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arXiv:2412.13820 · Physical Review Letters

Exceptional Stationary State in a Dephasing Many-Body Open Quantum System

Alice Marché, Gianluca Morettini, Leonardo Mazza, Lorenzo Gotta, Luca Capizzi

Published Jun 23, 2025• 8 citations
arXivHTMLPDF

Abstract

We study a dephasing many-body open quantum system that hosts, together with the infinite-temperature state, another additional stationary state, that is associated with a nonextensive strong symmetry. This state, that is a pure dark state, is exceptional in that it retains memory of the initial condition, whereas any orthogonal state evolves toward the infinite-temperature state erasing any information on the initial state. We discuss the approach to stationarity of the model focusing in particular on the fate of interfaces between the two states. A simple model based on a membrane picture helps developing an effective large-scale theory, which is different from the usual hydrodynamics since no extensive conserved quantities are present. The fact that the model reaches stationary properties on timescales that diverge with the system size, while the Lindbladian gap is finite, is duly highlighted. We point out the reasons for considering these exceptional stationary states as quantum many-body scars in the open system framework.

DOI
10.1103/zn9v-k73w

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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 a quantum many-body scar of the dual Hamiltonian $\mathcal{H}$. This scar eigenstate has a form of a thermofield double state for a single-body conserved quantity entering the detailed balance conditions. A remarkable feature of this thermofield scar is a tunable single-layer 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 eigenoperators $Q$ of the Lindbladian superoperator. The expectation values of the corresponding observables exhibit a simple exponential decay, ${\ensuremath{\langle}Q\ensuremath{\rangle}}_{t}={e}^{\ensuremath{-}\mathrm{\ensuremath{\Gamma}}t}{\ensuremath{\langle}Q\ensuremath{\rangle}}_{0}$, irrespectively of the initial state. Under the above duality, these eigenoperators 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 harboring scars and provide an example thereof.

