The Anderson impurity problem in a uniform fractal host

We report our solution of the Anderson model for a magnetic impurity weakly hybridizing with host electrons whose density of states (DOS) is described by a symmetric uniform Cantor set. This fractal DOS naturally lends itself to a logarithmic binning of the band energy. Discretized in this way, the original problem can be mapped onto Anderson Hamiltonian on a Wilson tight-binding chain with exponentially decaying nearest-neighbor hoppings. The resulting problem is entirely equivalent to one arising in the numerical renormalization-group (NRG) treatment of the Anderson model with a smooth (non-fractal) band diverging in a power-law fashion at the Fermi energy, in which the system will always flow toward a strong-coupling regime and the impurity makes negative contributions to the magnetic susceptibility and the entropy. Decreasing the width of the logarithmic energy bins in a series of steps to approach the original continuum limit of the problem resolves more structures in the fractal and manifests periodicity in scaled hoppings on the Wilson chain. Full NRG calculations confirm that all members of this series of discretizations describe the same asymptotic divergence of the DOS, described by a power connected to the fractal dimension of the uniform Cantor set.