Strictly Localized States on the Socolar Dodecagonal Lattice

TL;DR

This paper investigates the strictly localized states in the Socolar dodecagonal lattice, quantifying their frequency, types, and properties, and compares these features with other well-known quasicrystal lattices.

Contribution

It provides a detailed classification and frequency analysis of localized states in the Socolar dodecagonal lattice, including new bounds and properties of these states.

Findings

Localized states span approximately 7.61% of the Hilbert space.

Identified 18 independent localized state types with a combined lower bound frequency of about 7.58%.

Discovered sites forbidden for localized states, with a lower bound frequency of about 3.90%.

Abstract

Socolar dodecagonal lattice is a quasicrystal closely related to the better-known Ammann-Beenker and Penrose lattices. The cut and project method generates this twelve-fold rotationally symmetric lattice from the six-dimensional simple cubic lattice. We consider the vertex tight-binding model on this lattice and use the acceptance domains of the vertices in perpendicular space to count the frequency of strictly localized states. We numerically find that these states span $f_{\mathrm{Num}}\simeq 7.61$ \% of the Hilbert space. We give 18 independent localized state types and calculate their frequencies. These localized state types provide a lower bound of $f_{\mathrm{LS}} =\frac{10919-6304\sqrt{3}}{2} \simeq 0.075854$, accounting for more than $99 \%$ of the zero-energy manifold. Numerical evidence points to larger localized state types with smaller frequencies, similar to the Ammann-Beenker lattice. On the other hand, we find sites forbidden by local connectivity to host localized states. Forbidden sites do not exist for the Ammann-Beenker lattice but are common in the Penrose lattice. We find a lower bound of $f_{\mathrm{Forbid}}\simeq 0.038955$ for the frequency of forbidden sites. Finally, all the localized state types we find can be chosen to have constant density and alternating signs over their support, another feature shared with the Ammann-Beenker lattice.