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I'm looking for the two-site Fermi-Hubbard Hamiltonian in phase density representation in linearized form and I don't know how to derive it. I then want to derive the equations of motion from that.

For the Bose-Hubbard model, the Hamiltonian to second order looks like that:

$$H_2=(J_0/n_0+U)\Delta n^2+J_0n_0\theta^2 $$

where $J_0$ is the tunneling Energy, $U$ the interaction Energy of the site and $\Delta n$ is the density imbalance of the two sites.

If you could help me that would be great. Thank you very much in advance!

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