Local topological markers for Chern insulators in ribbon geometry

Local topological markers are used to characterize Chern insulators in the presence of spatial inhomogeneities, such as boundaries and disorder. In this paper, we study the local Chern marker in systems with partial translational symmetry. We express the local Chern marker in the hybrid position-momentum basis for both open and periodic boundary conditions. We calculate the local Chern marker for a Haldane model ribbon. We show that the behavior at the two boundaries is qualitatively different from fully open geometries. We further compare the local Chern marker with the local St\v{r}eda marker and show agreement in the bulk and small deviations at the boundaries that diminish with increasing system size. The correspondence between the two markers remains good if disorder is introduced, provided its magnitude remains below large values that cause substantial change of the Chern number due to Anderson physics. Finally, by exploiting the numerical efficiency due to partial translational symmetry, we study equilibrium critical behavior and the Kibble-Zurek mechanism in a weakly disordered Qi-Wu-Zhang Chern insulator. We extract relevant scaling exponents from the local Chern marker configuration and show that they converge to the analytically predicted values with increasing system size.