Electric Polarization in a Chern Insulator

We extend the Berry-phase concept of polarization to insulators having a nonzero value of the Chern invariant. The generalization to such Chern insulators requires special care because of the partial occupation of chiral edge states. We show how the integrated bulk current arising from an adiabatic evolution can be related to a difference of bulk polarizations. We also show how the surface charge can be related to the bulk polarization, but only with a knowledge of the wave vector at which the occupancy of the edge state is discontinuous. Furthermore, we present numerical calculations on a model Hamiltonian to provide additional support for our analytic arguments.

Figure 1

(a) Sketch of ${\theta }_1(k_2)$ in a Chern insulator ($C=+1$). Solid black frame and dashed gray (red) frame indicate reciprocal-cell origin chosen at ${\kappa }_2$ and ${\kappa }_2+\Delta k$, respectively. Dotted lines indicate corresponding averages, proportional to $P_1$. (b) Computed $P_1(\alpha )$ and ${\mathcal{P}}_1(\alpha )$ for the modified Haldane model, in units of $-0.01e/S$, for adiabatic (dashed lines) and thermal (solid line and symbols) filling. See text.

Figure 2

Sketch of a band structure of a finite ribbon of a Chern insulator. Solid regions indicate projected bulk bands; thin solid lines are edge states. Black and gray (red) correspond to $\alpha ={\alpha }_i$ and $\alpha ={\alpha }_f$, respectively; corresponding values of $k_2^{\times }$ are indicated. Thick lines indicate filling of edge states as dictated by $k_2^{*}$, chosen to illustrate system thermalized at ${\alpha }_i$ and then carried adiabatically to ${\alpha }_f$. Inset: Edge states associated with left (green) and right (blue) surfaces.

Figure 3

Black dots show ribbon HWF centers ${\varrho }_{j{k}_2}^{[k_2^{*}]}$ and gray (red) triangles show bulk HWF centers ${\rho }_{k_2}^{[{\kappa }_2]}$ and its periodic images as a function of $k_2$. Dashed frame corresponds to choice of origin at discontinuity in ${\varrho }_{j{k}_2}^{[k_2^{*}]},k_2^{*}$.