Abstract
We construct the most general real space renormalization for the two-dimensional kinetic Ising model on the triangular lattice which, to second order in the high-temperature expansion, conserves detailed balance and avoids fast transients for the cell spins. We show the corresponding dynamical recursion relations (as well as the exponentz) to be unaltered with respect to the ones found, in a previous paper, for a completely different class of transformations. This finding resolves long-standing confusions and controversies.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Haake, F., Lewenstein, M., Wilkens, M.: Z. Phys. B — Condensed Matter54, 333 (1984)
Barred variables are to be summed over all possible configurations
v is the correlation length exponent
The violation of detailed balance can be understood as an initial slip; see, e.g.,
Haake, F., Lewenstein, M.: Phys. Rev. A28, 3606 (1983);
Geigenmüller, U., Titulaer, U.M., Felderhof, B.U.: Physica119A, 41 (1983);119A, 53 (1953)
Graham, R.: Springer Tracts in Modern Physics. Vol. 66. Berlin, Heidelberg, New York: Springer 1973
Kampen, N.G. van: Stochastic processes in physics and chemistry. Amsterdam: North Holland 1981
Mazenko, G.F.: Preprint 1983
Mazenko, G.F., Nolan, M.J., Valls, O.T.: Phys. Rev. B22, 1263 (1980); B22, 1275 (1980)
In (2.6) and several formulas below we write matrix products without displaying indices
The corresponding result forr which we shall not need can be obtained from (2.7) by first replacing all matricesL (0),L (1), andQ by their transposes and then transposing the full expression (2.7)
Betts, D., Cuthiell, D., Plieschke, M.: Physics98A, 27 (1979)
Our site summations count all equivalent pairsij (and triplesijk) once; the sum\(\sum\limits_{ij,kl}^{1n,1n} {\sigma _i \sigma _j \sigma _k \sigma _l } \) is that part of\(\left( {\sum\limits_{ij}^{1n} {\sigma _i \sigma _j } } \right)^2 \) in which the two bondsij andkl don't overlap
IfL is to be written as a matrix the rules\(1_{\sigma \sigma '} \tfrac{1}{2}(1 + \sigma \sigma '),\sigma _{\sigma \sigma '} = \tfrac{1}{2}(\sigma + \sigma '),A_{\sigma \sigma '} = \tfrac{1}{2}\sigma \sigma '\), and\((A\sigma )_{\sigma \sigma '} = \tfrac{1}{2}\sigma \) can be used [1] site per site
Takano, H., Suzuki, M.: Prog. Theor. Phys.67, 1332 (1982)
There is one more solution in which the four-spin term in (7.13) is replaced by μν μλ μρ μσ − μν′ μλ′ μρ′ μσ′ this solution cannot be excluded on symmetry grounds. We must rather reject it because it would entail a nonlocal generatorL″
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Haake, F., Lewenstein, M. & Wilkens, M. The irrelevance of detailed balance for the dynamical critical exponent. Z. Physik B - Condensed Matter 55, 211–218 (1984). https://doi.org/10.1007/BF01329013
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01329013