Topological phase transition in a disordered elastic quantum spin Hall system

We investigate the effect of disorder on topologically nontrivial states in a two-dimensional (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.

Figure 1

(a) Schematic of the unit cell of a spring-mass-based discrete structure. (b) Band structure of the expanded unit cell. (c) Berry curvature of the lower band of the spin-up channel. (d) Configuration of a finite size lattice structure with fixed boundary conditions. Green dots stand for the mass particles whose displacements are set to zero. (e) Eigenfrequencies of the finite size structure. (f) Eigenvalues of the spin projector operator $P_z$.

Figure 2

(a) Dependence of spin Bott index $B_S$ on the disorder strength under random discrete disorder. (b) Eigenfrequencies of a topologically nontrivial structure with periodic boundary conditions under different levels of disorder. (c) Disorder-averaged band gap edges and relative band gap width.

Figure 3

(a) Numerical setup of the rhombic structure. The interface between two kinds of topologically distinct lattices is indicated by the black dashed line. (b) Transportation of the pseudospin-dependent wave package along the interface at a different time in a clean system. Color intensity represents the power spectral density of the particle displacements. (c)–(f) Schematics of the disordered inner core under random discrete disorder with strength $W=0.2, W=0.4, W=0.6$, and $W=0.8$, respectively. (g)–(j) Snapshots of transient simulated results under pseudospin excitation in the corresponding disordered structures. Color bar represents the magnitude of the nondimensionalized displacement given as $20\text{ln}\frac{u_j}{A_0}$, where $u_j$ is the particle displacement and $A_0$ is the amplitude of the displacement input.

Figure 4

(a) Detailed view of the design of the phononic crystal. (b) A top view of the unit cell and a mapping relation to the discrete model. (c) Relationship between the width $d$ of air tubes and the effective spring stiffness. (d) Schematic of the finite element simulation setup. A U-shape interface is created by putting substructures with two types of unit cells adjacently. The red star indicates the excitation point. (e)–(i) Simulated wave field with a harmonic excitation along the interface under discrete disorder with disorder strength $W=0$, 0.2, 0.4, 0.6, and 0.8, respectively.

Figure 5

Berry curvature and spin Chern number of four pseudospin channels for $K_{\mathrm{in}}<K_{\mathrm{out}}$. (a) The lower band of pseudospin up. (b) The upper band of pseudospin up. (c) The lower band of pseudospin down. (d) The upper band of pseudospin down.

Figure 6

(a) Eigenfrequency spectrum of the topologically trivial (red circles) or nontrivial (blue circles) structure. (b)–(e) Eigenmodes of the finite structure corresponding to the markers in (a).

Figure 7

Effect of (a) disorder ratio and (b) disorder type on the topological phase transition.

Figure 8

(a)–(d) Transient simulated wave fields at $t=300$ under pseudospin excitation for two more realizations under random discrete disorder with disorder strength $W=0.2, W=0.4, W=0.6$, and $W=0.8$, respectively.

Figure 9

(a) Unit cell of the discrete model. (b) Top view of the unit cell of a phononic crystal. (c),(d) Comparison of unit-cell dispersion curves with a double Dirac point or a band gap obtained from the spring-mass model (solid lines) and the phononic crystal (red dots).

Figure 10

Procedure to generate the disordered phononic crystal.

Figure 11

(a)–(d) Steady state wave field under a harmonic excitation in the presence of discrete disorder with disorder strength $W=0.2$, 0.4, 0.6, and 0.8, respectively.

Figure 12

(a)–(d) Propagation of pseudospin dependent waves along a mixed topological interface that is composed of both zigzag and armchair type boundaries with disorder strength $W=0, W=0.2, W=0.4$, and $W=0.6$, respectively.