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Prog. Theor. Phys. Vol. 5 No. 2 (1950) pp. 177-186

[ Full Text PDF : FREE ACCESS (589K) ]

The Statistics of Honeycomb and Triangular Lattice. I

Kodi Husimi and Itiro Syôzi

Department of Physics, Osaka University

(Received December 1, 1949)

Abstract:

After the matrix method in crystal statistics was introduced by R. Kubo Kramers and Wannier, and others, Onsager, using the abstract algebraic method, got the exact solution in the case of plane square net. His rather complicated method was simplified by Nambu considerably. We, independenty, attained the simplification of Onsager's method and have applied to the honeycomb lattice. And the exact solution has been obtained. From this, by the so called dual transformation we can get the partition function for triangular net.
The Curie point occurs at ch 2H = 2 in the honeycomb lattice and at exp (4H) = 3 in the triangular net. These coincide with the results obtained by the dual and star-triangle transformations. The specific heat becomes logarithmically infinite but the energy itself remains continuous at this temperature. In the antiferromagnetic case, the honeycomb lattice behaves similar to the former case but the triangular net exhibits no phase change.


URL : http://ptp.ipap.jp/link?PTP/5/177/
DOI : 10.1143/PTP.5.177

[ Full Text PDF : FREE ACCESS (589K) ] Citation:

References:

  1. R. Kubo, Busseiron Kenkyu 1 (1943).
  2. H. A. Kramers and G. H. Wannier, Phys. Rev. 60 (1941), 252[APS].
    N. W. Montroll, J. Chem. Phys. 9 (1941), 706[CrossRef]; ibid. 10 (1942), 61[CrossRef].
    E. N. Lassettre and J. P. Howe, J. Chem. Phys. 9 (1941), 747[CrossRef]; ibid. 9 (1941), 801[CrossRef].
  3. L. Onsager, Phys. Rev. 65 (1944), 117[APS].
  4. G. H. Wannier, Rev. Mod. Phys. 17 (1945), 50[APS].
  5. K. Husimi, The Quantum Statistical Mechanics, (1948), 336.
  6. Y. Nambu, Prog. Theor. Phys. 5 (1950), 1[PTP].

Citing Article(s) :

  1. Journal of the Physical Society of Japan 60 (1991) pp. 1513-1522 :
    Structure of Ground States of Antiferromagnetic Ising Model with General-Spin on Triangular Lattice
    Tsuyoshi Horiguchi, Ojiro Nagai and Seiji Miyashita
  2. Journal of the Physical Society of Japan 60 (1991) pp. 1523-1532 :
    Determination of the Critical Points of Antiferromagnetic Ising Model with Next Nearest Neighbour Interactions on the Triangular Lattice
    Seiji Miyashita, Hidetsugu Kitatani and Yasumasa Kanada
  3. Journal of the Physical Society of Japan 61 (1992) pp. 64-69 :
    Low-Temperature Behavior of Antiferromagnetic Ising Model on Triangular Lattice
    Tsuyoshi Horiguchi, Kazuyuki Tanaka and Tohru Morita
  4. Journal of the Physical Society of Japan 61 (1992) pp. 3114-3124 :
    Low-Temperature Behaviors of Antiferromagnetic Ising Model of Infinite-Spin on Triangular Lattice
    Tsuyoshi Horiguchi, Ojiro Nagai, Seiji Miyashita, Yukio Miyatake and Yoshiki Seo
  5. Journal of the Physical Society of Japan 62 (1993) pp. 880-887 :
    Thermodynamic Properties of Antiferromagnetic Quantum XY-Model on the Triangular Lattice
    Shigeo Homma, Hirotsugu Matsuda, Naofumi Ogita and Kazuhiro Sano
  6. Journal of the Physical Society of Japan 64 (1995) pp. 4609-4627 :
    New Partially Disordered Phases with Commensurate Spin Density Wave in Frustrated Triangular Lattice
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  7. Journal of the Physical Society of Japan 65 (1996) pp. 725-731 :
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  8. Journal of the Physical Society of Japan 76 (2007) 013708 (4 pages) :
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  9. Journal of the Physical Society of Japan 77 (2008) 074002 (10 pages) :
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  10. Progress of Theoretical Physics Vol. 5 No. 1 (1950) pp. 1-13 :
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  11. Progress of Theoretical Physics Vol. 6 No. 1 (1951) pp. 1-16 :
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  12. Progress of Theoretical Physics Vol. 6 No. 1 (1951) pp. 17-26 :
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  13. Progress of Theoretical Physics Vol. 6 No. 4 (1951) pp. 533-542 :
    On the Crystal Statistics of Two-Dimensional Ising Ferromagnets
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  14. Progress of Theoretical Physics Supplement No.87 (1986) pp. 112-126 :
    The Ground State and Thermodynamic Properties of Generalized Heisenberg Models on the Triangular Lattice
    Seiji Miyashita

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