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Heisenberg interaction Hamiltonian for square lattice 
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#1
Nov812, 09:28 AM

P: 23

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( ^{Si+}S_{i+δ}^{+} +S_{i+δ}^{+}S_{}^{} ) + S_{i}^{z}S_{i+δ}z and use S_{i}^{z} = Sa_{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_{i}a_{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? 


#2
Nov2012, 08:27 PM

P: 26

Firstly, let's correct your terminology a little bit. The Heisenberg interaction is just:
[tex]\mathcal{H}=\mathcal{J}\sum_{i} S_i \cdot S_{i+\delta} [/tex] You have rewritten it in terms of [itex]S^z, S^+[/itex] and [itex]S^[/itex] operators which is fine. Your next step is to write it with respect to bosonic operators [itex]a, a^\dagger[/itex] in the HolsteinPrimakoff representation, in which case the bosonic operators create and destroy spin waves. It appears you have taken [itex]\mathcal{J}[/itex] 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 HP representation. So far your bosonic operators are in the position representation. When you work all this out, you get terms with [itex]a^\dagger_i a^\dagger_{i+\delta}[/itex]. 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 [itex]a^\dagger_i a^\dagger_{i+\delta}[/itex] terms. Google this if you don't know what it is. 


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