Abstract
We investigate finite lattice approximations to the Wilson renormalization group in models of unconstrained spins. We discuss first the properties of the renormalization group transformation (RGT) that control the accuracy of this type of approximation and explain different methods and techniques to practically identify them. We also discuss how to determine the anomalous dimension of the field. We apply our considerations to a linear sigma model in two dimensions in the domain of attraction of the Ising fixed point using a Bell–Wilson RGT. We are able to identify optimal RGTs which allow accurate computations of quantities such as critical exponents, fixed-point couplings and eigenvectors with modest statistics. We finally discuss the advantages and limitations of this type of approach.
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REFERENCES
K. G. Wilson and J. Kogut, Phys. Rep. 12:75 (1974).
R. H. Swendsen, Phys. Rev. Lett. 42:829 (1979).
A. Gonzalez-Arroyo and M. Okawa, Phys. Rev. D 35:672 (1987).
T. L. Bell and K. G. Wilson, Phys. Rev. B 11:3431 (1975).
P. Hasenfratz and F. Niedermayer, Nucl. Phys. B 414:785 (1994).
A. Travesset, Ph.D. thesis, Universitat de Barcelona (1997), unpublished.
R. H. Swendsen, Phys. Rev. Lett. 52:2321 (1984).
A. Hasenfratz et al., Phys. Lett. B 140:76 (1984).
H. Shenker and J. Tobochnik, Phys. Rev. B 22:4462 (1980); J. Hirsch and H. Shenker, Phys. Rev. B 27:1736 (1983).
F. Wegner, in Phase Transitions and Critical Phenomena, C. Domb and M. S. Green, eds. (1987), Vol. 6, p. 7.
A. Travesset, Nucl. Phys. B 63A-C (Proc. Suppl.):640 (1998).
G. Murthy and R. Shankar, Phys. Rev. B 32:5851 (1985).
A. Gocksch and U. M. Heller, Nucl. Phys. B 275:219 (1986).
R. Brower and P. Tamayo, Phys. Rev. Lett. 62:1087 (1989).
U. Wolff, Phys. Rev. Lett. 62:361 (1989).
D. Espriu and A. Travesset, Phys. Lett. B 356:329 (1995).
G. S. Pawley et al., Phys Rev. B 29:4030 (1984).
C. Baillie et al., Phys. Rev. B 45:10438 (1992).
R. Gupta and P. Tamayo, cond-mat 9601048.
R. Guidaa and J. Zinn-Justin, cond-mat 9803240.
Baker and J. M. Kincaid, J. Stat. Phys. 24:469 (1981).
D. S. Gaunt and A. J. Guttmann, in Phase Transitions and Critical Phenomena, C. Domb and M. S. Green, eds. (1974), Vol. 3.
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Cacciuto, A., Gregory, E. & Travesset, A. The Renormalization Group and Its Finite Lattice Approximations. Journal of Statistical Physics 97, 541–574 (1999). https://doi.org/10.1023/A:1004659124079
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DOI: https://doi.org/10.1023/A:1004659124079