Conduction in quasiperiodic and quasirandom lattices: Fibonacci, Riemann, and Anderson models

We study the ground state conduction properties of noninteracting electrons in aperiodic but nonrandom one-dimensional models with chiral symmetry and make comparisons against Anderson models with nondeterministic disorder. The first model we consider is the Fibonacci lattice, which is a paradigmatic model of quasicrystals; the second is the Riemann lattice, which we define inspired by Dyson's proposal on the possible connection between the Riemann hypothesis and a suitably defined quasicrystal. Our analysis is based on Kohn's many-particle localization tensor defined within the modern theory of the insulating state. In the Fibonacci quasicrystal, where all single-particle eigenstates are critical (i.e., intermediate between ergodic and localized), the noninteracting electron gas is found to be an insulator, due to spectral gaps, at various specific fillings rho, including the values rho = 1/g(n), where g is the golden ratio and n is any integer; however away from these spectral anomalies, the system is found to be a conductor, including the half-filled case. In the Riemann lattice metallic behavior is found at half filling as well; however, in contrast to the Fibonacci quasicrystal, the Riemann lattice is generically an insulator due to single-particle eigenstate localization, likely at all other fillings. Its behavior turns out to be alike that of the off-diagonal Anderson model, albeit with different system-size scaling of the band-center anomalies. The advantages of analyzing the Kohn's localization tensor instead of other measures of localization familiar from the theory of Anderson insulators (such as the participation ratio or the Lyapunov exponent) are highlighted.