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:2407.13617 · Physical Review X

Fast Scrambling at the Boundary

Ancel Larzul, Anirvan M. Sengupta, Antoine Georges, Marco Schirò

Published Jan 20, 2026• 0 citations
arXivHTMLPDF

Abstract

Many-body systems that saturate the quantum bound on chaos are attracting interest across a wide range of fields. Notable examples include the Sachdev-Ye-Kitaev model and its variations, all characterized by some form or randomness and all-to-all couplings. Here, we study many-body quantum chaos in a quantum impurity model showing non-Fermi-liquid physics, the overscreened multichannel S U ( N ) Kondo model. We exactly compute the low-temperature behavior of the out-of-time order correlator in the limit of large N and large number of channels, K , at a fixed ratio γ = K / N . Because of strong correlations at the impurity site, the spin fractionalizes in auxiliary fermions and bosons. We show that all the degrees of freedom of our theory acquire a Lyapunov exponent that is linear in temperature as T → 0 , with a prefactor that depends on γ . Remarkably, for N = K , the impurity spin displays maximal chaos, while bosons and fermions only reach half of the maximal Lyapunov exponent. Our results highlight two key features: a nondisordered model that is maximally chaotic due to strong correlations at its boundary, with the maximal chaos appearing in a composite gauge-invariant operator, the impurity spin, and not in the auxiliary single-particle degrees of freedom.

DOI
10.1103/4ys9-ct98

Related papers

Quantum chaos of the Bose-Fermi Kondo model at intermediate temperature

Xinloong Han, Zuodong Yu

We study the quantum chaos in the Bose-Fermi Kondo model in which the impurity spin interacts with conduction electrons and a bosonic bath at the intermediate temperature in the large-$N$ limit. The out-of-time-ordered correlator is calculated based on the Bethe-Salpeter equation and the Lyapunov exponent ${\ensuremath{\lambda}}_{L}$ is extracted. Our calculation shows that the Lyapunov exponent monotonically increases as the Kondo coupling ${J}_{K}$ increases, and it can reach an order of ${\ensuremath{\lambda}}_{L}\ensuremath{\approx}T$ as ${J}_{K}$ approaches the multichannel Kondo fixed point ($MCK$). Furthermore, we also demonstrate that ${\ensuremath{\lambda}}_{L}$ decreases monotonically as the impurity and bosonic bath coupling $g$ increases, which is contrary to the general expectation that the most chaotic property occurs at the quantum critical point with the non-Fermi-liquid nature.

arXiv:2103.11929•Physical review. B./Physical review. B•Aug 20, 2021•6 citations

Quantum chaos and phase transition in the Yukawa–Sachdev-Ye-Kitaev model

Andrew Davis, Yuxuan Wang

We analyze the relation between quantum chaotic behavior and phase transitions of the Yukawa--Sachdev-Ye-Kitaev model as a function of filling and temperature, which describes random Yukawa interactions between $N$ complex fermions and $M$ bosons in zero spatial dimensions for both the non-Fermi liquid and insulating states at finite temperature and chemical potential. We solve the ladder equations for the out-of-time-order correlator (OTOC) for both bosons and fermions. Despite the appearance of the chemical potential in the Hamiltonian, which explicitly introduces an additional energy scale, the OTOCs for the fermions and bosons in the non-Fermi liquid state turn out to be unaffected, and the Lyapunov exponents that diagnose chaos remain maximal. As the chemical potential increases, the system is known to experience a first-order transition from a critical phase to a gapped insulating phase. We postulate that the boundary of the region in parameter space where each phase is (meta)stable coincides with the curve on which the Lyapunov exponent is maximal. By calculating the exponent in the insulating phase and comparing to numerical results on the boundaries of stability, we show that this is plausible.

arXiv:2212.03265•Physical review. B./Physical review. B•May 11, 2023•14 citations

Signatures of the interplay between chaos and local criticality on the dynamics of scrambling in many-body systems

Felix Meier, Mathias Steinhuber, Juan Diego Urbina +2 more

Fast scrambling, quantified by the exponential initial growth of out-of-time-ordered correlators (OTOCs), is the ability to efficiently spread quantum correlations among the degrees of freedom of interacting systems and constitutes a characteristic signature of local unstable dynamics. As such, it may equally manifest both in systems displaying chaos or in integrable systems around criticality. Here we go beyond these extreme regimes with an exhaustive study of the interplay between local criticality and chaos right at the intricate phase-space region where the integrability-chaos transition first appears. We address systems with a well-defined classical (mean-field) limit, as coupled large spins and Bose-Hubbard chains, thus allowing for semiclassical analysis. Our aim is to investigate the dependence of the exponential growth of the OTOCs, defining the quantum Lyapunov exponent λ_{q} on quantities derived from the classical system with mixed phase space, specifically the local stability exponent of a fixed point λ_{loc} as well as the maximal Lyapunov exponent λ_{L} of the chaotic region around it. By extensive numerical simulations covering a wide range of parameters we give support to a conjectured linear dependence 2λ_{q}=aλ_{L}+bλ_{loc}, providing a simple route to characterize scrambling at the border between chaos and integrability.

