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Localized States in Local Isomorphism Classes of Pentagonal Quasicrystals

A family of pentagonal quasicrystals can be defined by projecting a section of the five-dimensional cubic lattice to two dimensions. A single parameter, the sum of intercepts $Γ=\sum_j γ_j$, describes this family. Each value of $0\le Γ\le \frac{1}{2}$ defines a unique local isomorphism class for these quasicrystals, with $Γ=0$ giving the Penrose lattice. Except for a few special values of $Γ$, these lattices lack simple inflation-deflation rules making it hard to count how frequently a given local configuration is repeated. We consider the vertex-tight-binding model on these quasicrystals and investigate the strictly localized states (LS) for all values of $Γ$. We count the frequency of localized states both by numerical exact diagonalization on lattices of $~10^5$ sites and by identifying localized state types and calculating their perpendicular space images. While the imbalance between the number of sites of the sublattices of the bipartite quasicrystal grows monotonically with $Γ$, we find that the localized state fraction first decreases and then increases as the distance from the Penrose lattice grows. The highest LS fraction of $~10.17\%$ is attained at $Γ=0.5$ while the minimum is $~4.5\%$ at $Γ\simeq 0.12$. The LS on the even sublattice are generally concentrated near sites with high symmetry, while the LS on the odd sublattice are more uniformly distributed. The odd sublattice has a higher LS fraction, having almost three times the LS frequency of the even sublattice at $Γ=0.5$. We identify 20 LS types on the even sublattice, and their total frequency agrees well with the numerical exact diagonalization result for all values of $Γ$. For the odd sublattice, we identify 45 LS types. However, their total frequency remains significantly below the numerical calculation, indicating the possibility of more independent LS types.