Universality of dimensional crossovers in topological insulators

Abstract: We investigate dimensional crossovers in minimal tight-binding models of three-dimensional (3D) topological insulators subject to geometric confinement. While thin films are commonly understood to host a crossover from a 3D strong topological insulator to a two-dimensional (2D) quantum spin Hall phase via hybridization of surface states, we demonstrate that this picture is incomplete once bulk confinement effects and boundary termination are fully taken into account. Using lattice models, we show that reducing the system size induces a strongly non-monotonic dependence of the topology on thickness and microscopic parameters, leading to a sequence of topological phase transitions that is highly sensitive to surface termination. In particular, we find a cascade of dimensional reduction from a 3D topological insulator to a 2D quantum spin Hall phase and ultimately to a one-dimensional phase consisting of end states of Kramers pairs protected by inversion symmetry. Remarkably, we show that both the 2D and 1D topological phases can emerge even when the corresponding 3D bulk phase is topologically trivial. Our results reveal an unexpected universality in the phase diagrams of 3D-to-2D and 2D-to-1D crossovers, pointing toward a unified framework for topology under dimensional reduction.