Renormalization-group analysis on fractals: Ising spin-glass and the Schrödinger equation

A renormalization-group analysis of random frustrated Ising models on a d=2 Sierpinski gasket is carried out. A numerical study of the recursion relations shows that in the spin-glass phase, the width of the probability distribution of the renormalized exchange couplings and the characteristic energy sensitivity to the boundary conditions decay algebraically with the size of the system. In the paramagnetic phase the two quantities decay exponentially. An exact renormalization-group analysis of the tight-binding Hamiltonian on a d-dimensional Sierpinski gasket is carried out. In any d, for almost all energies, the hopping matrix element renormalizes to zero faster than exponentially, showing that the corresponding eigenstates, if any, are localized.