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We investigate the dynamical properties of one-dimensional dissipativeFermi-Hubbard models, which are described by the Lindblad master equations withsite-dependent jump operators. The corresponding non-Hermitian effectiveHamiltonians with pure loss terms possess parity-time ($\mathcal{PT}$) symmetryif we compensate the system additionally an overall gain term. By solving thetwo-site Lindblad equation with fixed dissipation exactly, we find that thedynamics of rescaled density matrix shows an instability as the interactionincreases over a threshold, which can be equivalently described in the schemeof non-Hermitian effective Hamiltonians. This instability is also observed inmulti-site systems and closely related to the $\mathcal{PT}$ symmetry breakingaccompanied by appearance of complex eigenvalues of the effective Hamiltonian.Moreover, we unveil that the dynamical instability of the anti-ferromagneticMott phase comes from the $\mathcal{PT}$ symmetry breaking in highly excitedbands, although the low-energy effective model of the non-Hermitian Hubbardmodel in the strongly interacting regime is always Hermitian. We also provide aquantitative estimation of the time for the observation of dynamical$\mathcal{PT}$ symmetry breaking which could be probed in experiments.

Interaction induced dynamical $\mathcal{PT}$ symmetry breaking in dissipative Fermi-Hubbard models