We propose the Sachdev-Ye-Kitaev Lindbladian as a paradigmatic solvable model of dissipative many-body quantum chaos. It describes $N$ strongly coupled Majorana fermions with random all-to-all interactions, with unitary evolution given by a quartic Hamiltonian and the coupling to the environment described by $M$ quadratic jump operators, rendering the full Lindbladian quartic in the Majorana operators. Analytical progress is possible by developing a dynamical mean-field theory for the Liouvillian time evolution on the Keldysh contour. By disorder-averaging the interactions, we derive an (exact) effective action for two collective fields (Green's function and self-energy). In the large-$N$, large-$M$ limit, we obtain the saddle-point equations satisfied by the collective fields, which determine the typical timescales of the dissipative evolution, particularly the spectral gap that rules the relaxation of the system to its steady state. We solve the saddle-point equations numerically and find that, for strong or intermediate dissipation, the system relaxes exponentially, with a spectral gap that can be computed analytically, while for weak dissipation, there are oscillatory corrections to the exponential relaxation. In this letter, we illustrate the feasibility of analytical calculations in strongly correlated dissipative quantum matter.

• Received 6 January 2022
• Accepted 23 May 2022

DOI:https://doi.org/10.1103/PhysRevResearch.4.L022068

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Published by the American Physical Society