Free fermions with dephasing and boundary driving: Bethe Ansatz results

By employing the Lindblad equation, we derive the evolution of the two-point correlator for a free-fermion chain of length $L$ subject to bulk dephasing and boundary losses. We use the Bethe ansatz to diagonalize the Liouvillian ${\mathcal L}^{\scriptscriptstyle(2)}$ governing the dynamics of the correlator. The majority of its energy levels are complex. Precisely, $L(L-1)/2$ complex energies do not depend on dephasing, apart for a trivial shift. The remaining complex levels are perturbatively related to the dephasing-independent ones for large $L$. The long-time dynamics is governed by a band of real energies, which contains an extensive number of levels. They give rise to diffusive scaling at intermediate times, when boundaries can be neglected. Moreover, they encode the breaking of diffusion at asymptotically long times. Interestingly, for large loss rate two boundary modes appear in the spectrum. The real energies correspond to string solutions of the Bethe equations, and can be treated effectively for large chains. This allows us to derive compact formulas for the dynamics of the fermionic density. We check our results against exact diagonalization, finding perfect agreement.