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

Physical review. E

Local versus translationally invariant slowest operators in quantum Ising spin chains

Ekaterina Izotova

Published Aug 24, 2023• 1 citation
Publisher

Abstract

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.

DOI
10.1103/physreve.108.024138

Related papers

Local vs. translationally-invariant slowest operators in quantum Ising spin chains

Ekaterina Izotova

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 $σ_{x}$ and $σ_{z}$ and slow rate of delocalization.

arXiv:2302.07995•Feb 16, 2023•0 citations

Slowest local operators in quantum spin chains

HyungWon Kim, Mari Carmen Bañuls, J. I. Cirac +2 more

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.

arXiv:1410.4186•Physical Review E•Jul 22, 2015•58 citations

Slow dynamics in translation-invariant quantum lattice models

Alexios A. Michailidis, Marko Žnidarič, Mariya V. Medvedyeva +3 more

Many-body quantum systems typically display fast dynamics and ballistic spreading of information. Here we address the open problem of how slow the dynamics can be after a generic breaking of integrability by local interactions. We develop a method based on degenerate perturbation theory that reveals slow dynamical regimes and delocalization processes in general translation invariant models, along with accurate estimates of their delocalization time scales. Our results shed light on the fundamental questions of the robustness of quantum integrable systems and the possibility of many-body localization without disorder. As an example, we construct a large class of one-dimensional lattice models where, despite the absence of asymptotic localization, the transient dynamics is exceptionally slow, i.e., the dynamics is indistinguishable from that of many-body localized systems for the system sizes and time scales accessible in experiments and numerical simulations.

arXiv:1706.05026•Physical review. B./Physical review. B•Mar 19, 2018•42 citations

Long time dynamics following a quench in an integrable quantum spin chain: Local versus nonlocal operators and effective thermal behavior

Davide Rossini, Sei Suzuki, Giuseppe Mussardo +2 more

We study the dynamics of the quantum Ising chain following a zero-temperature quench of the transverse field strength. Focusing on the behavior of two-point spin correlation functions, we show that the correlators of the order parameter display an effective asymptotic thermal behavior; i.e., they decay exponentially to zero with a phase coherence rate and a correlation length dictated by the equilibrium law with an effective temperature set by the energy of the initial state. On the contrary, the two-point correlation functions of the transverse magnetization or the density-of-kinks operator decay as a power law and do not exhibit thermal behavior. We argue that the different behavior is linked to the locality of the corresponding operator with respect to the quasiparticles of the model: nonlocal operators, such as the order parameter, behave thermally, while local ones do not. We study which features of the two-point correlators are a consequence of the integrability of the model by analyzing their robustness with respect to a sufficiently strong integrability-breaking term.

arXiv:1002.2842•Physical Review B•Oct 11, 2010•158 citations

Eigenstate thermalization for local versus translationally invariant observables

Rohit Patil, Marcos Rigol

Local observables and their translationally invariant counterparts are generally thought to provide the same predictions for experiments. While this equivalence holds for expectation values in clean systems (up to finite-size effects), it is often assumed to extend to correlation functions, where it need not hold. We examine this assumption from the perspective of the eigenstate thermalization hypothesis. Specifically, we explore the spectral functions of local and translationally invariant observables in the spin-1 tilted-field Ising chain with periodic and open boundary conditions. We identify the contexts in which these observables and boundary conditions differ and those in which they are interchangeable. We unveil an off-diagonal eigenstate thermalization in translationally invariant systems for matrix elements between energy eigenstates with different quasimomenta.

quant-pharXiv:2602.09087•Physical review. B./Physical review. B•Feb 11, 2026•0 citations

Finding local integrals of motion in quantum lattice models in the thermodynamic limit

Jakub Pawłowski, J. Herbrych, Marcin Mierzejewski

Local integrals of motion (LIOMs) play a key role in understanding the long-time properties of closed macroscopic systems. They were found for selected integrable systems via complex analytical calculations. The existence of LIOMs and their structure can also be studied via numerical methods, which, however, involve exact diagonalization of Hamiltonians, posing a bottleneck for such studies. We show that finding LIOMs in translationally invariant lattice models or unitary quantum circuits can be reduced to a problem for which one may numerically find an exact solution in the thermodynamic limit. We develop a simple algorithm and demonstrate its efficiency by calculating LIOMs and bounds on correlations (the Mazur bounds) for infinite integrable spin chains and unitary circuits. Finally, we demonstrate that this approach identifies slow modes in nearly integrable spin models and estimates their relaxation times.

arXiv:2505.05882•Physical review. B./Physical review. B•Sep 24, 2025•4 citations
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.
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.
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.
Many-body quantum systems typically display fast dynamics and ballistic spreading of information. Here we address the open problem of how slow the dynamics can be after a generic breaking of integrability by local interactions. We develop a method based on degenerate perturbation theory that reveals slow dynamical regimes and delocalization processes in general translation invariant models, along with accurate estimates of their delocalization time scales. Our results shed light on the fundamental questions of the robustness of quantum integrable systems and the possibility of many-body localization without disorder. As an example, we construct a large class of one-dimensional lattice models where, despite the absence of asymptotic localization, the transient dynamics is exceptionally slow, i.e., the dynamics is indistinguishable from that of many-body localized systems for the system sizes and time scales accessible in experiments and numerical simulations.
We study the dynamics of the quantum Ising chain following a zero-temperature quench of the transverse field strength. Focusing on the behavior of two-point spin correlation functions, we show that the correlators of the order parameter display an effective asymptotic thermal behavior; i.e., they decay exponentially to zero with a phase coherence rate and a correlation length dictated by the equilibrium law with an effective temperature set by the energy of the initial state. On the contrary, the two-point correlation functions of the transverse magnetization or the density-of-kinks operator decay as a power law and do not exhibit thermal behavior. We argue that the different behavior is linked to the locality of the corresponding operator with respect to the quasiparticles of the model: nonlocal operators, such as the order parameter, behave thermally, while local ones do not. We study which features of the two-point correlators are a consequence of the integrability of the model by analyzing their robustness with respect to a sufficiently strong integrability-breaking term.
Local observables and their translationally invariant counterparts are generally thought to provide the same predictions for experiments. While this equivalence holds for expectation values in clean systems (up to finite-size effects), it is often assumed to extend to correlation functions, where it need not hold. We examine this assumption from the perspective of the eigenstate thermalization hypothesis. Specifically, we explore the spectral functions of local and translationally invariant observables in the spin-1 tilted-field Ising chain with periodic and open boundary conditions. We identify the contexts in which these observables and boundary conditions differ and those in which they are interchangeable. We unveil an off-diagonal eigenstate thermalization in translationally invariant systems for matrix elements between energy eigenstates with different quasimomenta.
Local integrals of motion (LIOMs) play a key role in understanding the long-time properties of closed macroscopic systems. They were found for selected integrable systems via complex analytical calculations. The existence of LIOMs and their structure can also be studied via numerical methods, which, however, involve exact diagonalization of Hamiltonians, posing a bottleneck for such studies. We show that finding LIOMs in translationally invariant lattice models or unitary quantum circuits can be reduced to a problem for which one may numerically find an exact solution in the thermodynamic limit. We develop a simple algorithm and demonstrate its efficiency by calculating LIOMs and bounds on correlations (the Mazur bounds) for infinite integrable spin chains and unitary circuits. Finally, we demonstrate that this approach identifies slow modes in nearly integrable spin models and estimates their relaxation times.