Journal of Experimental and Theoretical Physics

, Volume 89, Issue 6, pp 1107–1113 | Cite as

Berezinskii-Kosterlitz-Thouless phase transitions in two-dimensional systems with internal symmetries

  • S. A. Bulgadaev
Solids: Structure


The Berezinskii-Kosterlitz-Thouless (BKT) phase transitions in two-dimensional systems with internal continuous Abelian symmetries are investigated. In order for phase transitions to occur, the kinetic part of the action of the system must have conformal invariance, and the vacuum manifold must be degenerate and have a discrete Abelian homotopy group π1. In this case topological excitations have a logarithmically divergent energy and can be described by effective theories that generalize the two-dimensional Euclidean sine-Gordon theory, which is an effective theory of the original XY model. In particular, the effective actions are found for chiral models on the maximal Abelian tori TG of the simple compact Lie groups G. The critical properties of the possible effective theories are found, and it is shown that they are characterized by the Coxeter numbers hG of lattices of the \(\mathbb{A},\mathbb{D},\mathbb{E}\) and ℤ series and can be interpreted as properties of conformal theories with an integer central charge C=n, where n is the rank of the groups π1 and G. The possibility of reconstructing the complete symmetry of G in the massive phase is also discussed.


Homotopy Group Effective Theory Internal Symmetry Conformal Theory Massive Phase 
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  1. 1.
    H. E. Stanley and T. A. Kaplan, Phys. Rev. Lett. 17, 913 (1966).CrossRefADSMathSciNetGoogle Scholar
  2. 2.
    L. D. Landau, Zh. Éksp. Teor. Fiz. 7, 627 (1937).Google Scholar
  3. 3.
    R. E. Peierls, Ann. Inst. Henri Poincare 5, 177 (1935).zbMATHMathSciNetGoogle Scholar
  4. 4.
    N. N. Bogolyubov, Selected Works [in Russian], Naukova Dumka, Kiev (1971), Vol. 3.Google Scholar
  5. 5.
    J. Goldstone, Nuovo Cimento 19, 154 (1961).zbMATHMathSciNetGoogle Scholar
  6. 6.
    N. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).ADSGoogle Scholar
  7. 7.
    P. C. Hohenberg, Phys. Rev. 158, 383 (1967).CrossRefADSGoogle Scholar
  8. 8.
    T. M. Rice, Phys. Rev. 140, 1889 (1965).CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    B. Jancovici, Phys. Rev. Lett. 19, 20 (1967).CrossRefADSGoogle Scholar
  10. 10.
    V. L. Berezinskii, Zh. Éksp. Teor. Fiz. 59, 907 (1970) [Sov. Phys. JETP 32, 493 (1970)]; Zh. Éksp. Teor. Fiz. 61, 1144 (1971) [Sov. Phys. JETP 34, 610 (1972)].Google Scholar
  11. 11.
    V. N. Popov, Functional Integrals in Quantum Field Theory and Statistical Physics, Reidel, Dordrecht (1983) [Russ. original, Atomizdat, Moscow (1976)].Google Scholar
  12. 12.
    J. M. Kosterlitz and J. P. Thouless, J. Phys. C 6, 118 (1973).CrossRefGoogle Scholar
  13. 13.
    J. M. Kosterlitz, J. Phys. C 7, 1046 (1974).CrossRefADSGoogle Scholar
  14. 14.
    R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, New York (1982) [Russ. transl., Mir, Moscow (1985)].Google Scholar
  15. 15.
    J. Jose, L. Kadanoff, S. Kirkpatrick, and D. Nelson, Phys. Rev. B 16, 1217 (1977).ADSGoogle Scholar
  16. 16.
    P. B. Wiegmann, J. Phys. C 11, 1583 (1978).CrossRefADSGoogle Scholar
  17. 17.
    T. Ohta, Prog. Theor. Phys. 60, 968 (1978).ADSGoogle Scholar
