$\eta $-pairing states in the Hubbard model with non-uniform Hubbard interaction

The existence of $\eta$-pairing eigenstates in the fermionic Hubbard model is fundamentally rooted in the $\eta$-pairing symmetry, which may hold for systems with non-uniform Hubbard interaction $U$. In this work, we present a generalized Hubbard model containing a variety of pseudo-spin terms that break the SO$_{4}$ symmetry but retain the $\eta$-pairing symmetry. This allows us to construct a variety of correlated systems possessing $\eta$% -pairing eigenstates.\ We exemplify our findings by considering a modified Hubbard model associated with alternative magnetic fields and on-site repulsion. We find that the same quasi-$\eta$-pairing eigenstate exhibits two distinct dynamic behaviors in the two models. Numerical results of the time evolution driven by several typical Hamiltonians accord with the analytic predictions and provide a way of the control of an $\eta$-pairing wavepacket with the aid of a time-dependent Hamiltonian.