arXiv:2211.12147•Physical review. E•May 2, 2023•5 citations

Scrambling dynamics and many-body chaos in a random dipolar spin model

Ahmet Keleş, Erhai Zhao, W. Vincent Liu

Is there a quantum many-body system that scrambles information as fast as a black hole? The Sachev-Ye-Kitaev model can saturate the conjectured bound for chaos, but it requires random all-to-all couplings of Majorana fermions that are hard to realize in experiments. Here we examine a quantum spin model of randomly oriented dipoles where the spin exchange is governed by dipole-dipole interactions. The model is inspired by recent experiments on dipolar spin systems of magnetic atoms, dipolar molecules, and nitrogen-vacancy centers. We map out the phase diagram of this model by computing the energy-level statistics, spectral form factor, and out-of-time-order correlation (OTOC) functions. We find a broad regime of many-body chaos where the energy levels obey Wigner-Dyson statistics and the OTOC shows distinctive behaviors at different times: Its early-time dynamics is characterized by an exponential growth, while the approach to its saturated value at late times obeys a power law. The temperature scaling of the Lyapunov exponent ${\ensuremath{\lambda}}_{L}$ shows that while it is well below the conjectured bound $2\ensuremath{\pi}T$ at high temperatures, ${\ensuremath{\lambda}}_{L}$ approaches the bound at low temperatures and for large number of spins.

arXiv:1810.03815•Physical Review A•May 29, 2019•15 citations

Prethermalization and transient dynamics of multichannel Kondo systems under generic quantum quenches: Insights from large- N Schwinger-Keldysh approach

Iksu Jang, Po-Yao Chang

Understanding out-of-equilibrium many-body quantum systems is an essential task in contemporary physics. While advanced numerical methods have been developed, capturing the universal dynamics of generic many-body quantum systems still remains a significant challenge. In this study, we focus on the multichannel Kondo impurity (MCKI) model, an intriguing theoretical model hosting an overscreened Kondo state with non-Fermi-liquid characteristics, which serves as a theoretical platform for investigating universal properties in many-body quantum dynamics. Utilizing the large-$N$ Schwinger-Keldysh approach, we systematically investigate both transient dynamics and long-time quasiequilibrium properties in the MCKI model following a sudden change of the Kondo coupling. Our investigations encompass two distinct initial states: the overscreened Kondo state and the high-temperature Fermi-liquid state. For the overscreened Kondo state initial condition, we observe oscillations in various physical observables, including spin-spin correlations and the Kondo order parameter. These oscillations signify the quantum revival of the entangled state that characterizes the overscreened Kondo state. On the other hand, in the case of the high-temperature Fermi-liquid initial state, the absence of oscillations can be attributed to the dephasing mechanism. Furthermore, we discover that the system reaches a quasiequilibrium on an $O(1)$ timescale. This quasiequilibrium manifests as incoherent thermalization between the impurity and the conduction electrons, in which we observe the nonvanishing effective temperature difference between the Abrikosov fermion representing the impurity spin and the composite boson formed by the Abrikosov fermion and the conduction electrons at the impurity site. This quasiequilibrium can be interpreted as prethermalization. Incorporating the $1/N$ correction, we demonstrate that the system attains complete thermalization on an $O(N)$ timescale. Additionally, we discuss the quantum cooling effect and quantum Boltzmann equations. Our comprehensive study establishes a foundation for investigating quantum many-body systems using large-$N$ quantum field-theory treatment, while our findings reveal several universal properties of quantum dynamics and provide a different perspective on prethermalization.

Physical review. B./Physical review. B•Apr 18, 2024•3 citations

The relation between classical and quantum Lyapunov exponent and the bound on chaos in classically chaotic quantum systems

