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

$Z_{N}(K,L)=2e^{N(K+L)}\sum _{P\subset \Lambda _{D}}e^{-2Lr-2Ks}$

defines a low temperature expansion of $Z_{N}(K,L)$.

High-temperature expansion

Since $\sigma \sigma '=\pm 1$ one has

$e^{K\sigma \sigma '}=coshK+sinhK(\sigma \sigma ')=coshK(1+tanhK(\sigma \sigma '))$

Therefore

$Z_{N}(K,L)=(coshKcoshL)^{N}\sum _{\{\sigma \}}\prod _{\langle ij\rangle _{H}}(1+v\sigma _{i}\sigma _{j})\prod _{\langle ij\rangle _{V}}(1+w\sigma _{i}\sigma _{j})$

where $v=tanhK$ and $w=tanhL$. 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 assocaiting 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

$\sum _{\sigma _{i}=\pm 1}\sigma _{i}^{n}={\begin{cases}0\ if\ n\ odd\\2\ if\ n\ even\end{cases}}$

Shows that only configurations with an even number of lines at each vertex (Polygons) will contribute to the partition function, giving

$Z_{N}(K,L)=2^{N}(coshKcoshL)^{N}\sum _{P\subset \Lambda }v^{r}w^{s}$

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.