isolated system. Projecting such states onto fixed particle number gives a very good approximation to the true ground state of an isolated superconductor. In previous work, we showed that so projecting the prototypical Kitaev wave function of a single wire retains key features of the mean-field description, such as the zero-energy single-particle spectral peaks near the wire edges. Here we consider the case of multiple wires with conserved total charge. Again, using the projected Kitaev wave functions as the template, we identify the many-body counterparts of the mean-field topologically degenerate ground states. We then demonstrate how the quantum gates equivalent to braiding of Majorana zero modes can be implemented, albeit with reduced protection from noise and disorder.