# A fixed-point equation for the high-temperature phase of discrete lattice spin systems

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## Abstract

A fixed-point equation on an infinite-dimensional space is proposed as an alternative to the usual definition of the infinite-volume limit in discrete lattice spin systems in the high-temperature phase. It is argued heuristically that the free energy and correlation functions one obtains by solving this equation agree with the usual definitions of these quantities. A theorem is then proved that says that if a certain finite-volume condition is satisfied, then this fixed-point equation has a solution and the resulting free energy is analytic in the parameters in the Hamiltonian. For particular values of the temperature this finite-volume condition may be checked with the help of a computer. The two-dimensional Ising model is considered as a test case, and it is shown that the finite-volume condition is satisfied for*β⩽0.77β*_{critical}.

## Key words

Finite volume condition high temperature phase lattice spin system## Preview

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## References

- 1.I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Rigorous results on valence-bond ground states in antiferromagnets,
*Phys. Rev. Lett.***59**:799–802 (1987); Valence-bond ground states in isotropic quantum antiferromagnets,*Commun. Math. Phys.***115**:477–528 (1988).Google Scholar - 2.A. M. Aizenman, Rigorous studies of critical behavior I and II, in
*Applications of Field Theory in Statistical Mechanics*, L. Garrido, ed. (Springer-Verlag, 1984), in*Statistical Physics and Dynamical Systems: Rigorous Results*, D. Szesz and E. Retz, eds. (Birkhauser, 1984);*Commun. Math. Phys.***86**:1 (1982).Google Scholar - 3.D. P. Arovas, A. Auerbach, and F. D. M. Haldane, Extended Heisenberg models of antiferromagnetism: Analogies to the fractional quantum Hall effect,
*Phys. Rev. Lett.***60**:531–534 (1988).Google Scholar - 4.K. Decker, A. Hasenfratz, and P. Hasenfratz, Singular renormalization group transformations and first order phase transitions (II),
*Nucl. Phys. B***295**:21–35 (1988).Google Scholar - 5.R. L. Dobrushin and S. B. Shlosman, Completely analytic Gibbs fields, in
*Statistical Physics and Dynamical Systems*(Birkhauser, 1985); Constructive criterion for the uniqueness of Gibbs field, in*Statistical Physics and Dynamical Systems*(Birkhauser, 1985); Completely analytical interactions: Constructive description,*J. Stat. Phys.***46**:983 (1987).Google Scholar - 6.R. L. Dobrushin, J. Kolafa, and S. B. Shlosman, Phase diagram of the two-dimensional Ising antiferromagnet,
*Commun. Math. Phys.***102**:89 (1985).Google Scholar - 7.T. C. Dorlas and A. C. D. van Enter, Non-Gibbsian limit for large block majority-spin transformations,
*J. Stat. Phys.***55**:171–182 (1989).Google Scholar - 8.R. B. Griffiths and P. A. Pearce, Position-space renormalization-group transformations: Some proofs and some problems,
*Phys. Rev. Lett.***41**:917–920 (1978); Mathematical properties of position-space renormalization-group transformations,*J. Stat. Phys.***20**:499–545 (1979).Google Scholar - 9.R. B. Griffiths, Mathematical properties of renormalization group transformations,
*Physica***106A**:59–69 (1981).Google Scholar - 10.A. Hasenfratz and P. Hasenfratz, Singular renormalization group transformations and first order phase transitions (I),
*Nucl. Phys. B***295**:1–20 (1988).Google Scholar - 11.T. Kennedy, E. H. Lieb, and H. Tasaki, A two dimensional isotropic quantum antiferromagnet with unique disordered ground state,
*J. Stat. Phys.***53**:383–415 (1988).Google Scholar - 12.Th. Niemeijer and M. J. van Leeuwen, Renormalization theory for Ising-like spin systems, in
*Phase Transitions and Critical Phenomena*, Vol. 6, C. Domb and M. S. Green, eds. (Academic Press, 1976).Google Scholar - 13.B. Simon, Correlation inequalities and the decay of correlations in ferromagnets,
*Commun. Math. Phys.***77**:111 (1980).Google Scholar