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:2208.05509 · PRX Quantum

Disordered Lieb-Robinson Bounds in One Dimension

Christopher L. Baldwin, Adam Ehrenberg, Andrew Y. Guo, Alexey Vyacheslavovich Gorshkov

Published Jun 22, 2023• 9 citations
arXivHTMLPDF

Abstract

By tightening the conventional Lieb-Robinson bounds to better handle systems that lack translation invariance, we determine the extent to which "weak links" suppress operator growth in disordered onedimensional spin chains.In particular, we prove that ballistic growth is impossible when the distribution of coupling strengths μ(J ) has a sufficiently heavy tail at small J and we identify the correct dynamical exponent to use instead.Furthermore, through a detailed analysis of the special case in which the couplings are genuinely random and independent, we find that the standard formulation of Lieb-Robinson bounds is insufficient to capture the complexity of the dynamics-we must distinguish between bounds that hold for all sites of the chain and bounds that hold for a subsequence of sites and we show by explicit example that these two can have dramatically different behaviors.All the same, our result for the dynamical exponent is tight, in that we prove by counterexample that there cannot exist any Lieb-Robinson bound with a smaller exponent.We close by discussing the implications of our results, both major and minor, for numerous applications ranging from quench dynamics to the structure of ground states.

DOI
10.1103/prxquantum.4.020349

Related papers

Subballistic operator growth in spin chains with heavy-tailed random fields

Christopher L. Baldwin

An active area of research in quantum many-body physics is the extent to which quenched disorder impedes ``natural'' behaviors, such as transport and thermalization. This is an inherently challenging problem to study, and thus it is particularly valuable to have rigorous results that unambiguously prove the absence of transport in certain situations. This work provides such a result: the author uses here the machinery of Lieb-Robinson bounds to prove that ballistic and even diffusive motion is indeed impossible when the distribution of on-site disorder is sufficiently heavy tailed, regardless of most other details of the Hamiltonian.

arXiv:2409.17242•Physical review. B./Physical review. B•May 28, 2025•3 citations

Lieb-Robinson Bounds with Exponential-in-Volume Tails

Ben T. McDonough, Chao Yin, Andrew Lucas +1 more

Lieb-Robinson bounds demonstrate the emergence of locality in many-body quantum systems. Intuitively, Lieb-Robinson bounds state that, with local or exponentially decaying interactions, the correlation that can be built up between two sites separated by distance r after a time t decays as exp  ( v t − r ) , where v is the emergent Lieb-Robinson velocity. In many problems, it is important to also capture how much of an operator grows to act on r d sites in d spatial dimensions. Perturbation theory and cluster expansion methods suggest that, at short times, these volume-filling operators are suppressed as exp  ( − r d ) . We confirm this intuition, showing that, for r > v t , the volume-filling operator is suppressed by exp  ( − ( r − v t ) d / ( v t ) d − 1 ) . This closes a conceptual and practical gap between the cluster expansion and the Lieb-Robinson bound. We then present two very different applications of this new bound. Firstly, we obtain improved bounds on the classical computational resources necessary to simulate many-body dynamics with error tolerance ε for any finite time t : as ε becomes sufficiently small, only ε − O ( t d − 1 ) resources are needed. A protocol that likely saturates this bound is given. Secondly, we prove that disorder operators have volume-law suppression near the “solvable (Ising) point” in quantum phases with spontaneous symmetry breaking, which implies a new diagnostic for distinguishing many-body phases of quantum matter.

arXiv:2502.02652•PRX Quantum•Aug 20, 2025•1 citation

A Lieb–Robinson bound for quantum spin chains with strong on-site impurities

Martin Gebert, Alvin Moon, Bruno Nachtergaele

We consider a quantum spin chain with nearest neighbor interactions and sparsely distributed on-site impurities. We prove commutator bounds for its Heisenberg dynamics which incorporate the coupling strengths of the impurities. The impurities are assumed to satisfy a minimum spacing, and each impurity has a non-degenerate spectrum. Our results are proven in a broadly applicable setting, both in finite volume and in thermodynamic limit. We apply our results to improve Lieb–Robinson bounds for the Heisenberg spin chain with a random, sparse transverse field drawn from a heavy-tailed distribution.

arXiv:2104.00968•Reviews in Mathematical Physics•Jan 15, 2022•4 citations

New Anomalous Lieb-Robinson Bounds in Quasiperiodic XY Chains

David Damanik, Marius Lemm, Milivoje Lukić +1 more

We announce and sketch the rigorous proof of a new kind of anomalous (or sub-ballistic) Lieb-Robinson (LR) bound for an isotropic $XY$ chain in a quasiperiodic transversal magnetic field. Instead of the usual effective light cone $|x|\ensuremath{\le}v|t|$, we obtain $|x|\ensuremath{\le}v|t{|}^{\ensuremath{\alpha}}$ for some $0<\ensuremath{\alpha}<1$. We can characterize the allowed values of $\ensuremath{\alpha}$ exactly as those exceeding the upper transport exponent ${\ensuremath{\alpha}}_{u}^{+}$ of a one-body Schr\"odinger operator. To our knowledge, this is the first rigorous derivation of anomalous quantum many-body transport. We also discuss anomalous LR bounds with power-law tails for a random dimer field.

