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:1608.03280 · Physical review. B./Physical review. B

Slow scrambling in disordered quantum systems

Brian G. Swingle, Debanjan Chowdhury

Published Feb 21, 2017• 229 citations
arXivHTMLPDF

Abstract

Understanding the spread of quantum entanglement and scrambling of information across a quantum many-body system is a fundamental problem in quantum dynamics. A necessary part of the spread of the entanglement is the growth in time of Heisenberg operators of initially localized operators. This operator growth can be diagnosed by studying the space-time structure of commutators of well-separated operators. Previous studies in this area focused on translation invariant models, but the effects of disorder, which can radically alter the motion of heat and charge, had not been studied. Here, the authors study the growth of operators in interacting systems with quenched disorder. In the localized phase, they find that operator sizes grow logarithmically in time similar to other measures of entanglement spreading. In the disordered metal, they find that operator sizes grow linearly with time, i.e., ballistically, in contrast to the diffusive motion of charge and heat. The ballistic growth of operators is quantified by the butterfly velocity which the authors relate to the charge diffusion constant and the interaction-induced inelastic scattering rate. When the diffusion of charge is slow, the resulting butterfly velocity is much smaller than the maximum speed allowed by microscopic causality constraints.

DOI
10.1103/physrevb.95.060201

Related papers

Scrambling Dynamics across a Thermalization-Localization Quantum Phase Transition

Subhayan Sahu, Shenglong Xu, Brian G. Swingle

We study quantum information scrambling, specifically the growth of Heisenberg operators, in large disordered spin chains using matrix product operator dynamics to scan across the thermalization-localization quantum phase transition. We observe ballistic operator growth for weak disorder, and a sharp transition to a phase with subballistic operator spreading. The critical disorder strength for the ballistic to subballistic transition is well below the many body localization phase transition, as determined from finite size scaling of energy eigenstate entanglement entropy in small chains. In contrast, we find that the transition from subballistic to logarithmic behavior at the actual eigenstate localization transition is not resolved in our finite numerics. These data are discussed in the context of a universal form for the growing operator shape and substantiated with a simple phenomenological model of rare regions.

arXiv:1807.06086•Physical Review Letters•Oct 18, 2019•54 citations

Information Scrambling in Quantum Many-Body Systems

Yongliang Zhang

A closed quantum system never forgets its initial state, but the encoded information can get scrambled and become inaccessible without measuring a large fraction of all the system degrees of freedom. This scrambling can be diagnosed by studying the spatial spreading of initially local operators under the Heisenberg time evolution, and the decay of the out-of-time-ordered correlators (OTOC). What insights can OTOCs provide to understand the dynamics of quantum many-body systems? What are the characteristic behaviors of OTOCs during the time evolution? How is information scrambling affected by the dissipation in open quantum many-body systems? We first study slow scrambling in many-body localized systems via calculating various correlators, two-point retarded correlators and OTOCs. Comparing with retarded correlators, OTOCs provide more information about the dynamics. We find that disorder slows and partially halts the onset of information scrambling. Instead of ballistic spreading, propagation of information forms a logarithmic light cone. Next, we study the finite-size scaling of OTOCs at late times in generic thermalizing quantum many-body systems. When energy is conserved, the late-time saturation value of the OTOC of generic traceless local operators scales as an inverse polynomial in the system size. This is in contrast to the inverse exponential scaling expected for chaotic dynamics without energy conservation. We also study information scrambling in open quantum many-body systems. We define a dissipative version of OTOC and study its behaviors in a prototypical chaotic quantum chain with dissipation. We find that dissipation leads to not only the overall decay of the scrambled information due to leaking, but also structural changes so that the information light cone can only reach a finite distance even when the effect of overall decay is removed. Finally, we construct a family of local Hamiltonians for understanding the asymmetric information scrambling. Our models live on a one-dimensional lattice and exhibit asymmetric butterfly light cone between the left and right spatial directions.

