We study correlations, transport and chaos in a Heisenberg magnet as a classical model many-body system. By varying temperature and dimensionality, we can tune between settings with and without symmetry breaking and accompanying collective modes or quasiparticles. We analyse both conventional and out-of-time-ordered spin correlators (`decorrelators') to track the spreading of a spatiotemporally localised perturbation -- the wingbeat of the butterfly -- as well as transport coefficients and Lyapunov exponents. We identify a number of qualitatively different regimes. Trivially, at $T=0$, there is no dynamics at all. In the limit of low temperature, $T=0^+$, integrability emerges, with infinitely long-lived magnons; here the wavepacket created by the perturbation propagates ballistically, yielding a lightcone at the spin wave velocity which thus subsumes the butterfly velocity; inside the lightcone, a pattern characteristic of the free spin wave spectrum is visible at short times. On top of this, residual interactionslead to spin wave lifetimes which, while divergent in this limit, remain finite at any nonzero $T$. At the longest times, this leads to a `standard' chaotic regime; for this regime, we show that the Lyapunov exponent is simply proportional to the inverse spin-wave lifetime. Visibly strikingly, between this and the `short-time' integrable regimes, a scarred regime emerges: here, the decorrelator is spatiotemporally highly non-uniform, being dominated by rare and random scattering events seeding secondary lightcones. As the spin correlation length decreases with increasing $T$, the distinction between these regimes disappears and at high temperature the previously studied chaotic paramagnetic regime emerges. For this, we elucidate how, somewhat counterintuitively, the ballistic butterfly velocity arises from a diffusive spin dynamics.
In the short wavelength limit the Bogoliubov quasiparticles of trapped Bose-Einstein condensates can be described as classical particles and antiparticles with dynamics in a mixed phase-space. For anisotropic parabolic traps we determine the location of the resonances and study the influence of the sharpness of the condensate surface on the appearance of chaos as the energy of the quasiparticles is lowered from values much larger than to values comparable with the chemical potential.
Chaos is an important characterization of classical dynamical systems. How is chaos linked to the long-time dynamics of collective modes across phases and phase transitions? We address this by studying chaos across Ising and Kosterlitz-Thouless transitions in classical XXZ model. We show that spatio-temporal chaotic properties have crossovers across the transitions and distinct temperature dependence in the high and low-temperature phases which show normal and anomalous diffusions, respectively. Our results also provide new insights into the dynamics of interacting quantum systems in the semic…
How many-body chaos emerges in the presence of quasiparticles
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（国际理论科学中心，塔塔基础研究所）；Max-Planck-Institut für Physik komplexer Systeme（马克斯·普朗克复杂系统物理研究所）
We extend the results of two of our papers [Phys. Rev. A 94, 041603R (2016) and Phys. Rev. B 97, 060303R (2018)] that touch upon the intimately connected topics of quantum chaos and thermalization. In the first, we argued that when the initial state of isolated lattice many-body quantum systems is chaotic, the power-law decay of the survival probability is caused by the bounds in the spectrum, and thus anticipates thermalization. In the current work, we provide stronger numerical support for the onset of these algebraic behaviors. In the second paper, we demonstrated that the correlation hole,…
Local measurements in quantum systems are projective operations which act to counteract the spread of quantum entanglement. Recent work has shown that local, random measurements applied to a generic volume-law entanglement generating many-body system are able to force a transition into an area-law phase. This work shows that projective operations can also force a similar classical phase transition; we show that local projections in a chaotic system can freeze information dynamics. In rough analogy with measurement-induced phase transitions, this is characterized by an absence of information sp…
The order-by-order renormalization of the self-consistent mean-field potential in many-body perturbation theory for normal Fermi systems is investigated in detail. Building on previous work mainly by Balian and de Dominicis, as a key result we derive a thermodynamic perturbation series that manifests the consistency of the adiabatic zero-temperature formalism with perturbative statistical mechanics---for both isotropic and anisotropic systems---and satisfies at each order and for all temperatures the thermodynamic relations associated with Fermiliquid theory. These properties are proved to all…
Despite its importance to experiments, numerical simulations, and the development of theoretical models, self-averaging in many-body quantum systems out of equilibrium remains underinvestigated. Usually, in the chaotic regime, self-averaging is taken for granted. The numerical and analytical results presented here force us to rethink these expectations. They demonstrate that self-averaging properties depend on the quantity and also on the time scale considered. We show analytically that the survival probability in chaotic systems is not self-averaging at any time scale, even when evolved under…
In this work we revisit the problem of equilibration in isolated many-body interacting quantum systems. We pay particular attention to quantum chaotic Hamiltonians, and rather than focusing on the properties of the asymptotic states and how they adhere to the predictions of the Eigenstate Thermalization Hypothesis, we focus on the equilibration process itself, i.e., \emph{the road to equilibrium}. Along the road to equilibrium the diagonal ensembles obey an emergent form of the second law of thermodynamics and we provide an information theoretic proof of this fact. With this proof at hand we s…
This paper considers a type of generalized large $q$ SYK models which include multi-body interactions between Majorana fermions. We derive an effective action in the limit of large $N$ and large $q$ (with ${~q^2\over N} $ small), and find a universal expression for thermodynamical quantities. We also consider the chaos exponent using the retarded kernel method and find an efficient way to calculate the Lyapunov exponent for generalized large $q$ SYK models numerically.
