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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}= 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_{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?

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# Heisenberg interaction Hamiltonian for square lattice

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