Identification and properties of topological states in the bulk of quasicrystals

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.

💡 Research Summary

This paper investigates a novel class of topological states that reside in the interior of a quasicrystalline system rather than at its edges. The authors focus on the Hofstadter model defined on the Ammann‑Beenker (AB) tiling, a two‑dimensional quasicrystal with eight‑fold rotational symmetry but no translational periodicity. In conventional topological insulators, bulk‑boundary correspondence guarantees that a non‑trivial bulk invariant (e.g., a Chern number) manifests as conducting edge states whose number scales with the system’s perimeter. In contrast, the authors identify bulk‑localized transport (BLT) states whose number scales with the total number of sites, i.e., the system volume, indicating a fundamentally different scaling behavior.

To detect and characterize these BLT states, the authors introduce a physically motivated “cross‑hair marker.” The marker is defined by applying a line‑shaped electric field (non‑zero only along y = R_y) and measuring the Hall current crossing a perpendicular line (x = R_x). The resulting quantity, C_R(r,E), is evaluated for each lattice site r at a given Fermi energy E. Mathematically it reads
C_R(r,E)=4π Im Tr_r