Higher-dimensional Hofstadter butterfly on the Penrose lattice

It is now possible to use quasicrystals to search for novel topological phenomena enhanced by their peculiar structure characterized by an irrational number and high-dimensional primitive vectors. Here, we extend the concept of a topological insulator with an emerging staggered local magnetic flux (i.e., without external fields), similar to Haldane's honeycomb model, to the Penrose lattice as a quasicrystal. The Penrose lattice consists of two different tiles, where the ratio of the numbers of tiles corresponds to an irrational number. Contrary to periodic lattices, the periodicity of the energy spectrum with respect to the magnetic flux no longer exists, reflecting the irrational number in the Penrose lattice. Calculating the Bott index as a topological invariant, we find topological phases appearing in a fractal energy spectrum similar to the Hofstadter butterfly. More intriguingly, by folding the one-dimensional aperiodic magnetic flux into a two-dimensional periodic flux space, the fractal structure of the energy spectrum is extended to a higher dimension, whose section corresponds to the Hofstadter butterfly.

Figure 1

(a) Penrose lattice constructed by fat (red) and thin (blue) tiles. (b) A $g=1$ approximant of the Penrose lattice, where the first to 11th green sites are linked by green bonds. Owing to the approximant periodicity, we regard the lattice as a unit cell. The magenta edges linking green to magenta sites denote the bonds bridging between neighboring unit cells, corresponding to a connection of boundaries with a periodic boundary condition. The number colored in red (blue) in each tile counts red (blue) tiles in the unit cell, and the number of red (blue) tiles amounts to 7 (4).

Figure 2

Energy spectrum of a $g=7$ Penrose lattice ($N_{\mathrm{r}}=2207$ and $N_{\mathrm{b}}=1364$) with anormalized staggered magnetic flux $n$. We set the hopping integral $t=1$ as an energy unit. The position of the colored symbols denotes the chemical potential $\mu$ (corresponding to the value on the energy axis) and the normalized magnetic flux $n$, and the colored indicators show the Bott index (6) obtained with the parameters $(\mu ,n)$. To calculate the Bott index, we choose the chemical potential corresponding to the center of the gap. The gaps are painted with colors representing the Bott index (shown at the bottom of the panel) together with the upper and lower triangles. The vertical dashed lines represent multiples of $N_{\mathrm{r}}=2207$ (red) and $N_{\mathrm{b}}=1364$ (blue).

Figure 3

The path of the possible region of the staggered magnetic fluxes $({\varphi }_{\mathrm{r}},{\varphi }_{\mathrm{b}})$ for $g=1$, 2, and 3 generations. Since the magnetic flux ${\varphi }_{\mathrm{r}}$ (${\varphi }_{\mathrm{b}}$) is unique modulo $2\pi$, the parameter space of the two fluxes corresponds to a 2-torus, where the toroidal (poloidal) axis denotes ${\varphi }_{\mathrm{r}}$ (${\varphi }_{\mathrm{b}}$). With increasing the generation, the path densely covers the whole region of the parameter space, due to approaching the irrational ratio of the fluxes.

Figure 4

Three-dimensional Hofstadter butterfly structure. (a) Gap structure and (b) Bott index in the energy spectrum as a function of two independent magnetic fluxes ${\varphi }_{\mathrm{r}}$ and ${\varphi }_{\mathrm{b}}$ in a $g=7$ Penrose lattice. In (a), only gaps with a size larger than $0.01t$ are shown. In (b), the Bott index is shown for gaps with a size larger than $0.05t$. The colors in (b) are the same as in Fig. 2.