Topological phases in quasicrystals: A general principle of construction

Quasicrystals are projections of higher-dimensional crystals on lower-dimensional branes of irrational inclination. They feature unique structural properties, such as five- and eight-fold rotational symmetries, which are forbidden in crystals. Therefore, quasicrystals constitute a unique platform to harness topological states of dimensionality d>3 as well as crystal-forbidden topological phases of matter. Here we will demonstrate a general theoretical approach to construct topological phases in quasicrystals by combining their structural dimensional descending with renormalized Hilbert space in terms of an effective Hamiltonian within the quasicrystalline brane. Specifically, we focus on the 2D square lattice Chern insulator and demonstrate its signature on 1D Fibonacci quasicrystals in terms of the hallmark edge states, dislocation modes, and Bott index. We also construct hybrid Weyl semimetals by stacking Fibonacci Chern insulators, featuring robust Weyl nodes, Fermi arc, and also a chiral anomaly in the presence of an external magnetic field.