Fabian Haneder, Gerrit Caspari, Juan Diego Urbina +1 more

Out-of-Time-Ordered Commutators (OTOCs), representing a key diagnostic for scrambling as a facet of short-time quantum chaos, have attracted wide-ranging interest, from many-body physics to quantum gravity. By means of a suitable form of the Wigner-Moyal expansion, and invoking ensemble equivalence in statistical physics, we provide a consistent approach to the growth rate of the OTOC for many-body systems with chaotic classical limit where both the classical Lyapunov exponent and the quantum nature of the density of states enter. Applying this construction to quantized high-dimensional hyperbolic motion, i.e., a quantum chaotic system that exhibits gravity-like correlation functions in the late-time regime, we compute the OTOC growth rate $Λ$ as a function of the number of degrees of freedom, $f$, and inverse temperature, $β$. We show that the scaled growth rate, $Λ/f$, can be described by a universal function of $f β$ and displays a cross-over from classical to quantum behavior as we increase $f$ and/or lower the temperature. In the deep quantum regime of infinite $f$, we find maximally fast scrambling in the sense of the Maldacena-Shenker-Stanford bound on chaos. This elucidates the non-perturbative mechanism underlying the saturation of the bound via quantum contributions to the mean density of states, and it provides further support for this dynamical system as a dual to two-dimensional quantum gravity. In this way, we present first evidence of maximally fast scrambling in a quantum chaotic system with a well-defined classical Hamiltonian limit, without invoking any external mechanism such as (disorder) averaging.

