Abstract
Two-dimensionalO(N)-invariant hierarchical Heisenberg models (of Dyson-Wilson type) are investigated by the real space renormalization group method. It is established that ifN≧2, then the effective actions at long distance scales are driven into the high temperature region by the iterative use of the block spin transformations thus concluding the non-existence of phase transitions in the system. The correlation functions are also obtained and they decay at the speed of massive gaussian field model. The driving force is geometrical and an a priori one and stronger than the boundary effect which is bounded byO(1) in the present system. Thus the hierarchical formulas fail to exhibit the Kosterlitz-Thouless transitions (at least forN=2).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Wilson, K. G.: Confinement of quarks. Phys. Rev.D10, 2445 (1974)
Wilson, K. G.: Renormalization group and critical phenomena. Rev. Mod. Phys.55, 583 (1983) and references cited therein
Polyakov, A. M.: Interactions of Goldstone Bosons and restoration of symmetries in two dimensions. Phys. Lett.59B, 79 (1975)
Wilson, K. G. and Kogut, K.: Renormalization group and ε-expansion. Phys. Rep.12C (1974)
Ma, S. K.: Modern theory of critical phenomena, Chap. 8. Reading, MA.: Benjamin 1976
Gawedzki, K., Kupiainen, A.: Non-Gaussian fixed points of the block spin transformation, Hierarchical model approximation. Commun. Math. Phys.89, 191 (1983); Asymptotic freedom beyond perturbation theory, in Les Houches 1984, Critical Phenomena, random systems, gauge theories Part 1. Course 4, Amsterdam-New York: North Holland 1984 Osterwalder, K, Stora, R. (eds.)
Bleher, P. M., Major, P.: The Large scale limit of Dyson's hierarchical vector valued model of low temperature, Commun. Math. Phys.125, 43 (1989); The large scale limit of Dyson's hierarchical vector valued model of low temperature, the non-Gaussian case. Ann. l'Inst. Henri Poincaré, Theor. Phys.,49, 7 (1988)
Ito, K. R.: Origin of asymptotic freedom in non-Abelian field theories. Phys. Rev. Lett.58, 439 (1987)
Ito, K. R.: Renormalization group on hierarchical lattices and beyond, Prog. Theor. Phys. [Suppl.] No.92, 46 (1987)
Petrov, V. V.: Limit theorem for large deviations violating Cramer's condition I, II. Am. Math. Trans.. (S)7, 235–253, 254–280 (1967) Am. Math. Soc.
Richter, W.: A more precise form of inequality of B. N. Bernstein for large deviation. Am. Math. Soc. Transl. (S)4, 225–231, (1965), Am. Math. Soc.
Simon, B., Aizenman, M.: Ward Identities and Decay of Correlations. Commun. Math. Phys.77, 137 (1980)
Seiler, E., Stamatescu, I. O., Linke, V., Patrasciou, A.: Critical behavior, scaling and universality in some two-dimensional spin models. Nucl. Phys.B305 [FS23] 623 (1988)
Patrasciou, A.: Mass gap in non-abelian sigma models and gauge theories. Phys. Rev. Lett.58, 2285 (1987)
Ito, K. R.: Kosterlitz-Thouless Type Transitions in Two-Dimensional Non-Abelian field theories. Phys. Rev. Lett.64, 2839 (1990)
Heller, U. M.: Monte-Carlo renormalization group investigation of the two-dimensionalO(4) sigma model. Phys. Rev. Lett.60, 2235 (1988)
Wolff, U.: Continuum behaviors in the latticeO(3) non-linear sigma model. Nucl. Phys.B334, 581 (1990)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J. Frohlich
Rights and permissions
About this article
Cite this article
Ito, K.R. Renormalization group flow of two-dimensional hierarchical Heisenberg model of Dyson-Wilson type. Commun.Math. Phys. 139, 45–70 (1991). https://doi.org/10.1007/BF02102728
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02102728