Regularized out-of-time-ordered correlators (OTOCs) are frequently used as a measure of chaos, i.e., nonintegrability, in quantum many-body systems. Here, the authors compute the regularized OTOCs of both fermion and boson operators at the fixed point of the spin-fermion model found by Schlief, Lunts, and Lee. They find that, despite being a strongly coupled theory, the degrees of nonintegrability of the fermion and boson sectors are parametrically different, and both are parametrically smaller than the upper bound for the growth rate of regularized OTOCs.
Regularized out-of-time-ordered correlators (OTOCs) are frequently used as a measure of chaos, i.e., nonintegrability, in quantum many-body systems. Here, the authors compute the regularized OTOCs of both fermion and boson operators at the fixed point of the spin-fermion model found by Schlief, Lunts, and Lee. They find that, despite being a strongly coupled theory, the degrees of nonintegrability of the fermion and boson sectors are parametrically different, and both are parametrically smaller than the upper bound for the growth rate of regularized OTOCs.
The out-of-time-ordered correlators (OTOCs) have been proposed and widely used recently as a tool to define and describe many-body quantum chaos. Here, we develop the Keldysh nonlinear sigma model technique to calculate these correlators in interacting disordered metals. In particular, we focus on the regularized and unregularized OTOCs, defined as ${f}^{(r)}(t)=\mathrm{Tr}\phantom{\rule{0.16em}{0ex}}\left[\sqrt{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}(t)\sqrt{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}}^{\ifmmode\dagger\else\textdagger\fi{}}(t)\right]$ and ${f}^{(u)}(t)=\mathrm{Tr}\phantom{\rule{0.16em}{0ex}}\left[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}(t){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}}^{\ifmmode\dagger\else\textdagger\fi{}}(t)\right]$, respectively (where $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}(t)={\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\psi}}(\mathrm{r},t),{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\psi}}}^{\ifmmode\dagger\else\textdagger\fi{}}(0,0)}$ is the anticommutator of fermion field operators and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}$ is the thermal density matrix). The calculation of the rate of OTOCs' exponential growth is reminiscent to that of the Altshuler-Aronov-Khmelnitskii dephasing rate in interacting metals, but here it involves two replicas of the system (two ``worlds''). The intraworld contributions reproduce the Altshuler-Aronov-Khmelnitskii dephasing (that would correspond to a decay of the correlator), while the interworld terms provide a term of the opposite sign that exceeds dephasing. Consequently, both regularized and unregularized OTOCs grow exponentially in time, but surprisingly we find that the corresponding many-body Lyapunov exponents are different. For the regularized correlator, we reproduce an earlier perturbation theory result for the Lyapunov exponent that satisfies the Maldacena-Shenker-Stanford bound, ${\ensuremath{\lambda}}^{(r)}\ensuremath{\le}2\ensuremath{\pi}{k}_{\mathsf{B}}T/\ensuremath{\hbar}$. However, the Lyapunov exponent of the unregularized correlator parametrically exceeds the bound, ${\ensuremath{\lambda}}^{(u)}\ensuremath{\gg}2\ensuremath{\pi}{k}_{\mathsf{B}}T/\ensuremath{\hbar}$. We argue that ${\ensuremath{\lambda}}^{(u)}$ is not a reliable indicator of many-body quantum chaos as it contains additional contributions from elastic scattering events due to virtual processes that should not contribute to many-body chaos. These results bring up an important general question of the physical meaning of the OTOCs often used in calculations and proofs. We briefly discuss possible connections of the OTOCs to observables in quantum interference effects and level statistics via a generalization of the Bohigas-Giannoni-Schmit conjecture to many-body chaotic systems.
Out-of-time-ordered correlators (OTOC) have been proposed to characterize quantum chaos in generic systems. However, they can also show interesting behavior in integrable models, resembling the OTOC in chaotic systems in some aspects. Here we study the OTOC for different operators in the exactly-solvable one-dimensional quantum Ising spin chain. The OTOC for spin operators that are local in terms of the Jordan-Wigner fermions has a ``shell-like'' structure: After the wavefront passes, the OTOC approaches its original value in the long-time limit, showing no signature of scrambling; the approach is described by a ${t}^{\ensuremath{-}1}$ power law at long time t. On the other hand, the OTOC for spin operators that are nonlocal in the Jordan-Wigner fermions has a ``ball-like'' structure, with its value reaching zero in the long-time limit, looking like a signature of scrambling; the approach to zero, however, is described by a slow power law ${t}^{\ensuremath{-}1/4}$ for the Ising model at the critical coupling. These long-time power-law behaviors in the lattice model are not captured by conformal field theory calculations. The mixed OTOC with both local and nonlocal operators in the Jordan-Wigner fermions also has a ``ball-like'' structure, but the limiting values and the decay behavior appear to be nonuniversal. In all cases, we are not able to define a parametrically large window around the wavefront to extract the Lyapunov exponent.
