Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1412.3073

Local integrals of motion and the logarithmic lightcone in many-body localized systems

Isaac H. Kim, Anushya Chandran, Dmitry A. Abanin

Published Dec 9, 2014• 14 citations
arXivHTMLPDF

Abstract

We propose to define full many-body localization in terms of the recently introduced integrals of motion[Chandran et al., arXiv:1407.8480], which characterize the time-averaged response of the system to a local perturbation. The quasi-locality of such integrals of motion implies an effective lightcone that grows logarithmically in time. This subsequently implies that (i) the average entanglement entropy can grow at most logarithmically in time for a global quench from a product state, and (ii) with high probability, the time evolution of a local operator for a time interval $|t|$ can be classically simulated with a resource that scales polynomially in $|t|$.

DOI
10.48550/arxiv.1412.3073

Related papers

Dynamics of many-body localized systems: logarithmic lightcones and $\log \, t$-law of $α$-Rényi entropies

Daniele Toniolo, Sougato Bose

In the context of the Many-Body-Localization phenomenology we consider arbitrarily large one-dimensional local spin systems, the XXZ model with random magnetic field is a prototypical example. Without assuming the existence of exponentially localized integrals of motion (LIOM), but assuming instead that the system's dynamics gives rise to a Lieb-Robinson bound (L-R) with a logarithmic lightcone, we rigorously evaluate the dynamical generation, starting from a generic product state, of $ α$-Rényi entropies, with $ α$ close to one, obtaining a $\log \, t$-law, that denotes a slow spread of entanglement. This is in sharp contrast with Anderson localized phases that show no dynamically generated entanglement. To prove this result we apply a general theory recently developed by us in arXiv:2408.00743 that quantitatively relates the L-R bounds of a local Hamiltonian with the dynamical generation of entanglement. Assuming instead the existence of LIOM we provide new independent proofs of the known facts that the L-R bound of the system's dynamics has a logarithmic lightcone and show that the dynamical generation of the von Neumann entropy has for large times a $ \log \, t$-shape. L-R bounds, that quantify the dynamical spreading of local operators, may be easier to measure in experiments in comparison to global quantities such as entanglement.

arXiv:2408.02016•Aug 4, 2024•0 citations

Logarithmic entanglement lightcone in many-body localized systems

Dong‐Ling Deng, Xiaopeng Li, J. H. Pixley +2 more

We theoretically study the response of a many-body localized system to a local quench from a quantum information perspective. We find that the local quench triggers entanglement growth throughout the whole system, giving rise to a logarithmic lightcone. This saturates the modified Lieb-Robinson bound for quantum information propagation in many-body localized systems previously conjectured based on the existence of local integrals of motion. In addition, near the localization-delocalization transition, we find that the final states after the local quench exhibit volume-law entanglement. We also show that the local quench induces a deterministic orthogonality catastrophe for highly excited eigenstates, where the typical wave-function overlap between the pre- and postquench eigenstates decays exponentially with the system size.

arXiv:1607.08611•Physical review. B./Physical review. B•Jan 10, 2017•49 citations

Logarithmic entanglement light cone from eigenstate correlations in the many-body localized phase

Ratul Thakur, Bikram Pain, Sthitadhi Roy

We investigate the operator entanglement of the time-evolution operator through the framework of eigenstate correlations. Focusing on strongly disordered quantum many-body systems in the many-body localized (MBL) regime, we analyze the operator entanglement across various spatiotemporal cuts, revealing the logarithmic lightcone of entanglement spreading. We demonstrate that this logarithmic lightcone arises directly from a hierarchy of energyscales and lengthscales encoded in eigenstate correlations. By characterizing the statistics of these hierarchical scales, we develop a microscopic theory for the spatiotemporal structure of entanglement spreading in MBL systems---without invoking phenomenological constructs such as $\ensuremath{\ell}$-bits. This approach reveals the fundamental connection between eigenstate correlations and the emergent entanglement structure in MBL systems.

