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:1901.10446

Chaos in a quantum rotor model

Gong Cheng, Brian G. Swingle

Published Jan 29, 2019• 5 citations
arXivHTMLPDF

Abstract

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 $T$.

DOI
10.48550/arxiv.1901.10446

Related papers

Slow scrambling and hidden integrability in a random rotor model

Dan Mao, Debanjan Chowdhury, T. Senthil

We analyze the out-of-time-order correlation functions of a solvable model of a large number $N$ of $M$-component quantum rotors coupled by Gaussian-distributed random, infinite-range exchange interactions. We focus on the growth of commutators of operators at a temperature $T$ above the zero temperature quantum critical point separating the spin-glass and paramagnetic phases. In the large $N,\phantom{\rule{0.16em}{0ex}}M$ limit, the squared commutators of the rotor fields do not display any exponential growth of commutators, in spite of the absence of any sharp quasiparticlelike excitations in the disorder-averaged theory. We show that in this limit, the problem is integrable and points out interesting connections to random-matrix theory. At leading order in $1/M$, there are no modifications to the critical behavior but an irrelevant term in the fixed-point action leads to a small exponential growth of the squared commutator. We also introduce and comment on a generalized model involving $p$-pair rotor interactions.

arXiv:1903.10499•Physical review. B./Physical review. B•Sep 17, 2020•6 citations

Chaos in a classical limit of the Sachdev-Ye-Kitaev model

Thomas Scaffidi, Ehud Altman

We study chaos in a classical limit of the Sachdev-Ye-Kitaev (SYK) model obtained in a suitably defined large-$S$ limit. The low-temperature Lyapunov exponent is found to depend linearly on temperature, with a slope that is parametrically different than in the quantum case: It is proportional to $N/S$. The classical dynamics can be understood as the rotation of an $N$-dimensional body with a random inertia tensor, corresponding to the random couplings of the SYK Hamiltonian. This allows us to find an extensive number of fixed points, corresponding to the body's principal axes of rotation. The thermodynamics is mapped to the $p$-spin model with $p=2$, which exhibits a spin glass phase at low temperature whose presence does not preclude the existence of chaos.

arXiv:1711.04768•Physical review. B./Physical review. B•Oct 15, 2019•40 citations

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

Debanjan Chowdhury, Brian G. Swingle

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.

arXiv:1703.02545•Physical review. D/Physical review. D.•Sep 12, 2017•139 citations

Lyapunov growth in quantum spin chains

Ben Craps, Marine De Clerck, D. Janssens +2 more

The Ising spin chain with longitudinal and transverse magnetic fields is often used in studies of quantum chaos, displaying both chaotic and integrable regions in its parameter space. However, even at a strongly chaotic point this model does not exhibit Lyapunov growth of the commutator squared of spin operators, as this observable saturates before exponential growth can manifest itself (even in situations where a spatial suppression factor makes the initial commutator small). We extend this model from the spin 1/2 Ising model to higher spins, demonstrate numerically that a window of exponential growth opens up for sufficiently large spin, and extract a quantity which corresponds to a notion of a Lyapunov exponent. In the classical infinite-spin limit, we identify and compute the appropriate classical analog of the commutator squared, and show that the corresponding exponent agrees with the infinite-spin limit extracted from the quantum spin chain.

arXiv:1908.08059•Physical review. B./Physical review. B•May 28, 2020•50 citations

Landau theory of quantum spin glasses of rotors and Ising spins

N. Read, Subir Sachdev A, J. Ye

We consider quantum rotors or Ising spins in a transverse field on a d-dimensional lattice, with random, frustrating, short-range, exchange interactions. The quantum dynamics are associated with a finite moment of inertia for the rotors, and with the transverse field for the Ising spins. For a suitable distribution of exchange constants, these models display spin-glass and quantum paramagnet phases and a zero-temperature (T) quantum transition between them. An earlier exact solution for the critical properties of a model with infinite-range interactions cna be reproduced by minimization of a Landau effective-action functional for the model in finite d with short-range interactions. The functional is expressed in terms of a composite spin field which is bilocal in time. The mean-field phase diagram near the T=0 critical point is mapped out as a function of T, strength of the quantum coupling, and applied fields. The spin-glass phase has replica symmetry breaking; but, as in the classical Ising spin glass, the order parameter becomes replica symmetric as T\ensuremath{\rightarrow}0. Next we examine the consequences of fluctuations about the mean field for the critical properties. Above d=8, and with certain restrictions on the values of the Landau couplings, we find that the transition is controlled by a Gaussian fixed point with mean-field critical exponents. For couplings not attracted by the Gaussian fixed point above d=8, and for all physical couplings below d=8, we find runaway renormalization-group flows to strong coupling. General scaling relations that should be valid even at the strong-coupling fixed point are proposed and compared with Monte Carlo simulations.

arXiv:cond-mat/9412032•Physical review. B, Condensed matter•Jul 1, 1995•165 citations

