Abstract
We study a classical spin model (more precisely a class of models) with O(N) symmetry that can be viewed as a simplified D dimensional lattice model. It is equivalent to a non-translationinvariant one dimensional model and contains the dimensionality D as a parameter that need not be an integer. The critical dimension turns out to be 2, just as in the usual translation invariant models. We study the phase structure, critical phenomena and spontaneous symmetry breaking. Furthermore we compute the perturbation expansion to low order with various boundary conditions. In our simplified models a number of questions can be answered that remain controversial in the translation invariant models, such as the asymptoticity of the perturbation expansion and the role of super-instantons. We find that perturbation theory produces the right asymptotic expansion in dimension D≤2 only with special boundary conditions. Finally the model allows a test of the percolation ideas of Patrascioiu and Seiler.
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REFERENCES
R. L. Dobrushin and S. B. Shlosman, Commun. Math. Phys. 42:31(1975).
A. Patrascioiu and E. Seiler, J. Statist. Phys. 69:573(1992).
H.-O. Georgii, Gibbs Measures and Phase Transitions, Studies in Mathematics, Vol. 9 (Walter de Gruyter Verlag, Berlin, 1988).
Y. Brihaye and P. Rossi, Nucl. Phys. B 235:226(1984).
A. Patrascioiu, Phys. Rev. Lett. 54:2292(1985).
R. L. Dobrushin, Funct. Anal. Appl. 2:31(1968); R. L. Dobrushin, Funct. Anal. Appl. 3:27(1969); O. E. Lanford and D. Ruelle, Commun. Math. Phys. 13:194(1968).
B. Simon, The Statistical Mechanics of Lattice Gases, Vol. I (Princeton University Press, Princeton, N.J., 1995).
A. Patrascioiu and E. Seiler, Nucl. Phys. B (Proc. Suppl.) 30:184(1993).
A. Patrascioiu and E. Seiler, J. Statist. Phys. 106:811(2002).
K. Yildirim, Kritische Eigenschaften eines Quasi-D-Dimensionalen, Nicht-Translationsinvarianten Spin-Modells, Doctoral Dissertation (Technical University Munich, 1998).
A. Patrascioiu and E. Seiler, Phys. Rev. Lett. 74:1920(1995).
N. D. Mermin, J. Math. Phys. 8:1061(1967); N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17:1133(1966).
A. Cucchieri, T. Mendes, A. Pelissetto, and A. Sokal, J. Statist. Phys. 86:581(1997).
E. Seiler and K. Yildirim, J. Math. Phys. 38:4872(1997).
W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, Berlin, 1966).
J. N. Vilenkin, Special Functions and the Theory of Group Representations, AMS translations, Vol. 22 (Providence, R.I., 1968).
M. Reed and B. Simon, Methods of Modern Mathematical Phyics, Vol. IV (Academic Press, New York etc., 1978).
E. Brézin and J. Zinn-Justin, Phys. Rev. B 14:3110(1976).
J. Bricmont, J.-R. Fontaine, J. L. Lebowitz, E. H. Lieb, and T. Spencer, Commun. Math. Phys. 78:545(1981).
P. Hasenfratz, Phys. Lett. B 141:385(1984).
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Seiler, E., Yildirim, K. Critical Behavior in a Quasi D Dimensional Spin Model. Journal of Statistical Physics 112, 457–495 (2003). https://doi.org/10.1023/A:1023815706708
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DOI: https://doi.org/10.1023/A:1023815706708