arXiv:2504.02815•Physical review. B./Physical review. B•May 20, 2025•2 citations

Constructing local integrals of motion in the many-body localized phase

Anushya Chandran, Isaac H. Kim, Guifré Vidal +1 more

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.

arXiv:1407.8480•Physical Review B•Feb 24, 2015•301 citations

Simulation Complexity of Many-Body Localized Systems

Adam Ehrenberg, Abhinav Deshpande, Christopher L. Baldwin +2 more

We use complexity theory to rigorously investigate the difficulty of classically simulating evolution under many-body localized (MBL) Hamiltonians. Using the defining feature that MBL systems have a complete set of quasilocal integrals of motion (LIOMs), we demonstrate a transition in the classical complexity of simulating such systems as a function of evolution time. On one side, we construct a quasipolynomial-time tensor-network-inspired algorithm for strong simulation of 1D MBL systems (i.e., calculating the expectation value of arbitrary products of local observables) evolved for any time polynomial in the system size. On the other side, we prove that even weak simulation, i.e. sampling, becomes formally hard after an exponentially long evolution time, assuming widely believed conjectures in complexity theory. Finally, using the consequences of our classical simulation results, we also show that the quantum circuit complexity for MBL systems is sublinear in evolution time. This result is a counterpart to a recent proof that the complexity of random quantum circuits grows linearly in time.

arXiv:2205.12967•May 25, 2022•1 citation

Local constants of motion imply information propagation

M Friesdorf, A H Werner, M Goihl +2 more

Interacting quantum many-body systems are expected to thermalize, in the sense that the evolution of local expectation values approaches a stationary value resembling a thermal ensemble. This intuition is notably contradicted in systems exhibiting many-body localisation (MBL). In stark contrast to the non-interacting case of Anderson localisation, the entanglement of states grows without limit over time, albeit slowly. In this work, we establish a novel link between quantum information theory and notions of condensed matter physics, capturing this phenomenon in the Heisenberg picture. We show that the mere existence of local constants of motion, often taken as the defining property of MBL, together with a generic spectrum of the Hamiltonian, is already sufficient to rigorously prove information propagation: these systems can be used to send a classical bit over arbitrary distances, in that the impact of a local perturbation can be detected arbitrarily far away. This counterintuitive result is compatible with and further corroborates the intuition of a slow entanglement growth following global quenches in MBL systems. We perform a detailed perturbation analysis of quasi-local constants of motion and also show that they indeed can be used to construct efficient spectral tensor networks, as recently suggested. Our results provide a detailed and at the same time model-independent picture of information propagation in MBL systems.

