Critical behavior of the Gaussian model on fractal lattices in external magnetic field

Abstract

For inhomogeneous lattices we generalize the classical Gaussian model, i.e. it is proposed that the Gaussian type distribution constant and the external magnetic field of site i in this model depend on the coordination number q i of site i , and that the relation b q i /b q j =q i q j holds among b q i 's, where b q i is the Gaussian type distribution constant of site i . Using the decimation real-space renormalization group following the spin-rescaling method, the critical points and critical exponents of the Gaussian model are calculated on some Koch type curves and a family of the diamond-type hierarchical (or DH) lattices. At the critical points, it is found that the nearest-neighbor interaction and the magnetic field of site i can be expressed in the form K * = b q i / q i and h q 1 * =0, respectively. It is also found that most critical exponents depend on the fractal dimensionality of a fractal system. For the family of the DH lattices, the results are identical with the exact results on translation symmetric lattices, and if the fractal dimensionality d f =4, the Gaussian model and the mean field theories give the same results.