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Europhys. Lett.
Volume 73, Number 5, March 2006
Page(s) 747 - 751
Section Condensed matter: structural, mechanical and thermal properties
Published online 20 January 2006
Europhys. Lett., 73 (5), pp. 747-751 (2006)
DOI: 10.1209/epl/i2005-10451-1

The 2D Ising square lattice with nearest- and next-nearest-neighbor interactions

H. J. W. Zandvliet

Solid State Physics Group and MESA+ Research Institute for Nanotechnology University of Twente - P.O. Box 217, 7500 AE Enschede, The Netherlands

received 5 July 2005; accepted in final form 5 January 2006
published online 20 January 2006

An analytical expression for the boundary free energy of the Ising square lattice with nearest- and next-nearest-neighbor interactions is derived. The Ising square lattice with anisotropic nearest-neighbor (Jx and Jy) and isotropic next-nearest-neighbor (Jd) interactions has an order-disorder phase transition at a temperature T=Tc given by the condition $\mth{e^{-2J_x /k_b T_c}+e^{\ensuremath{-2J_y /k_b T_c} }+e^{-2(J_x +J_y )/k_b T_c}\left(2-e^{-4J_d /k_b T_c}\right)=e^{4J_d /k_b T_c}}$. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase is in excellent agreement with series expansion, finite scaling of transfer matrix and Monte Carlo results. For a vanishing next-nearest-neighbor interaction Onsager's famous result, i.e. $\sinh (2J_{x} /k_{b} T_{c})\sinh (2J_{y} /k_{b} T_{c})=1$, is recaptured.

68.35.Rh - Phase transitions and critical phenomena.
75.40.-s - Critical-point effects, specific heats, short-range order.

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