Abstract
There has been a surprising amount of theoretical and experimental interest recently in one of the simplest problems in the area of critical dynamics. The systems under scrutiny are two-dimensional Ising-like systems with a nonconserved order parameter. I have chosen to focus on this problem in these lectures for three reasons:
-
(i)
These systems appear superficially to be very simple. There is only one dynamical process that is important for long times and we understand the equilibrium properties in considerable detail.
-
(ii)
There exist natural examples of such systems (usually two-dimensional Ising antiferromagnets) and there are some interesting recent experiments on such systems which have been somewhat provocative.
-
(iii)
Essentially every known calculational method for critical dynamics has been applied to this problem. Thus we can use this system to review conveniently and compare these various methods.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. Glauber, J. Math. Phys. (N.Y.) 2, 263 (1966).
K. Kawasaki, in Phase Transitions and Critical Phenomena, ed. by C. Domb and M.S. Green (Academic, N.Y., 1972), Vol. 2.
See for example the review by J. Gunton, M. San Miguel, and P. Sahni, to be published.
In adsorbed systems one must allow for the observed n x m orderings. This requires studying a model with more structure than the nearest-neighbor Ising model. See for example P.S. Sahni and J.D. Gunton, Phys. Rev. Lett. 47 1754 (1981) or E. Oguz, Preprint.
B. Halperin, P. Hohenberg, and S. Ma, Phys. Rev. B10, 139 (1974).
See, for example, Monte Carlo Methods in Statistical Physics, ed. by K. Binder, Springer-Verlag, Berlin (1979).
See, for example, the discussion of G.F. Mazenko in Correlation Functions and Quasiparticie Interactions, ed. by J.W. Halley (Plenum, N.Y., 1978).
G.F. Mazenko and O.T. Valls, Phys. Rev. B24, 1419 (1981).
See, for example, P.C. Hohenberg and B.I. Hal perin, Rev. Mod. P-ys. 49, 435 (1977).
S. De Dominicis, E. Brezin, and J. Zinn-Justin, Phys. Rev. B12, 4945 (1975).
R. Bausch, V. Dohm, H.K. Janssen, and R.K.P. Zia, Phys. Rev. Lett. 47, 1837 (1981).
H. Yahata and M. Suzuki, J. Phys. Soc. Jpn. 27, 1421 (1969).
H. Yahata, J. Phys. Soc. Jpn. 30, 657 (1971).
Z. Racz and M.F. Collins, Phys. Rev. B13, 3074 (1976).
E. Stoll, K. Binder, and T. Schneider, Phys. Rev. B8, 3266 (1973).
M.C. Yalabik and J.D. Gunton, Prog. Theor. Phys. 62, 1573 (1979).
M.P. Nightengale and H.W.J. Blöte, Physica 104A, 352 (1980).
R. Pandit, G. Forgacs, and P. Rujan, Phys. Rev. B24, 1576 (1981).
J.C. d’Auriac, R. Maynard, and R. Rammal, J. Stat. Phys. 28, 307 (1982).
H. Takano, Prog. Theo. Phys. 68, 493 (1982).
Y. Achaim and M. Kosterlitz, Phy. Rev. Lett. 41, 128 (1978).
G.F. Mazenko, M. Nolan, O.T. Valls, Phys. Rev. Lett. 41, 500 (1978).
W. Kinzel, Z. Phys. B29, 361 (1978).
Y. Achaim, J. Phys. A11, L129 (1978).
M. Suzuki, K. Sogo, I. Matsuba, H. Ikada, T. Chikama, and H. Takano, Prog. Theor. Phys. 61, 864 (1979).
S.T. Chui, G. Forgacs, and H.L. Frisch, Phys. Rev. B20, 243 (1979).
M. Droz, Phys. Lett. 73A, 407 (1979).
M. Suzuki, in Dynamical Critical Phenomena and Related Topics, ed. by C.P. Enz, Springer-Verlag, Berlin (1979).
G.F. Mazenko, in Dynamical Critical Phenomena and Related Topics, ed. by C.P. Enz, Springer-Verlag, Berlin (1979).
