Emergent fractals in dirty topological crystals

Abstract

Non-trivial geometry of electronic Bloch states gives rise to topological insulators which are robust against sufficiently weak randomness inevitably present in any quantum material. However, increasing disorder triggers a quantum phase transition into a featureless normal insulator. As the underlying quantum critical point is approached from the topological side, small scattered droplets of normal insulators start to develop in the system and their coherent nucleation causes ultimate condensation into a trivial insulator. Unless disorder is too strong, the normal insulator accommodates disjoint tiny topological puddles. Furthermore, in the close vicinity of such a transition the emergent islands of topological and trivial insulators display spatial fractal structures, a feature that is revealed only by local topological markers. Here we showcase this (possibly) generic phenomenon that should be apposite to dirty topological crystals of any symmetry class in any dimension from the Bott index and local Chern marker for a square-lattice-based disordered Chern insulator model.

Figures (3)

• Figure 1: (a) Disorder-averaged BI $\langle B \rangle$ as a function of disorder strength ($W$), obtained from exact diagonalization (ED) and kernel polynomial method (KPM), cross at $W \approx 6.0$ for different $L$ (linear dimension of square lattice), where $\langle B \rangle \approx 0.5$, marking the critical disorder $W_c$ for the TI-NI insulator QPT. Inset: Data collapse for $\delta L^{1/\nu}$ vs $\langle B \rangle$ with $\nu=2.49$, where $\delta=(W-W_c)/W_c$ and $\nu$ is the correlation length exponent. (b) Spatial distribution of the LCM $C_{\rm loc}(\boldsymbol{r})$ on a $L=40$ square lattice is shown after excluding four sites from each end of the system in both directions for a single disorder realization for each $W$. The pair of numbers correspond to $(W, \langle B \rangle)$. (c) Fraction of such an interior area ($f$) occupied by regions with $C_{\rm loc}(\boldsymbol{r}) = 1.0 \pm 0.2$ (blue) and $0.0 \pm 0.2$ (red). (d) Disorder-averaged LCM $\langle {\mathcal{C}}_{\rm loc} \rangle$ in a square box of linear dimension $\ell=L/4$ at the center of the system as a function of $W$, which for all choice of $L$ cross roughly at $W=W_c$. (e) Data collapse for $\delta L^{1/\nu}$ vs $\langle {\mathcal{C}}_{\rm loc} \rangle$ with $\nu=2.51$ for $\ell=L/4$. (f) Variation of $\nu$ with $\ell/L$. (g) Distributions of $C_{\rm loc}(\boldsymbol{r})$ for a single disorder realization with $W=W_c$ on a $L=100$ square lattice is shown after excluding ten sites from each end of the system in both directions. Patches of (h) trivial [(i) topological] insulators with $C_{\rm loc}(\boldsymbol{r}) = 0.0 \pm 0.2$ [$1.0 \pm 0.2$] are shown in different colors, exhibiting fractal structures (see Fig. \ref{['fig:Fig2']}). LCM is always computed from ED. We average over 50 (200) disorder realizations far from (near) $W_c$.
• Figure 2: Computation of (a) hull fractal dimension $d_h$, (b) volume fractal dimension $d_v$, and (c) anomalous dimension $\eta$ and correlation length $\xi$ from the pair-connectivity $G(r)$ at critical disorder $W=W_c=6.0$ from the islands of NIs in a $L=120$ system (in black). Panels (d)-(f) same as (a)-(c), respectively, for the islands of TIs. Dependance of (g) $d_h$, (h) $d_v$, (i) $\eta$, and (j) $\xi$ on $L$ for a few $W \lesssim W_c$ (color coded) for the islands of NIs. Panels (k)-(n) are same as (g)-(j), respectively, for the islands of TIs. We average over 50 disorder realizations. For each each $W$ we quote (color coded) the values of $d_h$ [(a) and (d)], $d_v$ [(b) and (e)], and $\eta$ and $\xi$ [(c) and (f)]. Error bars capture bootstrap standard error SM.
• Figure 3: Distribution of the LCM inside a $L=120$ square lattice for a single disorder realization of strenght $W=5.25$ showing the topological (trivial) regions with $C_{\rm loc}(\boldsymbol{r}) \approx 1.0$ ($0.0$) in black (white). Local density of states $\rho_{\rm loc}(E)$ (b) at the edge and (c) in the interior of the topological region, computed over five green colored sites within the golden and red boxes, respectively, showing gapless and multi-gapped structures near the Fermi energy $E_F=0$. (d) Corresponding average density of state $D(E)$ as a function of energy $E$.