Definitions

square odd

Square-lattice Ising model

The two-dimensional square-lattice Ising model was solved by Lars Onsager in 1944 for the special case that the external field H = 0. The general case for H neq 0 has yet to be found.

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).

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(N-2s) +L(N-2r)}

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.

This reduces the partition function to

Z_N(K,L) = 2e^{N(K+L)} sum_{P subset Lambda_D} e^{-2Lr-2Ks}

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.

Low-temperature 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^{-2Lr-2Ks}

defines a low temperature expansion of Z_N(K,L) .

High-temperature 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+wsigma_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 Kramers-Wannier duality.

References

  • R.J. Baxter, Exactly solved models in statistical mechanics, London, Academic Press, 1982

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