Dimensional evolution between one- and two-dimensional topological phases

TL;DR

This paper systematically explores the transition from 2D to 1D topological phases, revealing oscillations, weak topological phases, and conditions for robust edge states in time-reversal invariant insulators.

Contribution

It introduces the concept of weak topological phases in 2D insulators derived from 1D topological insulators and analyzes their properties and experimental realizability.

Findings

Oscillating behavior during 2D to 1D crossover.

Existence of weak topological phases with Z2=0.

Conditions for robust edge states along specific boundaries.

Abstract

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.