Critical properties of the S4 model on diamond-type hierarchial lattices

We study the critical properties of the S4 model on diamond-type hierarchical lattices in the presence of an external magnetic field. It is assumed that for this type of inhomogenous fractal lattice, the Gaussian distribution constant, the four-spins interaction parameter and the external magnetic field in the S4 model depend on the coordination number of the site on the fractal lattices. By combining the real-space renormalization-group scheme with the cumulative expansion method, we obtain the critical points and further calculate critical exponents according to the scaling theory. The results show that on diamond-type hierarchical lattices with branches m>4 of 2 bonds, the critical point of the S4 model is just the Gaussian fixed point, and therefore critical exponents are in full agreement with those of the Gaussian model, and that on those with m⩽4, the Wilson–Fisher fixed point as well as the Gaussian fixed point is obtained. The Wilson–Fisher fixed point has a decisive influence on the critical behavior of the system, and critical exponents are related to the fractal dimensionality. We found that the S4 model on the fractal lattices and that on the translation symmetric lattices show similar behaviors in the dependence of the critical properties on the dimensionality.