Topology in the Sierpiński-Hofstadter problem

Using the Sierpiński carpet and gasket, we investigate whether fractal lattices embedded in two-dimensional space can support topological phases when subjected to a homogeneous external magnetic field. To this end, we study the localization property of eigenstates, the Chern number, and the evolution of energy level statistics when disorder is introduced. Combining these theoretical tools, we identify regions in the phase diagram of both the carpet and the gasket, for which the systems exhibit properties normally associated with gapless topological phases with a mobility edge.

Figure 1

Sierpiński (a) carpet and (b) gasket at iteration $n=4$ and $n=6$, respectively. Black squares correspond to kept sites from underlying square (in case of SC) and triangular (SG) lattices. The summation regions included in real-space Chern number calculations are marked with A, B, C.

Figure 2

[(a), (d)] Density of states in the energy-flux plane, [(b), (e)] localization of the eigenstates, and [(c), (f)] edge-locality marker for the SC at iteration $n=4$ and the SG at iteration $n=6$, respectively. Darker regions are related to smaller density. Energy spectra reveal two gaps at $E=0$ for SC and numerous gaps in the case of SG. Electronic densities presented in (b) and (e) correspond to the time-reversal symmetric point ($\alpha =1/2$) indicated by a framed part of the spectrum. The color scale corresponds to the square modulus $|{\psi }_i{|}^2$ of the wave function normalized by its maximum value. (c) and (f) show how the ${\mathcal{B}}_{\lambda ,l}$ marker changes between consecutive eigenstates at fixed flux. Low DOS regions are associated with largely varying localization properties of the eigenstates.

Figure 3

[(a), (e)] Density of states, [(b), (f)] scaling exponent $\nu$ of the DOS with system size as a function of $E$, [(c), (g)] the Chern number as a function of $E$, and [(d), (h)] variance of level spacings in the energy-disorder plane at fixed flux $\alpha =1/4$ for the SC and the SG, respectively. Gray rectangles are drawn to guide the eye. We identify spectral gaps to be topologically trivial for both lattices. Regions with quantized values of the Chern number close to 1 are separated by a delocalized state from the Anderson insulator limit, as is the case for quantum Hall states [blue arrows in (d), (h)]. In contrast, a direct transition to a fully localized phase is observed for states carrying trivial Chern number as a function of $W$ [white arrows in (d), (h)]. States with $\mathcal{C}\ne 0$ are characterized by a DOS scaling exponent $\nu$ smaller than $d_{\mathrm{H}}$ in (b). A deviation from that behavior is caused by singular peaks of the DOS.

Figure 4

Distribution of IPR for SC at (a) $\alpha =0$ and (b) $\alpha =0.02$. Inset: A close-up of carpet with internal edges of different hierarchies indicated by distinct colors.

Figure 5

Statistics of IPR for SG at (a) $\alpha =0$ and (b) $\alpha =0.02$. Inset: A close-up of gasket with highlighted edges of different hierarchies.

Figure 6

Real-space Chern number calculations in the case of SC as a function of the half of the diagonal of a rectangular patch $R$ for two Fermi levels $E_F=-1.2$ and $E_F=-1.0$ at fixed flux $\alpha =1/4$. For energy regions with large DOS, in which we dominantly observe nonquantized $\mathcal{C}$, the value of $\mathcal{C}$ depends sensitivity on the size and shape of the summation region. For spectral regions exhibiting stable plateaus with quantized $\mathcal{C}$, this quantization is observed for a large range of patch sizes. $R=9.9$ corresponds to the size of the patch shown in Fig. 1.

Figure 7

(a) Energy levels for $n=3$ carpet in the absence of disorder and phase diagram at $\alpha =1/4$. (b)–(j) Distribution of the level spacings. Histograms are related to the spacings around [(h), (i), (j)] $\epsilon =-2.5$, [(e), (f), (g)] to $\epsilon =-1.5$, and [(b), (c), (d)] to $\epsilon =0$. Calculations were performed for three disorder strengths $W=1$ [(b), (e), (h)], $W=3$ [(c), (f), (i)], and $W=5$ [(d), (g), (j)].

Figure 8

(a) Energy levels for clean $n=5$ gasket together with phase diagram at $\alpha =1/4$. (b)–(j) Distribution of the level spacings. Histograms are related to the spacings around [(h), (i), (j)] $\epsilon =-2.5$, [(e), (f), (g)] $\epsilon =-1.5$, and [(b), (c), (d)] $\epsilon =0$. Calculations were performed for three disorder strengths $W=1$ [(b), (e), (h)], $W=3$ [(c), (f), (i)], and $W=5$ [(d), (g), (j)].

Figure 9

Variance of the level spacings for a [(a), (b), (c)] square lattice and [(d), (e), (f)] Sierpiński carpet at [(a), (d)] $W=1$, [(b), (e)] $W=3$, and [(c), (f)] $W=5$ in the energy-flux plane.

Figure 10

Variance of the level spacings for [(a), (b), (c)] triangular lattice and [(d), (e), (f)] Sierpiński gasket at [(a), (d)] $W=1$, [(b), (e)] $W=3$, and [(c), (f)] $W=5$ in the energy-flux plane.