Topological insulators avoid the parity anomaly

Abstract

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 is accomplished from two complementary perspectives: topological quantization conditions and regularization. We first discuss the conditions under which (continuous) large gauge transformations may exist when the theory lives on a boundary of a higher-dimensional spacetime. Next, we show how the higher-dimensional bulk theory is essential in providing a parity-invariant regularization of the theory living on the lower-dimensional boundary or defect.

Three-dimensional topological insulators and bosonization

“…Equation (2.17) establishes the bulk-boundary cancellation of the Z 2 anomaly in (3 + 1)-dimensional topological insulators [42][43][44]. Actually, this should rather be called a boundary-boundary cancellation: the theta term does not imply local effects in the bulk; its role is that of 'transporting' the anomalous action S CS [A] from one surface to the other.…”

Functional approach to electrodynamics of media

“…It is therefore desirable to allow more generally for external and induced fields which do not vanish before any initial time t 0 . We are thus led to consider the more general case of fields with a nonvanishing static contribution 52) and analogously for the vector potential. The solution (4.48) can be written in the limit t 0 → −∞ as…”

“…Further systems with magnetoelectric coupling include the prototype material Cr 2 O 3 [34][35][36], single-phase multiferroics [37][38][39][40], composite multiferroics and multiferroic interfaces [41][42][43][44] as well as systems with Rashba spin-orbit coupling [45][46][47]. Topological insulators show the topological magnetoelectric effect [48][49][50], which becomes observable if the surface response is separated from the bulk contribution by breaking the time-reversal symmetry [51,52], for example, through the introduction of ferromagnetism [53][54][55].…”

Interacting Surface States of Three-Dimensional Topological Insulators

“…On the surface of a 3DTI, where θ changes from 0 to π, the gauge invariance of this topological field theory demands the presence of a half-integer Chern-Simons term at the surface. In the absence of symmetry breaking, this Chern-Simons term is precisely what allows a single Dirac cone surface state to be gauge invariant and hence consistently defined on the 3DTI surface, whereas a single massless Dirac field in odd space-time dimensions would usually exhibit a parity anomaly [13][14][15].…”

“…On the surface of a 3DTI, where θ changes from 0 to π, the gauge invariance of this topological field theory demands the presence of a half-integer Chern-Simons term at the surface. In the absence of symmetry breaking, this Chern-Simons term is precisely what allows a single Dirac cone surface state to be gauge invariant and hence consistently defined on the 3DTI surface, whereas a single massless Dirac field in odd space-time dimensions would usually exhibit a parity anomaly [13][14][15].When the protecting U(1) particle number symmetry is broken, such as by a superconducting proximity effect, the 3DTI surface yields an unconventional gapped s-wave superconductor with Majorana modes in its vortex cores [16]. Upon breaking TRS, such as by a magnetic coating on the surface, the single surface Dirac cone gaps out, and the Chern-Simons boundary term of the axion bulk action manifests itself as a ν = 1/2 quantum Hall effect [12] without fractionalized excitations.…”