Abstract
We present upper bounds on the critical temperature of one-dimensional Ising models with long-range,l/n α interactions, where 1<α≦2. In particular for the often studied case of α=2 we have an upper bound onT c which is less than theT c found by a number of approximation techniques. Also for the case where α is small, such as α=1.1, we obtain rigorous bounds which are extremely close, within 1.0%, to those found by approximation methods.
References
F. J. Dyson, Existence of a phase-transition in a one-dimensional Ising ferromagnet,Commun. Math. Phys. 12:91 (1969).
F. J. Dyson, An Ising ferromagnet with discontinuous long-range order,Commun. Math. Phys. 21:269 (1971).
D. Ruelle, Statistical mechanics of a one-dimensional lattice gas,Commun. Math. Phys. 9:267 (1968).
J. Frohlich and T. Spencer, The phase transition in the one-dimensional Ising model with 1/r 2 interaction energy84:87 (1982).
M. Aizenman, J. T. Chayes, L. Chayes, and C. M. Newman, Discontinuity of the magnetization in one-dimensional 1/|x−y|2 Ising and Potts models,J. Stat. Phys. 50:1 (1988).
J. Z. Imbrie and C. M. Newman, An intermediate phase with slow decay of correlations in one-dimensional 1/|x−y|2 percolation, Ising and Potts models,Commun. Math. Phys. 118:303 (1988).
J. O. Vigfusson, Improved upper bounds on the critical temperature of the 1/n 2 Ising spin chain,Phys. Rev. B 34:3466 (1986).
J. L. Monroe, Upper bound on the critical temperature for various Ising models.J. Stat. Phys. 40:249 (1985).
J. O. Vigfusson, New upper bounds for the magnetization in ferromagnetic one-component systems,Lett. Math. Phys. 10:71 (1985).
J. L. Monroe, Bethe lattice approximation of long-range interaction Ising models,Phys. Lett. A 171:427 (1992).
J. F. Nagle and J. C. Bonner, Numerical studies of the Ising chain with long-range ferromagnetic interactions,J. Phys. C 3:352 (1970).
B. G. S. Doman, A cluster approach to the Ising linear chain with long range interactions,Phys. Stat. Sol. (b) 103:K169 (1981).
J. L. Monroe, R. Lucente, and J. P. Hourlland, The coherent anomaly method and long-range one-dimensional Ising models,J. Phys. A: Math. Gen. 23:2555 (1990).
Z. Glumac and K. Uzelac, Finite-range scaling study of the 1D long-range model,J. Phys. A: Math. Gen. 22:4439 (1989).
G. V. Matvienko, Critical behavior of the ferromagnetic Ising chain with interactionJ ij |i−j| −2,Teor. Mat. Fiz. 63:465 (1985).
J. Bhattacharjee, S. Chakravarthy, J. L. Richardson, and D. J. Scalapino, Some properties of a one-dimensional Ising chain with an inverse-square interaction,Phys. Rev. B 24:3862 (1981).
P. W. Anderson and G. Yuval, Some numerical results on the Kondo problem and the inverse square one-dimensional Ising model,J. Phys. C 4:607 (1971).
R. Mainieri, Thermodynamic ξ functions for Ising models with long-range interactions,Phys. Rev. A 45:3580 (1992).
M. J. Wragg and G. A. Gehring, The Ising model with long-range ferromagnetic interactions,J. Phys. A: Math. Gen. 23:2157 (1990).
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Monroe, J.L. Upper bounds onT c for one-dimensional Ising systems. J Stat Phys 76, 1505–1510 (1994). https://doi.org/10.1007/BF02187074
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DOI: https://doi.org/10.1007/BF02187074