The Anisotropic 3D Ising model

Henricus J.W. Zandvliet, A. Saedi, C. Hoede

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

An expression for the free energy of an (001) oriented domain wall of the anisotropic 3D Ising model is derived. The order--disorder transition takes place when the domain wall free energy vanishes. In the anisotropic limit, where two of the three exchange energies (e.g. Jx and Jy) are small compared to the third exchange energy (Jz), the following asymptotically exact equation for the critical temperature is derived, sinh(2Jz/kBTc)sinh(2(Jx + Jy)/kBTc)) = 1. This expression is in perfect agreement with a mathematically rigorous result (kBTc/Jz = 2[ln(Jz/(Jx + Jy))-ln(ln(Jz/(Jx + Jy)) + O(1)]-1) derived earlier by Weng, Griffiths and Fisher (Phys. Rev. 162, 475 (1967)) using an approach that relies on a refinement of the Peierls argument. The constant that was left undetermined in the Weng et al. result is estimated to vary from ∼0.84 at ((Hx + Hy)/Hz) = 10-2 to ∼0.76 at ((Hx + Hy)/Hz) = 10-20.
Original languageUndefined
Pages (from-to)981-986
Number of pages6
JournalPhase transitions
Volume80
Issue number9
DOIs
Publication statusPublished - 2007

Keywords

  • METIS-242115
  • IR-73058

Cite this

Zandvliet, Henricus J.W. ; Saedi, A. ; Hoede, C. / The Anisotropic 3D Ising model. In: Phase transitions. 2007 ; Vol. 80, No. 9. pp. 981-986.
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The Anisotropic 3D Ising model. / Zandvliet, Henricus J.W.; Saedi, A.; Hoede, C.

In: Phase transitions, Vol. 80, No. 9, 2007, p. 981-986.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - The Anisotropic 3D Ising model

AU - Zandvliet, Henricus J.W.

AU - Saedi, A.

AU - Hoede, C.

PY - 2007

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N2 - An expression for the free energy of an (001) oriented domain wall of the anisotropic 3D Ising model is derived. The order--disorder transition takes place when the domain wall free energy vanishes. In the anisotropic limit, where two of the three exchange energies (e.g. Jx and Jy) are small compared to the third exchange energy (Jz), the following asymptotically exact equation for the critical temperature is derived, sinh(2Jz/kBTc)sinh(2(Jx + Jy)/kBTc)) = 1. This expression is in perfect agreement with a mathematically rigorous result (kBTc/Jz = 2[ln(Jz/(Jx + Jy))-ln(ln(Jz/(Jx + Jy)) + O(1)]-1) derived earlier by Weng, Griffiths and Fisher (Phys. Rev. 162, 475 (1967)) using an approach that relies on a refinement of the Peierls argument. The constant that was left undetermined in the Weng et al. result is estimated to vary from ∼0.84 at ((Hx + Hy)/Hz) = 10-2 to ∼0.76 at ((Hx + Hy)/Hz) = 10-20.

AB - An expression for the free energy of an (001) oriented domain wall of the anisotropic 3D Ising model is derived. The order--disorder transition takes place when the domain wall free energy vanishes. In the anisotropic limit, where two of the three exchange energies (e.g. Jx and Jy) are small compared to the third exchange energy (Jz), the following asymptotically exact equation for the critical temperature is derived, sinh(2Jz/kBTc)sinh(2(Jx + Jy)/kBTc)) = 1. This expression is in perfect agreement with a mathematically rigorous result (kBTc/Jz = 2[ln(Jz/(Jx + Jy))-ln(ln(Jz/(Jx + Jy)) + O(1)]-1) derived earlier by Weng, Griffiths and Fisher (Phys. Rev. 162, 475 (1967)) using an approach that relies on a refinement of the Peierls argument. The constant that was left undetermined in the Weng et al. result is estimated to vary from ∼0.84 at ((Hx + Hy)/Hz) = 10-2 to ∼0.76 at ((Hx + Hy)/Hz) = 10-20.

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