General theoretic approach to classical Loschmidt echoes in chaotic systems with many degrees of freedom is developed. For perturbations which affect essentially all degrees of freedom we find a doubly exponential decay with the rate determined by the largest Lyapunov exponent. The scaling of the decay rate on the perturbation strength depends on whether the initial phase-space density is continuous or not.
We demonstrate that stability and chaotic-transport features of paradigmatic nonequilibrium many-body systems, i.e., periodically kicked and interacting particles, can deviate significantly from the expected ones of full instability and normal chaotic diffusion for arbitrarily strong chaos, arbitrary number of particles, and different interaction cases. We rigorously show that under the latter general conditions there exist {\em fully stable} orbits, accelerator-mode (AM) fixed points, performing ballistic motion in momentum. It is numerically shown that an {\em "isolated chaotic zone"} (ICZ),…
A profound quest of statistical mechanics is the origin of irreversibility - the arrow of time. New stimulants have been provided, thanks to unprecedented degree of control reached in experiments with isolated quantum systems and rapid theoretical developments of manybody localization in disordered interacting systems. The proposal of (many-body) eigenstate thermalization (ET) for these systems reinforces the common belief that either interaction or extrinsic randomness is required for thermalization. Here, we unveil a quantum thermalization mechanism challenging this belief. We find that, pro…
Linking thermodynamic variables like temperature $T$ and the measure of chaos, the Lyapunov exponents $λ$, is a question of fundamental importance in many-body systems. By using nonlinear fluid equations in one and three dimensions, we prove that in thermalised flows $λ\propto \sqrt{T}$, in agreement with results from frustrated spin systems. This reveals an underlying universality and provides evidence for recent conjectures on the thermal scaling of $λ$. We also reconcile seemingly disparate effects -- equilibration on one hand and pushing systems out-of-equilibrium on the other -- of many-b…
Systems of interacting bosons in triple-well potentials are of significant theoretical and experimental interest. They are explored in contexts that range from quantum phase transitions and quantum dynamics to semiclassical analysis. Here, we systematically investigate the onset of quantum chaos in a triple-well model that moves away from integrability as its potential gets tilted. Even in its deepest chaotic regime, the system presents features reminiscent of integrability. Our studies are based on level spacing distribution and spectral form factor, structure of the eigenstates, diagonal and…
We investigate the effect of kinetic constraints on classical many-body chaos in a translationally-invariant Heisenberg spin chain using a classical counterpart of the out-of-time-ordered correlator (OTOC). The strength of the constraint drives a 'dynamical phase transition' separating a delocalised phase, where the classical OTOC propagates ballistically, from a localised phase, where the OTOC does not propagate at all and the entire system freezes. This is unexpected given that all spins configurations are dynamically connected to each other. We show that localisation arises due to the dynam…
Is the hydrodynamics of an interacting many-body system fundamentally limited by basic principles of quantum mechanics? Starting with the conjecture that viscosity is at least as large as entropy density (as measured in fundamental units), there has been a long search for a precise answer to this question. In this work, we identify a simple relationship between hydrodynamics and many-body quantum chaos in a broad class of experimentally realizable systems. Consistency with the quantum butterfly effect leads to upper bounds on the speed of sound and diffusion constants of hydrodynamics. These b…
We consider a non-interacting many-fermion system populating levels of a unitary random matrix ensemble (equivalent to the q=2 complex Sachdev-Ye-Kitaev model) - a generic model of single-particle quantum chaos. We study the corresponding many-particle level statistics by calculating the spectral form factor analytically using algebraic methods of random matrix theory, and match it with an exact numerical simulation. Despite the integrability of the theory, the many-body spectral rigidity is found to have a surprisingly rich landscape. In particular, we find a residual repulsion of distant man…
