Square-lattice Ising model

In statistical mechanics, the two-dimensional square-lattice 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 ${\displaystyle H\neq 0}$ has yet to be found.

Consider the 2D Ising model on a square lattice ${\displaystyle \Lambda }$ 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

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