Phase diagram of amorphous quantum spin Hall insulators

TL;DR

The paper investigates how structural disorder affects a 2D quantum spin Hall insulator modeled by the BHZ Hamiltonian in amorphous settings. It develops a real-space $\mathbb{Z}_2$ marker to diagnose topology under disorder and maps phase diagrams as functions of mass $M$, disorder strength $\sigma$, and hopping cutoff $R$, including no-cutoff limit. The key findings are disorder-induced topological transitions and re-entrant behavior for certain $R$ values, with validation from edge-state spectra and two-terminal transport calculations that demonstrate bulk-boundary correspondence. The work highlights the sensitivity of amorphous topological phases to the connectivity range and provides a framework for exploring topology beyond crystalline symmetry.

Abstract

In light of recent progress in the study of amorphous topological phases, we investigate the effects of structural disorder on the topological properties of a two-dimensional quantum spin Hall insulator modeled by the Bernevig-Hughes-Zhang Hamiltonian. Using a real-space formulation of the Z2 invariant for Dirac-type Hamiltonian, we map out the phase diagram as a function of disorder strength and the mass parameter. Our results reveal that under the influence of structural disorder, a system can either undergo a phase transition from a topologically non-trivial to a topologically trivial phase or from a trivial to non-trivial phase. Remarkably, in certain parameter regimes, the system exhibits a re-entrant behaviour: a topologically non-trivial phase in the perfect lattice undergoes a transition to a trivial state under the influence of weak disorder but re-emerges as the disorder strength is further increased. We corroborate these findings through analysis of the bulk-boundary correspondence and transport calculations.

Figures (9)

• Figure 1: Phase diagram for perfect lattice with periodic boundary conditions for (a) $R=1.5$, system size: $20 \times 20$, and (b) $R=2.0$, system size: $24 \times 24$. At $\lambda=0.8$, phase transition occurs at $M \approx 2.62$ for $R=1.5$ and at $M \approx 4.115$ for $R=2.0$.
• Figure 2: Phase diagram as a function of $M$ and $\sigma$ at $\lambda=0.8$ with periodic boundary conditions. System size is $40 \times 40$. Top row: (a) $R=2.03$, (b) $R=3.03$, (c) $R=3.70$. Bottom row: (d) $R=1.70$, (e) $R=2.50$, (f) $R=3.38$.
• Figure 3: Top panel :Topological phase diagram for $R=24.03$ ($\approx$ no cutoff limit) Bottom panel : Average bulk gap as a function of $\sigma$ for $M=10.92$ (blue), 10.94 (green), 11.04 (red) and 11.10 (purple).
• Figure 4: Phase diagram for the amorphous BHZ model based on two-terminal conductance. Top row: (a) $R=2.03$, (b) $R=3.03$, (c) $R=3.70$. Bottom row: (d) $R=1.70$, (e) $R=2.50$, (f) $R=3.38$.
• Figure 5: Left panel: Spectrum for $R=2.03$ with periodic and open boundary conditions at different values of $\sigma$ for $M=3.80$ close to $E=0$ for a single configuration. Red dots denote the states for periodic boundary conditions, while blue dots denote states for open boundary conditions. Right panel: Plot of $|\Psi|^2$ for the state marked by a star in the figure to its right. (a,d) $\sigma=0.02$, (b,e) $\sigma=0.10$, (c,f) $\sigma=0.30$. The colour and size of the blobs are proportional to the wavefunction density at each site.