We study numerically and analytically the time dependence and saturation of out-of-time ordered correlators (OTOC) in chaotic few-body quantum-mechanical systems: quantum Henon-Heiles system (weakly chaotic), BMN matrix quantum mechanics (strongly chaotic) and Gaussian random matrix ensembles. The growth pattern of quantum-mechanical OTOC is complex and nonuniversal, with no clear exponential regime at relevant timescales in any of the examples studied (which is not in contradiction to the exponential growth found in the literature for many-body systems, i.e. fields). On the other hand, the pl…
We study the dynamics of the entanglement between two qubits coupled to a common chaotic environment, described by the quantum kicked rotator model. We show that the kicked rotator, which is a single-particle deterministic dynamical system, can reproduce the effects of a pure dephasing many-body bath. Indeed, in the semiclassical limit the interaction with the kicked rotator can be described as a random phase-kick, so that decoherence is induced in the two-qubit system. We also show that our model can efficiently simulate non-Markovian environments.
We introduce well-defined characterizations of prethermal states in realistic periodically driven many-body systems with unbounded chaotic diffusion of the kinetic energy. These systems, interacting arrays of periodically kicked rotors, are paradigmatic models of many-body chaos theory. We show that the prethermal states in these systems are well described by a generalized Gibbs ensemble based essentially on the average Hamiltonian. The latter is the quasi-conserved quantity in the prethermal state and the ensemble is characterized by the temperature of the state. An explicit exact expression …
We study properties of chaos in generic one-dimensional nonlinear Hamiltonian lattices comprised of weakly coupled nonlinear oscillators, by numerical simulations of continuous-time systems and symplectic maps. For small coupling, the measure of chaos is found to be proportional to the coupling strength and lattice length, with the typical maximal Lyapunov exponent being proportional to the square root of coupling. This strong chaos appears as a result of triplet resonances between nearby modes. In addition to strong chaos we observe a weakly chaotic component having much smaller Lyapunov expo…
We show that the onset of quantum chaos at infinite temperature in two many-body one-dimensional lattice models, the perturbed spin-1/2 XXZ and Anderson models, is characterized by universal behavior. Specifically, we show that the onset of quantum chaos is marked by maxima of the typical fidelity susceptibilities that scale with the square of the inverse average level spacing, saturating their upper bound, and that the strength of the integrability- or localization-breaking perturbation at these maxima decreases with increasing system size. We also show that the spectral function below the ``…
This paper summarises a numerical investigation of the statistical properties of orbits evolved in `frozen,' time-independent N-body realisations of smooth, time-independent density distributions, allowing for 10^(2.5)<N<10^(5.5). Two principal conclusions were reached: (1) In the limit of a nearly `unsoftened' two-body kernel, the value of the largest Lyapunov exponent does NOT appear to decrease systematically with increasing N. (2) Nevertheless, there is a clear quantifiable sense in which, on the average, as N increases chaotic orbits in the frozen-N systems come to more closely approximat…
Discrete time crystals (DTCs) are new phases of matter characterized by the presence of an observable evolving with $nT$ periodicity under a $T$-periodic Hamiltonian, where $n>1$ is an integer insensitive to small parameter variations. In particular, DTCs with $n=2$ have been extensively studied in periodically quenched and kicked spin systems in recent years. In this paper, we study the emergence of DTCs from the many-body quantum chaos perspective, using a rather simple model depicting a harmonically driven spin chain. We advocate to first employ a semiclassical approximation to arrive at a …
Statistical structural properties of many-body chaotic eigenfunctions and applications
University of Science and Technology of China（中国科学技术大学）；CAS Key Laboratory of Microscale Magnetic Resonance, University of Science and Technology of China（中国科学院微观磁学共振重点实验室，中国科学技术大学）；Anhui Center for fundamental sciences in theoretical physics（安徽基础科学前沿中心）；University of Osnabrück（奥斯纳布吕克大学）
We numerically investigate the minimum number of interacting particles, which is required for the onset of strong chaos in quantum systems on a one-dimensional lattice with short-range and long-range interactions. We consider multiple system sizes which are at least three times larger than the number of particles and find that robust signatures of quantum chaos emerge for as few as 4 particles in the case of short-range interactions and as few as 3 particles for long-range interactions, and without any apparent dependence on the size of the system.
