March 15, 2019
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 note. 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 $\pm\frac{1}{2}$ (in units of $e$) which 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 $S_{\text{eff}}[A]$ which 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 $\frac{1}{2}$, as is expected from path integral treatments of the parity anomaly. Finally, we propose the study of a bosonic analogue of the parity anomaly as a topic for future work.
Similar papers 1
November 8, 2018
We present a derivation of the recently discovered duality between the free massless (2+1)-dimensional Dirac fermion and QED$_3$. Our derivation is based on a regularized lattice model of the Dirac fermion and is similar to the more familiar derivation of the boson-vortex duality. It also highlights the important role played by the parity anomaly, which is somewhat less obvious in other discussions of this duality in the literature.
January 17, 2013
The surface of a 3+1d topological insulator hosts an odd number of gapless Dirac fermions when charge conjugation and time-reversal symmetries are preserved. Viewed as a purely 2+1d system, this surface theory would necessarily explicitly break parity and time-reversal when coupled to a fluctuating gauge field. Here we explain why such a state can exist on the boundary of a 3+1d system without breaking these symmetries, even if the number of boundary components is odd. This i...
August 12, 2022
Boundary conditions for a massless Dirac fermion in 2+1 dimensions where the space is a half-plane are discussed in detail. It is argued that linear boundary conditions that leave the Hamiltonian Hermitian generically break $C$ $P$ and $T$ symmetries as well as Lorentz and conformal symmetry. We show that there is essentially one special case where a single species of fermion has $CPT$ and the full Poincare and conformal symmetry of the boundary. We show that, with doubled fe...
March 11, 2013
The strong time-reversal symmetric (TRS) topological insulator (TI) in three space dimensions features gapless surface states in the form of massless Dirac fermions. We study these surface states with the method of bosonization, and find that the resulting bosonic theory has a topological contribution due to the parity anomaly of the surface Dirac fermions. We argue that the presence of a quantum anomaly is, in fact, the main reason for the existence of a surface state, by th...
June 30, 2000
We present an alternative derivation of the parity anomaly for a massless Dirac field in 2+1 dimensions coupled to a gauge field. The anomaly functional, a Chern-Simons action for the gauge field, is obtained from the non-trivial Jacobian corresponding to a non local symmetry of the Pauli-Villars regularized action. That Jacobian is well-defined, finite, and yields the standard Chern-Simons term when the cutoff tends to infinity.
February 25, 2018
We review the parity anomaly and a duality web in 2+1 dimensions. An odd dimensional non-interacting Dirac fermion theory is not parity invariant at quantum level. We demonstrate the parity anomaly in a three dimensional non-interacting Dirac fermion theory and a one dimensional non-interacting Dirac fermion theory. These theories can generate non-gauge invariant Abelian Chern-Simons terms at a finite temperature through an effective action. The parity anomaly also leads us t...
December 14, 2007
Fermion-number fractionalization without breaking of time-reversal symmetry was recently demonstrated for a field theory in $(2+1)$-dimensional space and time that describes the couplings between massive Dirac fermions, a complex-valued Higgs field carrying an axial gauge charge of 2, and a U(1) axial gauge field. Charge fractionalization occurs whenever the Higgs field either supports vortices by itself, or when these vortices are accompanied by half-vortices in the axial ga...
February 9, 2005
It is a well known feature of odd space-time dimensions $d$ that there exist two inequivalent fundamental representations $A$ and $B$ of the Dirac gamma matrices. Moreover, the parity transformation swaps the fermion fields living in $A$ and $B$. As a consequence, a parity invariant Lagrangian can only be constructed by incorporating both the representations. Based upon these ideas and contrary to long held belief, we show that in addition to a discrete exchange symmetry for ...
July 17, 2023
In this article, we extend our study on a new class of modular Hamiltonians on an interval attached to the origin on the semi-infinite line, introduced in a recent work dedicated to scalar fields. Here, we shift our attention to fermions and similarly to the scalar case, we investigate the modular Hamiltonians of theories which are obtained through dimensional reduction, this time, of a free massless Dirac field in $d$ dimensions. By following the same methodology, we perform...
August 19, 2012
It is shown that parity operator plays an interesting role in Dirac equation in (1+2) dimensions and can be used for defining chiral currents. It is shown that the "anomalous" current induced by an external gauge field can be related to the anomalous divergence of an axial vector current which arises due to quantum radiative corrections provided by triangular loop Feynman diagrams in analogy with the corresponding axial anomaly in (1+3) dimensions. It is shown that the non-co...