一簇金刚石晶格上S~4模型的相变

采用重整化群和累积展开的方法,研究了一簇金刚石晶格上S~4模型的相变,求得了系统的临界点.结果表明:当分支数m=2和m 12时,该系统只存在一个Gauss不动点K~*=b_2/2, u_2~*=0;当分支数3≤m≤12时,该系统不仅有Gauss不动点,还存在一个Wilson-Fisher不动点,并且后一个不动点对系统的临界特性产生决定性的影响.

Phase transition of S4 model on a family of diamond lattice

The fractal is a kind of geometric figure with self-similar character. Phase transition and critical phenomenon of spin model on fractal lattice have been widely studied and many interesting results have been obtained. The S⁴ model regarded as an extension of the Ising model, can take a continuous spin value. Research of the S⁴ model can give a better understanding of the phase transition in the real ferromagnetic system in nature. In previous work, the phase transition of the S⁴ model on the translation symmetry lattice has been studied with the momentum space renormalization group technique. It is found that the number of the fixed points is related to the space dimensionality. In this paper, we generate a family of diamond hierarchical lattices. The lattice is a typical inhomogenous fractal with self-similar character, whose fractal dimensionality and the order of ramification are d_f = 1 + ln m/ln 3 and R = ∞, respectively. In order to discuss the phase transition of the S⁴ model on the lattice, we assume that the Gaussian distribution constant b_i and the fourth-order interaction parameter u_i depend on the coordination number q_i of the site on the fractal lattices, and the relation b_i/b_j = u_i/u_j = q_i/q_j is satisfied. Using the renormalization group and the cumulative expansion method, we study the phase transition of the S⁴ model on a family of diamond lattices of m branches. Removing the inner sites, we obtain the system recursion relation and the system corresponding critical point. Furthermore, we find that if the number of branches is m = 2 or m > 12(fractal dimensionality d_f = 1.63 or d_f > 3.26), the system only has the Gaussian fixed point of K^* = b₂/2,u₂^* = 0. The critical point of the system is in agreement with that from the Gaussian model on the fractal lattice, which predicts that the two systems belong to the same university class. We also find that under the condition of 3 ≤ m ≤ 12 (fractal dimensionality 2 ≤ d_f ≤ 3.26), both the Gaussian fixed point and the Wilson-Fisher fixed point can be obtained in the system, and the Wilson-Fisher fixed point plays a leading role in the critical properties of the system. According to the real space renormalization group transformation and scaling theory, we obtain the critical exponent of the correlation length. Finally, we find that the critical points of the S⁴ model on a family of diamond lattices depend on the value of the fractal dimensionality. The above result is similar to that obtained from the S⁴ model on the translation symmetry lattice.