Spherical Model on a Cayley Tree: Large Deviations

• Published in: Journal of statistical physics 2017, Vol.166 (1), p.45-71
• Main Author: Patrick, A. E.
• Format: Article
• Language: English
• Subjects: Analysis; Boundary conditions; Critical temperature; Eigenvectors; Ferromagnetic phases; Ferromagnetism; Magnetization; Mathematical and Computational Physics; Mathematical models; Order parameters; Phase transitions; Physical Chemistry; Physics; Physics and Astronomy; Quantum Physics; Statistical Physics and Dynamical Systems; Theoretical; Thermodynamics
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-016-1696-4

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 c = 6 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 c , the equilibrium values of the order parameter are given by ± ρ ∗ , where ρ ∗ = 2 π ( 2 - 1 ) 2 1 - T T c . One more notable temperature in the model is the penetration temperature T p = J W Cayley ( 3 / 2 ) 1 - 1 2 h 2 J 2 . Below T 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.