Real-space renormalization-group treatment of quadratic chains

Abstract

We have recently proposed a one-dimensional nonperiodic chain with lattice positions at 02 d, 12 d, 22 d, ... with length d a constant. The spectrum is singular-continuous, and for weak potential, the states are all extended apart from a trivial set of localized states. In this study, we obtain the exact extended-state spectrum of the quadratic chain in a nearest-neighbor tight-binding model where the quadratic modulation is in the onsite matrix elements. Then, a real-space renormalization-group method (RSRG) is used by decimation to reduce the transfer matrix for the chain into self-similar matrix products. The RSRG decimation scheme is used here to organize the calculation and facilitate numerical computation. The extended-state spectrum appears as minibands broken by numerous gaps. Previous work on quadratic chains shows that the structure factor is singular-continuous and given by a dense set of states with wavevectors with scaling exponent γ(k) = 2 as in periodic and quasi-periodic chains. The origin of extended states in this nonperiodic lattice appears to arise from a type of mechanism not yet identified in deterministic nonperiodic lattices, and is based on a hidden symmetry giving rise to an energy-dependent translational invariance of the transfer matrix.