Abstract
An analysis is made of various methods of phenomenological renormalization based on finite-dimensional scaling equations for inverse correlation lengths, the singular part of the free energy density, and their derivatives. The analysis is made using two-dimensional Ising and Potts lattices and the three-dimensional Ising model. Variants of equations for the phenomenological renormalization group are obtained which ensure more rapid convergence than the conventionally used Nightingale phenomenological renormalization scheme. An estimate is obtained for the critical finite-dimensional scaling amplitude of the internal energy in the three-dimensional Ising model. It is shown that the two-dimensional Ising and Potts models contain no finite-dimensional corrections to the internal energy so that the positions of the critical points for these models can be determined exactly from solutions for strips of finite width. It is also found that for the two-dimensional Ising model the scaling finite-dimensional equation for the derivative of the inverse correlation length with respect to temperature gives the exact value of the thermal critical index.
Similar content being viewed by others
References
S. Ma, Modern Theory of Critical Phenomena (Benjamin, Reading, Mass., 1976; Mir, Moscow, 1980).
M. P. Nightingale, Physica A (Amsterdam) 83, 561 (1976).
M. P. Nightingale, Proc. K. Ned. Akad. Wet., Ser. B: Paleontol., Geol., Phys., Chem., Anthropol. 82, 235 (1979).
P. Nightingale, J. Appl. Phys. 53, 7927 (1982).
M. P. Nightingale, in Finite Size Scaling and Numerical Simulation of Statistical Systems, Ed. by V. Privman (World Scientific, Singapore, 1990), p. 287.
R. R. dos Santos and L. Sneddon, Phys. Rev. B 23, 3541 (1981).
K. Binder, Phys. Rev. Lett. 47, 693 (1981).
K. Binder, Z. Phys. B 43, 119 (1981).
M. Itakura, cond-mat/9611174.
M. Fisher, in International School of “Enrico Fermi” on Critical Phenomena, 1970, Ed. by M. S. Green (Academic, New York, 1971); F. J. Dyson, E. W. Montroll, M. Kac, and M. Fisher, Stability and Phase Transition (Mir, Moscow, 1973), p. 245.
M. E. Fisher and M. N. Barber, Phys. Rev. Lett. 28, 1516 (1972).
M. N. Barber, in Phase Transitions and Critical Phenomena, Ed. by C. Domb and J. L. Lebowitz (Academic, London, 1983), Vol. 8, p. 145.
V. Privman, in Finite Size Scaling and Numerical Simulation of Statistical Systems, Ed. by V. Privman (World Scientific, Singapore, 1990), p. 1.
W. Kinzel and M. Schick, Phys. Rev. B 23, 3435 (1981).
M. A. Yurishchev, hep-lat/9908019; submitted to Nucl. Phys. B (Proc. Suppl.) (2000).
M. A. Yurishchev, Phys. Rev. B 50, 13533 (1994).
M. A. Yurishchev, Phys. Rev. E 55, 3915 (1997).
R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic, New York, 1982; Mir, Moscow, 1985).
F. Y. Wu, Rev. Mod. Phys. 54, 235 (1982).
L. Onsager, Phys. Rev. 65, 117 (1944).
H. Saleur and B. Derrida, J. Phys. (Paris) 46, 1043 (1985).
B. Kaufman, Phys. Rev. 76, 1232 (1949).
K. Huang, Statistical Mechanics (Wiley, New York, 1963; Mir, Moscow, 1966).
R. B. Potts, Proc. Cambridge Philos. Soc. 48, 106 (1952).
R. J. Baxter, J. Phys. A 6, L445 (1973).
R. J. Baxter, Proc. R. Soc. London, Ser. A 383, 43 (1982).
J. Wosiek, Phys. Rev. B 49, 15023 (1994).
Z. Burda and J. Wosiek, Nucl. Phys. B (Proc. Suppl.) 34, 677 (1994).
A. Pelizzola, Phys. Rev. B 51, 12005 (1995).
V. Privman and M. E. Fisher, Phys. Rev. B 30, 322 (1984).
H. W. J. Blöte, L. N. Shchur, and A. L. Talapov, Int. J. Mod. Phys. C 10, 437 (1999).
K. K. Mon, Phys. Rev. B 39, 467 (1989).
M. Hasenbusch and K. Pinn, Nucl. Phys. B (Proc. Suppl.) 63, 619 (1998).
M. Hasenbusch and K. Pinn, J. Phys. A 31, 6157 (1998).
Author information
Authors and Affiliations
Additional information
__________
Translated from Zhurnal Éksperimental’no\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l}\) i Teoretichesko\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l}\) Fiziki, Vol. 118, No. 2, 2000, pp. 380–387.
Original Russian Text Copyright © 2000 by Yurishchev.
This work was partly carried out at the Landau Institute of Theoretical Physics, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia.
Rights and permissions
About this article
Cite this article
Yurishchev, M.A. Improved phenomenological renormalization schemes. J. Exp. Theor. Phys. 91, 332–337 (2000). https://doi.org/10.1134/1.1311992
Received:
Issue Date:
DOI: https://doi.org/10.1134/1.1311992