A rescaling approach to two- and three-dimensional lattice spectra

An exact real-space rescaling transformation is used to calculate the spectral properties of tight-binding Hamiltonians with an arbitrary range of interaction of two- and three-dimensional lattices. The intrinsic problem of the proliferation of initial short-range interactions under rescaling is shown to be tractable numerically. This method provides a direct way of obtaining the local Green functions and local density of states. Singularities of the spectral functions are associated with fixed points or limit cycles of the rescaling transformation. Band edge fixed points and exponents have been found for the square, triangular and cubic lattices. In two and three dimensions there is rapid convergence to accurate numerical results for the Green function outside the band, while good results have been obtained for the in-band (e.g. density of states) spectra in two dimensions by suitable choices of cluster size and attenuation factor. Insight into the relationship between these factors follows from an investigation of finite-cluster approximations to the exact transformation. It is found that the zeros of the Green functions in the complex plane are Julia sets which have a fractal structure. For a given cluster size the attenuation factors must lie outside these zeros in order for the procedure to be convergent.