S. Ma, Phys. Rev. B19, 4824 (1979).
M. Droz and A. Malaspinas, J. Phys. C13. 4365 (1980).
M. Droz and A. Malaspinas, Helv. Phys. Acta. 53, 214 (1980).
J.O. Indekeu and A.L. Stella, Phy. Lett. 78A, 160 (1980).
G.F. Mazenko, M. Nolan and O.T. Valls, Phys. Rev. B22, 1275 (1980).
U. Deker and F. Haake, Z. Phys. B36, 379 (1980).
A. Stella and R. Dekeyser, J. Stat. Phys. 25, 443 (1981).
G.F. Mazenko and O.T. Valls, Phys. Rev. B24, 443 (1981).
G.F. Mazenko and O.T. Valls, in Real Space Renormalization, ed. by J.M.J. van Leeuwen and T.N. Burkhardt, (Springer-Verlag, Berlin, 1982).
H. Takano and M. Suzuki, Prog. Theor. Phys. 67, 1322 (1982).
F. Haake and M. Lewenstein, Phys. Rev. B27, 5868 (1983).
I will follow here rather closely the arguments of G.F. Mazenko and O.T. Valls, preprint.
T. Neimeijer and J.M.J. van Leeuwen, in Phase Transitions and Critical Phenomena, ed. by C. Domb and M.S. Green (Academic, N.Y., 1979) Vol. 6.
J. Tobochnik, S. Sarker, and R. Cordery, Phys. Rev. Lett. 46, 1417 (1981).
S.L. Katz, J.D. Gunton, Phys. Rev. B25, 6008 (1982).
M.C. Yalabik and J.D. Gunton, Phys. Rev. B25, 534 (1982).
R.B. Stinchcombe, Phys. Rev. Lett. 50, 200 (1983).
Possible discrepancies are discussed in: G.A. Baker, Jr., Phys. Rev. B15, 1552 (1977); B.G. Nickel and B. Sharpe J. Phys. A. Math. Gen. 12, 1819 (1979); and B. Nickel, Physica, 106A, 48 (1981).
F. Wegner in Phase Transitions and Critical Phenomena, ed. by C. Domb and M.S. Green, (Academic, N.Y. 1976), Vol. 6.
The high-temperature expansion of the operator Q for the “minimal coupling” operator, (2.25), begins with 1 + 4u4/3 + ….
One unsettling aspect of the treatment in this paper is the introduction of a flipping rate \(\widetilde\alpha \sim \alpha \;(1 - 3{{\rm{u}}^2})\) and the writing of the recursion relation in terms of \(\widetilde\alpha\) rather than α′. Using \({\rm{z}} = - \ell {\rm{n(}}\tilde \alpha '/\tilde \alpha )/\ell {\rm{nb}}\), they obtain z = 2.23, but if one uses \({\rm{z}} = - \ell {\rm{n(}}\alpha '/\tilde \alpha )/\ell {\rm{nb}}\), it looks as if one finds a significant change in z.
The parameter p of Ref. (40) is closely related to the parameter φ in Ref. (35). It was concluded there that the existence of this parameter invalidated the whole procedure, while in Ref. (40) it was decided that an intelligent choice of φ makes the method acceptable. In our own method (Refs. 37, 38) we have no such free parameter.
G.F. Mazenko and J. Luscombe, Ann. Phys. (N.Y.) 132, 121 (1981).
G.F. Mazenko, J. Hirsch, M. Nolan, and O.T. Valls, Phys. Rev. Lett. 44, 1083 (1980).
M. Fisher and R. Burford, Phys. Rev. 156, 583 (1967).
S. Ma, Phys. Rev. Lett. 37, 461 (1976).
J. Bartel and M.J. Nolan, Bull. Am. Phy. Soc. 28, 265 (1983).
M.T. Hutchings, H. Ikeda, and E. Janke, Phys. Rev. Lett. 49, 386 (1982).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1983 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mazenko, G.F. (1983). Critical Dynamics in Simple Ising-Like Systems. In: Ausloos, M., Elliott, R.J. (eds) Magnetic Phase Transitions. Springer Series in Solid-State Sciences, vol 48. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-82138-7_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-82138-7_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-82140-0
Online ISBN: 978-3-642-82138-7
eBook Packages: Springer Book Archive