Spherical Model on a Cayley Tree: Large Deviations

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• 作者：A. E. Patrick
• 关键词：Critical temperature ; Order parameter ; Phase transition ; Spherical model
• 刊名：Journal of Statistical Physics
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：166
• 期：1
• 页码：45-71
• 全文大小：
• 刊物类别：Physics and Astronomy
• 刊物主题：Statistical Physics and Dynamical Systems; Theoretical, Mathematical and Computational Physics; Physical Chemistry; Quantum Physics;
• 出版者：Springer US
• ISSN：1572-9613
• 卷排序：166

文摘

We study the spherical model of a ferromagnet on a Cayley tree and show that in the case of empty boundary conditions a ferromagnetic phase transition takes place at the critical temperature \(T_\mathrm{c} =\frac{6\sqrt{2}}{5}J\), where J is the interaction strength. For any temperature the equilibrium magnetization, \(m_n\), tends to zero in the thermodynamic limit, and the true order parameter is the renormalized magnetization \(r_n=n^{3/2}m_n\), where n is the number of generations in the Cayley tree. Below \(T_\mathrm{c}\), the equilibrium values of the order parameter are given by \(\pm \rho ^*\), where $$\begin{aligned} \rho ^*=\frac{2\pi }{(\sqrt{2}-1)^2}\sqrt{1-\frac{T}{T_\mathrm{c}}}. \end{aligned}$$One more notable temperature in the model is the penetration temperature $$\begin{aligned} T_\mathrm{p}=\frac{J}{W_\mathrm{Cayley}(3/2)}\left( 1-\frac{1}{\sqrt{2}}\left( \frac{h}{2J}\right) ^2\right) . \end{aligned}$$Below \(T_\mathrm{p}\) the influence of homogeneous boundary field of magnitude h penetrates throughout the tree. The main new technical result of the paper is a complete set of orthonormal eigenvectors for the discrete Laplace operator on a Cayley tree.KeywordsCritical temperatureOrder parameterPhase transitionSpherical modelReferences1.Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, New York (1982)MATHGoogle Scholar2.Berlin, T.H., Kac, M.: The spherical model of a ferromagnet. Phys. Rev. 86, 821–835 (1952)ADSCrossRefMATHMathSciNetGoogle Scholar3.Bleher, P., Ruiz, J., Schonmann, R.H., Shlosman, S., Zagrebnov, V.: Rigidity of the critical phases on a Cayley tree. Mosc. Math. J. 1, 345–363 (2001)MATHMathSciNetGoogle Scholar4.Bleher, P.M., Ruiz, J., Zagrebnov, V.A.: On the purity of the limiting Gibbs state for the Ising model on the Bethe Lattice. J. Stat. Phys. 79, 473–482 (1995)ADSCrossRefMATHMathSciNetGoogle Scholar5.Eggarter, T.P.: Cayley trees, the Ising problem, and the thermodynamic limit. Phys. Rev. B 9, 2989–2992 (1974)ADSCrossRefGoogle Scholar6.Gersch, H.A., Berlin, T.H.: Spherical lattice gas. Phys. Rev. 127, 2276–2283 (1962)ADSCrossRefMATHGoogle Scholar7.Matsuda, H.: Infinite susceptibility without spontaneous magnetization. Prog. Theor. Phys. 51, 1053–1063 (1974)ADSCrossRefGoogle Scholar8.Morita, T., Horiguchi, T.: Susceptibility and correlation function of the Ising model on the Cayley tree. Prog. Theor. Phys. 54, 982–998 (1975)ADSCrossRefGoogle Scholar9.Patrick, A.E.: The influence of external boundary conditions on the spherical model of a ferromagnet. I. Magnetization profiles. J. Stat. Phys. 75, 253–295 (1994)ADSCrossRefGoogle Scholar10.Pressman, W., Keller, J.B.: Equation of state and phase transition of the spherical lattice gas. Phys. Rev. 120, 22–32 (1960)ADSCrossRefMATHMathSciNetGoogle Scholar11.van den Berg, M., Dorlas, T.C., Priezzhev, V.B.: The boson gas on a Cayley tree. J. Stat. Phys. 69, 307–328 (1992)ADSCrossRefMATHMathSciNetGoogle ScholarCopyright information© Springer Science+Business Media New York 2016Authors and AffiliationsA. E. Patrick1Email authorView author's OrcID profile1.Laboratory of Theoretical Physics, Joint Institute for Nuclear ResearchDubnaRussia About this article CrossMark Publisher Name Springer US Print ISSN 0022-4715 Online ISSN 1572-9613 About this journal Reprints and Permissions Article actions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. More information Accept Over 10 million scientific documents at your fingertips