•Feb 28, 2020•5 citations

Operator Spreading and the Emergence of Dissipative Hydrodynamics under Unitary Evolution with Conservation Laws

Vedika Khemani, Ashvin Vishwanath, David A. Huse

We study the scrambling of local quantum information in chaotic many-body systems in the presence of a locally conserved quantity like charge or energy that moves diffusively.The interplay between conservation laws and scrambling sheds light on the mechanism by which unitary quantum dynamics, which is reversible, gives rise to diffusive hydrodynamics, which is a slow dissipative process.We obtain our results in a random quantum circuit model that is constrained to have a conservation law.We find that a generic spreading operator consists of two parts: (i) a conserved part which comprises the weight of the spreading operator on the local conserved densities, whose dynamics is described by diffusive charge spreading; this conserved part also acts as a source that steadily emits a flux of (ii) nonconserved operators.This emission leads to dissipation in the operator hydrodynamics, with the dissipative process being the slow conversion of operator weight from local conserved operators to nonconserved, at a rate set by the local diffusion current.The emitted nonconserved parts then spread ballistically at a butterfly speed, thus becoming highly nonlocal and, hence, essentially nonobservable, thereby acting as the "reservoir" that facilitates the dissipation.In addition, we find that the nonconserved component develops a power-law tail behind its leading ballistic front due to the slow dynamics of the conserved components.This implies that the out-of-time-order commutator between two initially separated operators grows sharply upon the arrival of the ballistic front, but, in contrast to systems with no conservation laws, it develops a diffusive tail and approaches its asymptotic late-time value only as a power of time instead of exponentially.We also derive these results within an effective hydrodynamic description which contains multiple coupled diffusion equations.

arXiv:1710.09835•Physical Review X•Sep 7, 2018•502 citations

Operator and entanglement growth in nonthermalizing systems: Many-body localization and the random singlet phase

Ian MacCormack, Mao Tian Tan, Jonah Kudler-Flam +1 more

We characterize the growth and spreading of operators and entanglement in two paradigmatic nonthermalizing phases---the many-body localized (MBL) phase and the random singlet phase (RSP)---using the entanglement contour and multipartite operator entanglement measures. The entanglement contour characterizes the spacetime spreading of entanglement and reveals logarithmically growing entanglement light cones in the MBL and RSP phases, sharply contrasting the linear light cones of clean, thermalizing systems. The operator entanglement characterizes scrambling, i.e., the delocalization of information. We find slow scrambling behavior in the MBL phase; the late-time value of the tripartite mutual information scales linearly with system size, but is submaximal. The tripartite logarithmic negativity is also negative and nonzero, but smaller in magnitude, revealing an intriguing distinction between classical and quantum information scrambling in the MBL phase. This is in contrast with the RSP, which, as a noninteracting model, is nonscrambling.

arXiv:2001.08222•Physical review. B./Physical review. B•Dec 2, 2021•50 citations

Tripartite Information and Scrambling in Quantum Many-Body Systems

Niklas Bölter

This thesis contributes to the study of quantum chaos and thermalization of closed quantum systems by applying tools from quantum information theory. We study scrambling – the delocalization of quantum information – in one-dimensional quantum many-body systems using the observable-independent tripartite information 𝐼3. We consider closed quantum systems that undergo unitary time evolution governed by a Hamiltonian, and compare them to the strong scrambling caused by random unitary operators (Haar scrambling). The tripartite information is a tool from quantum information theory that allows us to quantify correlations in space and time by considering the unitary operator evolving the system as a state |𝑈 (𝑡)⟩ in a doubled Hilbert space H ⊗ H. We found a ballistic spreading of information in translation invariant systems of spinless fermions that causes Haar scrambling in interacting models. For strongly disordered versions of the XXZ chain and the transverse-field Ising chain we found a complete breakdown of in- formation transport in non-interacting Anderson localized systems, as well as a logarithmically slow spreading of the information signal in systems that are said to be many-body localized (MBL).

