Electronic structure of topological superconductors in the presence of a vortex lattice

TL;DR

This paper studies the formation and properties of Majorana zero mode bands in dense vortex lattices of topological superconductors, using analytical and numerical methods on specific models, with implications for quantum computing.

Contribution

It provides a detailed analysis of Majorana band structures in two models, revealing their description via tight binding models with non-trivial Z2 gauge flux and identifying a flat band at the neutrality point.

Findings

Majorana bands are described by tight binding models with Z2 gauge flux.

The Fu-Kane model exhibits a flat Majorana band at the neutrality point.

Low energy physics at the neutrality point involves four-fermion interactions.

Abstract

Certain types of topological superconductors and superfluids are known to host protected Majorana zero modes in cores of Abrikosov vortices. When such vortices are arranged in a dense periodic lattice one expects zero modes from neighboring vortices to hybridize and form dispersing bands. Understanding the structure of these bands is essential for the schemes that aim to employ the zero modes in quantum computation applications and in studies of their strongly interacting phases. We investigate here the band formation phenomenon in two concrete models, describing a two dimensional p + ip superconductor and a superconducting surface of a three-dimensional strong topological insulator (Fu-Kane model), using a combination of analytical and numerical techniques. We find that the physics of the Majorana bands is well described by tight binding models of Majorana fermions coupled to a static Z2 gauge field with a non-trivial gauge flux through each plaquette, in accord with expectations based on very general arguments. In the case of the Fu-Kane model we also find that, irrespective of the lattice geometry, the Majorana band becomes completely flat at the so called neutrality point (chemical potential coincident with the Dirac point) where the model exhibits an extra chiral symmetry. In this limit the low energy physics will be dominated by four-fermion interaction terms which are permitted by symmetries and may arise from the Coulomb interaction between the constituent electron degrees of freedom.