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Journal of Physics Conference Series

Critical temperature in 2D square Ising model with anisotropic coupling constant using Kramers-Wannier duality

L A Aviles, Rayda Gammag

Published Jul 1, 2023• 0 citations
Publisher

Abstract

Abstract To solve the 2D square Ising model incorporating anisotropy, we discuss a graphical representation of Kramers-Wannier duality. The method yields an equation relating the critical temperature (Tc ) to the horizontal and vertical coupling constants. The result reproduces the known exact solution by Onsager. The equation derived was utilized to plot Tc as a function of anisotropy by introducing the ratio (α) of the coupling constants. A phase diagram is then obtained showing Tc as an increasing function of α In the limit as α approaches zero, the critical temperature asymptotically goes to zero. This agrees with the exact solution of the 1D model.

DOI
10.1088/1742-6596/2543/1/012006

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Abstract To solve the 2D square Ising model incorporating anisotropy, we discuss a graphical representation of Kramers-Wannier duality. The method yields an equation relating the critical temperature (Tc ) to the horizontal and vertical coupling constants. The result reproduces the known exact solution by Onsager. The equation derived was utilized to plot Tc as a function of anisotropy by introducing the ratio (α) of the coupling constants. A phase diagram is then obtained showing Tc as an increasing function of α In the limit as α approaches zero, the critical temperature asymptotically goes to zero. This agrees with the exact solution of the 1D model.
The Kramers-Wannier duality is shown to hold for all the even number spin correlation functions of the two dimensional square lattice Ising model in the sense that the high temperature (T>Tc)(T>T_{c})(T>Tc​) expressions for these correlation functions are transformed into the low temperature $(T
For the 2D Ising model, we analyzed dependences of thermodynamic characteristics on number of spins by means of computer simulations. We compared experimental data obtained using the Fisher-Kasteleyn algorithm on a square lattice with N = l × l spins and the asymptotic Onsager solution ( N → ∞). We derived empirical expressions for critical parameters as functions of N and generalized the Onsager solution on the case of a finite-size lattice. Our analytical expressions for the free energy and its derivatives (the internal energy, the energy dispersion and the heat capacity) describe accurately the results of computer simulations. We showed that when N increased the heat capacity in the critical point increased as ln N . We specified restrictions on the accuracy of the critical temperature due to finite size of our system. Also in the finite-dimensional case, we obtained expressions describing temperature dependences of the magnetization and the correlation length. They are in a good qualitative agreement with the results of computer simulations by means of the dynamic Metropolis Monte Carlo method.
The anisotropic Heisenberg model has been investigated by interpolating anisotropy between the Ising and isotropic Heisenberg limits via general spin high temperature series (HTS) for the zero-field susceptibility of the ferromagnets for the square lattice. This is done by locating the critical temperature ( T c (2) ) and the susceptibility exponent (γ) from the analysis of the series by Pade approximant and ratio methods. The critical temperature is found to be spin dependent. It decreases as anisotropy changes from the Ising limit towards the isotropic limit and remains finite (lower than that of Stanley and Kaplan) for all the spins except for S =1/2 at the isotropic limit. Exponent γ is almost Ising like for the anisotropic system and anomalous jump in it occurs for the isotropic case. In addition, expressions for the magnetization are derived by using spin-wave theory. The limit of complete isotropy has been found not to be a special case for the susceptibility, in contrast to magnetization.
A new graphical method is developed to calculate the critical temperature of 2- and 3-dimensional Ising models as well as that of the 2-dimensional Potts models. This method is based on the transfer matrix method and using the limited lattice for the calculation. The reduced internal energy per site has been accurately calculated for different 2-D Ising and Potts models using different size-limited lattices. All calculated energies intersect at a single point when plotted versus the reduced temperature. The reduced temperature at the intersection is 0.4407, 0.2746, and 0.6585 for the square, triangular, and honeycombs Ising lattices and 1.0050, 0.6309, and 1.4848 for the square, triangular, and honeycombs Potts lattices, respectively. These values are exactly the same as the critical temperatures reported in the literature, except for the honeycomb Potts lattice. For the two-dimensional Ising model, we have shown that the existence of such an intersection point is due to the duality relation. The method is then extended to the simple cubic Ising model, in which the intersection point is found to be dependent on the lattice sizes. We have found a linear relation between the lattice size and the intersection point. This relation is used to obtain the critical temperature of the unlimited simple cubic lattice. The obtained result, 0.221(2), is in a good agreement with the accurate value of 0.22165 reported by others.
We propose a new approach which permits us to obtain exact results for the mixed spin-1/2 and Spin- S ( S > 1/2) Ising models with single ion anisotropy Δ on the square lattice. We drive an explicit expression for the critical temperature for arbitrary values of S. We determine the exact phase diagrams for different values of S , and we show that there is no tricritical point. For S integer, there is no long range order when the anisotropy exceeds a critical value which is independent of S . Furthermore, the exact Ising transition temperature T C is always recovered, for any values of S, in the limit of Δ → − ∞ . These exact results are based on a conjecture which extend the analyticity of the n-spin-1/2 correlations functions (except at T = T C (Ising)) for any finite number n, to n → ∞ . This is confirmed by our calculations since our results are practically similar to those obtained by Monte Carlo simulations for S = 1, 2 and 3/2.
The quantum version of the XY model in the presence of anisotropy is presented in the boson space and solved exactly in the spherical limit. We obtain the temperature (T) - anisotropy (D) phase diagram of the model for the dimension d>dc{\mathrm{d}}_{\mathrm{c}}dc​=2, where dc{\mathrm{d}}_{\mathrm{c}}dc​ is the lower critical dimension. It is shown that the critical temperature Tc{\mathrm{T}}_{\mathrm{c}}Tc​ below which a ferromagnetic phase appears increases with increasing the anisotropy parameter, and remains finite even for anisotropy parameter D\ensuremath{\gg}:1.