Universal Multifractality at the Topological Anderson Insulator Transition

Abstract

Disorder is ubiquitous in quantum materials, and its interplay with topology can generate phases absent in the clean limit. Using the Haldane model as a minimal setting, we show that disorder not only shifts topological boundaries but also stabilizes a topological Anderson insulator (TAI) between trivial and Chern insulating regimes. Employing the local Chern marker as a real-space topological probe, we map the full phase diagram and demonstrate that the TAI forms a finite domain bounded by trivial and Anderson insulators. Multifractal analysis of low-energy eigenstates at the boundary reveals universal critical spectra, independent of whether disorder generates or destroys topology. These results place topology, localization, and criticality within a unified framework and provide clear benchmarks for real-space diagnostics of disordered topological phases.

Figures (4)

• Figure 1: Schematic of the disorder-induced topological Anderson insulator in the Haldane model on a honeycomb lattice. Sublattice staggering $\pm M$ breaks inversion symmetry (blue/cyan sites), while complex next-nearest-neighbour hopping breaks time-reversal symmetry (indicated by arrows in the inset). The upper surface illustrates a representative real-space probability density $|\psi(\mathbf{r})|^2$, and the lower plane shows the (local) density of states at energy $E=0$, highlighting edge-localized spectral weight consistent with a nontrivial Chern phase ($C \neq 0$).
• Figure 2: Energy dispersion of a disorder-free ($W=0$) Haldane nanoribbon with 12 unit cells across. Panels (a)–(c) show the armchair edge for $M=1.48$, $t_{2}=0$ (trivial insulator), $M=1.48$, $t_{2}=1/3$ (Chern insulator), and $M=1.89$, $t_{2}=1/3$ (trivial insulator). Panels (d)–(f) display the corresponding spectra for the zigzag edge. The top valence and bottom conduction bands are highlighted in purple.
• Figure 3: Disorder dependence of the local Chern marker $c(\mathbf{r})$, computed on a $120\times120$ Haldane sample with $t_{2}=1/3$ for (a) $M=1.48$ and (b) $M=1.89$. The colour map is clipped to $c(\mathbf{r}) = \pm 2$. (c) Corresponding phase diagram as a function of $W$ and $M$, obtained from the integrated LCM $\bar{c}$ along the edges of the finite sample and averaged over 10 disorder realizations. Values of $\bar{c}$ are normalized by the mean across the phase diagram. To emphasize the contrast between trivial ($\bar{c}=0$) and topological ($\bar{c}=1$) regions, the colormap is clipped at $\bar{c}<1$. The guiding line indicates the approximate phase boundary separating the band insulator (BI) and Anderson insulator (AI) from the Chern insulator (CI) and topological Anderson insulator (TAI).
• Figure 4: (a) Map of the correlation dimension $D_{2}$ of Haldane-model eigenstates in the $(M,W)$ plane. The dashed curve indicates the disorder-driven phase boundary extracted from the LCM analysis in Fig. \ref{['fig:chern_h']}(c). (b,c) Disorder-averaged multifractal spectra $\tau(q)$ and generalized dimensions $D_{q}$ evaluated at representative points in the BI (orange hexagons), CI (green diamonds), and TAI (red circles and purple triangles) phases. Dashed curves show best fits to the parabolic approximation, Eq. \ref{['eq:WZNW']}. (d,e) Same as (b,c), but for three representative parameter sets along the phase boundary, demonstrating the collapse of $\tau(q)$ and $D_q$ onto universal critical curves. In (b-e), the errorbars are included but due to their insignificance appear invisible and hidden behind the markers.