Solvable spin glass of quantum rotors

J. Ye, Subir Sachdev A, N. Read

We examine a model of M-component quantum rotors coupled by Gaussian-distributed random, infinite-range exchange interactions. A complete solution is obtained at M=\ensuremath{\infty} in the spin-glass and quantum-disordered phases. The quantum phase transition separating them is found to possess logarithmic violations of scaling, with no further modifications to the leading critical behavior at any order in 1/M; this suggests that the critical properties of the transverse-field Ising model (believed to be identical to the M\ensuremath{\rightarrow}1 limit) are the same as those of the M=\ensuremath{\infty} quantum rotors.

arXiv:cond-mat/9212027•Physical Review Letters•Jun 21, 1993•145 citations
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 analyze the out-of-time-order correlation functions of a solvable model of a large number NNN of MMM-component quantum rotors coupled by Gaussian-distributed random, infinite-range exchange interactions. We focus on the growth of commutators of operators at a temperature TTT above the zero temperature quantum critical point separating the spin-glass and paramagnetic phases. In the large N,MN,\phantom{\rule{0.16em}{0ex}}MN,M limit, the squared commutators of the rotor fields do not display any exponential growth of commutators, in spite of the absence of any sharp quasiparticlelike excitations in the disorder-averaged theory. We show that in this limit, the problem is integrable and points out interesting connections to random-matrix theory. At leading order in 1/M1/M1/M, there are no modifications to the critical behavior but an irrelevant term in the fixed-point action leads to a small exponential growth of the squared commutator. We also introduce and comment on a generalized model involving ppp-pair rotor interactions.
We study chaos in a classical limit of the Sachdev-Ye-Kitaev (SYK) model obtained in a suitably defined large-SSS limit. The low-temperature Lyapunov exponent is found to depend linearly on temperature, with a slope that is parametrically different than in the quantum case: It is proportional to N/SN/SN/S. The classical dynamics can be understood as the rotation of an NNN-dimensional body with a random inertia tensor, corresponding to the random couplings of the SYK Hamiltonian. This allows us to find an extensive number of fixed points, corresponding to the body's principal axes of rotation. The thermodynamics is mapped to the ppp-spin model with p=2p=2p=2, which exhibits a spin glass phase at low temperature whose presence does not preclude the existence of chaos.
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 ${\ensuremath{\lambda}}_{L}$. At large NNN 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 ccc is the Lorentz-invariant speed of particle excitations in the system. We briefly comment on the behavior of ${\ensuremath{\lambda}}_{L}$ and vB{v}_{B}vB​ in the neighboring symmetry broken and unbroken phases.
The Ising spin chain with longitudinal and transverse magnetic fields is often used in studies of quantum chaos, displaying both chaotic and integrable regions in its parameter space. However, even at a strongly chaotic point this model does not exhibit Lyapunov growth of the commutator squared of spin operators, as this observable saturates before exponential growth can manifest itself (even in situations where a spatial suppression factor makes the initial commutator small). We extend this model from the spin 1/2 Ising model to higher spins, demonstrate numerically that a window of exponential growth opens up for sufficiently large spin, and extract a quantity which corresponds to a notion of a Lyapunov exponent. In the classical infinite-spin limit, we identify and compute the appropriate classical analog of the commutator squared, and show that the corresponding exponent agrees with the infinite-spin limit extracted from the quantum spin chain.
We consider quantum rotors or Ising spins in a transverse field on a d-dimensional lattice, with random, frustrating, short-range, exchange interactions. The quantum dynamics are associated with a finite moment of inertia for the rotors, and with the transverse field for the Ising spins. For a suitable distribution of exchange constants, these models display spin-glass and quantum paramagnet phases and a zero-temperature (T) quantum transition between them. An earlier exact solution for the critical properties of a model with infinite-range interactions cna be reproduced by minimization of a Landau effective-action functional for the model in finite d with short-range interactions. The functional is expressed in terms of a composite spin field which is bilocal in time. The mean-field phase diagram near the T=0 critical point is mapped out as a function of T, strength of the quantum coupling, and applied fields. The spin-glass phase has replica symmetry breaking; but, as in the classical Ising spin glass, the order parameter becomes replica symmetric as T\ensuremath{\rightarrow}0. Next we examine the consequences of fluctuations about the mean field for the critical properties. Above d=8, and with certain restrictions on the values of the Landau couplings, we find that the transition is controlled by a Gaussian fixed point with mean-field critical exponents. For couplings not attracted by the Gaussian fixed point above d=8, and for all physical couplings below d=8, we find runaway renormalization-group flows to strong coupling. General scaling relations that should be valid even at the strong-coupling fixed point are proposed and compared with Monte Carlo simulations.
We examine a model of M-component quantum rotors coupled by Gaussian-distributed random, infinite-range exchange interactions. A complete solution is obtained at M=\ensuremath{\infty} in the spin-glass and quantum-disordered phases. The quantum phase transition separating them is found to possess logarithmic violations of scaling, with no further modifications to the leading critical behavior at any order in 1/M; this suggests that the critical properties of the transverse-field Ising model (believed to be identical to the M\ensuremath{\rightarrow}1 limit) are the same as those of the M=\ensuremath{\infty} quantum rotors.