Abstract
Critical exponents of weakly dilute Ising-like systems are computed for non-integer space dimensionalities in the range 2</d⩽4. The calculations are performed in the framework of the Callan-Symanzik field-theoretic approach. Two-loop renormalization group functions are obtained as renormalized perturbation theory series expansions directly in noninteger dimensions. The values of the critical exponents are estimated with the use of the two-variable Borel resummation method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Harris, M. Plischke, and M. J. Zuckermann, New model for amorphous magnetism,Phys. Rev. Lett. 31:160 (1973).
A. B. Harris, R. G. Caflisch, and J. R. Banavar, Random-anisotropy-axis magnet with infinite anisotropy,Phys. Rev. B 35:4929 (1987).
K. E. Newman and E. K. Riedel, CubicN-vector model and randomly dilute Ising model in general dimensions,Phys. Rev. B 25:264 (1982).
J. C. Le Guillou and J. Zinn-Justin, Accurate critical exponents from theɛ-expansions,J. Phys. Lett. 46:L-137 (1985).
J. C. Le Guillou and J. Zinn-Justin, Accurate critical exponents for Ising like systems in non-integer dimensions,J. Phys. (Paris)48:19 (1987).
R. Brout, Statistical mechanical theory of a random ferromagnetic system,Phys. Rev. 115:824 (1959).
G. Grinstein and A. Luther, Application of the renormalization group to phase transitions in disordered systems,Phys. Rev. B 13:1329 (1976).
A. B. Harris, Effect of random defects on the critical behavior of Ising models,J. Phys. C 7:1671 (1974).
A. B. Harris and T. C. Lubensky, Renormalization-group approach to the critical behaviour of random-spin models,Phys. Rev. Lett. 33:1540 (1974).
T. C. Lubensky, Critical properties of random-spin models from theɛ-expansion,Phys. Rev. B 11:3573 (1975).
D. E. Khmelnitskii, Phase transition of the second kind in inhomogeneous bodies,Zh. Eksp. Teor. Fiz. 68:1960 (1975).
C. Jayaprakash and H. J. Katz, Higher-order corrections to theɛ 1/2-expansion of the critical behavior of the random Ising system,Phys. Rev. B 66:3987 (1977).
B. N. Shalaev, Phase transition in a weakly disordered uniaxial ferromagnet,Zh. Eksp. Teor. Fiz. 73:3201 (1977).
K. De'Bell and D. J. W. Geldart, Coefficients toO(ɛ 3) of the mixed fixed point of themn-component field model,Phys. Rev. B 32:4763 (1985).
G. Parisi, Field-theoretic approach to second-order phase transitions in two- and three-dimensional systems,J. Stat. Phys. 23:49 (1980).
G. Jug, Critical behaviour of disordered spin systems in two and three dimensions,Phys. Rev. B 27:609 (1983).
I. O. Mayer and A. I. Sokolov, Critical indices of impure Ising model,Fiz. Tverd. Tela 26:3454 (1984); On the critical behaviour of cubic crystals undergoing structural phase transitions,Izv. Akad. Nauk SSSR 51:2103 (1987).
N. A. Shpot, Critical behaviour of themn-component field model in three dimensions. II. Three-loop results;Phys. Lett. A 142:474 (1989); On the critical behaviour of themn-component field model in three dimensions: Three-loop approximation, Preprint ITP-88-140R, Kiev (1988).
I. O. Mayer, A. I. Sokolov, and B. N. Shalaev, Critical exponents for cubic and impure uniaxial crystals: Most accurate theoretical values,Ferroeleclricity 95:93 (1989); I. O. Mayer, Critical exponents of the dilute Ising model from four-loop expansions,J. Phys. A 22:2815 (1989).
B. N. Shalaev, Phase transition in two-dimensional ferromagnet with cubic anisotropy,Fiz. Tverd. Tela 30:895 (1988); Correlation function and susceptibility of two-dimensional ferromagnet with cubic anisotropy,Fiz. Tverd. Tela 31:93 (1989).
Yu. V. Holovatch, Phase transition in the anisotropic cubic model,Ferroelect. Lett. 8:11 (1987); Critical behaviour of a model with non-spherical symmetry, Preprint ITP-86-143E, Kiev (1986).
Yu. V. Holovatch, Critical temperature of the site dilute model by the approximate renormalization group approach,Ferroelect. Lett. 11:111 (1990); Pecularities of the critical behaviour of the dilutem-vector model in percolation region, Preprint 88-153E, Kiev (1988).
Yu. V. Holovatch and N. A. Shpot, Critical indices of randomm-vector model by approximate renormalization group approach,Acta Phys. Pol. A 78:369 (1990); Approximate renormalization group transformation for the dilutem-vector model, Preprint ITP-87-39E, Kiev (1987).
E. Brézin, J. C. Le Guillou, and J. Zinn-Justin, Field theoretical approach to critical phenomena, inPhase Transitions and Critical Phenomena, Vol. VI, C. Domb and M. S. Green, eds. (Academic Press, New York, 1976); D. J. Amit,Field Theory, The Renormalization Group and Critical Phenomena (McGraw-Hill, New York, 1978).
N. A. Shpot, Critical behaviour of themn-component field model in three dimensions,Phys. Lett. A 133:125 (1988).
J. S. R. Chisholm, Rational approximants defined from double power series,Math. Comput. 27:841 (1973).
M. Abramowitz and A. I. Stegun, eds.,Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (National Bureau of Standards, 1964).
E. Brézin, J. C. Le Guillou, and J. Zinn-Justin, Critical exponents for then-vector model in three dimensions from field theory,Phys. Rev. Lett. 39:95 (1977).
Yu. Holovatch and N. Shpot, Critical behaviour of dilute Ising-like systems in non-integer dimensions, Preprint ITP-90-48E, Kiev (1990).
S. G. Gorishny, S. A. Larin and F. V. Tkachov,ɛ-Expansion for critical exponents: TheO(ɛ 5) approximation,Phys. Lett. A 101:120 (1984).
C. Bervillier, Estimate of a universal critical amplitude ratio from itsɛ-expansion up toɛ 2,Phys. Rev. B 34:8141 (1986).
N. A. Shpot, Equation of state and universal combinations of thermodynamic critical amplitudes of dilute Ising model,Zh. Eksp. Teor. Fiz. 98:1762 (1990).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Holovatch, Y., Shpot, M. Critical exponents of random Ising-like systems in general dimensions. J Stat Phys 66, 867–883 (1992). https://doi.org/10.1007/BF01055706
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01055706