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:1703.02545 · Physical review. D/Physical review. D.

Onset of many-body chaos in the O(N) model

Debanjan Chowdhury, Brian G. Swingle

Published Sep 12, 2017• 139 citations
arXivHTMLPDF

Abstract

The growth of commutators of initially commuting local operators diagnoses the onset of chaos in quantum many-body systems. We compute such commutators of local field operators with $N$ components in the ($2+1$)-dimensional $O(N)$ nonlinear sigma model to leading order in $1/N$. The system is taken to be in thermal equilibrium at a temperature $T$ above the zero temperature quantum critical point separating the symmetry broken and unbroken phases. The commutator grows exponentially in time with a rate denoted ${\ensuremath{\lambda}}_{L}$. At large $N$ the growth of chaos as measured by ${\ensuremath{\lambda}}_{L}$ is slow because the model is weakly interacting, and we find ${\ensuremath{\lambda}}_{L}\ensuremath{\approx}3.2T/N$. The scaling with temperature is dictated by conformal invariance of the underlying quantum critical point. We also show that operators grow ballistically in space with a ``butterfly velocity'' given by ${v}_{B}/c\ensuremath{\approx}1$ where $c$ is the Lorentz-invariant speed of particle excitations in the system. We briefly comment on the behavior of ${\ensuremath{\lambda}}_{L}$ and ${v}_{B}$ in the neighboring symmetry broken and unbroken phases.

DOI
10.1103/physrevd.96.065005

Related papers

Onset of many-body chaos in the O(N) model

Brian G. Swingle, Debanjan Chowdhury

The growth of commutators of initially commuting local operators diagnoses the onset of chaos in quantum many-body systems. We compute such commutators of local field operators with N components in the (2+1)-dimensional O(N) nonlinear sigma model to leading order in 1/N. The system is taken to be in thermal equilibrium at a temperature T above the zero temperature quantum critical point separating the symmetry broken and unbroken phases. The commutator grows exponentially in time with a rate denoted λ[subscript L]. At large N the growth of chaos as measured by λ[subscript L] is slow because the model is weakly interacting, and we find λ[subscript L]≈3.2T/N. The scaling with temperature is dictated by conformal invariance of the underlying quantum critical point. We also show that operators grow ballistically in space with a “butterfly velocity” given by v[subscript B]/c≈1 where c is the Lorentz-invariant speed of particle excitations in the system. We briefly comment on the behavior of λ[subscript L] and v_{B} in the neighboring symmetry broken and unbroken phases.

Physical Review Letters•Sep 1, 2017•14 citations

Thermalization and chaos in QED3

Julia Steinberg, Brian G. Swingle

We study the real time dynamics of ${N}_{f}$ flavors of fermions coupled to a $U(1)$ gauge field in $2+1$ dimensions to leading order in a $1/{N}_{f}$ expansion. For large enough ${N}_{f}$, this is an interacting conformal field theory and describes the low energy properties of the Dirac spin liquid. We focus on thermalization and the onset of many-body quantum chaos which can be diagnosed from the growth of initially anticommuting fermion field operators. We compute such anticommutators in this gauge theory to leading order in $1/{N}_{f}$. We find that the anticommutator grows exponentially in time and compute the quantum Lyapunov exponent. We briefly comment on chaos, locality, and gauge invariance.

arXiv:1901.04984•Physical review. D/Physical review. D.•Apr 11, 2019•26 citations

Chaos in a quantum rotor model

Gong Cheng, Brian G. Swingle

We study scrambling in a model consisting of a number $N$ of $M$-component quantum rotors coupled by random infinite-range interactions. This model is known to have both a paramagnetic phase and a spin glass phase separated by second order phase transition. We calculate in perturbation theory the squared commutator of rotor fields at different sites in the paramagnetic phase, to leading non-trivial order at large $N$ and large $M$. This quantity diagnoses the onset of quantum chaos in this system, and we show that the squared commutator grows exponentially with time, with a Lyapunov exponent proportional to $\frac{1}{M}$. At high temperature, the Lyapunov exponent limits to a value set by the microscopic couplings, while at low temperature, the exponent exhibits a $T^4$ dependence on temperature $T$.

