Critical Properties of $S^{4}$ System Restudied via Generalized Migdal-Kadanoff Bond-moving Renormalization

We study the critical properties of the spin-continuous $S^{4}$ system on the typical translational invariant triangular lattices by combining the recently-developed generalized Migdal-Kadanoff bond-moving recursion procedures with the cumulative expansion technique. In three different cases of nearest-neighbor, next nearest neighbor and external field we obtain the critical points and further calculate the critical exponents according to the scaling theory. In all case it is found that there exists three fixed points. The correlation length critical exponents obtained near the Wilson-Fisher fixed points are found getting smaller and smaller with the increasing of the system complexity. Others are found similar to the results of the classical Gaussian model and different from those of the Ising system.