Nature of protected zero-energy states in Penrose quasicrystals
Ezra Day-Roberts, Rafael M. Fernandes, Alex Kamenev
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 $\mathrm{\ensuremath{\Delta}}n$ 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 $\mathrm{\ensuremath{\Delta}}n(\mathbf{r})$. Although $\mathrm{\ensuremath{\Delta}}n(\mathbf{r})$ 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 $Z\ensuremath{\propto}{N}^{\ensuremath{-}\ensuremath{\eta}}$ with $\ensuremath{\eta}=1\ensuremath{-}ln2/2ln\ensuremath{\tau}\ensuremath{\approx}0.2798$ (where $\ensuremath{\tau}$ is the golden ratio). It also reproduces the known total fraction of zero-energy states, $81\ensuremath{-}50\ensuremath{\tau}\ensuremath{\approx}0.0983$. 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, and random dilution of the lattice. We attribute this robustness to the hidden domain structure.