Abstract
We compute hierarchical renormalization-group fixed points as solutions to an algebraic equation for the coupling constants. This method does not rely on an iteration of renormalization-group transformations and therefore avoids the problem of fine tuning. We solve truncated versions of the fixed-point equation numerically for different values of the dimension parameter in the range 2<d<4 and different orders of truncations. The method is well suited even for multicritical fixed points with any number of unstable directions. Precise numerical data are presented for the first three nontrivial fixed points and their critical indices. We also develop an ε-expansion for the hierarchical models using computer algebra. The numerical results are compared with the ε-expansion.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K. G. Wilson,Physica 73:119 (1974); K. G. Wilson and J. Kougut,Phys. Rep. C 12:12 (1974).
F. J. Dyson,Commun. Math. Phys. 12:91 (1969); K. Wilson,Phys. Rev. B 4:3184 (1971); G. A. Baker,Phys. Rev. B 5:2622 (1972); P. Collet and J.-P. Eckmann, A renormalization group analysis of the hierarchical model in statistical mechanics, inLecture Notes in Physics, Vol. 74. (Springer, Berlin, 1978); P. M. Bleher and J. G. Siani,Commun. Math. Phys. 33:23 (1973); H. Kunz and G. Zumbach,J. Phys. A. 23:999 (1990); K. Pinn and C. Wieczerkowski,Int. J. Mod. Phys. A. 6:463 (1991); A. Pordt,Helv. Phys. Acta 66:105 (1993).
G. R. Golner,Phys. Rev. B 8:339 (1973).
J. Glimm and A. Jaffe,Quantum Physics, 2nd ed. (Springer, Berlin, 1987); K. Gawedzki and A. Kupiainen,Commun. Math. Phys. 99:197 (1985); G. Gallavotti,Rev. Mod. Phys. 57:471 (1985); A. Pordt and G. Mack,Commun. Math. Phys. 97:267 (1985); J. Feldman, J. Magnen, V. Rivasseau, and R. Sénéor,Commun. Math. Phys. 109:137 (1987); D. Brydges and A. Yau,Commun. Math. Phys. 129:351 (1990).
H. Koch and P. Wittwer,Commun. Math. Phys. 106:495 (1986);138:537 (1991).
A. Pordt and C. Wieczerkowski, Preprint MS-TPI-94-04 (May 1994).
G. Felder,Commun. Math. Phys. 111:101 (1987).
P. Collet and J.-P. Eckmann,Commun. Math. Phys. 55:67 (1977).
Numerical Algorithms Group Ltd., Mark 15 ed. (1991).
M. Le Bellac,Quantum and Statistical Field Theory (Clarendon Press, Oxford, 1991).
C. F. Baillie, R. Gupta, K. A. Hawick, and G. S. Pawley,Phys. Rev. B 45:10438 (1992); J.-C. Le Guillou and J. Zinn-Justin,Phys. Rev. B 21:3976 (1980); J. Zinn-Justin, inPhase Transitions, M. Lévyet al., eds. (Plenum Press, New York, 1982).
J. Zinn-Justin,Quantum Field Theory and Critical Phenomena (Clarendon Press, Oxford, 1989).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pinn, K., Pordt, A. & Wieczerkowski, C. Computation of hierarchical renormalization-group fixed points and their ε-expansions. J Stat Phys 77, 977–1005 (1994). https://doi.org/10.1007/BF02183147
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02183147