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The renormalization group and the large n limit
1.Extensive references on the subject of renormalization group, both old and new, can be found in K. G. Wilson and J. Kogut, “The Renormalization Group and the ε Expansion,” Phys. Rep. (to be published).
2.The ground work was laid out by Wilson in Phys. Rev. B 4, 3174 (1971). Further numerical investigation has been carried out by many authors.
2.See, for example, M. K. Grover, L. P. Kadanoff, and F. J. Wegner, Phys. Rev. B 6, 311 (1972);
2.G. Golner, Phys. Rev. B 8, 339 (1973).
3.K. G. Wilson and M. E. Fisher, Phys. Rev. Lett. 28, 240 (1972).
4.K. G. Wilson, Phys. Rev. Lett. 28, 548 (1972);
4.E. Brezin, D. Wallace, and K. G. Wilson, Phys. Rev. Lett. 29, 591 (1972);
4.B. G. Nickel, Phys. Rev. (to be published);
4.E. Brezin, J. C. LeGuillou, and J. Zinh‐Justin, Phys. Rev. B 8, 5530 (1973).
5.H. E. Stanley, Phys. Rev. 176, 718 (1968).
6.See, for example, R. Abe, Prog. Theor. Phys. 48, 1414 (1972);
6.R. Abe and S. Hikami, Phys. Lett. A 42, 419 (1973);
6.and R. Abe and S. Hikami, Prog. Theor. Phys. 49, 442 (1973);
6.K. G. Wilson, Phys. Rev. D 7, 2911 (1973);
6.R. A. Ferrell and D. J. Scallapino, Phys. Rev. Lett. 29, 413 (1972);
6.S. Ma, Phys. Rev. Lett. 29, 1311 (1972);
6.M. Suzuki, Phys. Lett. A 42, 5 (1972),
6.M. Suzuki, Prog. Theor. Phys. 49, 424, 1017 (1973).
7.S. Ma, Phys. Rev. A, Phys. Rev. A 7, 2172 (1973).
8.E. Brezin and D. Wallace, Phys. Rev. B 7, 1967 (1973).
9.Much of the elementary discussions here was later used (by consent of the editor) as a part of a review, S. Ma, Rev. Mod. Phys. 45, 589 (1973). To preserve continuity and completeness of the present paper, we have made no attempt to eliminate duplication of material.
10.For a review of earlier work on critical phenomena, see M. E. Fisher, Rep. Prog. Phys. 30, 615 (1967);
10.L. P. Kadanoff et al., Rev. Mod. Phys. 39, 395 (1967).
11.In other words, we assume can be expanded in a Taylor series and also assume that
12.The here differs from that defined by (4.21) by a numerical factor so that can have a simpler appearance. [See (5.29)].
13.A. Erdelyi et al., Higher Transcendental Functions, (McGraw‐Hill, New York, 1953) Vol. I, pp. 27, 30.
14.Details of calculating can be found in Ref. 7. Note the difference in cutoff.
15.F. J. Wegner, Phys. Rev. B 5, 4529 (1972).
16.For simplicity of notation, we shall not write out the component labels i explicitly.
17.The scaling law follows from (8.12). The exponent α is defined by the behavior of the specific heat near taken as
18.For d very close to 4, (8.20) holds. The second inequality should be generalized to
19.See the work of Abe and Hikami (Ref. 6) for details in this direction. See also Ref. 7, Appendix C.
20.A more general discussion can be found in S. Ma, “Scaling Variables and Dimensions” (to be published).
21.In subsequent formulas, shall mean i.e., with
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