Identification and properties of topological states in the bulk of quasicrystals

Abstract

In contrast to the usual bulk-boundary correspondence, topological states localized within the bulk of the system have been numerically identified in quasicrystalline structures, termed bulk localized transport (BLT) states. These states exhibit properties different from edge states, one example being that the number of BLT states scales with system size, while the number of edge states scales with system perimeter. Here, we define a procedure to identify BLT states, which is based on the physically motivated crosshair marker and robustness analyses. Applying the procedure to the Hofstadter model on the Ammann-Beenker tiling, we find that the BLT states appear mainly for magnetic fluxes within a specific interval. While edge states appear at low densities of states, we find that BLT states can appear at many different densities of states. Many of the BLT states are found to have real-space localization that follows geometric patterns characteristic of the given quasicrystal. Furthermore, BLT states can appear both isolated and in groups within the energy spectrum which could imply greater robustness for the states within such groups. The spatial localization of the states within a certain group can change depending on the Fermi energy.

Figures (13)

• Figure 1: Instance of Ammann-Beenker tiling with $89$ sites centered at the point of eightfold rotational symmetry. We choose a scale such that the distance between any two vertices connected by an edge is $1/\sqrt{2}$.
• Figure 2: Spectrum of the Hofstadter model a circular cutout of the AB-tiling with radius $34$ centered at the point with coordinates $(17,16)$ to avoid rotational symmetry. This system consists of $8772$ sites. The spectrum is calculated for $\phi \in (0,2)$ where darker colors symbolize larger density of states. Due to chiral symmetry the energy spectrum is mirror symmetric around $E=0$.
• Figure 3: Spectral flow of the model with $\phi=0.7375$ chosen in accordance with later results. The graph is a circular cut-out whose origin has been shifted by the vector $(x,y)=(17,16)$ to avoid rotational symmetry. (a) Spectral flow on a graph of radius $24$ corresponding to $4364$ sites. (b) Spectral flow on a graph of radius $34$ corresponding to $8772$ sites. The states shown in graph (b) are among those later identified as BLT states. Both graphs show an avoided crossing but the size of the gap is the same for both systems.
• Figure 4: Visualization of topological measures for a square lattice of $8586$ sites cut out in a circle of radius $34$ and flux $\phi=0.69$ at a Fermi energy $E_f=2.413$. (a) The cumulated crosshair marker $\Lambda(r, E_f)$ as a function of radius. The plateau of the graph indicates a Chern number of two. (b) Spectral flow of the adiabatic charge pump showing two states transported past the Fermi energy highlighted as a dashed line in accordance with (a).
• Figure 5: Maximal value of the cumulated crosshair marker $Q_{\text{dis}}(E_i)$ as a function of the strength of disorder $W$ for a sample of 35 states. The calculations are done for the system $(R_0, \phi, x_0, y_0) = (34, 0.69, 17, 16)$. At low $W$ the non-robust states are too close to the robust ones to distinguish accurately. At high $W$ the states bundle together according to energy due to the disorder erasing finer details. $W=0.3$ marks the chosen disorder between these limiting cases.