Abstract
In the Local Potential Approximation, renormalization group equations reduce to a semilinear parabolic partial differential equation. Felder [8] has derived this equation and has constructed a family of non-trivial fixed pointsu *2n (n=2,3,4,...) which have the form ofn-well potentials and exist in the ranges of dimensions 2<d<2+2/n−1. In this paper we show that ifd≧4, then these non-trivial fixed points disappear, and if 3≦d<4 then we have only theu *4 fixed point.
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Wilson, K.G.: Renormalization group and critical phenomena I, II. Phys. Rev.B4, 3174–3183 (1971)
Wilson, K.G., Kogut, J.B.: The renormalization group and the ɛ-expansion. Phys. Rep.12, 76–199 (1974)
Wilson, K.G., Fisher, M.E.: Critical exponents in 3.99 dimensions. Phys. Rev. Lett.28, 240–243 (1972)
Dyson, F.: Existence of a phase transition in one-dimensional Ising ferromagnetic. Commun. Math. Phys.12, 91–107 (1969); An Ising ferromagnetic with discontinuous long range order. Commun. Math. Phys.21, 269–283 (1971)
Bleher, P.M. and Sinai, Ya.G.: Investigation of the critical point in models of the type Dyson's hierarchical model. Commun. Math. Phys.33, 23–42 (1973); Collet, P., Eckmann, J.P.: The ɛ-expansion for the hierarchical model. Commun. Math. Phys.55, 67–96 (1977); A renormalization group analysis of the hierarchical model in statistical mechanics. Lecture Notes in Physics, Vol.74, Berlin, Heidelberg, New York: Springer, 1978
Gawedzki, K., Kupiainen, A.: Renormalization group for a critical lattice model. Commun. Math. Phys.88, 77–94 (1983)
Aizenman, M.: Geometric analysis of φ4 fields and Ising models. Part I and II. Commun. Math. Phys.86, 1–48 (1982)
Felder, G.: Renormalization group in the local potential approximation. Commun. Math. Phys.111, 101–121 (1987)
Lima, P.C.: Ph.D. thesis, Courant Institute of Mathematical Sciences, N.Y.U., 1990
Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. Robert E. Krieger Publishing Company, 1955, Theorem 1.1, p 208
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Communicated by G. Felder
Research supported by CNPq, Brazil
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Lima, P.C. Renormalization group fixed points in the local potential approximation ford≧3. Commun.Math. Phys. 170, 529–539 (1995). https://doi.org/10.1007/BF02099148
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DOI: https://doi.org/10.1007/BF02099148