Dimensional evolution between one- and two-dimensional topological phases

Dimensional evolution between one- (1D) and two-dimensional (2D) topological phases is investigated systematically. The crossover from a 2D topological insulator to its 1D limit shows oscillating behavior between a 1D ordinary insulator and a 1D topological insulator. By constructing a 2D topological system from a 1D topological insulator, it is shown that there exist possibly weak topological phases in 2D time-reversal invariant band insulators, one of which can be realized in anisotropic systems. The topological invariant of the phase is ${Z}_{2}=0$. However, the edge states may appear along specific boundaries. It can be interpreted as arranged 1D topological phases, and have a symmetry-protecting nature as does the corresponding 1D topological phase. Robust edge states can exist under specific conditions. These results provide further understanding of 2D time-reversal invariant insulators, and can be realized experimentally.