Topological phase transition in disordered elastic quantum spin Hall system

TL;DR

This paper explores how disorder affects topological states in a 2D mechanical system, demonstrating a transition from a quantum spin Hall insulator to a trivial insulator through analytical, numerical, and design-based methods.

Contribution

It introduces a mechanical QSH insulator model, uses the spin Bott index to characterize disorder effects, and verifies topological phase transitions in both discrete and continuous systems.

Findings

Disorder induces a transition from topologically nontrivial to trivial states.

The spin Bott index effectively characterizes topological changes under disorder.

Topologically protected boundary states are verified in phononic crystal models.

Abstract

We investigate the effect of disorder on topologically nontrivial states in a two dimension (2D) mechanical system. We first propose a quantum spin Hall (QSH) insulator based on an out-of-plane spring-mass model and analytically study the interplay between the disorder and topology in both topologically trivial and nontrivial systems. We adopt the spin Bott index to characterize the topological property in disordered mechanical systems. By tracking the evolution of the spin Bott index with the increase of disorders, we quantitatively demonstrate the disorder induced transition from a topologically nontrivial QSH insulator to a trivial insulator. We then validate the topological phase transition through transient analysis in discrete lattices. Finally, we design a phononic crystal based on the discrete spring-mass model and numerically verify the topologically protected states along the boundary between the trivial insulator and disordered topological QSH insulator in a continuous system. This work puts a step forward in understanding the role of disorder in a 2D topological classical system.