Abstract
A rigorous derivation is given for the “constant-magnetization” free energy density of the classical, anisotropic Heisenberg model with long-range Kac interactions. The derivation involves bounding arguments similar to those used for a classical fluid by Lebowitz and Penrose. The present work is carried out in a constant-magnetization ensemble. The free energy density is determined exactly under a quadruple-limiting process. The limits involved are a Lebowitz-Penrose type of triple-limiting process, followed by a final limit,x→ 0, wherex is a parameter which represents the range over which each component of the net spin density can vary. Explicit equations of state are determined for the special case of zero short-range interactions plus pure Kac-type long-range interactions.
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References
N. D. Mermin and H. Wagner,Phys. Rev. Letters 17:1133 (1966).
G. F. Newell and E. W. Montroll,Rev. Mod. Phys. 25:353 (1953).
K. Millard and H. S. Leff,J. Stat. Phys. 6:133 (1972).
K. Millard and H. S. Leff,J. Math. Phys. 12:1000 (1971); E. H. Lieb,Commun. Math. Phys. 31:327 (1973).
A. B. Bortz and R. B. Griffiths,Comm. Math. Phys. 26:102 (1972).
J. L. Lebowitz and O. Penrose,J. Math. Phys. 7:98 (1966).
H. S. Leff, M. Flicker, and K. Millard,Phys. Letters 35A:137 (1971).
T. M. Apostol,Mathematical Analysis, Addison-Wesley, Reading, Massachusetts (1957).
G. H. Hardy, J. E. Littlewood, and G. Pólya,Inequalities, Cambridge University Press, Cambridge (1967), Chapter III.
M. E. Fisher,Rep. Prog. Phys. 30:615 (1967).
K. Millard, Ph.D. Dissertation, Case Western Reserve University, January 1971, Chapter III.
E. Lieb,J. Math. Phys. 6:1016 (1966).
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This research was carried out at the Department of Physics, Case Western Reserve University, Cleveland, Ohio, and was supported in part by the U.S. Atomic Energy Commission. This work is based on portions of a dissertation by K. M. submitted in partial fulfillment of the Ph.D. degree, Case Western Reserve University.
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Millard, K., Leff, H.S. An exact solution to the classical, anisotropic Heisenberg model with long-range Kac interactions. J Stat Phys 10, 205–235 (1974). https://doi.org/10.1007/BF01016177
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DOI: https://doi.org/10.1007/BF01016177