Topological models in rotationally symmetric quasicrystals

We investigate the physics of quasicrystalline models in the presence of a uniform magnetic field, focusing on the presence and construction of topological states. This is done by using the Hofstadter model but with the sites and couplings denoted by the vertex model of the quasicrystal, giving the Hofstadter vertex model. We specifically consider two-dimensional quasicrystals made from tilings of two tiles with incommensurate areas, focusing on the fivefold Penrose and the eightfold Ammann-Beenker tilings. This introduces two competing scales: the uniform magnetic field and the incommensurate scale of the cells of the tiling. Due to these competing scales, the periodicity of the Hofstadter butterfly is destroyed. We observe the presence of topological edge states on the boundary of the system via the Bott index that exhibit two-way transport along the edge. For the eightfold tiling, we also observe internal edgelike states with nonzero Bott index, which exhibit two-way transport along this internal edge. The presence of these internal edge states is a characteristic of quasicrystalline models which leads to open questions on their properties and future applications. We then move on to considering interacting systems. This is challenging, in part because exact diagonalization on a few tens of sites is not expected to be enough to accurately capture the physics of the quasicrystalline system and in part because it is not clear how to construct topological flatbands having a large number of states. We show that these problems can be circumvented by building the models analytically, and in this way we construct models with Laughlin-type fractional quantum Hall ground states.

Figure 1

Examples of the rhombus Penrose tiling obtained from the Robinson triangle decomposition method. (a) A zoomed-in portion of the Penrose tiling showing the two types of rhombuses by shading. (b) A larger portion of the tiling showing the self-similar structure.

Figure 2

Examples of the rhombus Ammann-Beenker tiling obtained from the cut-and-project method. (a) A zoomed-in portion of the Ammann-Beenker tiling showing the two types of rhombuses by shading. (b) A larger portion of the tiling showing the self-similar structure.

Figure 3

Examples of the quasilattice structures considered in this paper. Solid lines show the couplings of the vertex model. (a) The fivefold symmetric quasilattice of the vertex model on the Penrose tiling. Shown is an example with 86 sites. (b) The eightfold symmetric quasilattice of the vertex model on the Ammann-Beenker tiling. Shown is an example with 73 sites.

Figure 4

Energy-flux plane for the Hofstadter vertex model on the fivefold Penrose tiling with 3448 sites. (a) The standard flux 0 to 1 plot. (b) A larger region of flux showing the incommensurate nature in the energy-flux plane.

Figure 5

Energy-flux plane for the Hofstadter vertex model on the eightfold Ammann-Beenker tiling with 3345 sites. (a) The standard flux 0 to 1 plot. (b) A larger region of flux showing the incommensurate nature in the energy-flux plane.

Figure 6

Illustration of the staggered flux square lattice Hofstadter model considered, with ${\varphi }_1$ and ${\varphi }_2$ two different fluxes that are not necessarily commensurate with each other.

Figure 7

Energy-flux plane for the Hofstadter vertex model on the square lattice with incommensurate cells and 1600 sites with ${\varphi }_1=\tau$ and ${\varphi }_2=1$. (a) The standard flux 0 to 1 plot. (b) A larger region of flux showing the incommensurate nature in the energy-flux plane.

Figure 8

Vertex Hofstadter model for the Penrose fivefold ($N=1241$) and Ammann-Beenker eightfold ($N=1273$) quasilattices. (a) The energy spectrum for the fivefold quasilattice with $\varphi =0$ shown by a solid (black) line and $\varphi /{\varphi }_0=0.1$ shown by a dashed (red) line. (b) Same as (a) but for the eightfold quasilattice with $\varphi =0$ shown by a solid (black) line and $\varphi /{\varphi }_0=0.69$ shown by a dashed (red) line. (c) The Bott index shown with circles (black) and the Bott index estimate with a solid (red) line for the fivefold quasilattice and $\varphi /{\varphi }_0=0.1$. (d) Same as (c) but for the eightfold quasilattice with $\varphi /{\varphi }_0=0.69$.

Figure 9

Example states (probability densities) for the Hofstadter vertex model on the fivefold Penrose quasilattice with 1241 sites and $\varphi /{\varphi }_0=0.1$, corresponding to Figs. 8 and 8. The example states are (a) the ground state, (b) state 107 with Bott index $-1$, (c) state 213 with Bott index $-2$, and (d) state 215 with Bott index $-2$.

Figure 10

Example states (probability densities) for the Hofstadter vertex model on the eightfold Ammann-Beenker quasilattice with 1273 sites and $\varphi /{\varphi }_0=0.69$, corresponding to Figs. 8 and 8. The example states are (a) the ground state, (b) state 246 with Bott index 1, (c) state 483 with Bott index 1, and (d) state 491 with Bott index 1.

Figure 11

Focused plots of the internal edge state shown in Fig. 10. (a) A repeated plot of the state. (b), (c) Focused portions of the state showing its structure of being located on an almost full eightfold star of the Ammann-Beenker tiling.

Figure 12

Exciting the edge states at the finite-size edge for the eightfold Ammann-Beenker quasilattice with 1273 sites for $\varphi /{\varphi }_0=0.69$. An example of the edge states on the finite size edge is shown in (a). The dynamics of a single site excitation on the edge is shown in (b)–(f) with (b) showing the state at $t=0J^{-1}$, (c) $t=25J^{-1}$, (d) $t=50J^{-1}$, (e) $t=75J^{-1}$, and (f) $t=100J^{-1}$.

Figure 13

Exciting the edge states at the internal edge for the eightfold Ammann-Beenker quasilattice with 1273 sites for $\varphi /{\varphi }_0=0.69$. An example of the internal edge states in the bulk of the quasilattice is shown in (a). The dynamics of a single site excitation on the internal edge is shown in (b)–(f) with (b) showing the state at $t=0J^{-1}$, (c) $t=100J^{-1}$, (d) $t=200J^{-1}$, (e) $t=300J^{-1}$, and (f) $t=400J^{-1}$.

Figure 14

Examples of small topological flat bands in the Hofstadter vertex model for (a) the fivefold Penrose quasilattice with 1241 sites and $\varphi /{\varphi }_0=0.034$ and (b) the eightfold Ammann-Beenker quasilattice with 1273 sites and $\varphi /{\varphi }_0=0.05$. The filling of the circles shows the value of the Bott index, with empty circles having a nonzero Bott index of $-1$ and filled circles having a Bott index of zero.

Figure 15

Density profiles $\rho \left(z_i\right)$ from Eq. (20) on the (a) fivefold ($N=381$) and on the (b) eightfold quasilattices ($N=273$) with one quasihole and one quasielectron in the system. The position of the quasihole is shown by a (red) star and the position of the quasielectron by a (blue) pentagon.

Figure 16

The excess charge distributions $Q_k\left(r\right)$ from Eq. (21) plotted as a function of the radial distance $r$ from each anyon's position in (a) the fivefold and (b) the eightfold quasilattices. We note that the anyons are well screened and approach the right charges $\simeq \pm 1/3$. We find an error of ${10}^{-4}$ arising from the Monte Carlo simulations.