We consider the fate of 1/N expansions in unstable many-body quantum systems, as realized by a quench across criticality, and show the emergence of e^{2λt}/N as a renormalized parameter ruling the quantum-classical transition and accounting nonperturbatively for the local divergence rate λ of mean-field solutions. In terms of e^{2λt}/N, quasiclassical expansions of paradigmatic examples of criticality, like the self-trapping transition in an integrable Bose-Hubbard dimer and the generic instability of attractive bosonic systems toward soliton formation, are pushed to arbitrarily high orders. The agreement with numerical simulations supports the general nature of our results in the appropriately combined long-time λt→∞ quasiclassical N→∞ regime, out of reach of expansions in the bare parameter 1/N. For scrambling in many-body hyperbolic systems, our results provide formal grounds to a conjectured multiexponential form of out-of-time-ordered correlators.
We consider the fate of 1/N expansions in unstable many-body quantum systems, as realized by a quench across criticality, and show the emergence of e^{2λt}/N as a renormalized parameter ruling the quantum-classical transition and accounting nonperturbatively for the local divergence rate λ of mean-field solutions. In terms of e^{2λt}/N, quasiclassical expansions of paradigmatic examples of criticality, like the self-trapping transition in an integrable Bose-Hubbard dimer and the generic instability of attractive bosonic systems toward soliton formation, are pushed to arbitrarily high orders. The agreement with numerical simulations supports the general nature of our results in the appropriately combined long-time λt→∞ quasiclassical N→∞ regime, out of reach of expansions in the bare parameter 1/N. For scrambling in many-body hyperbolic systems, our results provide formal grounds to a conjectured multiexponential form of out-of-time-ordered correlators.
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.
Quantum chaotic interacting N-particle systems are assumed to show fast and irreversible spreading of quantum information on short (Ehrenfest) time scales ∼logN. Here, we show that, near criticality, certain many-body systems exhibit fast initial scrambling, followed subsequently by oscillatory behavior between reentrant localization and delocalization of information in Hilbert space. We consider both integrable and nonintegrable quantum critical bosonic systems with attractive contact interaction that exhibit locally unstable dynamics in the corresponding many-body phase space of the large-N limit. Semiclassical quantization of the latter accounts for many-body correlations in excellent agreement with simulations. Most notably, it predicts an asymptotically constant local level spacing ℏ/τ, again given by τ∼logN. This unique timescale governs the long-time behavior of out-of-time-order correlators that feature quasiperiodic recurrences indicating reversibility.
Fast scrambling of quantum correlations, reflected by the exponential growth of out-of-time-order correlators (OTOCs) on short pre-Ehrenfest time scales, is commonly considered as a major quantum signature of unstable dynamics in quantum systems with a classical limit. In two recent works [Phys. Rev. Lett. 123, 160401 (2019)0031-900710.1103/PhysRevLett.123.160401] and [Phys. Rev. Lett. 124, 140602 (2020)10.1103/PhysRevLett.124.140602], a significant difference in the scrambling rate of integrable (many-body) systems was observed, depending on the initial state being semiclassically localized around unstable fixed points or fully delocalized (infinite temperature). Specifically, the quantum Lyapunov exponent λ_{q} quantifying the OTOC growth is given, respectively, by λ_{q}=2λ_{s} or λ_{q}=λ_{s} in terms of the stability exponent λ_{s} of the hyperbolic fixed point. Here we show that a wave packet, initially localized around this fixed point, features a distinct dynamical transition between these two regions. We present an analytical semiclassical approach providing a physical picture of this phenomenon, and support our findings by extensive numerical simulations in the whole parameter range of locally unstable dynamics of a Bose-Hubbard dimer. Our results suggest that the existence of this crossover is a hallmark of unstable separatrix dynamics in integrable systems, thus opening the possibility to distinguish the latter, on the basis of this particular observable, from genuine chaotic dynamics generally featuring uniform exponential growth of the OTOC.
We investigate the quantum dynamics generated by quantum quenches (QQs) of the Hamiltonian parameters in many-body systems, focusing on protocols that cross first-order and continuous quantum transitions, both in finite-size systems and in the thermodynamic limit. As a paradigmatic example, we consider the quantum Ising chain in the presence of homogeneous transverse (g) and longitudinal (h) magnetic fields. This model exhibits a continuous quantum transition (CQT) at g=gc and h=0, and first-order quantum transitions (FOQTs) driven by h along the line h=0 (g<gc). In the integrable limit h=0, the system can be mapped onto a quadratic fermionic theory; however, any nonvanishing longitudinal field breaks integrability and the spectrum of the resulting Hamiltonian is generally expected to enter a chaotic regime. We analyze QQs in which the longitudinal field is suddenly changed from a negative value hi<0 to a positive value hf>0. We focus on values of hf such that the spectrum of the post-QQ Hamiltonian H(g,hf) lies in the chaotic regime, where thermalization may emerge at asymptotically long times. We study the out-of-equilibrium dynamics for different values of g, finding qualitatively distinct behaviors for g>gc (where the chain is in the disordered phase), for g=gc (QQ across the CQT), and for g<gc (QQ across the FOQT line).
Out-of-time-order correlators (OTOCs) have been proposed as sensitive probes for chaos in interacting quantum systems. They exhibit a characteristic classical exponential growth, but saturate beyond the so-called scrambling or Ehrenfest time τ_{E} in the quantum correlated regime. Here we present a path-integral approach for the entire time evolution of OTOCs for bosonic N-particle systems. We first show how the growth of OTOCs up to τ_{E}=(1/λ)logN is related to the Lyapunov exponent λ of the corresponding chaotic mean-field dynamics in the semiclassical large-N limit. Beyond τ_{E}, where simple mean-field approaches break down, we identify the underlying quantum mechanism responsible for the saturation. To this end we express OTOCs by coherent sums over contributions from different mean-field solutions and compute the dominant many-body interference term amongst them. Our method further applies to the complementary semiclassical limit ℏ→0 for fixed N, including quantum-chaotic single- and few-particle systems.
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 fβ 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.