Hall conductance of a non-Hermitian Chern insulator

In this paper, we study the Hall conductance for a non-Hermitian Chern insulator and quantitatively describe how the Hall conductance deviates from a quantized value. We show the effects of the non-Hermitian terms on the Hall conductance from two aspects. On one hand, it broadens the density of states of each band, because of which there always exists a nonuniversal bulk contribution. On the other hand, it adds a decay term to the edge state, because of which the topological contribution also deviates from a quantized Chern number. We provide a simple formula for the topological contribution for a general two-band non-Hermitian Chern insulator, as a non-Hermitian version of the Thouless-Kohmoto-Nightingale-de Nijs formula. It shows that the derivation from quantized value increases either when the strength of the non-Hermitian term increases, or when the momentum dependence of the non-Hermitian term increases. Our results can be directly verified in synthetic non-Hermitian topological systems where the strength of the non-Hermitian terms can be controlled.

Figure 1

An illustration of a gap. According to the spectrum decomposition theorem, each eigenstate with a complex energy ${\epsilon }_i$ contributes significant density of states from $[{\epsilon }_i^{\mathrm{R}}-|{\epsilon }_i^{\mathrm{I}}|,{\epsilon }_i^{\mathrm{R}}+|{\epsilon }_i^{\mathrm{I}}\left|\right]$ as is illustrated above. Here we use a dark color to represent the region for the collections as complex eigen-values, and we use a light color region to present the broadening of the spectrum. Only when ${\epsilon }_{\text{min}}^{+}\gg {\epsilon }_{\text{max}}^{-}$, with ${\epsilon }_{\text{max}}^m$ and ${\epsilon }_{\text{min}}^m$ defined in Eqs. (6) and (7), there exisits an energy window between two bands where the density of states is negligible.

Figure 2

The Hall conducance for the non-Hermitian Hamiltonian given by Eq. (22). The total Hall conductivity ${\sigma }_H$ calculated numerically is given by the red solid lines and the topological contribution ${\sigma }_{\text{H}}^{\text{T}}$ given by Eq. (17) is shown by green dashed lines. All conductivity is in unit of $e^2/h$. (a) ${\sigma }_H$ and ${\sigma }_{\text{H}}^{\text{T}}$ as a function of $m$ varying from $-2.5$ to 2.5, with $f\left(\mathbf{k}\right)=1$ and $\gamma =0.05$. (b) ${\sigma }_H$ and ${\sigma }_{\text{H}}^{\text{T}}$ as a function of $\gamma$ varying from 0 to 1, with $f\left(\mathbf{k}\right)=1$ and $m=1$ fixed. (c) ${\sigma }_H$ and ${\sigma }_{\text{H}}^{\text{T}}$ as a function of $a$ varying from 0 to 0.5, when $f\left(\mathbf{k}\right)$ is fixed at $1-(sin\left(a\eta \left(k_x\right)\right)+sin\left(a\eta \left(k_y\right)\right))/2$, where $\eta \left(x\right)\equiv |x-\mathrm{sgn}\left(x\right)\frac{\pi }{2}|+\mathrm{sgn}\left(x\right)\frac{\pi }{2}$, and with $\gamma =0.05$ and $m=1$ fixed.