# Heisenberg interaction Hamiltonian for square lattice

 P: 26 Firstly, let's correct your terminology a little bit. The Heisenberg interaction is just: $$\mathcal{H}=\mathcal{J}\sum_{i} S_i \cdot S_{i+\delta}$$ You have rewritten it in terms of $S^z, S^+$ and $S^-$ operators which is fine. Your next step is to write it with respect to bosonic operators $a, a^\dagger$ in the Holstein-Primakoff representation, in which case the bosonic operators create and destroy spin waves. It appears you have taken $\mathcal{J}$ to be positive, in which case you have the antiferromagnetic model where spins on neighbouring sites prefer to be antiparallel. This is implicit in your choice of S and -S in the H-P representation. So far your bosonic operators are in the position representation. When you work all this out, you get terms with $a^\dagger_i a^\dagger_{i+\delta}$. These do not violate momentum conservation because they are still in the position representation - if you fourier transform them you'll see there is no problem. You are SUPPOSED to get them. This is what makes a ferromagnet (J<0) different from an antiferromagnet (J>0). In order to diagonalize the Hamiltonian, you must do two steps. 1. Fourier transform it. 2. Use a Bogoliubov transformation to get rid of the $a^\dagger_i a^\dagger_{i+\delta}$ terms. Google this if you don't know what it is.