arXiv:1408.1796•Physical Review Letters•Sep 18, 2014•23 citations

Applications of Lieb-Robinson Bounds to Quantum Dynamics with and without Disorder

Jake Reschke

This dissertation presents new results on two problems concerning the dynamics of certainclasses of interacting many-body systems in quantum mechanics. Chapter 1 is an introduction,\nwhich includes mathematical preliminaries and a summary of the results.\nIn Chapter 2 we introduce the notion of transmission time to study the dynamics of disordered\nquantum spin chains and prove results relating its behavior to many-body localization properties.\nWe also study two versions of the so-called Local Integrals of Motion (LIOM) representation of\nspin chain Hamiltonians and their relation to dynamical many-body localization. We prove that\nuniform-in-time dynamical localization expressed by a zero-velocity Lieb-Robinson bound implies\nthe existence of a LIOM representation of the dynamics as well as a weak converse of this statement.\nWe also prove that for a class of spin chains satisfying a form of exponential dynamical localization,\nsparse perturbations result in a dynamics in which transmission times diverge at least as a power\nlaw of distance, with a power for which we provide lower bound that diverges with increasing\nsparseness of the perturbation.\nIn Chapter 3 we introduce a class of UV-regularized two-body interactions for fermions in Rd\nand prove a Lieb-Robinson estimate for the dynamics of this class of many-body systems. As a\nstep toward this result, we also prove a propagation bound of Lieb-Robinson type for Schrodinger\noperators. We apply the propagation bound to prove the existence of infnite-volume dynamics as\na strongly continuous group of automorphisms on the CAR algebra.

•Jan 1, 2021•0 citations

On anomalous Lieb–Robinson bounds for the Fibonacci XY chain

David Damanik, Marius Lemm, Milivoje Lukić +1 more

We rigorously prove a new kind of anomalous (or sub-ballistic) Lieb–Robinson bound for the isotropic XY chain with Fibonacci external magnetic field at arbitrary coupling. It is anomalous in that the usual exponential decay in |x|-v|t| is replaced by exponential decay in |x|-v|t|^\alpha with 0<\alpha<1 . In fact, we can characterize the values of \alpha for which such a bound holds as those exceeding \alpha_u^+ , the upper transport exponent of the one-body Fibonacci Hamiltonian. Following the approach of [14], we relate Lieb–Robinson bounds to dynamical bounds for the one-body Hamiltonian corresponding to the XY chain via the Jordan–Wigner transformation; in our case the one-body Hamiltonian with Fibonacci potential. We can bound its dynamics by adapting techniques developed in [8, 9, 2, 3] to our purposes. To our knowledge, this is the first rigorous derivation of anomalous quantum many-body transport. Along the way, we prove a new result about the one-body Fibonacci Hamiltonian: the upper transport exponent agrees with the time-averaged upper transport exponent, see Corollary 2.9. We also explain why our method does not extend to yield anomalous Lieb–Robinson bounds of power-law type for the random dimer model.

