In statistical mechanics, the twodimensional squarelattice Ising model is a simple model of interacting magnetic spins. The model is notable for having nontrivial interactions, yet having an analytical solution. The model was solved by (Lars Onsager 1944) for the special case that the external magnetic field H = 0. An analytical solution for the general case for H \neq 0 has yet to be found.
Contents

Definition of the model 1

Critical temperature 2

Dual lattice 3

Lowtemperature expansion 4

Hightemperature expansion 5

Exact solution 6

References 7
Definition of the model
Consider the 2D Ising model on a square lattice \Lambda with N sites, with periodic boundary conditions in both the horizontal and vertical directions, which effectively reduces the geometry of the model to a torus. In a general case, the horizontal coupling J is not equal to the coupling in the vertical direction, J*. With an equal number of rows and columns in the lattice, there will be N of each. In terms of

K = \beta J

L = \beta J*
where \beta = 1/(kT) where T is absolute temperature and k is Boltzmann's constant, the partition function Z_N(K,L) is given by

Z_N(K,L) = \sum_{\{\sigma\}} \exp \left( K \sum_{\langle ij \rangle_H} \sigma_i \sigma_j + L \sum_{\langle ij \rangle_V} \sigma_i \sigma_j \right).
Critical temperature
The critical temperature T_c can be obtained from the Kramers–Wannier duality relation. Denoting the free energy per site as F(K,L), one has:

\beta F\left(K^{*}, L^{*}\right) = \beta F\left(K,L\right) + \frac{1}{2}\log\left[\sinh\left(2K\right)\sinh\left(2L\right)\right]
where

\sinh\left(2K^{*}\right)\sinh\left(2L\right)=1

\sinh\left(2L^{*}\right)\sinh\left(2K\right)=1
Assuming there is only one critical line in the (K,L) plane, the duality relation implies that this is given by:

\sinh\left(2 K\right)\sinh\left(2 L\right)= 1
For the isotropic case J = J^{*}, one finds the famous relation for the critical temperature T_{c}

k_B T_c/J = \frac{2}{\ln(1+\sqrt{2})} \approx 2.26918531421.
Dual lattice
Consider a configuration of spins \{ \sigma \} on the square lattice \Lambda . Let r and s denote the number of unlike neighbours in the vertical and horizontal directions respectively. Then the summand in Z_N corresponding to \{ \sigma \} is given by

e^{K(N2s) +L(N2r)}
Dual lattice
Construct a dual lattice \Lambda_D as depicted in the diagram. For every configuration \{ \sigma \} , a polygon is associated to the lattice by drawing a line on the edge of the dual lattice if the spins separated by the edge are unlike. Since by traversing a vertex of \Lambda the spins need to change an even number of times so that one arrives at the starting point with the same charge, every vertex of the dual lattice is connected to an even number of lines in the configuration, defining a polygon.
Spin configuration on a dual lattice
This reduces the partition function to

Z_N(K,L) = 2e^{N(K+L)} \sum_{P \subset \Lambda_D} e^{2Lr2Ks}
summing over all polygons in the dual lattice, where r and s are the number of horizontal and vertical lines in the polygon, with the factor of 2 arising from the inversion of spin configuration.
Lowtemperature expansion
At low temperatures, K, L approach infinity, so that as T \rightarrow 0, \ \ e^{K}, e^{L} \rightarrow 0 , so that

Z_N(K,L) = 2 e^{N(K+L)} \sum_{ P \subset \Lambda_D} e^{2Lr2Ks}
defines a low temperature expansion of Z_N(K,L) .
Hightemperature expansion
Since \sigma \sigma' = \pm 1 one has

e^{K \sigma \sigma'} = \cosh K + \sinh K(\sigma \sigma') = \cosh K(1+\tanh K(\sigma \sigma')).
Therefore

Z_N(K,L) = (\cosh K \cosh L)^N \sum_{\{ \sigma \}} \prod_{\langle ij \rangle_H} (1+v \sigma_i \sigma_j) \prod_{\langle ij \rangle_V}(1+w\sigma_i \sigma_j)
where v =\tanh K and w = \tanh L . Since there are N horizontal and vertical edges, there are a total of 2^{2N} terms in the expansion. Every term corresponds to a configuration of lines of the lattice, by associating a line connecting i and j if the term v \sigma_i \sigma_j (or w \sigma_i \sigma_j) is chosen in the product. Summing over the configurations, using

\sum_{\sigma_i = \pm 1} \sigma_i^n = \begin{cases} 0 & \mbox{for } n \mbox{ odd} \\ 2 & \mbox{for } n \mbox{ even} \end{cases}
shows that only configurations with an even number of lines at each vertex (polygons) will contribute to the partition function, giving

Z_N(K,L) = 2^N(\cosh K \cosh L)^N \sum_{P \subset \Lambda} v^r w^s
where the sum is over all polygons in the lattice. Since tanh K, tanh L \rightarrow 0 as T \rightarrow \infty , this gives the high temperature expansion of Z_N(K,L).
The two expansions can be related using the KramersWannier duality.
Exact solution
The free energy per site in the limit N\to\infty is given as follows. Define the parameter k as:

k =\frac{1}{\sinh\left(2 K\right)\sinh\left(2 L\right)}
The free energy per site F can be expressed as:

\beta F = \frac{\log(2)}{2} + \frac{1}{2\pi}\int_{0}^{\pi}\log\left[\cosh\left(2 K\right)\cosh\left(2 L\right)+\frac{1}{k}\sqrt{1+k^{2}2k\cos(2\theta)}\right]d\theta
For the isotropic case J = J^{*}, from the above expression one finds for the internal energy per site:

U =  J \coth(2 \beta J) \left[ 1 + \frac{2}{\pi} (2 \tanh^2(2 \beta J) 1) \int_0^{\pi/2} \frac{1}{\sqrt{1  4 k (1+k)^{2} \sin^2(\theta)}} d\theta \right]
and the spontaneous magnetization is, for T < T_c:

M = \left[ 1  \sinh^{4}(2 \beta J) \right]^{1/8}
References

Baxter, Rodney J. (1982), Exactly solved models in statistical mechanics, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers],


Stephen G. Brush (1967), History of the LenzIsing Model. Reviews of Modern Physics (American Physical Society) vol. 39, pp 883–893. doi:10.1103/RevModPhys.39.883


Ising, E. (1925), "Beitrag zur Theorie des Ferromagnetismus", Z. Phys. 31: 253–258,

Itzykson, Claude; Drouffe, JeanMichel (1989), Théorie statistique des champs, Volume 1, Savoirs actuels (

Itzykson, Claude; Drouffe, JeanMichel (1989), Statistical field theory, Volume 1: From Brownian motion to renormalization and lattice gauge theory, Cambridge University Press,

Barry M. McCoy and Tai Tsun Wu (1973), The TwoDimensional Ising Model. Harvard University Press, Cambridge Massachusetts, ISBN 0674914406

Montroll, Elliott W.; Potts, Renfrey B.; Ward, John C. (1963), "Correlations and spontaneous magnetization of the twodimensional Ising model",



John Palmer (2007), Planar Ising Correlations. Birkhäuser, Boston, ISBN 9780817642488.

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