arXiv:1901.10446•Jan 29, 2019•5 citations

Real-time correlators in chaotic quantum many-body systems

Adam Nahum, Sthitadhi Roy, Sagar Vijay +1 more

We study real-time correlators $\ensuremath{\langle}\mathcal{O}(\mathbf{x},t)\mathcal{O}(0,0)\ensuremath{\rangle}$ of local operators in chaotic quantum many-body systems. These correlators show universal structure at late times, determined by the geometry of the dominant operator-space Feynman trajectories for the evolving operator $\mathcal{O}(\mathbit{x},t)$. These trajectories, which involve the operator contracting to a point at both the initial and final time, are qualitatively different from those that dominate the out-of-time-order correlation function. In the absence of conservation laws, local correlations decay exponentially: $\ensuremath{\langle}\mathcal{O}(\mathbf{x},t)\mathcal{O}(0,0)\ensuremath{\rangle}\ensuremath{\sim}exp(\ensuremath{-}{s}_{\text{eq}}\phantom{\rule{0.16em}{0ex}}r(\mathbit{v})\phantom{\rule{0.16em}{0ex}}t)$, where $\mathbit{v}=\mathbit{x}/t$ defines a ray in spacetime, and $r(\mathbit{v})$ is a rate function associated with this ray. We express $r(\mathbit{v})$ in terms of cost functions for various spacetime structures. In $1+1\mathrm{D}$ the operator histories can exhibit a phase transition at a critical value ${v}_{c}$ of the ray velocity, which leads to a singular behavior in $r(\mathbit{v})$. At low velocities, the dominant Feynman histories are ``fat'': The operator grows to a size of order ${t}^{\ensuremath{\alpha}}$ (with $\ensuremath{\alpha}=1/2$ in the simplest case) before contracting to a point again. At high velocities the trajectories are ``thin'': The operator always remains of order-one size. In a Haar-random quantum circuit, this transition maps to a simple binding transition for a pair of random walks, which represent the left and right spatial boundaries of the operator. In higher-dimensional systems, thin trajectories always dominate. We discuss the circumstances in which the butterfly velocity ${v}_{B}$ can be deduced from a time-ordered two-point function, rather than the out-of-time ordered correlator. In the random circuit, correlators may also be computed in the framework of an effective Ising-like statistical mechanics model: we describe this calculation, as well as a special feature of the weights in the case of a $1+1\mathrm{D}$ Haar-random brickwork circuit. The present paper addresses lattice models, but also suggests the possibility of morphological phase transitions for real-time Feynman diagrams in continuum quantum field theories.

arXiv:2205.11544•Physical review. B./Physical review. B•Dec 26, 2022•17 citations

Universal properties of many-body quantum chaos at Gross-Neveu criticality

Shao-Kai Jian, Hong Yao

Quantum chaos in many-body systems may be characterized by the Lyapunov exponent defined as the exponential growth rate of out-of-time-order correlators (OTOC). So far Lyaponov exponents around various quantum critical points (QCP) remain largely unexplored. Here, we investigate the Lyapunov exponent around QCPs of the Gross-Neveu (GN) model with $N$ flavors of Dirac fermions in (2+1) dimensions. Around the GN quantum phase transition between a Dirac semimetal and a gapped insulator breaking $Z_2$ symmetry (e.g., inversion symmetry of the honeycomb lattice), we find that the Lyaponov exponent $λ_L \approx 3.5 T/N$ at temperature $T$ and to the leading order of $1/N$ in the large-$N$ expansion. We also obtain the quantum scattering rate of an excitation with energy $ε$, which is proportional to $\sqrt{εT}/N$ at low energy. We further discuss possible experimental relevances of the GN model in many-body systems.

arXiv:1805.12299•May 31, 2018•10 citations

Scrambling in the quantum Lifshitz model

Eugeniu Plamadeala, Eduardo Fradkin

Abstract We study signatures of chaos in the quantum Lifshitz model through out-of-time ordered correlators (OTOC) of current operators. This model is a free scalar field theory with dynamical critical exponent z = 2. It describes the quantum phase transition in 2D systems, such as quantum dimer models, between a phase with a uniform ground state to another one with spontaneously broken translation invariance. At the lowest temperatures the chaotic dynamics are dominated by a marginally irrelevant operator which induces a temperature dependent stiffness term. The numerical computations of OTOC exhibit a non-zero Lyapunov exponent (LE) in a wide range of temperatures and interaction strengths. The LE (in units of temperature) is a weakly temperature-dependent function; it vanishes at weak interaction and saturates for strong interaction. The Butterfly velocity increases monotonically with interaction strength in the studied region while remaining smaller than the interaction-induced velocity/stiffness.

