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 symmetry-protecting nature as the corresponding $1D$ topological phase. Robust edge states can exist under specific conditions. These results provide further understanding on $2D$ time-reversal invariant insulators, and can be realized experimentally.