Local Hall Conductivity in Disordered Topological Insulators

TL;DR

The work derives a real-space expression for the local Hall conductivity by coupling to a local vector potential, separating diamagnetic and paramagnetic contributions, and linking the result to velocity operator matrix elements and two-point state overlaps. Applying this framework to a minimal square-lattice Chern insulator with onsite disorder, the authors show that semimetallic patches can induce a topological transition, and that the topological phase space grows with increased disorder, especially when disorder is split into multiple patches. They demonstrate that local Hall conductivity fluctuations localize around disorder patches and can percolate to drive global topology, with correlations between patches enhancing the transition. The study provides a practical, experimentally relevant route to visualize Hall currents in disordered bulk systems and suggests local probes and Streda-based approaches for imaging topological order in real materials.

Abstract

We derive the expression for the local Hall conductivity for systems that lack translation symmetry and use it to study the local fluctuations of the Hall signal around disordered patches in magnetic insulators. We find that the regime in parameter space over which the system is a Chern insulating state increases upon inclusion of non-magnetic potential disorder. In addition, the phase space over which the topological Anderson insulator exists can be enhanced by breaking up a single disordered patch into multiple smaller patches with the same total amount of disorder. We expect our results will motivate the next generation of local scanning and local impedance spectroscopy experiments to visualize Hall currents around patches in the bulk of a disordered topological insulator.

Figures (8)

• Figure 1: Topological Transitions in a Clean System. (a) At half filling the system is insulating and can be in three distinct topological states with $C=\pm1$ or $C=0$. The phase boundaries as a function of $M_z/t$ and $M_x/t$ with $M_y=0$ are shown. (b) Band structure for critical $M_z$ on the phase-space line $M_x=M_y=0$. At these values of $\bm{M}$ the lower and upper bands touch at high symmetry points in the Brillouin zone.
• Figure 2: Phase Diagram of Chern Insulator Model for Single Patch of Potential Disorder. (a) Gaussian disorder profile on a typical patch length scale $L$ over which the system is semi-metallic and where the mass in equation \ref{['LocPerHop']} vanishes. (b) The parameter region over which the topological phase exists increases with the size of disorder patch. (c) Phase boundary remains circular, but the center $\bm{M}=M_{z0}\bm{\hat{z}}$ is shifted and radius $M_R$ increases non-linearly with the size of the disorder patch $L$. Calculations were performed on a system of $N_s=400$ sites.
• Figure 3: Local Hall conductivity for Different Disorder Patch Sizes. (a)-(b) Hall conductivity calculated for two systems with $\bm{M}_1=2t\bm{\hat{x}}+5t\bm{\hat{z}}$ and $\bm{M}_2=-3t\bm{\hat{x}}+-2t\bm{\hat{z}}$ in the presence of a semimetallic disorder patch of size $L$. With increasing $L$ there are topological phase transitions indicated by white regions that separate the colored regions with well defined Chern numbers; here $N_s=400$. (c)-(d) Local Hall conductivity maps for four different size patches in system $\bm{M}_1$ and system $\bm{M}_2$. Contour where the gradient of the mass is peaked is shown by dashed black line. Topological phase transitions occur when the local Hall conductivity fluctuations permeate throughout the system.
• Figure 4: Local Currents Across Disorder Patches. Local Hall conductivity for system with $\bm{M}=(2t,0,5t)$ for patch sizes $L=3.2a$ (a)-(b) and $L=5.4a$ (c)-(d). (b) and (d) show a line cut ($y=10.5a$) across local Hall conductivity distribution showing how positive and negative fluctuations across sample average to zero in the trivial Chern insulating phase with $L=3.2a$ and average to $e^2/h$ in the $C=1$ phase with $L=5.4a$.
• Figure 5: Phase Diagram of Chern Insulator Model for Different Percent of Trivial Potential Disorder. (a) Phase boundaries in $M_z/M_x$-plane for various amounts of disorder. Shown are the best fits averaged over 25 disorder configurations. Here $N_s=100$ and the defect size is fixed at $L=a$. (b) We find that the phase space over which the topological insulator exists increases with the percent of disorder; analogous to the behavior for a single patch of disorder shown in Fig.\ref{['Boundaries1']}.