Narrow Results

Tight-binding Hamiltonian on the prismatic pentagonal lattice is exactly solved to obtain the analytic expressions of dispersion relations and eigenvectors. This lattice is made of prismatic pentagon which is different from Cairo pentagon. Six different dispersion relations and total density of states are obtained. Dispersion relations are symmetric about the zero energy at a particular point in the parameter space. Although a large gap is found for the Cairo pentagonal lattice, no gap as well as no Dirac cone is found to appear in the tight-binding band structure for this prismatic pentagonal lattice. Instead, a pair of van Hove singularities has been identified at two different energy values in the band structure.

In this paper, we consider a pentagonal lattice and we investigate the rule matrix with null boundary condition for two-dimensional cellular automata with the field ${\mathbb{Z}}_p$ (the set of integers modulo $p$) and analyze their characteristics. Moreover, an algorithm of computing the rank of rule matrix with null boundary condition for von Neumann neighborhood is developed. Finally, necessary and sufficient conditions for the existence of Garden of Eden configurations for two-dimensional cellular automata are obtained.