Approximations of Interface Topological Invariants

SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5521-5582, August 2024.
Abstract. This paper concerns the asymmetric transport observed along interfaces separating two-dimensional bulk topological insulators modeled by (continuous) differential Hamiltonians and how such asymmetry persists after numerical discretization. We first demonstrate that a relevant edge current observable is quantized and robust to perturbations for a large class of elliptic Hamiltonians. We then establish a bulk edge correspondence stating that the observable equals an integer-valued bulk difference invariant depending solely on the bulk phases. We next show how to extend such results to periodized Hamiltonians amenable to standard numerical discretizations. A form of no-go theorem implies that the asymmetric transport of periodized Hamiltonians necessarily vanishes. We introduce a filtered version of the edge current observable and show that it is approximately stable against perturbations and converges to its quantized limit as the size of the computational domain increases. To illustrate the theoretical results, we finally present numerical simulations that approximate the infinite domain edge current with high accuracy and show that it is approximately quantized even in the presence of perturbations.