  18. 18.
    D. J. Amit, Y. Y. Goldschmidt, and G. Grinstein, J. Phys. A 13, 585 (1980).CrossRefADSMathSciNetGoogle Scholar
  19. 19.
    A. Z. Patashinskii and V. L. Pokrovskii, Fluctuation Theory of Phase Transitions, Pergamon Press, Oxford (1979) [newer ed. of Russ. original, Nauka, Moscow (1982)].Google Scholar
  20. 20.
    A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Nucl. Phys. B 241, 333 (1984); Vl. S. Dotsenko and V. A. Fateev, Nucl. Phys. B 240, 312 (1984);, 251, 691 (1985); D. Friedan, Z. Qiu, and S. H. Shenker, Phys. Rev. Lett. 53, 1575 (1984); G. E. Andrews, R. J. Baxter, and P. J. Forrester, J. Stat. Phys. 35, 193 (1984); D. A. Huse, Phys. Rev. B 30, 3908 (1984).CrossRefADSMathSciNetGoogle Scholar
  21. 21.
    S. A. Bulgadaev, JETP Lett. 63, 780 (1996); S. A. Bulgadaev, Landau Institute Preprint 29/05/97 (1997), e-print Archive hep-th/9901036; Zh. Éksp. Teor. Fiz. 116, 1131 (1999) [JETP 89, 603 (1999)].ADSMathSciNetGoogle Scholar
  22. 22.
    M. Green, J. H. Schwarz, and E. Witten, Superstring Theory, Cambridge Univ. Press, Cambridge-New York (1987), Vols. 1 and 2; Ya. I. Kogan, JETP Lett. 45, 709 (1987); D. Gross and I. Klebanov, in Proceedings of the Trieste Spring School “String Theory and Quantum Gravity’91”, World Scientific, Singapore (1991).Google Scholar
  23. 23.
    T. Banks, W. Fischler, S. H. Shenker, and L. Susskind, Phys. Rev. D 55, 5112 (1997); T. Banks and N. Seiberg, Nucl. Phys. B 497, 41 (1997).CrossRefADSMathSciNetGoogle Scholar
  24. 24.
    S. A. Bulgadaev, talk given at the International Conference “Conformal Field Theories and Integrable Models,” Chernogolovka, Russia (1996); Landau Institute Preprint 02/06/97 (1997).Google Scholar
  25. 25.
    S. A. Bulgadaev, Phys. Lett. A 86, 213 (1981); Teor. Mat. Fiz. 49, 7 (1981); Nucl. Phys. B 224, 349 (1983); JETP Lett. 63, 780 (1996); S. A. Bulgadaev, Nucl. Phys. B 224, 349 (1983); Landau Institute Preprint 27/05/97 (1997).CrossRefADSMathSciNetGoogle Scholar
  26. 26.
    G. H. Conway and N. J. A. Sloane, Sphere Packing, Lattices and Groups, Springer-Verlag (1988), Vols. 1 and 2.Google Scholar
  27. 27.
    D. R. Nelson, Phys. Rev. B 18, 2318 (1978); D. R. Nelson and B. I. Halperin, Phys. Rev. B 19, 2457 (1979).ADSGoogle Scholar
  28. 28.
    V. N. Kac, Infinite Dimensional Lie Algebras, Cambridge University Press (1990).Google Scholar
  29. 29.
    P. Goddard, A. Kent, and D. Olive, Phys. Lett. B 152, 88 (1985); Commun. Math. Phys. 103, 105 (1986); V. A. Fateev and S. L. Lukyanov, Int. J. Mod. Phys. A 13, 507 (1988); T. Eguchi and S. K. Yang, Phys. Lett. B 224, 373 (1989).ADSMathSciNetGoogle Scholar
  30. 30.
    A. M. Polyakov, Gauge Fields and Strings, Harwood Academic Publishers, Chur, Switzerland-New York (1987) [newer ed. of Russ. original, Inst. Teor. Fiz. im L. D. Landau, Moscow (1995)].Google Scholar
  31. 31.
    E. Ogievetski and P. B. Wiegmann, Phys. Lett. B 168, 360 (1986); C. Destri and H. J. de Vega, Preprint CERN-TH-4895/87 (1987).ADSMathSciNetGoogle Scholar

Copyright information

© American Institute of Physics 1999

Authors and Affiliations

  • S. A. Bulgadaev
    • 1
  1. 1.L. D. Landau Institute of Theoretical PhysicsRussian Academy of SciencesMoscowRussia

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