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 nonuniform Hubbard interaction $U$. In this work, we present a generalized Hubbard model containing a variety of pseudospin terms that break the ${\mathrm{SO}}_4$ 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 method for the control of an $\eta$-pairing wave packet with the aid of a time-dependent Hamiltonian.

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• Magnetism
• Superconductivity
• Superfluidity
• Fermionic condensates
• Quantum many-body systems
• Strongly correlated systems
• Exact solutions for many-body systems
• Hubbard model