Abstract
We study the ground state properties of a one-dimensional Ising chain with a nearest-neighbor ferromagnetic interactionJ1, and akth neighboranti-ferromagnetic interactionJ k . WhenJk/J1=−1/k, there exists a highly degenerate ground state with a residual entropy per spin. For the finite chain with free boundary conditions, we calculate the degeneracy of this state exactly, and find that it is proportional to the (N+k−1)th term in a generalized Fibonacci sequence defined by,F (k) N =F (k)N−1 +F (k)N−k . In addition, we show that this one-dimensional model is closely related to the following problems: (a) a fully frustrated two-dimensional Ising system with a periodic arrangement of nearest-neighbor ferro- and antiferromagnetic bonds, (b) close-packing of dimers on a ladder, a 2×∞ strip of the square lattice, and (c) “directed” self-avoiding walks on finite lattice strips.
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References
G. Toulouse,Commun. Phys. 2:115 (1977).
S. Kirkpatrick,Phys. Rev. B 16:4630 (1977).
S. Redner and H. E. Stanley,J. Phys. C 10:4765 (1977); S. Redner and H. E. Stanley,Phys. Rev. B 16:4901 (1977).
U. Enz,Physica (Utrecht) 26:698 (1960).
R. J. Elliott,Phys. Rev. 124:346 (1961).
R. M. Hornreich, M. Luban, and S. Shtrikman,Phys. Rev. Lett. 35:1678 (1975).
W. Selke,Z. Phys. B 29:133 (1978).
M. E. Fisher and W. Selke,Phys. Rev. Lett. 44:1502 (1980); M. E. Fisher and W. Selke, preprint.
Solitons and Condensed Matter Physics, A. R. Bishop and T. Schneider, eds. (Springer Verlag, Berlin, 1978), and references therein.
J. von Boehm and P. Bak,Phys. Rev. Lett. 42:122 (1979).
J. Stephenson,Can. J. Phys. 48:1724 (1970); R. M. Kornreich, R. Liebmann, H. G, Schuster, and W. Selke,Z. Phys. B 35:91 (1979).
G. André, R. Bidaux, J.-P. Carton, R. Conte, and L. deSeze,J. Phys. (Paris) 40:479 (1979).
J. Villain,J. Phys. C 10:1717 (1977).
M. E. Fisher and M. F. Sykes,Phys. Rev. 114:45 (1959).
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Work partially supported by grants from AFOSR and ARO.
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Redner, S. One-dimensional Ising chain with competing interactions: Exact results and connection with other statistical models. J Stat Phys 25, 15–23 (1981). https://doi.org/10.1007/BF01008476
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DOI: https://doi.org/10.1007/BF01008476