Solvable BCS-Hubbard Liouvillians in arbitrary dimensions

We construct a solvable Lindblad model in arbitrary dimensions, in which the Liouvillian can be mapped to a BCS-Hubbard model featuring an imaginary Hubbard interaction. The Hilbert space of the system can be divided into multiple sectors, each characterized by an on-site invariant configuration. The model exhibits bistable steady states in all spatial dimensions, which is guaranteed by the fermion-number parity. Notably, the Liouvillian gap exhibits a Zeno transition, below which the Liouvillian gap is linear with respect to the dissipation. We also uncover a generic dimension-dependent gap behavior: In one dimension, the gap originates from multiple sectors with spectral crossing; in higher dimensions, a single sector determines the gap.