Realization of a three-dimensional quantum Hall effect in a Zeeman-induced second-order topological insulator on a torus

We propose a realization of a quantum Hall effect (QHE) in a second-order topological insulator (SOTI) in three dimensions, which is mediated by hinge states on a torus surface. It results from the nontrivial interplay of the material structure, the Zeeman effect, and the surface curvature. In contrast to the conventional two-dimensional (2D)- and 3D-QHE, we show that the 3D-SOTI QHE is not affected by orbital effects of the applied magnetic field, and it exists in the presence of a Zeeman term only, induced, e.g., by magnetic doping. To explain the 3D-SOTI QHE, we analyze the boundary charge for a 3D-SOTI and establish its universal dependence on the Aharonov-Bohm flux threading through the torus hole. Exploiting the fundamental relation between the boundary charge and the Hall conductance, we demonstrate the universal quantization of the latter, as well as its stability against random disorder potentials and continuous deformations of the torus surface.