Quantum Conduction on a Cayley Tree

Abstract

Quantum resistance of a Cayley tree composed of one-dimensional random scatterers is considered. The strength of a scatterers is characterized by its typical resistance ${\ensuremath{\rho}}_{0}$. It is shown that a metal-insulator transition occurs at some critical scattering strength ${\ensuremath{\rho}}_{0c}$, which depends on the coordination number of the tree. The resistance of an infinite tree diverges for ${\ensuremath{\rho}}_{0}>{\ensuremath{\rho}}_{0c}$ and saturates at some finite value for ${\ensuremath{\rho}}_{0}<{\ensuremath{\rho}}_{0c}$.