Probing Complex Quantum Dynamics through
Out-of-time-ordered Correlators

We argue that the low temperature phase of a conjugate pair of uncoupled, quantum chaotic, nonhermitian systems such as the Sachdev-Ye-Kitaev (SYK) model or the Ginibre ensemble of random matrices are dominated by replica symmetry breaking (RSB) configurations with a nearly constant free energy that terminates in a first order phase transition. In the case of the SYK model, we show explicitly that the spectrum of the effective replica theory has a gap. The Out of Time Order Correlators are discussed as well. For a non-chaotic SYK, the results are qualitatively different: the spectrum is gapless in the low temperature phase and the partition function factorizes into a product over Matsubara frequencies each with a different critical temperature for a second order phase transition.

Chaotic many body quantum systems can be in a phase of (many body) localized or extended eigenstates. In recent years, the seemingly less enigmatic phase of extended states has become a subject of controversy. It has been suggested that quantum many body eigenstates may be Non Ergodic yet Extended (NEE) in Hilbert space, with exotic distributions including signatures of multifractal behavior. In this talk I discuss how a blend of concepts developed in different fields --- including matrix integral techniques, lessons learned from the SYK model, and concepts of quantum information applied to random many body states --- lead to an improved understanding of many body ergodic phases. Our bottom line will be that (i) the delocalized quantum states of chaotic systems (subject to long range interactions) are ergodically distributed over (ii) an energy shell nontrivially interlaced into Hilbert space. We do not see room for the emergence of an NEE phase. At the same time (iii), the ergodic states of many body systems show entanglement exceeding that of thermal states, and of the (Page) entanglement of pure random states. We will argue that these entanglement signatures are sensitive probes into the many body physics of chaotic chaos.

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. Using such an approach, we are also able to show that the OTOC oscillations present for small spin chains compatible with experiments [Li J. et al., Phys. Rev. X, 7 (2017) 031011] describe qualitatively well the integrability-to-chaos transition inherited from the infinite chain.

The concepts of order and chaos are central to our understanding of the statistical mechanics of dynamical systems. Starting from the fabled butterfly effect, this talk presents a brief survey of general properties of chaotic many-body dynamics, in particular showing how out-of-time-ordered correlators (OTOCs) appear naturally there. It addresses such dynamics in simple and familiar model systems exhibiting different degrees of order, and concomitant descriptions with and without quasiparticle excitations. It finally demonstrates how OTOCs can be used in practise to mitigate errors in a noisy quantum computing platform. This is used to extract the intrinsic signal of a new type of spatiotemporal eigenstate order realised in a many-body localised discrete time crystal.

In many-body quantum systems admiting a regime where typical actions are large, asymptotic analysis can be applied to the transition amplitudes in many-body (Fock) space resulting in the many-body version of the celebrated van Vleck-Gutzwiller propagator and in the corresponding many-body semiclassical approximation. In its many-body version, semiclassical techniques express quantum mechanical effects, being themselves much more than the bare mean field approximation, as the result of complex interference between amplitudes built from solutions of a suitable classical mean-field limit and allow for a precise definition of many-body quantum chaos as the signature of mean-field chaotic dynamics. Under this unifying umbrella, all aspects concerning the influence of chaos in the scrambling of quantum correlations should have an explanation in terms of interference among mean field solutions and their characteristic unstable and ergodic properties. In this talk I will present the results of such program and explain how many-body interference of unstable mean-field solutions is responsible, in different disguises, of both the short- and long-time behavior of OTOCs in chaotic and critical systems. At the end, I will report on recent efforts to carefully designing a critical system with unstable mean-field limit at low temperatures where the bound of chaos of Maldacena, Shenker and Stanford holds non-trivially thus opening the possibility of a semiclassical study.

Quantum chaos imposes universal spectral signatures that govern the thermofield dynamics of a many-body system in isolation. The fidelity between the initial and time-evolving thermofield double states exhibits as a function of time a decay, dip, ramp, and plateau. Sources of decoherence give rise to a nonunitary evolution and result in information loss. Energy dephasing gradually suppresses quantum noise fluctuations and the dip associated with spectral correlations. Decoherence further delays the appearance of the dip and shortens the span of the linear ramp associated with chaotic behavior. The interplay between signatures of quantum chaos and information loss is determined by the competition among the decoherence, dip and plateau characteristic times, as demonstrated in the stochastic Sachdev-Ye-Kitaev model.