nlin.CDCross-listedarXiv:2512.19869•Dec 24, 2025•0 citations
Many-body systems that saturate the quantum bound on chaos are attracting interest across a wide range of fields. Notable examples include the Sachdev-Ye-Kitaev model and its variations, all characterized by some form or randomness and all-to-all couplings. Here, we study many-body quantum chaos in a quantum impurity model showing non-Fermi-liquid physics, the overscreened multichannel S U ( N ) Kondo model. We exactly compute the low-temperature behavior of the out-of-time order correlator in the limit of large N and large number of channels, K , at a fixed ratio γ = K / N . Because of strong correlations at the impurity site, the spin fractionalizes in auxiliary fermions and bosons. We show that all the degrees of freedom of our theory acquire a Lyapunov exponent that is linear in temperature as T → 0 , with a prefactor that depends on γ . Remarkably, for N = K , the impurity spin displays maximal chaos, while bosons and fermions only reach half of the maximal Lyapunov exponent. Our results highlight two key features: a nondisordered model that is maximally chaotic due to strong correlations at its boundary, with the maximal chaos appearing in a composite gauge-invariant operator, the impurity spin, and not in the auxiliary single-particle degrees of freedom.
We study the quantum chaos in the Bose-Fermi Kondo model in which the impurity spin interacts with conduction electrons and a bosonic bath at the intermediate temperature in the large-NNN limit. The out-of-time-ordered correlator is calculated based on the Bethe-Salpeter equation and the Lyapunov exponent ${\ensuremath{\lambda}}_{L}$ is extracted. Our calculation shows that the Lyapunov exponent monotonically increases as the Kondo coupling JK{J}_{K}JK​ increases, and it can reach an order of ${\ensuremath{\lambda}}_{L}\ensuremath{\approx}T$ as JK{J}_{K}JK​ approaches the multichannel Kondo fixed point (MCKMCKMCK). Furthermore, we also demonstrate that ${\ensuremath{\lambda}}_{L}$ decreases monotonically as the impurity and bosonic bath coupling ggg increases, which is contrary to the general expectation that the most chaotic property occurs at the quantum critical point with the non-Fermi-liquid nature.
We analyze the relation between quantum chaotic behavior and phase transitions of the Yukawa--Sachdev-Ye-Kitaev model as a function of filling and temperature, which describes random Yukawa interactions between NNN complex fermions and MMM bosons in zero spatial dimensions for both the non-Fermi liquid and insulating states at finite temperature and chemical potential. We solve the ladder equations for the out-of-time-order correlator (OTOC) for both bosons and fermions. Despite the appearance of the chemical potential in the Hamiltonian, which explicitly introduces an additional energy scale, the OTOCs for the fermions and bosons in the non-Fermi liquid state turn out to be unaffected, and the Lyapunov exponents that diagnose chaos remain maximal. As the chemical potential increases, the system is known to experience a first-order transition from a critical phase to a gapped insulating phase. We postulate that the boundary of the region in parameter space where each phase is (meta)stable coincides with the curve on which the Lyapunov exponent is maximal. By calculating the exponent in the insulating phase and comparing to numerical results on the boundaries of stability, we show that this is plausible.
Fast scrambling, quantified by the exponential initial growth of out-of-time-ordered correlators (OTOCs), is the ability to efficiently spread quantum correlations among the degrees of freedom of interacting systems and constitutes a characteristic signature of local unstable dynamics. As such, it may equally manifest both in systems displaying chaos or in integrable systems around criticality. Here we go beyond these extreme regimes with an exhaustive study of the interplay between local criticality and chaos right at the intricate phase-space region where the integrability-chaos transition first appears. We address systems with a well-defined classical (mean-field) limit, as coupled large spins and Bose-Hubbard chains, thus allowing for semiclassical analysis. Our aim is to investigate the dependence of the exponential growth of the OTOCs, defining the quantum Lyapunov exponent λ_{q} on quantities derived from the classical system with mixed phase space, specifically the local stability exponent of a fixed point λ_{loc} as well as the maximal Lyapunov exponent λ_{L} of the chaotic region around it. By extensive numerical simulations covering a wide range of parameters we give support to a conjectured linear dependence 2λ_{q}=aλ_{L}+bλ_{loc}, providing a simple route to characterize scrambling at the border between chaos and integrability.
Is there a quantum many-body system that scrambles information as fast as a black hole? The Sachev-Ye-Kitaev model can saturate the conjectured bound for chaos, but it requires random all-to-all couplings of Majorana fermions that are hard to realize in experiments. Here we examine a quantum spin model of randomly oriented dipoles where the spin exchange is governed by dipole-dipole interactions. The model is inspired by recent experiments on dipolar spin systems of magnetic atoms, dipolar molecules, and nitrogen-vacancy centers. We map out the phase diagram of this model by computing the energy-level statistics, spectral form factor, and out-of-time-order correlation (OTOC) functions. We find a broad regime of many-body chaos where the energy levels obey Wigner-Dyson statistics and the OTOC shows distinctive behaviors at different times: Its early-time dynamics is characterized by an exponential growth, while the approach to its saturated value at late times obeys a power law. The temperature scaling of the Lyapunov exponent ${\ensuremath{\lambda}}_{L}$ shows that while it is well below the conjectured bound $2\ensuremath{\pi}T$ at high temperatures, ${\ensuremath{\lambda}}_{L}$ approaches the bound at low temperatures and for large number of spins.
Understanding out-of-equilibrium many-body quantum systems is an essential task in contemporary physics. While advanced numerical methods have been developed, capturing the universal dynamics of generic many-body quantum systems still remains a significant challenge. In this study, we focus on the multichannel Kondo impurity (MCKI) model, an intriguing theoretical model hosting an overscreened Kondo state with non-Fermi-liquid characteristics, which serves as a theoretical platform for investigating universal properties in many-body quantum dynamics. Utilizing the large-NNN Schwinger-Keldysh approach, we systematically investigate both transient dynamics and long-time quasiequilibrium properties in the MCKI model following a sudden change of the Kondo coupling. Our investigations encompass two distinct initial states: the overscreened Kondo state and the high-temperature Fermi-liquid state. For the overscreened Kondo state initial condition, we observe oscillations in various physical observables, including spin-spin correlations and the Kondo order parameter. These oscillations signify the quantum revival of the entangled state that characterizes the overscreened Kondo state. On the other hand, in the case of the high-temperature Fermi-liquid initial state, the absence of oscillations can be attributed to the dephasing mechanism. Furthermore, we discover that the system reaches a quasiequilibrium on an O(1)O(1)O(1) timescale. This quasiequilibrium manifests as incoherent thermalization between the impurity and the conduction electrons, in which we observe the nonvanishing effective temperature difference between the Abrikosov fermion representing the impurity spin and the composite boson formed by the Abrikosov fermion and the conduction electrons at the impurity site. This quasiequilibrium can be interpreted as prethermalization. Incorporating the 1/N1/N1/N correction, we demonstrate that the system attains complete thermalization on an O(N)O(N)O(N) timescale. Additionally, we discuss the quantum cooling effect and quantum Boltzmann equations. Our comprehensive study establishes a foundation for investigating quantum many-body systems using large-NNN quantum field-theory treatment, while our findings reveal several universal properties of quantum dynamics and provide a different perspective on prethermalization.
Out-of-Time-Ordered Commutators (OTOCs), representing a key diagnostic for scrambling as a facet of short-time quantum chaos, have attracted wide-ranging interest, from many-body physics to quantum gravity. By means of a suitable form of the Wigner-Moyal expansion, and invoking ensemble equivalence in statistical physics, we provide a consistent approach to the growth rate of the OTOC for many-body systems with chaotic classical limit where both the classical Lyapunov exponent and the quantum nature of the density of states enter. Applying this construction to quantized high-dimensional hyperbolic motion, i.e., a quantum chaotic system that exhibits gravity-like correlation functions in the late-time regime, we compute the OTOC growth rate ΛΛΛ as a function of the number of degrees of freedom, fff, and inverse temperature, βββ. We show that the scaled growth rate, Λ/fΛ/fΛ/f, can be described by a universal function of fβf βfβ and displays a cross-over from classical to quantum behavior as we increase fff and/or lower the temperature. In the deep quantum regime of infinite fff, we find maximally fast scrambling in the sense of the Maldacena-Shenker-Stanford bound on chaos. This elucidates the non-perturbative mechanism underlying the saturation of the bound via quantum contributions to the mean density of states, and it provides further support for this dynamical system as a dual to two-dimensional quantum gravity. In this way, we present first evidence of maximally fast scrambling in a quantum chaotic system with a well-defined classical Hamiltonian limit, without invoking any external mechanism such as (disorder) averaging.