This paper continues a numerical investigation of orbits evolved in `frozen,' time-independent N-body realisations of smooth time-independent density distributions corresponding to both integrable and nonintegrable potentials, allowing for N as large as 300,000. The principal focus is on distinguishing between, and quantifying, the effects of graininess on initial conditions corresponding, in the continuum limit, to regular and chaotic orbits. Ordinary Lyapunov exponents X do not provide a useful diagnostic for distinguishing between regular and chaotic behaviour. Frozen-N orbits corresponding…
Eigenstates of local many-body interacting systems that are far from spectral edges are thought to be ergodic and close to being random states. This is consistent with the eigenstate thermalization hypothesis and volume-law scaling of entanglement. We point out that systematic departures from complete randomness are generically present in mid-spectrum eigenstates, and focus on the departure of the entanglement entropy from the random-state prediction. We show that the departure is (partly) due to spatial correlations and due to orthogonality to the eigenstates at the spectral edge, which impos…
We study many-body chaos in a (2+1)D relativistic scalar field theory at high temperatures in the classical statistical approximation, which captures the quantum critical regime and the thermal phase transition from an ordered to a disordered phase. We evaluate out-of-time ordered correlation functions (OTOCs) and find that the associated Lyapunov exponent increases linearly with temperature in the quantum critical regime, and approaches the non-interacting limit algebraically in terms of a fluctuation parameter. OTOCs spread ballistically in all regimes, also at the thermal phase transition, …
The spectral form factor (SFF), characterizing statistics of energy eigenvalues, is a key diagnostic of many-body quantum chaos. In addition, partial spectral form factors (PSFFs) can be defined which refer to subsystems of the many-body system. They provide unique insights into energy eigenstate statistics of many-body systems, as we show in an analysis on the basis of random matrix theory and of the eigenstate thermalization hypothesis. We propose a protocol that allows the measurement of the SFF and PSFFs in quantum many-body spin models, within the framework of randomized measurements. Aim…
We study the time evolution of occupation numbers for interacting Fermi-particles in the situation when exact compound states are "chaotic". This situation is generic for highly excited many-particles states in heavy nuclei, complex atoms, quantum dots, spin systems and quantum computer models. Numerical data show perfect agreement with a simple theory for the onset of thermalization in close systems of interacting particles.
A heuristic generalization of the Boltzmann-Gibbs microcanonical entropy is proposed, able to describe meta-equilibrium features and evolution of macroscopic systems. Despite its simple-minded derivation, such a function of "collective parameters" characterizing the microscopic state of N-body systems, yields, at one time, a statistical interpretation of dynamic evolution, and dynamic insights on the basic assumption of statistical mechanics. Its natural (implicit) time dependence entails} a "Second Law-like" behaviour and allows moreover, to perform an elementary test of the Loschmidt reversi…
It is suggested that many-body quantum chaos appears as the spontaneous symmetry breaking of unitarity in interacting quantum many-body systems. It has been shown that many-body level statistics, probed by the spectral form factor (SFF) defined as $K(η,t)=\langle|{\rm Tr}\, \exp(-ηH + itH)|^2\rangle$, is dominated by a diffuson-type mode in a field theory analysis. The key finding of this Letter is that the "unitary" $η=0$ case is different from the $η\to 0^{\pm}$ limit, with the latter leading to a finite mass of these modes due to interactions. This mass suppresses a rapid exponential ramp i…
On the emergence of quantum many-body chaos for tunably-broken integrability
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（国际理论科学中心，塔塔基础研究所）；Max-Planck-Institut für Physik komplexer Systeme（马克斯·普朗克复杂系统物理研究所）
We study spectral statistics in spatially extended chaotic quantum many-body systems, using simple lattice Floquet models without time-reversal symmetry. Computing the spectral form factor $K(t)$ analytically and numerically, we show that it follows random matrix theory (RMT) at times longer than a many-body Thouless time, $t_{\rm Th}$. We obtain a striking dependence of $t_{\rm Th}$ on the spatial dimension $d$ and size of the system. For $d>1$, $t_{\rm Th}$ is finite in the thermodynamic limit and set by the inter-site coupling strength. By contrast, in one dimension $t_{\rm Th}$ diverges wi…
