Heisenberg interaction Hamiltonian for square lattice

[JVanUW]

Hi,

I just started self studying solid state and I'm having trouble figuring out what the hamiltonian for a square lattice would be when considering the Heisenberg interaction.

I reformulated the dot product into 1/2( ^S_i+S_i+δ⁺ +S_i+δ⁺S_-^- ) + S_i^zS_i+δz

and use

S_i^z = S-a_i⁺a_i
S_i⁺ = √2S]a_i
...
S_i+δ^z=-S+a_i+δ⁺a_i+δ
...

Etc.

But I'm getting for the terms of the hamiltonian

a_ia_i+δ +a_i+δ⁺a_i⁺ ...

but don't these terms violate momentum conservation?
What is the real Heisenberg interaction hamiltonian for the square lattice?

 

[OhYoungLions]

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.