Localized states in local isomorphism classes of pentagonal quasicrystals

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