Exact Solution of a Generalized ANNNI Model on a Cayley Tree

  • U. A. RozikovEmail author
  • H. Akin
  • S. Ug~uz


We consider the Ising model on a rooted Cayley tree of order two with nearest neighbor interactions and competing next nearest neighbor interactions restricted to spins belonging to the same branch of the tree. This model was studied by Vannimenus who found a new modulated phase, in addition to the paramagnetic, ferromagnetic, antiferromagnetic phases and a (+ + - -) periodic phase. Vannimenus’s results are based on an analysis of the recurrence equations (relating the partition function of an n − generation tree to the partition function of its subsystems containing (n −1) generations) and most results are obtained numerically. In this paper we analytically study the recurrence equations and obtain some exact results: critical temperatures and curves, number of phases, partition function.


Cayley tree Configuration Ising model Phase Gibbs measure 

Mathematics Subject Classifications (2010)

82B20 82B26 


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  1. 1.
    Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, London/New York (1982)zbMATHGoogle Scholar
  2. 2.
    Ganikhodjaev, N.N., Akin, H., Temir, S.: Potts model with two competing binary interactions. Turk. J. Math. 31, 229–238 (2007)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Ganikhodjaev, N.N., Akin, H., Ug~uz, S., Temir, S.: Phase diagram and extreme Gibbs measures of the Ising model on a Cayley tree in the presence of competing binary and ternary interactions. Phase Transit. 84, 1045–1063 (2011)CrossRefGoogle Scholar
  4. 4.
    Ganikhodjaev, N.N., Mukhamedov, F.M., Pah, C.H.: Phase diagram of the three states Potts model with next nearest neighbour interactions on the Bethe lattice. Phys. Lett. A 373, 33–38 (2008)CrossRefzbMATHMathSciNetADSGoogle Scholar
  5. 5.
    Ganikhodjaev, N.N., Temir, S., Akin, H.: Modulated phase of a Potts model with competing binary interactions on a Cayley tree. J. Stat. Phys. 137, 701–715 (2009)CrossRefzbMATHMathSciNetADSGoogle Scholar
  6. 6.
    Ganikhodjaev, N.N., Rozikov, U.A.: A description of periodic extremal Gibbs measures of some lattice models on the Cayley tree. Theor. Math. Phys. 111, 480–486 (1997)CrossRefGoogle Scholar
  7. 7.
    Ganikhodjaev, N.N., Rozikov, U.A.: On Ising model with four competing interactions on Cayley tree. Math. Phys. Anal. Geom. 12, 141–156 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Georgii, H.O.: Gibbs measures and phase transitions. Walter de Gruyter, Berlin (1988)CrossRefzbMATHGoogle Scholar
  9. 9.
    Inawashiro, S., Thompson, C.J.: Competing Ising interactions and chaotic glass-like behaviour on a Cayley tree. Phys. Lett. 97A, 245–248 (1983)CrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Katsura, S., Takizawa, M.: Bethe lattice and the Bethe approximation. Prog. Theor.Phys. 51, 82–98 (1974)CrossRefADSGoogle Scholar
  11. 11.
    Mariz, A., Tsallis, C., Albuquerque, E.L.: 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)CrossRefMathSciNetADSGoogle Scholar
  12. 12.
    Ohanyan, V.R.: L.N.Ananikyan, N.S.Ananikian, An exact solution on the ferromagnetic face-cubic spin model on a Bethe lattice. Physica A 377, 501–513 (2007)CrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Pemantle, R., Peres, Y.: The critical ising model on trees, concave recursions and nonlinear capacity. Ann. Probab. 38, 184–206 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Preston, C.: Gibbs States on Countable Sets. Cambridge University Press, London (1974)CrossRefzbMATHGoogle Scholar
  15. 15.
    Rozikov, U.A.: Partition structures of the Cayley tree and applications for describing periodic Gibbs distributions. Theor. Math. Phys. 112, 929–933 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Rozikov, U.A., Shoyusupov, Sh.A.: Fertile three state HC models on Cayley tree. Theor. Math. Phys. 156, 1319–1330 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Rozikov, U.A., Suhov, Y.M.: Gibbs measures for SOS model on a Cayley tree. Inf. Dim. Anal. Quant. Prob. Rel. Fields. 9, 471–488 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Rozikov, U.A.: A contour method on Cayley trees. J. Stat. Phys. 130, 801–813 (2008)CrossRefzbMATHMathSciNetADSGoogle Scholar
  19. 19.
    Sinai, Ya.G.: Theory of Phase Transitions: Rigorous Results. Pergamon, Oxford (1982)zbMATHGoogle Scholar
  20. 20.
    Suhov, Y.M., Rozikov, U.A.: A hard-core model on a Cayley tree: an example of a loss network. Queueing Syst. 46, 197–212 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Vannimenus, J.: Modulated phase of an Ising system with competing interactions on a Cayley tree. Z. Phys. B 43, 141–148 (1981)CrossRefMathSciNetADSGoogle Scholar

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© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Institute of MathematicsTashkentUzbekistan
  2. 2.Faculty of Education Department of Mathematics Kizilhisar CampusZirve UniversityGaziantepTurkey
  3. 3.Department of Mathematics Arts and Science FacultyHarran UniversitySanliurfaTurkey

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