Parity anomaly from the Hamiltonian point of view

Matthew F. Lapa
Phys. Rev. B 99, 235144 – Published 21 June 2019

Abstract

We review the parity anomaly of the massless Dirac fermion in 2+1 dimensions from the Hamiltonian, as opposed to the path integral, point of view. We have two main goals for this paper. First, we hope to make the parity anomaly more accessible to condensed matter physicists, who generally prefer to work within the Hamiltonian formalism. The parity anomaly plays an important role in modern condensed matter physics, as the massless Dirac fermion is the surface theory of the time-reversal invariant topological insulator (TI) in 3+1 dimensions. Our second goal is to clarify the relation between the time-reversal symmetry of the massless Dirac fermion and the fractional charge of ±12 (in units of e) that appears on the surface of the TI when a magnetic monopole is present in the bulk. To accomplish these goals we study the Dirac fermion in the Hamiltonian formalism using two different regularization schemes. One scheme is consistent with the time-reversal symmetry of the massless Dirac fermion, but leads to the aforementioned fractional charge. The second scheme does not lead to any fractionalization, but it does break time-reversal symmetry. For both regularization schemes we also compute the effective action Seff[A] that encodes the response of the Dirac fermion to a background electromagnetic field A. We find that the two effective actions differ by a Chern-Simons counterterm with fractional level equal to 12, as is expected from path-integral treatments of the parity anomaly. Finally, we propose the study of a bosonic analog of the parity anomaly as a topic for future work.

  • Received 9 April 2019

DOI:https://doi.org/10.1103/PhysRevB.99.235144

©2019 American Physical Society

Physics Subject Headings (PhySH)

Condensed Matter, Materials & Applied PhysicsParticles & Fields

Authors & Affiliations

Matthew F. Lapa*

  • Kadanoff Center for Theoretical Physics, University of Chicago, Chicago, Illinois 60637, USA

  • *mlapa@uchicago.edu

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Issue

Vol. 99, Iss. 23 — 15 June 2019

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