arXiv:1412.5605•New Journal of Physics•Nov 27, 2015•22 citations
We propose to define full many-body localization in terms of the recently introduced integrals of motion[Chandran et al., arXiv:1407.8480], which characterize the time-averaged response of the system to a local perturbation. The quasi-locality of such integrals of motion implies an effective lightcone that grows logarithmically in time. This subsequently implies that (i) the average entanglement entropy can grow at most logarithmically in time for a global quench from a product state, and (ii) with high probability, the time evolution of a local operator for a time interval ∣t∣|t|∣t∣ can be classically simulated with a resource that scales polynomially in ∣t∣|t|∣t∣.
Dynamics of many-body localized systems: logarithmic lightcones and log⁡ t\log \, tlogt-law of ααα-Rényi entropies
In the context of the Many-Body-Localization phenomenology we consider arbitrarily large one-dimensional local spin systems, the XXZ model with random magnetic field is a prototypical example. Without assuming the existence of exponentially localized integrals of motion (LIOM), but assuming instead that the system's dynamics gives rise to a Lieb-Robinson bound (L-R) with a logarithmic lightcone, we rigorously evaluate the dynamical generation, starting from a generic product state, of $ α$-Rényi entropies, with $ α$ close to one, obtaining a log⁡ t\log \, tlogt-law, that denotes a slow spread of entanglement. This is in sharp contrast with Anderson localized phases that show no dynamically generated entanglement. To prove this result we apply a general theory recently developed by us in arXiv:2408.00743 that quantitatively relates the L-R bounds of a local Hamiltonian with the dynamical generation of entanglement. Assuming instead the existence of LIOM we provide new independent proofs of the known facts that the L-R bound of the system's dynamics has a logarithmic lightcone and show that the dynamical generation of the von Neumann entropy has for large times a $ \log \, t$-shape. L-R bounds, that quantify the dynamical spreading of local operators, may be easier to measure in experiments in comparison to global quantities such as entanglement.
We theoretically study the response of a many-body localized system to a local quench from a quantum information perspective. We find that the local quench triggers entanglement growth throughout the whole system, giving rise to a logarithmic lightcone. This saturates the modified Lieb-Robinson bound for quantum information propagation in many-body localized systems previously conjectured based on the existence of local integrals of motion. In addition, near the localization-delocalization transition, we find that the final states after the local quench exhibit volume-law entanglement. We also show that the local quench induces a deterministic orthogonality catastrophe for highly excited eigenstates, where the typical wave-function overlap between the pre- and postquench eigenstates decays exponentially with the system size.
We investigate the operator entanglement of the time-evolution operator through the framework of eigenstate correlations. Focusing on strongly disordered quantum many-body systems in the many-body localized (MBL) regime, we analyze the operator entanglement across various spatiotemporal cuts, revealing the logarithmic lightcone of entanglement spreading. We demonstrate that this logarithmic lightcone arises directly from a hierarchy of energyscales and lengthscales encoded in eigenstate correlations. By characterizing the statistics of these hierarchical scales, we develop a microscopic theory for the spatiotemporal structure of entanglement spreading in MBL systems---without invoking phenomenological constructs such as $\ensuremath{\ell}$-bits. This approach reveals the fundamental connection between eigenstate correlations and the emergent entanglement structure in MBL systems.
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.
We use complexity theory to rigorously investigate the difficulty of classically simulating evolution under many-body localized (MBL) Hamiltonians. Using the defining feature that MBL systems have a complete set of quasilocal integrals of motion (LIOMs), we demonstrate a transition in the classical complexity of simulating such systems as a function of evolution time. On one side, we construct a quasipolynomial-time tensor-network-inspired algorithm for strong simulation of 1D MBL systems (i.e., calculating the expectation value of arbitrary products of local observables) evolved for any time polynomial in the system size. On the other side, we prove that even weak simulation, i.e. sampling, becomes formally hard after an exponentially long evolution time, assuming widely believed conjectures in complexity theory. Finally, using the consequences of our classical simulation results, we also show that the quantum circuit complexity for MBL systems is sublinear in evolution time. This result is a counterpart to a recent proof that the complexity of random quantum circuits grows linearly in time.
Interacting quantum many-body systems are expected to thermalize, in the sense that the evolution of local expectation values approaches a stationary value resembling a thermal ensemble. This intuition is notably contradicted in systems exhibiting many-body localisation (MBL). In stark contrast to the non-interacting case of Anderson localisation, the entanglement of states grows without limit over time, albeit slowly. In this work, we establish a novel link between quantum information theory and notions of condensed matter physics, capturing this phenomenon in the Heisenberg picture. We show that the mere existence of local constants of motion, often taken as the defining property of MBL, together with a generic spectrum of the Hamiltonian, is already sufficient to rigorously prove information propagation: these systems can be used to send a classical bit over arbitrary distances, in that the impact of a local perturbation can be detected arbitrarily far away. This counterintuitive result is compatible with and further corroborates the intuition of a slow entanglement growth following global quenches in MBL systems. We perform a detailed perturbation analysis of quasi-local constants of motion and also show that they indeed can be used to construct efficient spectral tensor networks, as recently suggested. Our results provide a detailed and at the same time model-independent picture of information propagation in MBL systems.