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arXiv:2506.02970 · Physical review. B./Physical review. B

Integrals of motion as slow modes in dissipative many-body operator dynamics

Tian‐Hua Yang, Dmitry A. Abanin

Published Aug 25, 2025• 1 citation
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Abstract

We consider Lindbladian operator dynamics in many-body quantum systems with one or more integrals of motion (IOMs), subject to weak local dissipation. We demonstrate that IOMs with small support become slow modes of these dynamics, in the sense that their Frobenius norms decay more slowly compared to generic operators. As a result, the eigenoperators of such Lindbladians with slowest decay rates have a large overlap with the IOMs of the underlying Hamiltonian. We demonstrate this correspondence between slow modes and IOMs numerically for a number of many-body models and further corroborate it with perturbative arguments. These results open up another method for the identification of IOMs and provide insights into the dissipative many-body dynamics.

DOI
10.1103/d9b4-wlv6

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We consider Lindbladian operator dynamics in many-body quantum systems with one or more integrals of motion (IOMs), subject to weak local dissipation. We demonstrate that IOMs with small support become slow modes of these dynamics, in the sense that their Frobenius norms decay more slowly compared to generic operators. As a result, the eigenoperators of such Lindbladians with slowest decay rates have a large overlap with the IOMs of the underlying Hamiltonian. We demonstrate this correspondence between slow modes and IOMs numerically for a number of many-body models and further corroborate it with perturbative arguments. These results open up another method for the identification of IOMs and provide insights into the dissipative many-body dynamics.
Understanding the dynamics of open quantum systems is intrinsically important to control and utilize quantum hardware in various applications. The generic behavior of purely dissipative open quantum many-body systems with kkk-local, normal dissipation processes can be investigated using random matrix theory, revealing different decay timescales of observables determined by their complexity, as shown by Wang et al. [K. Wang et al., Phys. Rev. Lett. 124, 100604 (2020)]. The time evolution of an open quantum system is entirely described by the eigenvalue spectrum of the Lindbladian, in which the hierarchy of decay timescales is reflected in the formation of well-separated eigenvalue clusters. Here, we extend the analysis of this spectrum to the presence of unitary dynamics using a random matrix model. In the case of strong dissipation, the unitary dynamics can be treated perturbatively and the kkk locality of the Hamiltonian determines how susceptible the spectrum is to such a perturbation. For the physically most relevant case of (dissipative) two-body interactions, the correction in the first order of the perturbation vanishes, leading to the relative robustness of the spectral features, which in turn implies the same robustness of the corresponding separation of decay timescales. We find the perturbative approach for the spectrum to yield an excellent agreement with exact-diagonalization results. For weak dissipation, the spectrum flows into clusters except for outlier eigenmodes, which we identify to be the local symmetries of the Hamiltonian. We support our analytic findings by numerical simulations and show that our generic model applies to a nonrandom Heisenberg chain.
Open quantum systems weakly coupled to the environment are modeled by completely positive, trace preserving semigroups of linear maps. The generators of such evolutions are called Lindbladians. In the setting of quantum many-body systems on a lattice it is natural to consider Lindbladians that decompose into a sum of local interactions with decreasing strength with respect to the size of their support. For both practical and theoretical reasons, it is crucial to estimate the impact that perturbations in the generating Lindbladian, arising as noise or errors, can have on the evolution. These local perturbations are potentially unbounded, but constrained to respect the underlying lattice structure. We show that even for polynomially decaying errors in the Lindbladian, local observables and correlation functions are stable if the unperturbed Lindbladian has a unique fixed point and a mixing time which scales logarithmically with the system size. The proof relies on Lieb-Robinson bounds, which describe a finite group velocity for propagation of information in local systems. As a main example, we prove that classical Glauber dynamics is stable under local perturbations, including perturbations in the transition rates which may not preserve detailed balance.
Many-body localization provides a generic mechanism of ergodicity breaking in quantum systems. In contrast to conventional ergodic systems, many-body-localized (MBL) systems are characterized by extensively many local integrals of motion (LIOM), which underlie the absence of transport and thermalization in these systems. Here we report a physically motivated construction of local integrals of motion in the MBL phase. We show that any local operator (e.g., a local particle number or a spin-flip operator), evolved with the system's Hamiltonian and averaged over time, becomes a LIOM in the MBL phase. Such operators have a clear physical meaning, describing the response of the MBL system to a local perturbation. In particular, when a local operator represents a density of some globally conserved quantity, the corresponding LIOM describes how this conserved quantity propagates through the MBL phase. Being uniquely defined and experimentally measurable, these LIOMs provide a natural tool for characterizing the properties of the MBL phase, in both experiments and numerical simulations. We demonstrate the latter by numerically constructing an extensive set of LIOMs in the MBL phase of a disordered spin-chain model. We show that the resulting LIOMs are quasilocal and use their decay to extract the localization length and establish the location of the transition between the MBL and ergodic phases.
In this paper we study one-dimensional quantum Ising spin chains in external magnetic field close to an integrable point. We concentrate on the dynamics of the slowest operator, that plays a key role at the final period of thermalization. We introduce two independent definitions of the slowest operator: local and translationally-invariant ones. We construct both operators numerically using tensor networks and extensively compare their physical properties. We find that the local operator has a significant overlap with energy flux, it does not correspond to an integral of motion, and, as one goes away from the integrable point, its revivals get suppressed and the rate of delocalization changes from extremely slow to slower than diffusion. The translationally-invariant operator corresponds to an integral of motion; as the system becomes less integrable, at some point this operator changes its nature: from no overlap with any magnetization and fast rate of delocalization, to non-zero overlap with magnetizations σxσ_{x}σx​ and σzσ_{z}σz​ and slow rate of delocalization.
In this paper we study one-dimensional quantum Ising spin chains in an external magnetic field close to an integrable point. We concentrate on the dynamics of the slowest operator that plays a key role at the final period of thermalization. We introduce two independent definitions of the slowest operator: local and translationally invariant ones. We construct both operators numerically using tensor networks and extensively compare their physical properties. We find that the local operator has a significant overlap with energy flux, it does not correspond to an integral of motion, and, as one goes away from the integrable point, its revivals get suppressed and the rate of delocalization changes from extremely slow to slower than diffusion. The translationally invariant operator corresponds to an integral of motion; as the system becomes less integrable, at some point this operator changes its nature: from no overlap with any magnetization and fast rate of delocalization, to nonzero overlap with magnetizations σ_{x} and σ_{z} and slow rate of delocalization.
We numerically construct slowly relaxing local operators in a nonintegrable spin-1/2 chain. Restricting the support of the operator to M consecutive spins along the chain, we exhaustively search for the operator that minimizes the Frobenius norm of the commutator with the Hamiltonian. We first show that the Frobenius norm bounds the time scale of relaxation of the operator at high temperatures. We find operators with significantly slower relaxation than the slowest simple "hydrodynamic" mode due to energy diffusion. Then we examine some properties of the nontrivial slow operators. Using both exhaustive search and tensor network techniques, we find similar slowly relaxing operators for a Floquet spin chain; this system is hydrodynamically "trivial," with no conservation laws restricting their dynamics. We argue that such slow relaxation may be a generic feature following from locality and unitarity.