Finite-Size Effects in the Dynamic Conductivity and Faraday Effect of Quantum Anomalous Hall Insulators

We theoretically study the finite-size effects in the dynamical response of a quantum anomalous Hall insulator in the disk geometry. Semianalytic and numerical results are obtained for the wave functions and energies of the disk within a continuum Dirac Hamiltonian description subject to a topological infinite mass boundary condition. Using the Kubo formula, we obtain the frequency-dependent longitudinal and Hall conductivities and find that optical transitions between edge states contribute dominantly to the real part of the dynamic Hall conductivity for frequency values both within and beyond the bulk band gap. We also find that the topological infinite mass boundary condition changes the low-frequency Hall conductivity to ${e}^{2}/h$ in a finite-size system from the well-known value ${e}^{2}/2h$ in an extended system. The magneto-optical Faraday rotation is then studied as a function of frequency for the setup of a quantum anomalous Hall insulator mounted on a dielectric substrate, showing both finite-size effects of the disk and Fabry-P\'erot resonances due to the substrate. Our work demonstrates the important role played by the boundary condition in the topological properties of finite-size systems through its effects on the electronic wave functions.