Nature of protected zero-energy states in Penrose quasicrystals

The electronic spectrum of the Penrose rhombus quasicrystal exhibits a macroscopic fraction of exactly degenerate zero-energy states. In contrast to other bipartite quasicrystals, such as the kite-and-dart one, these zero-energy states cannot be attributed to a global mismatch Δn between the number of sites in the two sublattices that form the quasicrystal. Here, we argue that these zero-energy states are instead related to a local mismatch Δn(r). Although Δn(r) averages to 0, its staggered average over self-organized domains gives the correct number of zero-energy states. Physically, the local mismatch is related to a hidden structure of nested self-similar domains that support the zero-energy states. This allows us to develop a real-space renormalization-group scheme, which yields the scaling law for the fraction of zero-energy states, Z, versus the size of their support domain, N, as Z∞N-η with η=1-ln2/2lnτ≈0.2798 (where τ is the golden ratio). It also reproduces the known total fraction of zero-energy states, 81-50τ≈0.0983. We also show that the exact degeneracy of these states is protected against a wide variety of local perturbations, such as irregular or random hopping amplitudes, magnetic field, and random dilution of the lattice. We attribute this robustness to the hidden domain structure.