Abstract
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 the physical field. We compute the renormalized polarization tensor in coordinate representation. We discuss then the induced Chern-Simons type action on the boundary and compare it to the effective action of a $(2+1)$ dimensional Dirac fermion.
Highlights
We study the polarization tensor of a Dirac field in (3 þ 1) dimensions confined to a half-space—a problem motivated by applications to the condensed matter physics, and to topological insulators in particular
Various applications to the physics of new materials sparked a lot of interest in quantum field theory (QFT) with boundaries or interfaces
We suggested a modification of the PV regularization scheme that consists in giving axial masses to the PV regulators that become infinite in the physical limit
Summary
Various applications to the physics of new materials sparked a lot of interest in quantum field theory (QFT) with boundaries or interfaces. The fermions in (2 þ 1) dimensions possess a parity anomaly [11,12,13] that leads to the Chern-Simons action of level k 1⁄4 1=2 for the electromagnetic field. After integration over the normal coordinates, this tensor will give a Hall-type conductivity near the boundary We shall compare this integrated tensor with the parity-odd part of the polarization tensor for a Dirac fermion in (2 þ 1) dimensions. The polarization tensor of electromagnetic field in the presence of boundaries was considered in the condensed matter literature, see e.g., [19,20] In these papers, nonrelativistic (non-Dirac) dispersion relations for quasiparticles were used. Some technicalities are contained in the Appendixes: the parity-odd part of the effective action is computed in Appendix A, while some useful formulas are collected in Appendix B
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