Duality between open systems and closed bilayer systems, and thermofield double states as quantum many-body scars

We establish a duality between open many-body systems governed by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation and satisfying the detailed balance condition on the one side, and closed bilayer systems with a self-adjoint Hamiltonian on the other side. Under this duality, the identity operator on the open system side maps to the thermofield double state which turns out to be a quantum many-body scar of the dual Hamiltonian $\mathcal H$. A remarkable feature of this thermofield scar is a tunable entanglement entropy controlled by the reservoir temperature on the open system side. Further, we identify broad classes of many-body open systems with nontrivial explicit eigen operators $Q$ of the Lindbladian superoperator. The expectation values of the corresponding observables exhibit a simple exponential decay, $\langle Q\rangle_t=e^{-\Gamma t} \langle Q \rangle_0$, irrespectively of the initial state. Under the above duality, these eigen operators give rise to additional (towers of) scars. Finally, we point out that more general superoperators (not necessarily of the GKSL form) can be mapped to self-adjoint Hamiltonians of bilayer systems harbouring scars, and provide an example thereof.
Comment: Comment on v2: v1 of this manuscript was split in two pieces. The first piece was posted separately as arXiv 2405.17310 and published as Phys. Rev. B 109, L140302 (2024). The second piece is presented in v2