arXiv:1802.07268•Journal of Statistical Mechanics Theory and Experiment•Jun 1, 2018•6 citations
The growth of commutators of initially commuting local operators diagnoses the onset of chaos in quantum many-body systems. We compute such commutators of local field operators with NNN components in the (2+12+12+1)-dimensional O(N)O(N)O(N) nonlinear sigma model to leading order in 1/N1/N1/N. The system is taken to be in thermal equilibrium at a temperature TTT above the zero temperature quantum critical point separating the symmetry broken and unbroken phases. The commutator grows exponentially in time with a rate denoted \ensuremathλL{\ensuremath{\lambda}}_{L}\ensuremathλL​. At large NNN the growth of chaos as measured by \ensuremathλL{\ensuremath{\lambda}}_{L}\ensuremathλL​ is slow because the model is weakly interacting, and we find \ensuremathλL\ensuremath≈3.2T/N{\ensuremath{\lambda}}_{L}\ensuremath{\approx}3.2T/N\ensuremathλL​\ensuremath≈3.2T/N. The scaling with temperature is dictated by conformal invariance of the underlying quantum critical point. We also show that operators grow ballistically in space with a ``butterfly velocity'' given by vB/c\ensuremath≈1{v}_{B}/c\ensuremath{\approx}1vB​/c\ensuremath≈1 where ccc is the Lorentz-invariant speed of particle excitations in the system. We briefly comment on the behavior of \ensuremathλL{\ensuremath{\lambda}}_{L}\ensuremathλL​ and vB{v}_{B}vB​ in the neighboring symmetry broken and unbroken phases.
The growth of commutators of initially commuting local operators diagnoses the onset of chaos in quantum many-body systems. We compute such commutators of local field operators with N components in the (2+1)-dimensional O(N) nonlinear sigma model to leading order in 1/N. The system is taken to be in thermal equilibrium at a temperature T above the zero temperature quantum critical point separating the symmetry broken and unbroken phases. The commutator grows exponentially in time with a rate denoted λ[subscript L]. At large N the growth of chaos as measured by λ[subscript L] is slow because the model is weakly interacting, and we find λ[subscript L]≈3.2T/N. The scaling with temperature is dictated by conformal invariance of the underlying quantum critical point. We also show that operators grow ballistically in space with a “butterfly velocity” given by v[subscript B]/c≈1 where c is the Lorentz-invariant speed of particle excitations in the system. We briefly comment on the behavior of λ[subscript L] and v_{B} in the neighboring symmetry broken and unbroken phases.
We study the real time dynamics of Nf{N}_{f}Nf​ flavors of fermions coupled to a U(1)U(1)U(1) gauge field in 2+12+12+1 dimensions to leading order in a 1/Nf1/{N}_{f}1/Nf​ expansion. For large enough Nf{N}_{f}Nf​, this is an interacting conformal field theory and describes the low energy properties of the Dirac spin liquid. We focus on thermalization and the onset of many-body quantum chaos which can be diagnosed from the growth of initially anticommuting fermion field operators. We compute such anticommutators in this gauge theory to leading order in 1/Nf1/{N}_{f}1/Nf​. We find that the anticommutator grows exponentially in time and compute the quantum Lyapunov exponent. We briefly comment on chaos, locality, and gauge invariance.
We study scrambling in a model consisting of a number NNN of MMM-component quantum rotors coupled by random infinite-range interactions. This model is known to have both a paramagnetic phase and a spin glass phase separated by second order phase transition. We calculate in perturbation theory the squared commutator of rotor fields at different sites in the paramagnetic phase, to leading non-trivial order at large NNN and large MMM. This quantity diagnoses the onset of quantum chaos in this system, and we show that the squared commutator grows exponentially with time, with a Lyapunov exponent proportional to 1M\frac{1}{M}M1​. At high temperature, the Lyapunov exponent limits to a value set by the microscopic couplings, while at low temperature, the exponent exhibits a T4T^4T4 dependence on temperature TTT.
