A note on the parity anomaly from the Hamiltonian point of view

We reinvestigate the classic example of chiral anomaly in (1+1) dimensional spacetime. By reviewing the derivation of charge conservation with the semiclassical Boltzmann equation, we argue that chiral anomalies could emerge in (1+1) dimensions without Berry curvature corrections to the kinetic theory. The pivotal step depends only on the asymptotic behavior of the distribution function of the quasiparticle, and thus its dispersion relation, in the limit of $|\mathbf p|\to\pm...

We give field theory descriptions of the time-reversal invariant quantum spin Hall insulator in 2+1 dimensions and the particle-hole symmetric insulator in 1+1 dimensions in terms of massive Dirac fermions. Integrating out the massive fermions we obtain a low-energy description in terms of a topological field theory, which is entirely determined by anomaly considerations. This description allows us to easily construct low-energy effective actions for the corresponding `fracti...

We develop a Dirac fermion theory for topological phases in magnetic topological insulator films. The theory is based on exact solutions of the energies and the wave functions for an effective model of the three-dimensional topological insulator (TI) film. It is found that the TI film consists of a pair of massless or massive Dirac fermions for the surface states, and a series of massive Dirac fermions for the bulk states. The massive Dirac fermion always carries zero or inte...

We note that in (2+1)-dimensional gauge theories with even number of massless fermions, there is anomalous $Z_2$ symmetry if theory is regularized in a parity-invariant way. We then consider a parity invariant $U(1)_V\times U(1)_A$ model, which induces a mutual Chern-Simons term in the effective action due to $Z_2$ anomaly. The effect of the discrete anomaly is studied in the induced spin and in the dynamical fermion mass.

We study the polarization tensor of a Dirac field in $(3+1)$ dimensions confined to a half space -- a problem motivated by applications to the condensed matter physics, and to Topological Insulators in particular. Although the Pauli-Villars regularization scheme has a number of advantages, like explicit gauge invariance and decoupling of heavy modes, it is not applicable on manifolds with boundaries. Here, we modify this scheme by giving an axial mass to the regulators and to...

Motivated by recent examples of three-dimensional lattice Hamiltonians with massless Dirac fermions in their (bulk) spectrum, I revisit the problem of fermion doubling on bipartite lattices. The number of components of the Dirac fermion in a time-reversal and parity invariant d-dimensional lattice system is determined by the minimal representation of the Clifford algebra of $d+1$ Hermitian Dirac matrices that allows a construction of the time-reversal operator with the square...

In a typical situation, gapless surface states of a three-dimensional (3D) weak topological insulator (WTI) appear only on sides, leaving the top and bottom surfaces gapped. To describe massless Dirac electrons emergent on such side surfaces of a WTI, a two-dimensional (2D) model consisting of a series of one-dimensional helical channels is usually employed. Yet, explicit derivation of such a model from a 3D bulk Hamiltonian has been lacking. Here, we explicitly derive an eff...

Particle-vortex duality is a powerful theoretical tool that has been used to study bosonic systems. Here we propose an analogous duality for Dirac fermions in 2+1 dimensions. The physics of a single Dirac cone is proposed to be described by a dual theory, QED3 with a dual Dirac fermion coupled to a gauge field. This duality is established by considering two alternate descriptions of the 3d topological insulator (TI) surface. The first description is the usual Dirac cone surfa...

In two spatial dimensions, spin characterizes how particle states re-phase under changes of frame that leave their momentum and energy invariant. Massless particles can in principle have non-trivial spin in this sense, but all existing field theories only describe the trivial case. This letter presents a field theory for a massless particle with non-trivial physical spin. These particles are the 2+1-dimensional analogues of "continuous-spin" particles in 3+1 dimensions, but h...

We study parity violation in $2+1$-dimensional gauge theories coupled to massive fermions. Using the $\zeta$-function regularization approach we evaluate the ground state fermion current in an arbitrary gauge field background, showing that it gets two different contributions which violate parity invariance and induce a Chern-Simons term in the gauge-field effective action. One is related to the well-known {\em classical} parity breaking produced by a fermion mass term in 3 di...