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

## TL;DR

This paper investigates how finite-size effects influence the dynamical conductivity and Faraday rotation in quantum anomalous Hall insulators, highlighting the importance of boundary conditions on topological properties and optical responses.

## Contribution

It provides a semi-analytic and numerical analysis of finite-size effects on edge states, conductivity, and Faraday rotation, emphasizing the role of boundary conditions in topological insulators.

## Key findings

- Edge state transitions dominate optical conductivity within and beyond the bulk gap.
- Finite-size boundary conditions alter the low-frequency Hall conductivity from e^2/2h to e^2/h.
- Finite-size effects and Fabry-Pérot resonances significantly impact the Faraday rotation spectrum.

## Abstract

We theoretically study the finite-size effects in the dynamical response of a quantum anomalous Hall insulator in the disk geometry. Semi-analytic and numerical results are obtained for the wavefunctions 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 wavefunctions.