Topological insulators between dimensions

Topological phases are typically classified within discrete dimensions, yet real materials often bridge these regimes as their geometry is tuned. We investigate how reducing a two-dimensional topological insulator toward the one-dimensional limit leads to the emergence of new symmetry-protected states beyond conventional classification schemes. Using tight-binding theory and symmetry analysis, we show that topological edge modes persist down to a critical width, below which the system transitions to a one-dimensional phase hosting symmetry-protected zero-dimensional end states. This dimensional crossover illustrates how topology can transition between dimensions. We show that the topological behavior depends non-monotonically on system width due to competing hybridization and symmetry effects, revealing distinct regimes of topological protection. Our results establish how the interplay between topological phases of matter and geometric confinement gives rise to new symmetry-protected phases of matter. Furthermore, we show how such states can be experimentally realized and controlled in buckled honeycomb systems such as germanene.