•Jan 1, 2023•0 citations

Operator spreading in the memory matrix formalism

Ewan McCulloch, Curt W. von Keyserlingk

Abstract The spread and scrambling of quantum information is a topic of considerable current interest. Numerous studies suggest that quantum information evolves according to hydrodynamical equations of motion, even though it is a starkly different quantity to better-known hydrodynamical variables such as charge and energy. In this work we show that the well-known memory matrix formalism for traditional hydrodynamics can be applied, with relatively little modification, to the question of operator growth in many-body quantum systems. On a conceptual level, this shores up the connection between information scrambling and hydrodynamics. At a practical level, it provides a framework for calculating quantities related to operator growth like the butterfly velocity and front diffusion constant, and for understanding how these quantities are constrained by microscopic symmetries. We apply this formalism to calculate operator-hydrodynamical coefficients perturbatively in a family of Floquet models. Our formalism allows us to identify the processes affecting information transport that arise from the spatiotemporal symmetries of the model.

arXiv:2111.08727•Journal of Physics A Mathematical and Theoretical•May 17, 2022•7 citations
Understanding the spread of quantum entanglement and scrambling of information across a quantum many-body system is a fundamental problem in quantum dynamics. A necessary part of the spread of the entanglement is the growth in time of Heisenberg operators of initially localized operators. This operator growth can be diagnosed by studying the space-time structure of commutators of well-separated operators. Previous studies in this area focused on translation invariant models, but the effects of disorder, which can radically alter the motion of heat and charge, had not been studied. Here, the authors study the growth of operators in interacting systems with quenched disorder. In the localized phase, they find that operator sizes grow logarithmically in time similar to other measures of entanglement spreading. In the disordered metal, they find that operator sizes grow linearly with time, i.e., ballistically, in contrast to the diffusive motion of charge and heat. The ballistic growth of operators is quantified by the butterfly velocity which the authors relate to the charge diffusion constant and the interaction-induced inelastic scattering rate. When the diffusion of charge is slow, the resulting butterfly velocity is much smaller than the maximum speed allowed by microscopic causality constraints.
We study quantum information scrambling, specifically the growth of Heisenberg operators, in large disordered spin chains using matrix product operator dynamics to scan across the thermalization-localization quantum phase transition. We observe ballistic operator growth for weak disorder, and a sharp transition to a phase with subballistic operator spreading. The critical disorder strength for the ballistic to subballistic transition is well below the many body localization phase transition, as determined from finite size scaling of energy eigenstate entanglement entropy in small chains. In contrast, we find that the transition from subballistic to logarithmic behavior at the actual eigenstate localization transition is not resolved in our finite numerics. These data are discussed in the context of a universal form for the growing operator shape and substantiated with a simple phenomenological model of rare regions.
A closed quantum system never forgets its initial state, but the encoded information can get scrambled and become inaccessible without measuring a large fraction of all the system degrees of freedom. This scrambling can be diagnosed by studying the spatial spreading of initially local operators under the Heisenberg time evolution, and the decay of the out-of-time-ordered correlators (OTOC). What insights can OTOCs provide to understand the dynamics of quantum many-body systems? What are the characteristic behaviors of OTOCs during the time evolution? How is information scrambling affected by the dissipation in open quantum many-body systems? We first study slow scrambling in many-body localized systems via calculating various correlators, two-point retarded correlators and OTOCs. Comparing with retarded correlators, OTOCs provide more information about the dynamics. We find that disorder slows and partially halts the onset of information scrambling. Instead of ballistic spreading, propagation of information forms a logarithmic light cone. Next, we study the finite-size scaling of OTOCs at late times in generic thermalizing quantum many-body systems. When energy is conserved, the late-time saturation value of the OTOC of generic traceless local operators scales as an inverse polynomial in the system size. This is in contrast to the inverse exponential scaling expected for chaotic dynamics without energy conservation. We also study information scrambling in open quantum many-body systems. We define a dissipative version of OTOC and study its behaviors in a prototypical chaotic quantum chain with dissipation. We find that dissipation leads to not only the overall decay of the scrambled information due to leaking, but also structural changes so that the information light cone can only reach a finite distance even when the effect of overall decay is removed. Finally, we construct a family of local Hamiltonians for understanding the asymmetric information scrambling. Our models live on a one-dimensional lattice and exhibit asymmetric butterfly light cone between the left and right spatial directions.
