I am given the two state Hamiltonian

$$ H = U \sum_{j \in \{L,R\}} n_{j \uparrow}n_{j \downarrow} - t \sum_{\sigma \in \{\uparrow,\downarrow\}}(a_{L \sigma}^{\dagger}a_{R \sigma} +a_{R \sigma}^{\dagger}a_{L \sigma}) $$

and I am supposed to calculate the eigenstate and eigenenergies. The only thing I know is what this operator does to an arbitrary two-spin state like $| \uparrow, \uparrow \rangle$. By playing around I found that $|\uparrow,\uparrow \rangle $ and $|\downarrow,\downarrow \rangle$ are mapped to zero, but how can I find all the eigenstates and eigenenergies?

  • $\begingroup$ What are the numbers of particles allowed on each site? If these are fermions, the number of states in your state space should be four or something similarly small, so you should be able to write down $H$ as a suitably small matrix and diagonalize it directly. $\endgroup$ – Emilio Pisanty Nov 29 '14 at 17:12
  • $\begingroup$ yes, we are dealing with fermions, so two on each side. could you explain how I get the matrix representation? $\endgroup$ – Xin Wang Nov 29 '14 at 21:14

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