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We study a dephasing many-body open quantum system that hosts, together with the infinite-temperature state, another additional stationary state, that is associated with a nonextensive strong symmetry. This state, that is a pure dark state, is exceptional in that it retains memory of the initial condition, whereas any orthogonal state evolves toward the infinite-temperature state erasing any information on the initial state. We discuss the approach to stationarity of the model focusing in particular on the fate of interfaces between the two states. A simple model based on a membrane picture helps developing an effective large-scale theory, which is different from the usual hydrodynamics since no extensive conserved quantities are present. The fact that the model reaches stationary properties on timescales that diverge with the system size, while the Lindbladian gap is finite, is duly highlighted. We point out the reasons for considering these exceptional stationary states as quantum many-body scars in the open system framework.
In this work, we undertake the problem of formally introducing a notion of quantum many-body scarring in open quantum systems governed by the Lindblad equation. To this goal, we rely on the commutant-algebra framework for the description of strong symmetries to introduce the unconventional strong-symmetry structure leading to the existence of anomalous stationary states, which we dub open-system quantum many-body scars (OSQMBS), besides a typical infinite-temperature state. We provide several benchmarks of the theoretical predictions on the stationary-state manifold and on convergence to stationarity, as well as describe the time-evolution of off-diagonal coherences among the Hilbert space symmetry sectors identified by OSQMBS and their orthogonal complement. Moreover, we investigate the existence of asymptotic OSQMBS (AOSQMBS), the latter being states that, despite converging to the typical infinite-temperature state in the large-time limit, display anomalously-large relaxation time scales, which we thoroughly describe by means of the behavior of their fidelity as a function of time through proper scaling Ansaetze.
Hydrodynamic relaxation in open quantum many-body systems can be understood in terms of diffusive Nambu--Goldstone (NG) modes associated with strong-to-weak spontaneous symmetry breaking (SWSSB). Here, we find a common structure underlying weak ergodicity breaking in closed quantum systems and diffusive hydrodynamics in open quantum systems. We establish a spectral correspondence between the parent Hamiltonian of rainbow quantum many-body scar (RQMBS) states in closed systems and the effective Lindbladian of strongly dissipative open systems with strong U(1)U(1)U(1) symmetry and local charge-dephasing Lindblad operators. For broad classes of spin, fermionic, and bosonic systems, we construct asymptotic RQMBS (ARQMBS) states. Within the enlarged scar subspace, the parent Hamiltonian admits a local Rokhsar--Kivelson-type frustration-free decomposition. A single-mode variational construction gives an O(k2)O(k^2)O(k2) upper bound on its lowest excitation energy and yields states satisfying the defining ARQMBS criteria: orthogonality, vanishing energy variance, and characteristic entanglement scaling. Under vectorization, RQMBS states are mapped to maximally mixed infinite-temperature states in fixed-charge sectors, which exhibit long-range SWSSB order under finite-density conditions. Assuming uniformly bounded positive transition rates, the effective Lindbladian has the same local projectors as the parent Hamiltonian but with positive transition-dependent weights, yielding two-sided spectral bounds and the same system-size scaling of their gaps. For uniform rates, the two operators are proportional to each other and their eigenmodes coincide. Therefore, an exact quadratic parent-Hamiltonian branch maps directly to a diffusive NG mode. Our results connect different mechanisms of slow relaxation in closed and open quantum many-body systems.
Quantum many-body scars are rare exceptions to thermalization: they sustain non-thermal stationary states without the protection of any local conservation law, and are generally expected to be fragile. Here we construct an analytically tractable random unitary circuit hosting a single scar, and derive from first principles the thermalization mechanism governing perturbations thereof - described by a picture of fluctuating interfaces. Surprisingly, despite being thermodynamically irrelevant for local observables, the scar leaves a sharp fingerprint in the entanglement dynamics, driving a transition as a function of perturbation strength that is not probed by any local measurement.
Abstract Given that any subsystem of a closed out-of-equilibrium quantum system is an open quantum system, its dynamics (reduced from the full system’s unitary evolution) can be either Markovian (memory-less) or non-Markovian, with the latter necessarily impeding the process of relaxation and thermalization. Seemingly independently, such non-ergodic dynamics occurs when an initial state has spectral weight on the so-called quantum many-body scar states, which are non-thermalizing eigenstates embedded deep in the spectrum of otherwise thermal eigenstates. In this article, we present numerical evidence that, in the class of systems which exhibit scars-induced entanglement oscillations, the presence of quantum scars is a microscopic ingredient that enables and enhances non-Markovianity of the dynamics of subsystems. We exemplify this with the PXP model and its deformations which either enhance or erase the signatures of scarred dynamics when quenched from simple product states with significant overlaps with the scarred states. The effect of thermalizing or scarring initial states is also similarly investigated. By probing information backflows with the dynamical behaviour of the distances between temporally-separated transient states of small subsystems, systematic signatures of subsystem non-Markovianity in these models are presented. It is seen that scarring-enhancing (erasing) deformations also exhibit enhanced (diminished) subsystem non-Markovianity. Likewise, results relating scarring (thermalizing) initial states to stronger (weaker) subsystem non-Markovianity are also presented. The retention of memory and revivals between transient subsystem states is a finer form of memory effect than captured by the revivals of full system’s fidelity with the initial states. This sheds new light on the dynamical memories associated with quantum scarring, and opens interesting new questions at the interface of quantum scarring and an open quantum systems approach to investigating far-from-equilibrium and non-thermalizing isolated quantum many-body systems.
We investigate the monitored dynamics of many-body quantum systems in which projective measurements of extensive operators are alternated with unitary evolution. Focusing on mean-field models characterized by all-to-all interactions, we develop a general framework that captures the thermodynamic limit, where a semiclassical description naturally emerges. Remarkably, we uncover novel stationary states, distinct from the conventional infinite-temperature state, that arise upon taking the infinite-volume limit. Counterintuitively, this phenomenon is not linked to the closing of the Lindbladian gap in that limit. We provide analytical explanation for this unexpected behavior.
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 a quantum many-body scar of the dual Hamiltonian H\mathcal{H}H. This scar eigenstate has a form of a thermofield double state for a single-body conserved quantity entering the detailed balance conditions. A remarkable feature of this thermofield scar is a tunable single-layer 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 eigenoperators QQQ of the Lindbladian superoperator. The expectation values of the corresponding observables exhibit a simple exponential decay, ${\ensuremath{\langle}Q\ensuremath{\rangle}}_{t}={e}^{\ensuremath{-}\mathrm{\ensuremath{\Gamma}}t}{\ensuremath{\langle}Q\ensuremath{\rangle}}_{0}$, irrespectively of the initial state. Under the above duality, these eigenoperators 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 harboring scars and provide an example thereof.