Material and size dependent corrections to conductance quantization in anomalous Hall effect from anomaly inflow

In quantum anomalous Hall (QAH) systems, the Hall conductance is quantized, and the corresponding effective topological theory of the system is the Chern-Simons theory. The conductance quantum is given by the universal constant ${e}^{2}/h$---the inverse von Klitzing constant---that is independent of the bulk gap, as well as the size of the system. This picture relies on the assumption that the edge modes are sharply localized at the edge, i.e., they have zero width. We show that, considering the physical case where the edge modes have finite localization length $b$, the effective action would not be topological in the bulk direction anymore. Because of nonzero $b$ the conductance quantum will be corrected as $(1\ensuremath{-}\ensuremath{\varepsilon}){e}^{2}/h$ where $\ensuremath{\varepsilon}$ encompasses the nonuniversal (i.e., material/sample dependent) part that is determined by the dimensionless ratios $\frac{gb}{\ensuremath{\hbar}{v}_{F}}$ and $\frac{b}{L}$ where $g,{v}_{F},L$ are the bulk gap, Fermi velocity, and sample length. To compute the nonuniversal correction $\ensuremath{\varepsilon}$ we use anomaly inflow framework according to which the bulk action produces the correct amount of anomaly inflow that would cancel the anomaly of the chiral edge modes. These corrections place limits on the precision of measurable quantization in units of the inverse von Klitzing constant for QAH systems with smaller sizes and/or smaller bulk gaps. Our result suggests that the failure of precision measurements to reproduce the exact conductance quantum ${e}^{2}/h$ is not an annoying sample quality issue, but it contains the quantitative physics of anomaly inflow that can be inferred by the systematic study of such corrections.