Abstract
We extend the self-consistent Ornstein–Zernike approximation (SCOZA), first formulated in the context of liquid-state theory, to the study of the random field Ising model. Within the replica formalism, we treat the quenched random field just as another spin variable, thereby avoiding the usual average over the random field distribution. This allows us to study the influence of the distribution on the phase diagram in finite dimensions. The thermodynamics and the correlation functions are obtained as solutions of a set a coupled partial differential equations with magnetization, temperature, and disorder strength as independent variables. A preliminary analysis based on high-temperature and 1/d series expansions shows that the theory can predict accurately the dependence of the critical temperature on disorder strength (no sharp transition, however, occurs for d≤4). For the bimodal distribution, we find a tricritical point which moves to weaker fields as the dimension is reduced. For the Gaussian distribution, a tricritical point may appear for d around 4.
Similar content being viewed by others
REFERENCES
T. Natterman and J. Villain, Phase Transition 11:5 (1988).
D. P. Belanger and A. P. Young, J. Magn. Magn. Mater 100:272 (1991).
T. Natterman, in Spin Glasses and Random Fields, A. P. Young, ed. (World Scientific, Singapore, 1997).
T. Schneider and E. Pytte, Phys. Rev. B 15:1519 (1977).
A. Aharony, Phys. Rev. B 18:3318 (1978).
A. P. Young and M. Nauenberg, Phys. Rev. Lett. 54:2429 (1985).
A. T. Ogielski and D. A. Huse, Phys. Rev. Lett. 56:1298 (1986).
H. Rieger and A. P. Young, J. Phys. A 26:5297 (1993).
H. Rieger, Phys. Rev. B 52:6659 (1995).
Y. Shapir and A. Aharony, J. Phys. C 15:1361 (1982).
A. Khurana, F. J. Seco, and A. Houghton, Phys. Rev. Lett. 54:357 (1985); A. Houghton, A. Khurana, and F. J. Seco, Phys. Rev. Lett. 55:856 (1985); Phys. Rev. B 34:1700 (1986).
M. Gofman, J. Adler, A. Aharony, A. B. Harris, and M. Schwartz, Phys. Rev. Lett. 71:1569 (1993), Phys. Rev. B 53:6362 (1996).
M. R. Swift, A. J. Bray, A. Maritan, M. Cieplack, and J. R. Banavar, Europhys. Lett. 38:273 (1997).
J.-C. Anglès d'Auriac and N. Sourlas, Europhys. Lett. 39:473 (1997).
J. R. L. de Almeida, and R. Bruinsma, Phys. Rev. B 35:7267 (1987).
M. Mézard and A. P. Young, Europhys. Lett. 18:653 (1992).
M. Mézard and R. Monasson, Phys. Rev. B 50:7199 (1994).
C. De Dominicis, H. Orland, and T. Temesvari, J. Physique I 5:987 (1996).
V. Dotsenko and M. Mézard, preprint cond-mat/9611017.
M. Cagnelli, E. Marinari, and G. Parisi, J. Phys. A 28:3359 (1995).
A. P. Young, J. Phys. A 10:L257 (1977).
G. Parisi and N. Sourlas, Phys. Rev. Lett. 43:744 (1979).
J. S. Hoye and G. Stell, J. Chem. Phys. 67:439 (1977); Mol. Phys. 52:1071 (1984); Int. J. Thermophys. 6:561 (1985).
R. Dickman and G. Stell, Phys. Rev. Lett. 77:996 (1996); D. Pini, G. Stell, and R. Dickman, Phys. Rev. E 57:2862 (1998).
E. Kierlik, M. L. Rosinberg, and G. Tarjus, J. Stat. Phys. 89:215 (1997).
E. Kierlik, M. L. Rosinberg, and G. Tarjus, in preparation.
W. G. Madden and E. D. Glandt, Stat. Phys. 51:537 (1988).
J. A. Given, Phys. Rev. A 45:816 (1992).
J. A. Given and G. Stell, J. Chem. Phys. 97:4573 (1992).
M. L. Rosinberg, G. Tarjus, and G. Stell, J. Chem. Phys. 100:5172 (1994).
G. Stell, Phys. Rev. 184:135 (1969).
E. Pitard, M. L. Rosinberg, G. Stell, and G. Tarjus, Phys. Rev. Lett. 74:4361 (1995).
E. Pitard, M. L. Rosinberg, and G. Tarjus, Molecular Simulation 17:339 (1996).
J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, New York, 1976).
G. S. Joyce, in Phase Transitions and Critical Phenomena, C. Domb and M. S. Green (Academic Press, London, 1972), Vol. 2. p. 375.
H. C. Andersen and D. Chandler, J. Chem. Phys. 57:1918 (1972).
J. L. Lebowitz and J. K. Percus, Phys. Rev. 144:251 (1966).
P. Lacour-Gayet and G. Toulouse, J. Phys. (Paris) 35:425 (1974).
J. Adler, in Annual Review of Computational Physics, D. Stauffer, ed., Vol. IV, p. 241 (1996).
A. Maritan, M. R. Swift, M. Cieplak, M. H. W. Chan, M. W. Cole, and J. R. Banavar, Phys. Rev. Lett. 67:1821 (1991).
J. S. Hoye and A. Borge, preprint.
M. Schwartz and A. Soffer, Phys. Rev. Lett. 55:2499 (1985); Phys. Rev. B 33:2059 (1986).
M. Schwartz, M. Gofman, and T. Nattermann, Physica A 178:6 (1991).
See, e.g., J. M. Luck, Systèmes désordonnés unidimensionnels, Collection Aléa Saclay (1992).
J. S. Hoye and G. Stell, Physica A 244:176 (1997).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kierlik, E., Rosinberg, M.L. & Tarjus, G. A Self-Consistent Ornstein–Zernike Approximation for the Random Field Ising Model. Journal of Statistical Physics 94, 805–836 (1999). https://doi.org/10.1023/A:1004526931714
Issue Date:
DOI: https://doi.org/10.1023/A:1004526931714