Abstract
Renormalization group recursion formulas for classical O(N) spin models in two dimensions are obtained. The main part of the recursion formulas is solved and yields the flows which are very close to those of the hierarchical model approximations of Dyson–Wilson type. Spontaneous mass generations also take place under our approximation.
Similar content being viewed by others
REFERENCES
D. Brydges, J. Fröhlich, and T. Spencer, The random walk representation of classical spin systems and correlation inequalities, Commun. Math. Phys. 83:123 (1982).
D. Brydges, J. Fröhlich, and A. Sokal, The random walk representation of classical spin systems and correlation inequalities II, Commun. Math. Phys. 91:117 (1985).
D. Brydges, A short course on cluster expansions, in Les Housch Summer School, Session XLIII (1984), K. Osterwalder et al., eds. (Elsevier Sci. Publ., 1986).
F. J. Dyson, Existence of a phase transition in a one-dimensional Ising ferromagnet, Commun. Math. Phys. 12:91 (1969).
S. Caracciolo, R. Edwards, A. Plisetto, and A. Sokal, Asymptotic scaling of in the twodimensional O(3) s model at correlation length 105, Phys. Rev. Lett. 74:2969 (1995); 75: 1891 (1996).
G. Gallavotti, Some aspects of the renormalization problems in statistical mechanics and field theory, Mem. Accad. Lincei 14:1 (1978).
K. Gawedzki and A. Kupiainen, A rigorous block spin approach to massless lattice theories, Commun. Math. Phys. 77:31 (1980); K. Gawedzki and A. Kupiainen, Massless lattice f44 theory, rigorous control of a renormalizable asymptotically free field theory, Commun. Math. Phys. 99:197 (1985).
K. Gawedzki and A. Kupiainen, Non-Gaussian fixed points of the block spin transformation: Hierarchical model approximations, Commun. Math. Phys. 89:191 (1983).
K. Gawedzki and A. Kupiainen, Continuum limit of the hierarchical O(N) non-linear s model, Commun. Math. Phys. 106:535 (1986).
J. Glimm, A. Jaffe, and T. Spencer, The particle structures of the weakly coupled P(F)2 models and other applications, Part II: The cluster expansion, in Constructive Quantum Field Theory, Lecture Notes in Physics, Vol. 25, G. Velo and A. Wightman, eds. (Springer Verlag, Heidelberg, 1973), p. 199.
K. R. Ito, Permanent quark confinement in 4D hierarchical LGT of Migdal-Kadanoff type, Phys. Rev. Lett. 55:558–561 (1985); Mass generations in 2D hierarchical Heisenberg model of Migdal-Kadanoff type, Commun. Math. Phys. 110:46–47 (1987).
K. R. Ito, Origin of asymptotic freedom in non-Abelian field theories, Phys. Rev. Lett. 58:439 (1987); K. R. Ito, Renormalization group flow of 2D hierarchical Heisenberg model of Dyson-Wilson type, Commun. Math. Phys. 137:45 (1991).
K. R. Ito, Paper in preparation (2001).
K. R. Ito and H. Tamura, N dependence of critical temperatures of 2D O(N) spin models, Commun. Math. Phys. 202:127 (1999).
C. Kopper, Mass generations in the large N non-linear s model, Commun. Math. Phys. 202:89 (1999).
S. K. Ma, The 1/n expansion, in Phase Transitions and Critical Phenomena, Vol. 6, C. Domb and M. S. Green, eds. (Academic Press, London, 1976), pp. 249–292.
A. Polyakov, Interactions of Goldstone bosons in two dimensions, Phys. Lett. B 59:79 (1975).
V. Rivasseau, Cluster expansion with small/large field conditions, in Mathematical Quantum Theory I: Field Theory and Many-Body Theory, CRM Proceedings and Lecture Notes, Vol. 7, J. Feldman et al., eds. (A.M.S., 1994).
K. Wilson, Confinement of quarks, Phys. Rev. D 10: 2455 (1974); Renormalization groups and critical phenomena, Rev. Mod. Phys. 55:583 (1983).
K. Wilson and J. Kogut, Renormalization group and critical phenomena, Phys. Rep. C 12:75 (1974).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ito, K.R. Renormalization Group Recursion Formulas and Flows of 2D O(N) Spin Models. Journal of Statistical Physics 107, 821–856 (2002). https://doi.org/10.1023/A:1014542315025
Issue Date:
DOI: https://doi.org/10.1023/A:1014542315025