We study the scrambling of local quantum information in chaotic many-body systems in the presence of a locally conserved quantity like charge or energy that moves diffusively.The interplay between conservation laws and scrambling sheds light on the mechanism by which unitary quantum dynamics, which is reversible, gives rise to diffusive hydrodynamics, which is a slow dissipative process.We obtain our results in a random quantum circuit model that is constrained to have a conservation law.We find that a generic spreading operator consists of two parts: (i) a conserved part which comprises the weight of the spreading operator on the local conserved densities, whose dynamics is described by diffusive charge spreading; this conserved part also acts as a source that steadily emits a flux of (ii) nonconserved operators.This emission leads to dissipation in the operator hydrodynamics, with the dissipative process being the slow conversion of operator weight from local conserved operators to nonconserved, at a rate set by the local diffusion current.The emitted nonconserved parts then spread ballistically at a butterfly speed, thus becoming highly nonlocal and, hence, essentially nonobservable, thereby acting as the "reservoir" that facilitates the dissipation.In addition, we find that the nonconserved component develops a power-law tail behind its leading ballistic front due to the slow dynamics of the conserved components.This implies that the out-of-time-order commutator between two initially separated operators grows sharply upon the arrival of the ballistic front, but, in contrast to systems with no conservation laws, it develops a diffusive tail and approaches its asymptotic late-time value only as a power of time instead of exponentially.We also derive these results within an effective hydrodynamic description which contains multiple coupled diffusion equations.
We characterize the growth and spreading of operators and entanglement in two paradigmatic nonthermalizing phases---the many-body localized (MBL) phase and the random singlet phase (RSP)---using the entanglement contour and multipartite operator entanglement measures. The entanglement contour characterizes the spacetime spreading of entanglement and reveals logarithmically growing entanglement light cones in the MBL and RSP phases, sharply contrasting the linear light cones of clean, thermalizing systems. The operator entanglement characterizes scrambling, i.e., the delocalization of information. We find slow scrambling behavior in the MBL phase; the late-time value of the tripartite mutual information scales linearly with system size, but is submaximal. The tripartite logarithmic negativity is also negative and nonzero, but smaller in magnitude, revealing an intriguing distinction between classical and quantum information scrambling in the MBL phase. This is in contrast with the RSP, which, as a noninteracting model, is nonscrambling.
This thesis contributes to the study of quantum chaos and thermalization of closed quantum systems by applying tools from quantum information theory. We study scrambling – the delocalization of quantum information – in one-dimensional quantum many-body systems using the observable-independent tripartite information 𝐼3. We consider closed quantum systems that undergo unitary time evolution governed by a Hamiltonian, and compare them to the strong scrambling caused by random unitary operators (Haar scrambling). The tripartite information is a tool from quantum information theory that allows us to quantify correlations in space and time by considering the unitary operator evolving the system as a state |𝑈 (𝑡)⟩ in a doubled Hilbert space H ⊗ H. We found a ballistic spreading of information in translation invariant systems of spinless fermions that causes Haar scrambling in interacting models. For strongly disordered versions of the XXZ chain and the transverse-field Ising chain we found a complete breakdown of in- formation transport in non-interacting Anderson localized systems, as well as a logarithmically slow spreading of the information signal in systems that are said to be many-body localized (MBL).
Abstract The spread and scrambling of quantum information is a topic of considerable current interest. Numerous studies suggest that quantum information evolves according to hydrodynamical equations of motion, even though it is a starkly different quantity to better-known hydrodynamical variables such as charge and energy. In this work we show that the well-known memory matrix formalism for traditional hydrodynamics can be applied, with relatively little modification, to the question of operator growth in many-body quantum systems. On a conceptual level, this shores up the connection between information scrambling and hydrodynamics. At a practical level, it provides a framework for calculating quantities related to operator growth like the butterfly velocity and front diffusion constant, and for understanding how these quantities are constrained by microscopic symmetries. We apply this formalism to calculate operator-hydrodynamical coefficients perturbatively in a family of Floquet models. Our formalism allows us to identify the processes affecting information transport that arise from the spatiotemporal symmetries of the model.