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 $\mathrm{\Delta }n$ 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 $\mathrm{\Delta }n\left(\mathbf{r}\right)$. Although $\mathrm{\Delta }n\left(\mathbf{r}\right)$ 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 $Z\propto N^{-\eta }$ with $\eta =1-ln2/2ln\tau \approx 0.2798$ (where $\tau$ is the golden ratio). It also reproduces the known total fraction of 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, and random dilution of the lattice. We attribute this robustness to the hidden domain structure.

Figure 1

A section of approximately 4500 sites of the Penrose rhombus lattice divided into domains showing the nesting of subdomains. In the red (blue) domains, the local sublattice mismatch is such that the $A$ ($B$) sublattice has more sites than the $B$ ($A$) sublattice. The domain walls connect sites belonging to different sublattices.

Figure 2

Kite-and-dart lattice. Sites with no amplitude of any of the zero-energy states are represented by the black dots. Note that all marked sites belong to one sublattice.

Figure 3

Density of states for the kite-and-dart lattice. The DOS is symmetric about 0 and has a macroscopic number ($\sim 10$%) of exactly zero-energy states.

Figure 4

The rhombus tiling is bipartite and has fivefold symmetry.

Figure 5

The DOS of the rhombus lattice shown here is similar to the DOS of the kite-and-dart lattice in Fig. 3, however, the origin of the zero-energy peak is completely different.

Figure 6

Fractions for the mismatch in the number of sites in the two sublattices (global and local, as defined in the text) and for the number of zero-energy states as a function of the total number of sites in the rhombus lattice.

Figure 7

Excluded sites for a portion of the rhombus lattice. They are shown in red or blue based on which of the two global sublattices they belong to. Bold links, connecting red and blue excluded sites, constitute the boundaries of the domains.

Figure 8

DOS for a Penrose lattice with random nearest-neighbor hopping amplitudes drawn uniformly from $t_{ij}\in [-1,1]$ (one realization).

Figure 9

DOS for a Penrose lattice in the presence of a perpendicular magnetic field of half-flux quanta per small rhombus.

Figure 10

Number of zero-energy states (ZES) removed upon the addition of random NNN links. A site was randomly chosen, and a hopping amplitude was added to a random NNN of this site. Twenty realizations were considered; here, we plot only the maximal and minimal reductions of zero-energy states. The dashed line has slope 1 and is just a guide for the eyes. The lattice contains 4581 sites and 441 zero-energy states.

Figure 11

Histogram of the diagonal entries of the projection matrix, $P_{ij}$, which determines the perturbative energy shift due to a local on-site potential (only the majority sublattice is kept).

Figure 12

Wave functions of the perturbed states for perturbations at four sites. The horizontal axis is the distance from the perturbation site in units of the rhombus edge length. The best-fit localization lengths, ${\xi }_k$, are shown.

Figure 13

The neighborhoods associated with each of the eight distinct types of sites. Each neighborhood uniquely identifies the central red site.

Figure 14

A new emergent domain, obtained after two inflations of the $S5$ neighborhood. Sites are labeled according to the convention in Fig. 13. All other domains are results of the repeated inflations of this patch. It is also used as the initial seed to generate Penrose rhombus lattices with a larger number of sites via the inflation process.

Figure 15

Local mismatch fraction, $L_k/N_k$, as a function of the total number of sites after $k$ inflations, $N_k={\sum }_i{\left({\vec{n}}_k\right)}_i$.

Figure 16

Plot of $G_l/{\tau }^{2l}$, the summand in Eq. (15), which is proportional to the relative contribution of a domain of size/age $l$ to the number of zero-energy states.

Figure 17

Boundaries for successive inflations. Note the $S3$ sites and $J$ sites on the inward-facing boundary on the left and right, respectively.

Figure 18

Domain D1 contains another domain, D2.