Quantum chaos is usually characterized through its statistical implications on the energy spectrum of a given system. We propose a decoherent mechanism for sensing quantum chaos. The chaotic nature of a many-body quantum system is sensed by studying the implications that the system produces in the long-time dynamics of a probe coupled to it under a dephasing interaction. By introducing the notion of an effective averaged decoherence factor, we show that the correction to the geometric phase acquired by the probe with respect to its unitary evolution can be exploited as a robust tool for sensing the integrable to chaos transition of the many-body quantum system to which it is coupled. This sensing mechanism is verified for several systems with different types of symmetries, disorder and even in the presence of long-range interactions, evidencing its universality.

I will describe results on the out-of-time-order correlators of a compressible quantum state without quasiparticle excitations.

I will overview our recent progress in using mathematical ideas from a many-body theory of quantum walks to prove bounds on quantum information propagation in: (1) the SYK model and other non-local models relevant to holography; (2) models with power-law interactions, where multiple notions of locality become relevant; (3) models of interacting bosons, such as the Bose-Hubbard model. In special (but often physically relevant) applications, our methods can dramatically improve upon Lieb-Robinson bounds and give qualitatively stronger results.

A generalized fluctuation-dissipation theorem for a bipartite out-of-time-ordered correlators (OTOCs) is discussed. The difference between the bipartite and physical OTOCs is given by the Wigner-Yanase skew information. Within this difference, the theorem describes a universal relation between chaotic behavior in quantum systems and a nonlinear-response function that involves a time-reversed process. We also discuss a one-parameter family of out-of-time-ordered correlators and prove that if the OTOC shows and exponential growth regardless of the choice of the regularization, then the growth rate is upper bounded by 2$\pi$ $k_B$ T/$\hbar$.

It is worth improving existing methods for estimating bulk transport coefficients from first principles; such methods enable theorists to make sharper predictions for existing experiments, aid in the exploration and discovery of new phenomena, and allow the design of future experimental systems with bespoke transport properties. Existing methods like exact diagonalization (ED), and time-evolved block decimation (TEBD) are not well suited for such calculations: ED is limited to small systems sizes, while estimates from TEBD are limited to just a few interaction times before memory requirements become impractical/truncation errors become significant. Recently, a number of methods have been proposed for sidestepping these issues. All of the methods rely on the intuition that much of the information content of a many-body density matrix (particularly high order correlations) can be thrown away without greatly affecting the calculation of transport coefficients. A detailed understanding of the size of systematic errors for these methods is lacking, however. In other words, if one does discard higher order correlations, what is the resulting size of error in the estimate of physical properties, such as the diffusion constant? This work aims to remedy this gap in the literature, focusing on the ``DAOE'' method developed by the present authors.

I will first present a (slight) reformulation of the proof by Tsuji, Shitara and Ueda (cf previous talk) of the bound to chaos, which emphasizes the central role of the fluctuation-dissipation theorem FDT. Next, I will show how FDT directly imposes a planckian timescale on more general problems.

Isolated many-body quantum systems quenched far from equilibrium can eventually equilibrate, but it is not yet clear how long they take to do so. To answer this question, we use exact numerical methods and analyze the entire evolution, from perturbation to equilibration, of paradigmatic many-body quantum systems. We investigate how the equilibration time depends on the system size, disordered strength, and observables. We consider the survival probability, entropies, spin autocorrelation function, connected spin-spin correlation function, and out-of-time ordered correlators, and show that if dynamical manifestations of spectral correlations in the form of the correlation hole (“ramp”) are taken into account, the time for equilibration scales exponentially with system size, while if they are neglected, the scaling is better described by a power law with system size.

I will discuss in depth the issues of many body localization and slow dynamics in interacting incommensurate lattices, considering different situations where the corresponding noninteracting single-particle models may or may not have single particle mobility edges.

Two topics have been gaining momentum in different fields of physics: At the intersection of condensed matter and high-energy physics lies the out-of-time-ordered correlator (OTOC). In quantum optics and quantum foundations, quasiprobabilities resemble probabilities but can become negative and nonreal. Such nonclassical values can signal nonclassical physics, such as the capacity for superclassical computation. I unite these two topics, showing that the OTOC equals an average over a quasiprobability distribution. The distribution, a set of numbers, contains more information than the OTOC, one number that follows from coarse-graining over the distribution. Aside from providing insight into the OTOC’s fundamental nature, the OTOC quasiprobability has several applications: Theoretically, the quasiprobability interrelates scrambling with uncertainty relations, with nonequilibrium statistical mechanics, and more closely with chaos. Experimentally, the quasiprobability points to a scheme for measuring the OTOC (via weak measurements, which refrain from disturbing the measured system much). The quasiprobability also signals false positives in attempts to measure scrambling of open systems. References: 1) NYH, Phys. Rev. A 95, 012120 (2017). https://journals.aps.org/pra/abstract/10.1103/PhysRevA.95.012120 2) NYH, Swingle, and Dressel, Phys. Rev. A 97, 042105 (2018). https://journals.aps.org/pra/abstract/10.1103/PhysRevA.97.042105 3) NYH, Bartolotta, and Pollack, Comms. Phys. 2, 92 (2019). https://dx.doi.org/10.1038/s42005-019-0179-8 4) Gonzàlez Alonso, NYH, and Dressel, Phys. Rev. Lett. 122, 040404 (2019). https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.122.040404 5) Arvidsson-Shukur, Drori-Chevalier, and NYH, J. Phys. A 54, 284001 (2021). https://iopscience.iop.org/article/10.1088/1751-8121/ac0289

