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?