Abstract
We employ high-temperature series to investigate a two-parameter class of renormalization group transformations for the two-dimensional Ising model on the triangular lattice. For the static case we identify an optimal organization of the high-temperature expansion and an optimal transformation matrix and thus find, in second order, ν=0.96 and the magnetic eigenvaluey=2-η/2=1.76.
From recursion relations for flip rates we find the dynamic exponent to be the same for all transformations in our two-parameter class,z=2.32.
Our fixed-point flip rates do not describe a Markov process even though the corresponding master equation for the single-event probability displays no explicit memory effects. The non-Markovian nature shows up only in a violation of the Markovian detailed balance conditions.
Similar content being viewed by others
References
Betts, D., Cuthiell, D., Plieschke, M.: Physica98A, 27 (1979)
Niemeijer, Th., Leeuwen, J.M.J. van: Physica71A, 17 (1974)
Kadanoff, L.P., Houghton, A.: Phys. Rev. B11, 377 (1975)
Deker, U., Haake, F.: Z. Phys. B-Condensed Matter36, 379 (1980)
Takano, H., Suzuki, M.: Prog. Theor. Phys.67, 1332 (1982)
The numerical result (1.2) is reasonably close to other recent results:
Tobochnik, J., Sarker, S., Cordery, R.: Phys. Rev. Lett.46, 1417 (1981);
Bausch, R., Dohm, V., Janssen, H.K., Zia, R.K.P.: Phys. Rev. Lett.47, 1837 (1981);
Chakrabarti, B.K., Baumgärtel, H.G., Stauffer, D.: Z. Phys. B-Condensed Matter44, 333 (1981);
Jan, N., Moseley, L.M., Stauffer, D.: J. Stat. Phys. (in press)
Haake, F., Lewenstein, M.: Phys. Rev. A28, 3606 (1983)
Geigenmüller, U., Titulaer, U.M., Felderhof, B.U.: Physica119A, 41 (1983);119A, 53 (1983)
Mazenko, G.F., Nolan, M.J., Valls, O.T.: Phys. Rev. B22, 1263 (1980); B22, 1275 (1980)
Achiam, Y., Kosterlitz, J.M.: Phys. Rev. Lett.41, 128 (1978);
Achiam, Y.: J. Physics A11, L129 (1978); Achiam, Y.: Phys. Rev. B19, 376 (1979)
Indekeu, J.O., Stella, A.L., Zhang, L.: Preprint 1983; see also Indekau, J.O., Stella, A.L.: Phys. Lett.78A, 160 (1980)
Haake, F., Lewenstein, M.: Phys. Rev. B27, 5868 (1983)
Glauber, R.J.: J. Math. Phys.4, 294 (1963)
Note thatR exp (Lt)\(\hat R\) is not, in general, a conditional probability. Actually, none of our arguments make use of (expLt)σ σ′ having that meaning
A simple analogy may serve to illustrate this point. Consider the everdamped harmonic oscillator,\(\dot q = p, \dot p = - \gamma p - q\), for γ≫1. The two eigenvalues are\( - \Gamma _ \pm = - (\gamma /2 \pm \sqrt {\gamma ^2 /4 - 1} )\). The ratio\(\dot q/q\) is somewhat similar in nature to the matrix quotientl(t) since the displacementq is the slow variable. The ratio\(\dot q/q\) assumes the time independent limit −ν − on the fast time scale, 1/ν +, and this limit is independent of the initial displacement as well as of the initial momentum. The asymptotic relation\(\dot q = - \Gamma _ - q\) may be understood as resulting from an adiabatic elimination of the momentum. Closer analogies exist with multidimensional stoachastic processes like the Ornstein-Uhlenbeck process [16] and the laser [17]
Haake, F.: Z. Phys. B-Condensed Matter48, 31 (1982)
Haake, F., Lewenstein, M.: Phys. Rev. A27, 1013 (1983)
Our summation convention is to count all equivalent pairs (and triples) of spins once
We could allot the configuration-independent flips of first-neighbor pairs of spins toL 1, i.e. takeψ 0 to be of first order. This modification is discussed in Sect. 9
For appropriate choices ofp andf these relations include the corresponding ones of [1, 5, 12]
See, e.g., Niemeijer, Th., Leeuwen, J.M.J., van: In: Physe transitions and critical phenomena. Domb, C., Green, M.S. (eds.). London: Academic Press 1976
It is because of the linearity of the magnetic recursion relations and because ofh * i =0 that we need not modify theh i
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Haake, F., Lewenstein, M. & Wilkens, M. Renormalization group and high-temperature series for the two-dimensional ising model. Z. Physik B - Condensed Matter 54, 333–350 (1984). https://doi.org/10.1007/BF01485831
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01485831