Effective action and electromagnetic response of topological superconductors and Majorana-mass Weyl fermions

TL;DR

This paper develops an effective action for Weyl fermions with Majorana mass using gauged Wess-Zumino-Witten functionals, revealing a reduced topological current and explaining phenomena like the chiral magnetic effect in topological superconductors.

Contribution

It introduces a novel method to compute the effective action and topological currents for Weyl fermions with Majorana mass, clarifying the anomaly-related current reduction.

Findings

Topological current is one-third of the expected gauge anomaly.

The anomaly remains unchanged despite the induced mass gap.

The chiral magnetic effect current is reduced to one-third, due to partial cancellation by the chiral vortical effect.

Abstract

Motivated by an apparent paradox in [X L. Qi, E.Witten, S-C. Zhang, Phys. Rev. B 87 134519 (2013)] we use the method of gauged Wess-Zumino-Witten functionals to construct an effective action for a Weyl fermion whose Majorana mass arises from coupling to a charged condensate. We obtain expressions for the current induced by an external gauge field and observe that the topological part of the current is only one-third of that that might have been expected from the gauge anomaly. The anomaly is not changed by the induced mass gap however. The topological current is supplemented by a conventional supercurrent that supplies the remaining two-thirds of the anomaly once the equation of motion for the Goldstone mode is satisfied. We apply our formula for the current to resolve the apparent paradox, and also to the chiral magnetic effect (CME) where it predicts a reduction of the CME current to one third of its value for a free Weyl gas in thermal equilibrium. We attribute this reduction to a partial cancelation of the CME by a chiral vortical effect (CVE) current arising from the persistent rotation of the fluid induced by the external magnetic field.