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?