Causal spectra and localization properties of electronic states on a hierarchical tree

We studied the non-interacting tight-binding model on a heterogenous tree structure. We defined the causal structure on the solved full energy spectrum of the normalized Hamiltonian. The degree of localization related to an eigenstate, measured by inverse participation ratios(IPR), is found directly related to a eigenvalue chain in the spectrum. The scaling behaviors of IPRs for eigenstates are determined by the tails of eigenvalue chains.