The exponential growth or decay with time of the out-of-time-order commutator (OTOC) is one widely used diagnostic of many-body chaos in spatially-extended systems. In studies of many-body classical chaos, it has been noted that one can define a velocity-dependent Lyapunov exponent, $λ({\bf v})$, which is the growth or decay rate along "rays" at that velocity. We examine the behavior of $λ({\bf v})$ for a variety of many-body systems, both chaotic and integrable. The so-called light cone for the spreading of operators is defined by $λ({\bf \hat n}v_B({\bf \hat n}))=0$, with a generally directi…
We study the consequences of having translational invariance in space and in time in many-body quantum chaotic systems. We consider an ensemble of random quantum circuits, composed of single-site random unitaries and nearest neighbour couplings, as a minimal model of translational invariant many-body quantum chaotic systems. We evaluate the spectral form factor (SFF) as a sum over many-body Feynman diagrams, which simplifies in the limit of large local Hilbert space dimension $q$. At sufficiently large $t$, diagrams corresponding to rigid translations dominate, reproducing the chaotic behavior…
Dissipation- versus Chaos-Induced Relaxation in Non-Markovian Quantum Many-Body Systems
Universidade de Lisboa（里斯本大学）；Beijing Computational Science Research Center（北京计算科学研究中心）；Technische Universität Dresden（德累斯顿工业大学）；University of Cambridge（剑桥大学）
Ergodicity is a fundamental principle of statistical mechanics underlying the behavior of generic quantum many-body systems. However, how this universal many-body quantum chaotic regime emerges due to interactions remains largely a puzzle. This paper demonstrates using both heuristic arguments and a microscopic calculation that a dephasing mechanism, similar to Altshuler-Aronov-Khmelnitskii dephasing in the theory of localization, underlies this transition to chaos. We focus on the behavior of the spectral form factor (SFF) as a function of "time", t, which characterizes level correlations in …
Integrable quantum systems of finite size are generically robust against weak enough integrability-breaking perturbations, but become quantum chaotic and thermalizing if the integrability-breaking is strong enough. We argue that the onset of quantum chaos can be described as a Fock-space delocalization process, with the eigenstates of the integrable system being taken as the "Fock states". The integrability-breaking perturbation introduces hopping in this Fock space, and chaos sets in when this hopping delocalizes the many-body eigenstates in this space. Depending on the range of the dominant …
Highly excited many-particle states in quantum systems (nuclei, atoms, quantum dots, spin systems, quantum computers) can be ``chaotic'' superpositions of mean-field basis states (Slater determinants, products of spin or qubit states). This is a result of the very high energy level density of many-body states which can be easily mixed by a residual interaction between particles. We consider the time dynamics of wave functions and increase of entropy in such chaotic systems. As an example we present the time evolution in a closed quantum computer. A time scale for the entropy S(t) increase is t…
For perturbative scalar field theories, the late-time-limit of the out-of-time-ordered correlation function that measures (quantum) chaos is shown to be equal to a Boltzmann-type kinetic equation that measures the total gross (instead of net) particle exchange between phase space cells, weighted by a function of energy. This derivation gives a concrete form to numerous attempts to derive chaotic many-body dynamics from ad hoc kinetic equations. A period of exponential growth in the total gross exchange determines the Lyapunov exponent of the chaotic system. Physically, the exponential growth i…
In a recent Letter [Phys. Rev. Lett. 125, 180604 (2020)], we introduced a closed-form analytic expression for the average bipartite von Neumann entanglement entropy of many-body eigenstates of random quadratic Hamiltonians. Namely, of Hamiltonians whose single-particle eigenstates have random coefficients in the position basis. A paradigmatic Hamiltonian for which the expression is valid is the quadratic Sachdev-Ye-Kitaev (SYK2) model in its Dirac fermion formulation. Here we show that the applicability of our result is much broader. Most prominently, it is also relevant for local Hamiltonians…
