Phase transition of <i>S</i> <sup>4</sup> model on a family of diamond lattice
Authors: Yin Xun-Chang , Liu Wan-Fang , Ma Ye-Wan , Kong Xiang-Mu , Wen Jun , Zhang Li-Hua ,   School of Physics and Electric Engineering, Anqing Normal University, Anqing 246011, China,   School of Physica and Physical Engineering, Qufu Normal University, Qufu 273165, China
Published: December 2020

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 <inline-formula><tex-math id="M8">\begin{document}${S^4}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M8.png"/></alternatives></inline-formula> model regarded as an extension of the Ising model, can take a continuous spin value. Research of the <inline-formula><tex-math id="M9">\begin{document}${S^4}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M9.png"/></alternatives></inline-formula> 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 <inline-formula><tex-math id="M10">\begin{document}${S^4}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M10.png"/></alternatives></inline-formula> 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 <inline-formula><tex-math id="M11">\begin{document}${d_{\rm{f}}} = {\rm{1}} + \ln m/\ln {\rm{3}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M11.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M12">\begin{document}$R = \infty $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M12.png"/></alternatives></inline-formula>, respectively. In order to discuss the phase transition of the <inline-formula><tex-math id="M13">\begin{document}${S^4}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M13.png"/></alternatives></inline-formula> model on the lattice, we assume that the Gaussian distribution constant <inline-formula><tex-math id="M14">\begin{document}${b_i}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M14.png"/></alternatives></inline-formula> and the fourth-order interaction parameter <inline-formula><tex-math id="M15">\begin{document}${u_i}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M15.png"/></alternatives></inline-formula> depend on the coordination number <inline-formula><tex-math id="M16">\begin{document}${q_i}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M16.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M16.png"/></alternatives></inline-formula> of the site on the fractal lattices, and the relation <inline-formula><tex-math id="M17">\begin{document}${b_i}/{b_j} = {u_i}/{u_j} = {q_i}/{q_j}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M17.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M17.png"/></alternatives></inline-formula> is satisfied. Using the renormalization group and the cumulative expansion method, we study the phase transition of the <inline-formula><tex-math id="M18">\begin{document}${S^4}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M18.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M18.png"/></alternatives></inline-formula> model on a family of diamond lattices of <inline-formula><tex-math id="M19">\begin{document}$m$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M19.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M19.png"/></alternatives></inline-formula> branches. Removing the inner sites, we obtain the system recursion relation and the system corresponding critical point. Furthermore, we find that if the number of branches is <inline-formula><tex-math id="M20">\begin{document}$m = 2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M20.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M20.png"/></alternatives></inline-formula> or <inline-formula><tex-math id="M21">\begin{document}$m > {\rm{1}}2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M21.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M21.png"/></alternatives></inline-formula>(fractal dimensionality<inline-formula><tex-math id="M22">\begin{document}${d_{\rm{f}}} = {\rm{1}}{\rm{.63}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M22.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M22.png"/></alternatives></inline-formula> or<inline-formula><tex-math id="M23">\begin{document}${d_{\rm{f}}} > {\rm{3}}{\rm{.26}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M23.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M23.png"/></alternatives></inline-formula>), the system only has the Gaussian fixed point of <inline-formula><tex-math id="M24">\begin{document}${K^ * } = {b_2}/2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M24.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M24.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M25">\begin{document}$u_2^ * = 0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M25.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M25.png"/></alternatives></inline-formula>. The critical point of the system is in agreement with that from the Gaussian model on the fractal lattice, which predicts that the two systems belong to the same university class. We also find that under the condition of <inline-formula><tex-math id="M26">\begin{document}${\rm{3}} \leqslant m \leqslant {\rm{1}}2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M26.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M26.png"/></alternatives></inline-formula> (fractal dimensionality<inline-formula><tex-math id="M27">\begin{document}${\rm{2}} \leqslant {d_{\rm f}} \leqslant {\rm{3}}{\rm{.26}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M27.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M27.png"/></alternatives></inline-formula>), both the Gaussian fixed point and the Wilson-Fisher fixed point can be obtained in the system, and the Wilson-Fisher fixed point plays a leading role in the critical properties of the system. According to the real space renormalization group transformation and scaling theory, we obtain the critical exponent of the correlation length. Finally, we find that the critical points of the <inline-formula><tex-math id="M28">\begin{document}${S^4}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M28.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M28.png"/></alternatives></inline-formula> model on a family of diamond lattices depend on the value of the fractal dimensionality. The above result is similar to that obtained from the <inline-formula><tex-math id="M29">\begin{document}${S^4}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M29.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20181315e-wen-revised_M29.png"/></alternatives></inline-formula> model on the translation symmetry lattice.