Exactly solvable dissipative spin liquid

Exactly solvable Hamiltonians with spin liquid ground states have proven to be extremely useful, not only because they unambiguously demonstrate that these phases can arise in systems of interacting spins but also as a pedagogical illustration of the concept and as a controlled starting point for further theoretical analysis. However, adding dissipative couplings to the environment---an important aspect for the realization of these phases---generically spoils the exact solvability. We here present and study a Lindbladian, describing a square-lattice spin-liquid with dissipative coupling to the environment, that admits an exact solution in terms of Majorana fermions coupled to static Z₂ gauge fields. This solution allows us to characterize the steady-state solutions as well as ``quasiparticle'' excitations within the Lindbladian spectrum. We uncover distinct types of quasiparticle excitations of the Lindbladian associated with parametrically different timescales governing the equilibration time of the expectation values of different classes of observables. Most notably, for small but nonzero dissipation, we find a separation into three different timescales associated with a three-step heating profile. On a more general level, our exactly solvable Lindbladian is expected to provide a starting point for a better understanding of the behavior of fractionalized systems under dissipative time evolution.