arXiv:1407.4924•Journal of Spectral Theory•Sep 15, 2016•12 citations
By tightening the conventional Lieb-Robinson bounds to better handle systems that lack translation invariance, we determine the extent to which "weak links" suppress operator growth in disordered onedimensional spin chains.In particular, we prove that ballistic growth is impossible when the distribution of coupling strengths μ(J ) has a sufficiently heavy tail at small J and we identify the correct dynamical exponent to use instead.Furthermore, through a detailed analysis of the special case in which the couplings are genuinely random and independent, we find that the standard formulation of Lieb-Robinson bounds is insufficient to capture the complexity of the dynamics-we must distinguish between bounds that hold for all sites of the chain and bounds that hold for a subsequence of sites and we show by explicit example that these two can have dramatically different behaviors.All the same, our result for the dynamical exponent is tight, in that we prove by counterexample that there cannot exist any Lieb-Robinson bound with a smaller exponent.We close by discussing the implications of our results, both major and minor, for numerous applications ranging from quench dynamics to the structure of ground states.
An active area of research in quantum many-body physics is the extent to which quenched disorder impedes ``natural'' behaviors, such as transport and thermalization. This is an inherently challenging problem to study, and thus it is particularly valuable to have rigorous results that unambiguously prove the absence of transport in certain situations. This work provides such a result: the author uses here the machinery of Lieb-Robinson bounds to prove that ballistic and even diffusive motion is indeed impossible when the distribution of on-site disorder is sufficiently heavy tailed, regardless of most other details of the Hamiltonian.
Lieb-Robinson bounds demonstrate the emergence of locality in many-body quantum systems. Intuitively, Lieb-Robinson bounds state that, with local or exponentially decaying interactions, the correlation that can be built up between two sites separated by distance r after a time t decays as exp  ( v t − r ) , where v is the emergent Lieb-Robinson velocity. In many problems, it is important to also capture how much of an operator grows to act on r d sites in d spatial dimensions. Perturbation theory and cluster expansion methods suggest that, at short times, these volume-filling operators are suppressed as exp  ( − r d ) . We confirm this intuition, showing that, for r > v t , the volume-filling operator is suppressed by exp  ( − ( r − v t ) d / ( v t ) d − 1 ) . This closes a conceptual and practical gap between the cluster expansion and the Lieb-Robinson bound. We then present two very different applications of this new bound. Firstly, we obtain improved bounds on the classical computational resources necessary to simulate many-body dynamics with error tolerance ε for any finite time t : as ε becomes sufficiently small, only ε − O ( t d − 1 ) resources are needed. A protocol that likely saturates this bound is given. Secondly, we prove that disorder operators have volume-law suppression near the “solvable (Ising) point” in quantum phases with spontaneous symmetry breaking, which implies a new diagnostic for distinguishing many-body phases of quantum matter.
We consider a quantum spin chain with nearest neighbor interactions and sparsely distributed on-site impurities. We prove commutator bounds for its Heisenberg dynamics which incorporate the coupling strengths of the impurities. The impurities are assumed to satisfy a minimum spacing, and each impurity has a non-degenerate spectrum. Our results are proven in a broadly applicable setting, both in finite volume and in thermodynamic limit. We apply our results to improve Lieb–Robinson bounds for the Heisenberg spin chain with a random, sparse transverse field drawn from a heavy-tailed distribution.
We announce and sketch the rigorous proof of a new kind of anomalous (or sub-ballistic) Lieb-Robinson (LR) bound for an isotropic XYXYXY chain in a quasiperiodic transversal magnetic field. Instead of the usual effective light cone $|x|\ensuremath{\le}v|t|$, we obtain $|x|\ensuremath{\le}v|t{|}^{\ensuremath{\alpha}}$ for some $0<\ensuremath{\alpha}<1$. We can characterize the allowed values of $\ensuremath{\alpha}$ exactly as those exceeding the upper transport exponent ${\ensuremath{\alpha}}_{u}^{+}$ of a one-body Schr\"odinger operator. To our knowledge, this is the first rigorous derivation of anomalous quantum many-body transport. We also discuss anomalous LR bounds with power-law tails for a random dimer field.
This dissertation presents new results on two problems concerning the dynamics of certainclasses of interacting many-body systems in quantum mechanics. Chapter 1 is an introduction,\nwhich includes mathematical preliminaries and a summary of the results.\nIn Chapter 2 we introduce the notion of transmission time to study the dynamics of disordered\nquantum spin chains and prove results relating its behavior to many-body localization properties.\nWe also study two versions of the so-called Local Integrals of Motion (LIOM) representation of\nspin chain Hamiltonians and their relation to dynamical many-body localization. We prove that\nuniform-in-time dynamical localization expressed by a zero-velocity Lieb-Robinson bound implies\nthe existence of a LIOM representation of the dynamics as well as a weak converse of this statement.\nWe also prove that for a class of spin chains satisfying a form of exponential dynamical localization,\nsparse perturbations result in a dynamics in which transmission times diverge at least as a power\nlaw of distance, with a power for which we provide lower bound that diverges with increasing\nsparseness of the perturbation.\nIn Chapter 3 we introduce a class of UV-regularized two-body interactions for fermions in Rd\nand prove a Lieb-Robinson estimate for the dynamics of this class of many-body systems. As a\nstep toward this result, we also prove a propagation bound of Lieb-Robinson type for Schrodinger\noperators. We apply the propagation bound to prove the existence of infnite-volume dynamics as\na strongly continuous group of automorphisms on the CAR algebra.
We rigorously prove a new kind of anomalous (or sub-ballistic) Lieb–Robinson bound for the isotropic XY chain with Fibonacci external magnetic field at arbitrary coupling. It is anomalous in that the usual exponential decay in |x|-v|t| is replaced by exponential decay in |x|-v|t|^\alpha with 0<\alpha<1 . In fact, we can characterize the values of \alpha for which such a bound holds as those exceeding \alpha_u^+ , the upper transport exponent of the one-body Fibonacci Hamiltonian. Following the approach of [14], we relate Lieb–Robinson bounds to dynamical bounds for the one-body Hamiltonian corresponding to the XY chain via the Jordan–Wigner transformation; in our case the one-body Hamiltonian with Fibonacci potential. We can bound its dynamics by adapting techniques developed in [8, 9, 2, 3] to our purposes. To our knowledge, this is the first rigorous derivation of anomalous quantum many-body transport. Along the way, we prove a new result about the one-body Fibonacci Hamiltonian: the upper transport exponent agrees with the time-averaged upper transport exponent, see Corollary 2.9. We also explain why our method does not extend to yield anomalous Lieb–Robinson bounds of power-law type for the random dimer model.