We study real-time correlators $\ensuremath{\langle}\mathcal{O}(\mathbf{x},t)\mathcal{O}(0,0)\ensuremath{\rangle}$ of local operators in chaotic quantum many-body systems. These correlators show universal structure at late times, determined by the geometry of the dominant operator-space Feynman trajectories for the evolving operator $\mathcal{O}(\mathbit{x},t)$. These trajectories, which involve the operator contracting to a point at both the initial and final time, are qualitatively different from those that dominate the out-of-time-order correlation function. In the absence of conservation laws, local correlations decay exponentially: $\ensuremath{\langle}\mathcal{O}(\mathbf{x},t)\mathcal{O}(0,0)\ensuremath{\rangle}\ensuremath{\sim}exp(\ensuremath{-}{s}_{\text{eq}}\phantom{\rule{0.16em}{0ex}}r(\mathbit{v})\phantom{\rule{0.16em}{0ex}}t)$, where $\mathbit{v}=\mathbit{x}/t$ defines a ray in spacetime, and $r(\mathbit{v})$ is a rate function associated with this ray. We express $r(\mathbit{v})$ in terms of cost functions for various spacetime structures. In 1+1D1+1\mathrm{D}1+1D the operator histories can exhibit a phase transition at a critical value vc{v}_{c}vc​ of the ray velocity, which leads to a singular behavior in $r(\mathbit{v})$. At low velocities, the dominant Feynman histories are ``fat'': The operator grows to a size of order ${t}^{\ensuremath{\alpha}}$ (with $\ensuremath{\alpha}=1/2$ in the simplest case) before contracting to a point again. At high velocities the trajectories are ``thin'': The operator always remains of order-one size. In a Haar-random quantum circuit, this transition maps to a simple binding transition for a pair of random walks, which represent the left and right spatial boundaries of the operator. In higher-dimensional systems, thin trajectories always dominate. We discuss the circumstances in which the butterfly velocity vB{v}_{B}vB​ can be deduced from a time-ordered two-point function, rather than the out-of-time ordered correlator. In the random circuit, correlators may also be computed in the framework of an effective Ising-like statistical mechanics model: we describe this calculation, as well as a special feature of the weights in the case of a 1+1D1+1\mathrm{D}1+1D Haar-random brickwork circuit. The present paper addresses lattice models, but also suggests the possibility of morphological phase transitions for real-time Feynman diagrams in continuum quantum field theories.
Quantum chaos in many-body systems may be characterized by the Lyapunov exponent defined as the exponential growth rate of out-of-time-order correlators (OTOC). So far Lyaponov exponents around various quantum critical points (QCP) remain largely unexplored. Here, we investigate the Lyapunov exponent around QCPs of the Gross-Neveu (GN) model with NNN flavors of Dirac fermions in (2+1) dimensions. Around the GN quantum phase transition between a Dirac semimetal and a gapped insulator breaking Z2Z_2Z2​ symmetry (e.g., inversion symmetry of the honeycomb lattice), we find that the Lyaponov exponent λL≈3.5T/Nλ_L \approx 3.5 T/NλL​≈3.5T/N at temperature TTT and to the leading order of 1/N1/N1/N in the large-NNN expansion. We also obtain the quantum scattering rate of an excitation with energy εεε, which is proportional to εT/N\sqrt{εT}/NεT​/N at low energy. We further discuss possible experimental relevances of the GN model in many-body systems.
Abstract We study signatures of chaos in the quantum Lifshitz model through out-of-time ordered correlators (OTOC) of current operators. This model is a free scalar field theory with dynamical critical exponent z = 2. It describes the quantum phase transition in 2D systems, such as quantum dimer models, between a phase with a uniform ground state to another one with spontaneously broken translation invariance. At the lowest temperatures the chaotic dynamics are dominated by a marginally irrelevant operator which induces a temperature dependent stiffness term. The numerical computations of OTOC exhibit a non-zero Lyapunov exponent (LE) in a wide range of temperatures and interaction strengths. The LE (in units of temperature) is a weakly temperature-dependent function; it vanishes at weak interaction and saturates for strong interaction. The Butterfly velocity increases monotonically with interaction strength in the studied region while remaining smaller than the interaction-induced velocity/stiffness.