Abstract
The Dirac mass of a two-dimensional QAH insulator is directly related to the parity anomaly of planar quantum electrodynamics, as shown initially in Phys. Rev. Lett. 52, 18 (1984). In this work, we connect the additional momentum-dependent Newtonian mass term of a QAH insulator to the parity anomaly. By calculating the effective action, we reveal that the Newtonian mass term acts like a parity-breaking element of a high-energy regularization scheme. As such, it is directly related to the parity anomaly. In addition, the calculation of the effective action allows us to determine the finite frequency correction to the DC Hall conductivity of a QAH insulator. We derive that the leading order AC correction contains a term proportional to the torsional Hall viscosity. This paves the way to measure this non-dissipative transport coefficient via electrical or magneto-optical experiments. Moreover, we prove that the Newtonian mass significantly changes the resonance structure of the AC Hall conductivity in comparison to pure Dirac systems like graphene.
Highlights
The discovery of Dirac materials, such as topological insulators [1,2,3,4,5,6,7,8,9,10] and Weyl or Dirac [11,12,13,14,15,16] semimetals yields the possibility to measure quantum anomalies in condensed matter systems
As their contribution to the Chern number does not vanish in the parity symmetric limit, both of these mass terms are directly related to the parity anomaly [18,26]
We show that the leading order AC correction contains a term which is proportional to the Chern number
Summary
The discovery of Dirac materials, such as topological insulators [1,2,3,4,5,6,7,8,9,10] and Weyl or Dirac [11,12,13,14,15,16] semimetals yields the possibility to measure quantum anomalies in condensed matter systems. It was shown that twodimensional quantum anomalous Hall (QAH) insulators, such as (Hg,Mn)Te quantum wells [17,18] or magnetically doped (Bi,Sb)Te thin films [19,20], are directly related to the parity anomaly of planar quantum electrodynamics (QED2+1) [21] In a nutshell, this anomaly implies that in odd space-time dimensions it is not possible to quantize an odd number of Dirac fermions at the same time in a gauge- and paritysymmetric manner [22,23,24]. Having discussed the particular influence of the mass terms on the band structure, let us emphasize one more time that both of them explicitly break the parity symmetry in 2 + 1 space-time dimensions, defined as invariance of the theory under P : (x0, x1, x2) → (x0, −x1, x2).
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