Diffusive metal in a percolating Chern insulator

TL;DR

The authors show that geometric disorder in a two-dimensional class D Chern insulator—implemented as random bond dilution with controlled stitching—generates a robust diffusive metal (DM) phase that carries charge and exhibits a nonquantized Hall response. The DM phase emerges when negative coupling $\alpha$ stitches broken bonds, creating randomly distributed $\mathbb{Z}_2$ flux plaquettes whose percolating zero-energy modes form conduction channels; this yields a metal-insulator transition with a critical exponent $\nu \approx 2$, distinct from the thermal-metal universality in disordered topological superconductors. AI-CI transitions retain Dirac universality ($\nu \approx 1$), while AI-DM and DM-CI transitions share the $\nu \approx 2$ criticality, signaling a unique universality class for geometrically disordered 2D SPTs. The work highlights the pivotal role of geometric disorder in engineering novel metallic phases in topological systems and suggests a broader relevance of flux-defect percolation to disordered quantum matter.

Abstract

Two-dimensional non-interacting fermions without any anti-unitary symmetries generically get Anderson localized in the presence of disorder. In contrast, topological superconductors with their inherent particle-hole symmetry can host a thermal metallic phase, which is non-universal and depends on the nature of microscopic disorder. In this work, we demonstrate that in the presence of geometric disorders, such as random bond dilution, a robust metal can emerge in a Chern insulator with particle-hole symmetry. The metallic phase is realized when the broken links are weakly stitched via concomitant insertion of $π$ fluxes in the plaquettes. These nucleate low-energy manifolds, which can provide percolating conduction pathways for fermions to elude localization. This diffusive metal, unlike those in superconductors, can carry charge current and even anomalous Hall current. We investigate the transport properties and show that while the topological insulator to Anderson insulator transition exhibits the expected Dirac universality, the metal insulator transition displays a different critical exponent $ν\approx 2$ compared to a disordered topological superconductor, where $ν\approx 1.4$. Our work emphasizes the unique role of geometric disorder in engineering novel phases and their transitions in topological quantum matter.

Figures (11)

• Figure 1: Phase diagram: Schematic phase diagram of a random bond-disordered Chern insulator. The inset shows the disorder protocol where the hopping strengths are either $t$ with probability $p$ or $\alpha t$ with probability $(1-p)$. Apart from Chern insulator (CI) and localized Anderson insulator (AI), the phase diagram features a diffusive metal (DM) phase characterized by longitudinal conductivity $\sigma_{xx} \sim \log L$ ($L$ is the system size), and non-quantized $\sigma_{xy}$. While for DM-AI and DM-CI transition critical exponent $\nu \approx 2$, for AI-CI transition $\nu \approx 1.$
• Figure 2: Characterizing different phases: (a) Configuration averaged $\sigma_{xx}$ phase diagram of the RBCI on a $40 \times 40$ lattice, computed over $200$ disorder realizations. Dashed lines denote the critical boundaries considering the virtual crystal approximation. (b) Along $p=0.5$, $\sigma_{xx}$, and $\sigma_{xy}$ distinguish DM, AI, and CI phases across $\alpha$. (c) The DM phase (point I: $p=0.5, \alpha = -0.6$) shows $\sigma_{xx}\sim \log L$ and finite $\sigma_{xy}$. Conductivities in (b, c) are averaged over 400 configurations; while for $\sigma_{xx}$ computation, width $W$ = $2L$, for $\sigma_{xy}$, $W = L/2$. (d) The disorder-averaged ($10^3$ configurations) density of states (DOS) for $L=512$ in the DM phase (point I) exhibits a logarithmic divergence.
• Figure 3: Probing phase transitions: (a) Finite size scaling of average $\sigma_{xx}$ across the DM-AI transition at $p=0.6$. Inset: Scaling collapse with $\alpha_c \approx -0.294$ and critical exponent $\nu \approx 2$. (b) Same as (a), but across the DM-CI transition with $p$ at $\alpha = -0.6$. Inset: scaling collapse with $p_c \approx 0.868$ and critical exponent $\nu \approx 2$. The standard error bars are within the width of the markers. See SM sm for the number of configurations considered to compute disorder averaging.
• Figure 4: Understanding the origin of metal: (a) Energy spectrum of a typical configuration of the RBCI model with small density of $\pi$-flux defects at $\alpha=-0.8, p=0.98$, shows sub-gap states near zero energy. (b) These states are localized at $\pi$-flux plaquettes in the configuration marked in crosses. (c) At $p=0.5$, local density of states of near-zero energy modes of a single configuration illustrate all three phases: CI edge state ($\alpha = 0.2)$), localized AI ($\alpha = -0.2$), and extended metallic state ($\alpha=-0.6$).
• Figure S1: (a) Chern number $\mathcal{C}$ with $\alpha$ for the $p=0$ clean limit Hamiltonian of RBCI (see Eq. (\ref{['eq_halpha']}). (b) Phase diagram of RBCI under virtual crystal approximation (see Hamiltonian in Eq. (\ref{['eq_hmf']})) in terms of $\mathcal{C}$ showing either trivial insulating or Chern insulating phase with $\mathcal{C}=1$.