A Diagonalization of Hubbard Hamiltonian

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Hi guys! I am starting to study Hubbard model with application in DFT and I have some doubts how to solve the Hubbard Hamiltonian. I have the DFT modeled to Hubbard, where the homogeneous Hamiltonian is

$$ H = -t\sum_{\langle i,j \rangle}\sigma (\hat{c}_{i\sigma}^{\dagger}\hat{c}_{j\sigma} + H.c.) + \sum_i v_i^{eff} \hat{c}_{i\sigma}^{\dagger}\hat{c}_{i\sigma} $$

How do I diagonalize it?

Thanks in advance.
 
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A. Neumaier

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Hi guys! I am starting to study Hubbard model with application in DFT and I have some doubts how to solve the Hubbard Hamiltonian. I have the DFT modeled to Hubbard, where the homogeneous Hamiltonian is

$$ H = -t\sum_{\langle i,j \rangle}\sigma (\hat{c}_{i\sigma}^{\dagger}\hat{c}_{j\sigma} + H.c.) + \sum_i v_i^{eff} \hat{c}_{i\sigma}^{\dagger}\hat{c}_{i\sigma} $$

How do I diagonalize it?
.
Make an arbitrary linear (Bogoliubov) transformation of creation and annihilation operators, work out the conditions that preserve the commutation rules and the conditions that make the resulting Hamiltonian diagonal, and you get a standard problem from linear algebra.
 

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