Topological Superconductor from the Quantum Hall Phase: Effective Field Theory Description

TL;DR

This paper develops effective field theories for quantum anomalous Hall and topological superconducting phases, revealing how strong pairing induces Majorana fermions and describing the edge states via orbifold conformal field theories.

Contribution

It introduces a novel effective field theory framework for topological superconductors derived from quantum Hall states, including the treatment of symmetry breaking and edge states.

Findings

Effective Chern-Simons action describes quantum Hall and topological superconducting phases.

Strong pairing leads to Majorana fermions in the topological superconductor.

Edge theory is a $U(1)/Z_2$ orbifold containing Majorana fermions.

Abstract

We derive low-energy effective field theories for the quantum anomalous Hall and topological superconducting phases. The quantum Hall phase is realized in terms of free fermions with nonrelativistic dispersion relation, possessing a global $U(1)$ symmetry. We couple this symmetry with a background gauge field and compute the effective action by integrating out the gapped fermions. In spite of the fact that the corresponding Dirac operator governing the dynamics of the original fermions is nonrelativistic, the leading contribution in the effective action is a usual Abelian $U(1)$ Chern-Simons term. The proximity to a conventional superconductor induces a pairing potential in the quantum Hall state, favoring the formation of Cooper pairs. When the pairing is strong enough, it drives the system to a topological superconducting phase, hosting Majorana fermions. Even though the continuum $U(1)$ symmetry is broken down to a $\mathbb{Z}_2$ one, we can forge fictitious $U(1)$ symmetries that enable us to derive the effective action for the topological superconducting phase, also given by a Chern-Simons theory. To eliminate spurious states coming from the artificial symmetry enlargement, we demand that the fields in the effective action are $O(2)$ instead of $U(1)$ gauge fields. In the $O(2)$ case we have to sum over the $\mathbb{Z}_2$ bundles in the partition function, which projects out the states that are not $\mathbb{Z}_2$ invariants. The corresponding edge theory is the $U(1)/\mathbb{Z}_2$ orbifold, which contains Majorana fermions in its operator content.