In recent times out-of-time-order correlators (OTOC) have been established as a tool to understand butterfly effects, quantum information scrambling, and many-body localization. They can also be useful in determining different phases of quantum critical systems. OTOCs can identify the quantum chaos within a system undergoing time evolution; and therefore, they can distinguish between chaotic and regular dynamics. This motivates us to study OTOCs in integrable and nonintegrable periodically kicked quantum spin models. A periodically kicked quantum Ising spin system, known as the quantum Ising Floquet system, is a variant of the transverse Ising model. In place of constant transverse magnetic fields in the transverse Ising system, time-periodic fields are applied in the form of delta pulses in the quantum Ising Floquet spin system. It provides very interesting and peculiar dynamics separate from that of the transverse Ising system.
Out-of-time-ordered correlation functions (OTOCs) are presently being extensively debated as quantifiers of dynamical chaos in interacting quantum many-body systems. We argue that in quantum spin and fermionic systems, where all local operators are bounded, an OTOC of local observables is bounded as well and thus its exponential growth is merely transient. As a better measure of quantum chaos in such systems, we propose, and study, the density of the OTOC of extensive sums of local observables, which can exhibit indefinite growth in the thermodynamic limit. We demonstrate this for the kicked quantum Ising model by using large-scale numerical results and an analytic solution in the integrable regime. In a generic case, we observe the growth of the OTOC density to be linear in time. We prove that this density in general, locally interacting, nonintegrable quantum spin and fermionic dynamical systems exhibits growth that is at most polynomial in time---a phenomenon, which we term weak quantum chaos. In the special case of the model being integrable and the observables under consideration quadratic, the OTOC density saturates to a plateau.
The idea of the out-of-time-order correlator (OTOC) has recently emerged in the study of both condensed matter systems and gravitational systems.It not only plays a key role in investigating the holographic duality between a strongly interacting quantum system and a gravitational system, it also diagnoses the chaotic behavior of many-body quantum systems and characterizes information scrambling.Based on OTOCs, three different concepts-quantum chaos, holographic duality, and information scrambling-are found to be intimately related to each other.Despite its theoretical importance, the experimental measurement of the OTOC is quite challenging, and thus far there is no experimental measurement of the OTOC for local operators.Here, we report the measurement of OTOCs of local operators for an Ising spin chain on a nuclear magnetic resonance quantum simulator.We observe that the OTOC behaves differently in the integrable and nonintegrable cases.Based on the recent discovered relationship between OTOCs and the growth of entanglement entropy in the many-body system, we extract the entanglement entropy from the measured OTOCs, which clearly shows that the information entropy oscillates in time for integrable models and scrambles for nonintgrable models.With the measured OTOCs, we also obtain the experimental result of the butterfly velocity, which measures the speed of correlation propagation.Our experiment paves a way for experimentally studying quantum chaos, holographic duality, and information scrambling in many-body quantum systems with quantum simulators.
Out-of-time-ordered correlators (OTOCs) have been proposed as a probe of chaos in quantum mechanics, on the basis of their short-time exponential growth found in some particular setups. However, it has been seen that this behavior is not universal. Therefore, we query other quantum chaos manifestations arising from the OTOCs, and we thus study their long-time behavior in systems of completely different nature: quantum maps, which are the simplest chaotic one-body system, and spin chains, which are many-body systems without a classical limit. It is shown that studying the long-time regime of the OTOCs it is possible to detect and gauge the transition between integrability and chaos, and we benchmark the transition with other indicators of quantum chaos based on the spectra and the eigenstates of the systems considered. For systems with a classical analog, we show that the proposed OTOC indicators have a very high accuracy that allow us to detect subtle features along the integrability-to-chaos transition.