Temperature and chemical potential dependence of the parity anomaly in quantum anomalous Hall insulators

The low-energy physics of two-dimensional quantum anomalous Hall insulators like (Hg,Mn)Te quantum wells or magnetically doped (Bi,Sb)Te thin films can be effectively described by two Chern insulators, including a Dirac as well as a momentum-dependent mass term. Each of those Chern insulators is directly related to the parity anomaly of planar quantum electrodynamics. In this work, we analyze the finite-temperature Hall conductivity of a single Chern insulator in $2+1$ space-time dimensions under the influence of a chemical potential and an out-of-plane magnetic field. At zero magnetic field, this nondissipative transport coefficient originates from the parity anomaly of planar quantum electrodynamics. We show that the parity anomaly itself is not renormalized by finite-temperature effects. However, it induces two terms of different physical origin in the effective action of a Chern insulator, which is proportional to the Hall conductivity. The first term is temperature and chemical potential independent, and solely encodes the intrinsic topological response. The second term specifies the nontopological thermal response of conduction and valence band states. In particular, we show that the relativistic mass of a Chern insulator counteracts finite-temperature effects, whereas its nonrelativistic mass enhances these corrections. Moreover, we extend our analysis to finite magnetic fields and relate the thermal response of a Chern insulator therein to the spectral asymmetry, which is a measure of the parity anomaly in orbital fields.

Figure 1

Schematic illustration of how a single Dirac fermion or Chern insulator realizes the parity anomaly in the Haldane (blue) or BHZ (red) model. In the Haldane model, both Dirac fermions at the $K$ and $K^{\prime }$ points of the hexagonal lattice structure contribute $\pm \frac{1}{2}$ to Chern number, whereas in the QAH phase of the BHZ model only one of the Chern insulators is topologically nontrivial and has a finite Chern number $C=\pm 1$. More explanations are given in the text.

Figure 2

Band structure of a Chern insulator for zero magnetic field and $D=0$. Red and blue curves encode topologically nontrivial phases with $m=A=1$, and $B=5$ (red, camel-back), or $m=1$, $A=3$, and $B=0.1$ (blue). The yellow curve corresponds to a topologically trivial phase with $m=1$, $A=5$, and $B=-0.1$. The minimal gap is either defined by $2\left|m\right|$ at the $\mathrm{\Gamma }$ point or by $\left|\mathrm{\Delta }\right|$ at $\alpha ={\alpha }_{\min }$, indicated by the black or green arrow, respectively.

Figure 3

Finite-temperature Hall conductivity of a Chern insulator with $A=1$ and $D=0$. In (a) and (c), we vary the Dirac mass while $B=\pm 0.1$, respectively. In (b) and (d), we vary the Newtonian mass for $m=1$. In all subfigures we consider zero chemical potential. (a), (b) Correspond to the topologically nontrivial regime, while (c) and (d) correspond to the topologically trivial regime.

Figure 4

Finite-temperature Hall conductivity of a Chern insulator in a magnetic field $H=3T$ with $A=1$, $m=-1$, $B=-0.1$, and $D=-0.05$. The response of each valence (blue) and conduction band (red) LL is shown separately. The zero LL response is illustrated in black, the parity anomaly related contribution is depicted in green. The combined signal is shown in orange. Dashed lines correspond to $k_{\mathrm{B}}T=0$, solid lines are associated to $k_{\mathrm{B}}T=0.01$. The Dirac mass gap is shown in gray.