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arXiv:1208.4892

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

Chun-Yang Wang, Wen-Xian Yang, Hong Du

Published Aug 24, 2012• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1208.4892

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Critical Properties of S4S^{4}S4 System Restudied via Generalized Migdal-Kadanoff Bond-moving Renormalization
We study the critical properties of the spin-continuous S4S^{4}S4 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.
Considering in symmetrical half-length bond operations, we present in this paper two types of newly-developed generalizations of the remarkable Migdal-Kadanoff bond-moving renormalization group transformation recursion procedures. The predominance in application of these generalized procedures are illustrated by making use of them to study the critical behavior of the spin-continuous Gaussian model constructed on the typical translational invariant lattices and fractals respectively. Results such as the correlation length critical exponents obtained by these means are found to be in good conformity with the classical results from other previous studies.
We report in a series of papers two types of generalized Migdal-Kadanoff bond-moving renormalization group transformation recursion procedures. In this first part the symmetrical operation of half length bonds on translational invariant lattices are considered. As an illustration of their predominance in application, the procedures are used to study the critical behavior of the spin-continuous Gaussian model constructed on the triangular lattices. Results such as the correlation length critical exponents obtained by this means are found to be in good conformity with the classical results from other studies.
The superiority of the generalized Migdal–Kadanoff bond-moving renormalization (GMKBR) method in studying phase transition of complex spin systems is revealed by a sampled application on the classical S4model. 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 the cases, it is shown that the method is of great convenience in obtaining results which are in good conformity with the previous ones.
The fixed-point topology of Kadanoff's lower-bound renormalization transformation is discussed for the d=2d=2d=2 square-lattice Ising model in the three-dimensional space of coupling constants for the nearest-neighbor, next-nearest neighbor, and four-spin interactions. An obstacle one encounters in trying to calculate a critical surface is pointed out. In the three-dimensional space of coupling constants Kadanoff's fixed point has two relevant vectors and can only be reached from a critical line, not from a critical surface. Another fixed point is reported which does have only one relevant vector and can be reached from a critical surface. Another fixed point is reported which does have only one relevant vector and can be reached from a critical surface. The critical exponents of this fixed point are close to the exact values, though not nearly so close as for Kadanoff's fixed point. The properties of both fixed points as a function of the variational parameter ppp are described. Critical values of the nearest-neighbor couplings leading to the fixed points are compared.
The spin 1/2 anisotropic Heisenberg model is studied by means of the real space renormalization transformations.Thermodynamic properties of one-dimensional systems are calculated by using an approximate decimation method.These results agree well with those obtained by Bonner and Fisher and by Katsura.Two-and three-dimensional systems are studied by generalizing the Migdal-Kadanoff renormalization transformations to quantum spin systems.For the Ising and Heisenberg models in two and three dimensions and for the X-V model in three dimensions, the critical properties thus obtained are in qualitative agreement with those obtained from high temperature series expansions.For the two-dimensional X-V model, our results suggest that this model undergoes some kind of phase transition and that the critical behaviour of this model is similar to that expected for the classical X-V and planar models.Contents I. Hiroshi Takano 2-3.Three-dimensional systems III.Decimation method for one-dimensional systems 3-1.Transformation function and approximation 3-2.Renormalization equations for scale factor Q= 2 3-3.Free energy IV.The Migdal-Kadanoff transformations for two-and three-dimensional systems 4-1.Fixed points and critical lines 4-2.Critical properties 4-3.Thermodynamic properties V. Summary and conclusions Appendix A. Real space renormalization transformations A-I.Renormalization transformations A-2.Critical properties A-3.Free energy
Using the renormalization-group transformation method together with the cumulative expansion technique,the phase transition and critical properties of the S4 model on a type of hierarchical lattice were studied,and its fixed points and critical exponents were obtained.The results show that there exists a Wilson-Fisher fixed point besides the Gaussian fixed point,compared with those of the S4 model on the special diamond-type of hierarchical lattices,the critical points and critical exponents have changed.It has been found that both of them belong to different university classes.