On the extendedness of eigenstates in a hierarchical lattice: a critical view

TL;DR

This paper critically examines the concept of extended eigenstates in hierarchical lattices with long-range interactions, showing that their transport properties depend on specific parameter correlations and can be tuned between conducting and insulating states.

Contribution

It provides an explicit construction of eigenstates with constant amplitudes in hierarchical systems and reveals how parameter correlations influence transport properties.

Findings

Eigenstates with constant amplitudes can exist at specific energies.

Transport properties depend on parameter correlations.

Tuning long-range interactions can switch between conducting and insulating states.

Abstract

We take a critical view at the basic definition of extended single particle states in a non-translationally invariant system. For this, we present the case of a hierarchical lattice and incorporate long range interactions that are also distributed in a hierarchical fashion. We show that it is possible to explicitly construct eigenstates with constant amplitudes (normalized to unity) at every lattice point for special values of the electron- energy. However, the end-to-end transmission, corresponding to the above energy of the electron in such a hierarchical system depends strongly on a special correlation between the numerical values of the parameters of the Hamiltonian. Keeping the energy and the distribution of the amplitudes invariant, one can transform the lattice from conducting to insulating simply by tunning the numerical values of the long range interaction. The values of these interactions themselves display a fractal character.