On Gibbs measures of models with competing ternary and binary interactions and corresponding von Neumann algebras

Journal of Statistical Physics

Volumn 114, Issue 3-4, 2004, Pages 825-848

  Mukhamedov, Farruh ^a   Rozikov, Utkir ^b  

^a NATIONAL UNIVERSITY OF UZBEKISTAN   (Uzbekistan)

^b Institute of Mathematics   (Uzbekistan)

Author keywords

Cayley tree; Competing interactions; Gibbs measure; GNS construction; Hamiltonian; Von Neumann algebra

Indexed keywords

EID: 3543001684     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1023/b:joss.0000012509.10642.83     Document Type: Article

References (45)

1. L. Accardi, On noncommutative Markov property, Funkt. anal. i ego proloj. 8:1-8 (1975).
2. L. Accardi and G. Frigerio, Markovian cocycles, Proc. R. Ir. Acad. 83A:245-273 (1983).
3. H. Araki, On quasi-free states of CAR and Bogolubov automorphisms, Publ. R.I.M.S. Kyoto Univ. 6:385-442 (1970/71).
4. H. Araki and E. J. Woods, A classification of factors, Publ. R.I.M.S. Kyoto Univ. 3:51-130 (1968).
5. H. Araki and W. Wyss, Representations of canonical anticommutations relations, Helv. Phys. Acta 37:136-159 (1964).
6. R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London/New York, 1982).
7. P. M. Bleher, Extremity of the disordered phase in the Ising model on the Bethe lattice, Comm. Math. Phys. 128:411-419 (1990).
8. P. M. Bleher and N. N. Ganikhodjaev, On pure phases of the Ising model on the Bethe lattice, Theor. Probab. Appl. 35:216-227 (1990).
9. P. M. Bleher, J. Ruiz, R. H. Schonmann, S. Shlosman, and V. A. Zagrebnov, Rigidity of the critical phases on a Cayley tree, Moscow Math. J. 3:345-362 (2001).
10. P. M. Bleher, J. Ruiz, and V. A. Zagrebnov, On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice, J. Statist. Phys. 79:473-482 (1995).
11. P. M. Bleher, J. Ruiz, and V. A. Zagrebnov, On the phase diagram of the random field Ising model on the Bethe lattice, J. Statist. Phys. 93:33-78 (1998).
12. O. Bratteli and D. Robinson, Operator Algebras and Quantum Statistical Mechanics, I (Springer-Verlag, Berlin/New York, 1979).
13. O. Bratteli and D. Robinson, Operator Algebras and Quantum Statistical Mechanics, II (Springer-Verlag, Berlin/New York, 1981).
14. A. Connes, Une classification des facteurs de type III, Ann. Ec. Norm. Sup. 6:133-252 (1973).
15. G. F. Dell'Antonio, Structure of algebras of some free systems, Comm. Math. Phys. 9:81-117 (1968).
16. N. N. Ganikhodjaev, Exact solution of the Ising model on the Cayley tree with competing ternary and binary interactions, Theor. Math. Phys. 130:419-424 (2002).
17. N. N. Ganikhodjaev, A group representations and automorphisms of Cayley tree, Dokl. Akad. Nauk. Rep. Uzb. 4:3-4 (1994).
18. N. N. Ganikhodjaev and F. M. Mukhamedov, Markov states on quantum lattice systems and its representations, Methods of Funct. Anal. and Topology 4:33-38 (1998).
19. N. N. Ganikhodjaev and U. A. Rozikov, Description of periodic extreme Gibbs measures of some lattice models on the Cayley tree, Theor. Math. Phys. 111:480-486 (1997).
20. N. N. Ganikhodjaev and U. A. Rozikov, On disordered phase in the ferromagnetic Potts model on the Bethe lattice, Osaka J. Math. 37:373-383 (2000).
21. H. O. Georgii, Gibbs Measures and Phase Transitions (Walter de Gruyter, Berlin, 1988).
22. V. Ya. Golodets and S. V. Neshveyev, Non-Bernoullian K-systems, Commun. Math. Phys. 195:213-232 (1998).
23. V. Ya. Golodets and G. N. Zholtkevich, On KMS-Markovian states, Theor. Math. Phys. 56:80-86 (1983).
24. U. Haagerup and C. Winslow, The Effros-Marechal topology in the space of von Neumann algebras, J. Funct. Anal. 171:401-431 (2000).
25. M. Mariz, C. Tsalis, and A. L. Albuquerque, Phase diagram of the Ising model on a Cayley tree in the presence of competing interactions and magnetic field, J. Stat. Phys. 40:577-592 (1985).
26. J. L. Monroe, Phase diagrams of Ising models on Husimi trees II. Pure multisite interaction systems, J. Stat. Phys. 67:1185-1200 (1992).
27. J. L. Monroe, A new criterion for the location of phase transitions for spin systems on recursive lattices, Phys. Lett. A 188:80-84 (1994).
28. F. M. Mukhamedov, Von Neumann algebras corresponding translation- invariant Gibbs states of Ising model on the Bethe lattice, Theor. Math. Phys. 123:489-493 (2000).
29. F. M. Mukhamedov and U. A. Rozikov, Von Neumann algebra corresponding to one phase of inhomogeneous Potts model on a Cayley tree, Theor. Math. Phys. 126:169-174 (2001).
30. T. Murakami and S. Yamagami, On types of quasifree representations of Clifford algebras, Publ. R.I.M.S. Kyoto Univ. 31:33-44 (1995).
31. R. Powers, Representation of uniformly hyperfinite algebras and their associated von Neumann rings, Ann. Math. 81:138-171 (1967).
32. R. Powers and E. Stormer, Free states of canonical anticommutation relations, Comm. Math. Phys. 16:1-33 (1970).
33. C. Preston, Gibbs States on Countable Sets (Cambridge University Press, London, 1974).
34. U. A. Rozikov, Description of limiting Gibbs measures for λ-models on the Bethe lattice, Siberian Math. J. 39:427-435 (1998).
35. U. A. Rozikov, Describtion uncountable number of Gibbs measures for inhomogeneous Ising model, Theor. Math. Phys. 118:95-104 (1999).
36. C. M. Series and Ya. G. Sinai, Ising models on the Lobachavsky plane, Commun. Math. Phys. 128:63-76 (1990).
37. A. N. Shiryaev, Probability (Nauka, Moscow, 1980).
38. C. R. da Silca and S. Coutinho, Ising model on the Bethe lattice with competing interactions up to the third-nearest-neighbor generation, Phys. Review B 34:7975-7985 (1986).
39. Ya. G. Sinai, Theory of Phase Transitions: Rigorous Results (Pergamon, Oxford, 1982).
40. E. Stormer, Automorphisms group and invariant states, Proc. Sym. Pure Math. 38:149-161 (1982).
41. S. Stratila, Modular Theory in Operator Algebras (Abacus Press, Bucuresti, 1981).
42. S. Stratila and D. Voiculescu, Representations of AF-Algebras and of the Group U(∞), Lectures Notes in Mathematics, Vol. 486 (Springer-Verlag, Berlin, 1975).
43. J. Vannimenus, Modulated phase of an Ising system with competing interactions on a Cayley tree, Z. Phys. B 43:141 (1981).
44. S. Zachary, Countable state space Markov random fields and Markov chains on trees, Ann. Prob. 11:894-903 (1983).
45. S. Zachary, Bounded attractive and repulsive Markov specifications on trees and on the one-dimensional lattice, Stochastic Process. Appl. 20:247-256 (1985).