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I'm learning about the 2D ferromagnetic Ising model in zero field and trying to verify what I know by calculating the ground-state energy for the state with all 'up' spins in a 3x3 lattice.

$$H = -J\sum_{<i,j>}s_{i}s_{j}$$

where the sum is over nearest neighbors.

My question is, how do you include the energy values for each spin? Do you just calculate the hamiltonian for each site and then add the sites together?

In that case, I get, assuming fixed boundaries, -4J for the central spin, -2J for each of the four spins on the corners (two nearest neighbors), and -3J for each of the four spins in the middle of the edge rows and columns (three nearest neighbors).

Adding it all together, I get H = -24J for the entire lattice.

In Chandler's stat mech book, he says that the lowest energy of the Ising model on a square lattice is given by -2NJ, where N is the number of spins. He doesn't specify what the boundary conditions are, but I assume he means fixed boundaries. Here, that means the lowest energy should be -18J. What am I doing wrong?

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    $\begingroup$ First, your answer should be divided by 2 as you are double counting each pair. Second, Chandler's answer is for an infinite lattice (or finite lattice with periodic boundary conditions. Every edge contributes -J. On a square lattice, every spin is connected to its neighbours by 4 edges. So the total contribution is $-4NJ/2$. The division by 2 is to account for the over counting as each edge appears twice (one for each spin at its ends). $\endgroup$
    – suresh
    Aug 29, 2014 at 0:28
  • $\begingroup$ @suresh that comment is an answer, you should make it one. $\endgroup$
    – alemi
    Aug 29, 2014 at 1:07

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Here is how Chandler does his counting: Take the (square) lattice to be of infinite extent or a finite lattice with periodic boundary conditions in both directions. The total number of edges is equal to $4N/2=2N$ if there are $N$ sites. The ground state corresponds to all spins being in the same state (all up or all down). The ground state energy is $-J$ for every edge and thus one obtains the total energy to be $-2NJ$. For a triangular lattice, the answer will be $-3NJ$.

(converting my comment to an answer following @alemi's suggestion)

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