Nature of protected zero-energy states in Penrose quasicrystals

Abstract

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 $\Delta n$ 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 $\Delta n(\bf r)$. Although $\Delta n(\bf r)$ averages to zero, 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 size of their support domain, $N$, as $Z\propto N^{-\eta}$ with $\eta =1-\ln 2/\ln(1+\tau) \approx 0.2798$ (where $\tau$ is the golden ratio). It also reproduces the known total fraction of the zero energy states, $81-50\tau\approx 0.0983$. 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, random dilution of the lattice, etc. We attribute this robustness to the hidden domain structure and speculate about its underlying topological origin.