Eta-pairing states in Hubbard models with bond-charge interactions on general graphs

We investigate Hubbard models with bond-charge interactions on general graphs. For a Hamiltonian $H$ of such a model, we provide the condition on its parameters under which the $\eta$-pairing method can be employed to construct its exact eigenstates. We arrive at this condition by finding that the requirement for the $\eta$-pairing state $(\eta^\dagger)^N |0\rangle$ to be an eigenstate of $H$ is identical to the requirement for it to be an eigenstate of a Hubbard-type Hamiltonian $H_m$. When the condition for $(\eta^\dagger)^N |0\rangle$ to be an eigenstate of the Hubbard-type Hamiltonian $H_m$ is satisfied, we demonstrate that there are additional states, distinct from $(\eta^\dagger)^N |0\rangle$, which are also exact eigenstates of $H_m$. Our results enhance the understanding of Hubbard models on general graphs, both with and without bond-charge interactions.