Topological states on fractal lattices

We investigate the fate of topological states on fractal lattices. Focusing on a spinless chiral $p$-wave paired superconductor, we find that this model supports two qualitatively distinct phases when defined on a Sierpinski gasket. While the trivial phase is characterized by a self-similar spectrum with infinitely many gaps and extended eigenstates, the “topological” phase has a gapless spectrum and hosts chiral states propagating along edges of the graph. Besides employing theoretical probes such as the real-space Chern number, inverse participation ratio, and energy-level statistics in the presence of disorder, we develop a simple physical picture capturing the essential features of the model on the gasket. Extending this picture to other fractal lattices and topological states, we show that the $p+ip$ state admits a gapped topological phase on the Sierpinski carpet and that a higher-order topological insulator placed on this lattice hosts gapless modes localized on corners.

Figure 1

(a) SG with “periodic” boundary conditions, such that all sites have coordination number four. (b) Regions $A,B,C$ considered in real-space Chern number calculations.

Figure 2

Gapless topological phase of the chiral $p+ip$ superconductor on the SG, with $g=5,\mathrm{\Delta }=1,t=0.5,\mu =0.5$. Energy spectrum and probability densities of eigenvectors at indicated energies are shown. Color scale indicates values of $x,y,z$ coordinates, with dot size indicating the magnitude of the probability density at that point.

Figure 3

Gapped trivial phase of the $p+ip$ superconductor on the SG, with $g=5,\mathrm{\Delta }=1,t=0.5,\mu =2$. Energy spectrum and probability densities of eigenvectors at indicated energies are shown. Color scale indicates values of $x,y,z$ coordinates, with dot size indicating the magnitude of the probability density at that point.

Figure 4

The real-space Chern number (black curve) as a function of Fermi energy $E_f$, with the corresponding spectrum shown in red in (a) the trivial phase $\left(\mu =2\right)$ and (b) the topological phase $\left(\mu =0.5\right)$. Here, $g=4,t=0.5$, and $\mathrm{\Delta }=0.5$.

Figure 5

${\mathrm{IPR}}_n$ (using open boundary conditions on a single SG), with $g=4,t=0.5,\mathrm{\Delta }=0.5$. All states are delocalized in the (a) trivial phase $\left(\mu =2\right)$ while the (b) topological phase $\left(\mu =0.5\right)$ exhibits states localized around edges.

Figure 6

${\mathrm{IPR}}_n$ with $g=5,t=0.5,\mathrm{\Delta }=0.5$. Comparison with Fig. 5 clearly shows that the number of localized states in the middle of the spectrum increases with $g$.

Figure 7

Distribution of normalized energy level spacings with disorder $W$, with $g=4,t=0.5,\mathrm{\Delta }=0.5$. Level statistics shown for weak $\left(W=2\right)$ and strong $\left(W=8\right)$ disorder for the trivial $\left(\mu =2\right)$ and topological $\left(\mu =0.5\right)$ phases.

Figure 8

Decimating a triangular lattice recursively to generate the SG. Blue (black) dots denote sites with boundary (bulk) coordination. Sites and bonds inside the red (green) triangle(s) are eliminated at the first (second) step.

Figure 9

(a) Spectrum and (b) nonpropagating corner modes in an HOTI defined on the SC $\left(g=3,\gamma =0.5,\lambda =1\right)$.

Figure 10

Chern number $C$ for the $p+ip$ superconductor on a triangular lattice. The plot shows that $C=1$ for $-6<\mu /t<2$ (the topological phase), and $C=0$ otherwise, i.e., in the trivial phase. The above holds for any $\mathrm{\Delta }\ne 0$.

Figure 11

Decimating a $g=1$ lattice to a $g=0$ lattice by using Eq. (B3).

Figure 12

The evolution of a wave packet created by projecting onto edge states close to zero energy within the energy range $\left(-0.3<\varepsilon <0.3\right)$ is shown. It can be seen that it moves exclusively on the edge of the system with definite chirality $\left(g=4,\mu =0.5,t=0.5,\mathrm{\Delta }=0.5\right)$.

Figure 13

The evolution of a wave packet initialized on an inner edge is shown. It can be seen that it moves exclusively on the corresponding inner edge of the system with chirality opposite to that of the outermost edge $\left(g=4,\mu =0.5,t=0.5,\mathrm{\Delta }=0.5\right)$.

Figure 14

Scaling of the gap as a function of the generation $g$ of the Sierpinski gasket in the topological phase. The gap decays to zero exponentially fast as $g\to \infty$. Here, $t=0.5,\mathrm{\Delta }=0.5$.

Figure 15

Generating an SC from a square lattice. The first (second) step results in inner edge mode(s), shown in red (green), when starting from the topological phase of the $p+ip$ superconductor on a square lattice. However, a finite number of bulk sites (black dots) remain even after the SC is generated, as shown in the zoomed-in image on the right.

Figure 16

$p+ip$ state on the Sierpinski carpet: The energy eigenvalue spectrum is shown for (a) the trivial phase $(t=0.5,\mu =1,)$, and (b) the topological phase $(t=0.5,\mu =0.25)$. (c) shows the real-space Chern number within the gap as function of $\mu /t$, clearly indicating the existence of a gapped topological phase on the SC.