Square lattice Ising model

In statistical mechanics, the two-dimensional square lattice Ising model is a simple lattice model of interacting magnetic spins, an example of the class of Ising models. The model is notable for having nontrivial interactions, yet having an analytical solution. The model was solved by Lars Onsager for the special case that the external magnetic field H = 0. An analytical solution for the general case for ${\displaystyle H\neq 0}$ has yet to be found.

Consider a 2D Ising model on a square lattice ${\displaystyle \Lambda }$ with N sites and periodic boundary conditions in both the horizontal and vertical directions, which effectively reduces the topology of the model to a torus. Generally, the horizontal coupling ${\displaystyle J}$ and the vertical coupling ${\displaystyle J^{*}}$ are not equal. With ${\displaystyle \textstyle \beta ={\frac {1}{kT}}}$ and absolute temperature ${\displaystyle T}$ and the Boltzmann constant ${\displaystyle k}$, the partition function

The critical temperature ${\displaystyle T_{\text{c}}}$ can be obtained from the Kramers–Wannier duality relation. Denoting the free energy per site as ${\displaystyle F(K,L)}$, one has:

where

Assuming that there is only one critical line in the (K, L) plane, the duality relation implies that this is given by:

For the isotropic case ${\displaystyle J=J^{*}}$, one finds the famous relation for the critical temperature ${\displaystyle T_{c}}$

Consider a configuration of spins ${\displaystyle \{\sigma \}}$ on the square lattice ${\displaystyle \Lambda }$. Let r and s denote the number of unlike neighbours in the vertical and horizontal directions respectively. Then the summand in ${\displaystyle Z_{N}}$ corresponding to ${\displaystyle \{\sigma \}}$ is given by

Construct a dual lattice ${\displaystyle \Lambda _{D}}$ as depicted in the diagram. For every configuration ${\displaystyle \{\sigma \}}$, 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 ${\displaystyle \Lambda }$ 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.