Robust phase transitions for Heisenberg and other models on general trees. (English) Zbl 0981.60096

Pemantle, Robin; Steif, Jeffrey E.

Ann. Probab. 27, No. 2, 876-912 (1999).

Summary: We study several statistical mechanical models on a general tree. Particular attention is devoted to the classical Heisenberg models, where the state space is the \(d\)-dimensional unit sphere and the interactions are proportional to the cosines of the angles between neighboring spins. The phenomenon of interest here is the classification of phase transition (non-uniqueness of the Gibbs state) according to whether it is robust. In many cases, including all of the Heisenberg and Potts models, occurrence of robust phase transition is determined by the geometry (branching number) of the tree in a way that parallels the situation with independent percolation and usual phase transition for the Ising model. The critical values for robust phase transition for the Heisenberg and Potts models are also calculated exactly. In some cases, such as the \(q\geq 3\) Potts model, robust phase transition and usual phase transition do not coincide, while in other cases, such as the Heisenberg models, we conjecture that robust phase transition and usual phase transition are equivalent. In addition, we show that symmetry breaking is equivalent to the existence of a phase transition, a fact believed but not known for the rotor model on \(\mathbb{Z}^2\).

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
82B05	Classical equilibrium statistical mechanics (general)
82B26	Phase transitions (general) in equilibrium statistical mechanics

Keywords:

phase transitions; symmetry breaking; Heisenberg models

References:

[1]	Adel’son-Vel’skii, G., Veisfeiler, B., Leman, A. and Faradzev, I. (1969). Example of a graph without a transitive automorphism group. Soviet Math. Dokl. 10 440-441. · Zbl 0208.52501
[2]	Aizenman, M., Chayes, J. T., Chayes, L. and Newman, C. M. (1988). Discontinuity of the magnetization in one-dimensional 1/ x y 2 Ising and Potts models, J. Statist. Phys. 50 1-40. · Zbl 1084.82514
[3]	Askey, R. (1975). Orthogonal Polynomials and Special Functions. Arrowsmith, Bristol, England. · Zbl 0298.33008
[4]	Brouwer, A., Cohen, A. and Neumaier, A. (1989). Distance Regular Graphs. Springer, New York. · Zbl 0747.05073
[5]	Biggs, N. (1993). Algebraic Graph Theory, 2nd ed. Cambridge Univ. Press. · Zbl 0805.68094
[6]	Cassi, D. (1992). Phase transition and random walks on graphs: a generalization of the Mermin-Wagner theorem to disordered lattices, fractals, and other discrete structures. Phys. Rev. Lett. 68 3631-3634.
[7]	Eisele, M. (1994). Phase transitions may be absent on graphs with transient random walks. Unpublished manuscript.
[8]	Evans, W., Kenyon, C., Peres, Y. and Schulman, L. J. (1998). Broadcasting on trees and the Ising model. Ann. Appl. Probab. · Zbl 1052.60076
[9]	Furstenberg, H. (1970). Intersections of Cantor sets and transversality of semigroups. In Problems in Analysis. Symposium in Honor of Salomon Bochner (R. C. Gunning, ed.) 41-59. Princeton Univ. Press. · Zbl 0208.32203
[10]	Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. de Gruyter, New York. · Zbl 0657.60122
[11]	Häggstr öm, O. (1996). The random-cluster model on a homogeneous tree. Probab. Theory Related Fields 104 231-253. · Zbl 0838.60086
[12]	Lebowitz, J. L. and Penrose, O. (1976). Thermodynamic limit of the free energy and correlation functions of spin systems. Acta Physica Austriaca, Suppl XVI 201-220.
[13]	Liggett, T. M. (1996). Multiple transition points for the contact process on a binary tree. Ann. Probab. 24 1675-1710. · Zbl 0871.60087
[14]	Lyons, R. (1989). The Ising model and percolation on trees and tree-like graphs. Comm. Math. Phys. 125 337-353. · Zbl 0679.60101
[15]	Lyons, R. (1990). Random walks and percolation on trees. Ann. Probab. 18 931-958. · Zbl 0714.60089
[16]	Merkl, F. and Wagner, H. (1994). Recurrent random walks and the absence of continuous symmetry breaking on graphs. J. Statist. Phys. 75 153-165. · Zbl 0828.60055
[17]	Monroe, J. L. and Pearce, P. A. (1979). Correlation inequalities for vector spin models. J. Statist. Phys. 21 615-633.
[18]	Natterer, F. (1986). The Mathematics of Computerized Tomography. Wiley, New York. · Zbl 0617.92001
[19]	Patrascioiu A. and Seiler, E. (1992). Phase structure of two-dimensional spin models and percolation. J. Statist. Phys. 69 573-595. · Zbl 0893.60076
[20]	Pemantle, R. (1992). The contact process on trees. Ann. Probab. 20 2089-2116. · Zbl 0762.60098
[21]	Pemantle, R. and Peres, Y. (1999). Recursions on trees and the Ising model. · Zbl 1197.60092
[22]	Rainville, E. D. (1960). Special Functions. MacMillan, New York. · Zbl 0092.06503
[23]	Stacey, A. (1996). The existence of an intermediate phase for the contact process on trees. Ann. Probab. 24 1711-1726. · Zbl 0878.60061
[24]	Terwilliger, P. (1998). Unpublished lecture notes.