In the first part, I will discuss a simple effective action for the OTOC contour based on work with Zhenbin Yang and Shunyu Yao. In the second part, I will review a problem that holographers have run into in regarding the spectral form factor and related quantities.

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. Here, we report the measurement of OTOCs of local operators for an Ising spin chain on a nuclear magnetic resonance (NMR) quantum simulator and observe that the OTOC behaves differently in the integrable and nonintegrable cases. We also observe the different behaviors of the OTOCs of local operators for different engineering Hamiltonians in solid NMR. Our experiment paves a way for experimentally studying quantum many-body complex dynamics, quantum chaos, holographic duality, and information scrambling in many-body quantum systems with quantum simulators.

Low angular momentum kicked tops present solvable models wherein the OTOC shows connections to the classical Lyapunov exponent. Reviewing these, we also discuss the case of weakly coupled bipartite systems wherein there are multiple regimes of scrambling with only the early phase being related to the classical Lyapunov exponent.

The far-from-equilibrium dynamics of generic interacting quantum systems is characterized by a handful of universal guiding principles, among them the ballistic spreading of initially local operators. Here, we show that in certain constrained many-body systems the structure of conservation laws can cause a drastic modification of this universal behavior. As an example, we study operator growth characterized by out-of-time-order correlations (OTOCs) in a dipole-conserving “fracton" chain. We identify a critical point with sub-ballistically moving OTOC front, that separates a ballistic from a dynamically frozen phase. This critical point is tied to an underlying localization transition and we use its associated scaling properties to derive an effective description of the moving operator front via a biased random walk with long waiting times. We support our arguments numerically using classically simulable automaton circuits. Johannes Feldmeier, Michael Knap, arXiv:2106.05292 Johannes Feldmeier, Pablo Sala, Giuseppe de Tomasi, Frank Pollmann, Michael Knap, Phys. Rev. Lett. 125, 245303 (2020).

Disordered quantum spin systems host many intriguing phenomena such as localization and glass-like relaxation. With cold atoms excited to Rydberg states we can experimentally realize a broad class of disordered spin models. The combination of experiments and numerical simulations of such models allows us to gain a better understanding of their dynamics far from equilibrium. In this talk I will report on recent results on glassy relaxation behavior in Heisenberg spin models with power law interactions and positional disorder. I will discuss possible schemes for Hamiltonian engineering including time-reversal of unitary dynamics allowing OTOC measurements.

In the long quest to identify and compensate the sources of decoherence in many-body systems far from the ground state, the varied family of Loschmidt echoes (LEs) became an invaluable tool in many experimental techniques. A LE involves the implementation of a time-reversal procedure to assess the effect of perturbations in a quantum excitation dynamics. However, when addressing microscopic systems one is repeatedly confronted with limitations that seem insurmountable. This led to the central hypothesis of irreversibility stating that the time-scale of decoherence, T_3, is proportional to the time-scale of the many-body interactions we reversed, T_2. In this work we evaluate the role of the perturbations in relation to T_2, quantifying the decay-time T_3. With this purpose, we implement two experimental schemes that modify the strength of the effective dipolar spin-spin coupling, i.e.~1/2T_2, while keeping constant the time-scale of the perturbations, T_Σ. This requires the application of solid pulses of off-resonance radio-frequency irradiation, to create a Floquet effective Hamiltonian that describes a spin-spin interaction reduced by a variable scale factor k. This factor, bearing positive and negative values, allows to vary the time scale of forward evolution, as well as that of the time-reversed one. Strikingly, we observe the superposition of the normalized Loschmidt echoes for the bigger values of k. This manifests the intrinsic dynamic dominance over the perturbation factors, even when the Loschmidt echo is devised to reverse those intrinsic dynamics. Thus, in the limit where the reversible interactions dominate over perturbations, the LE decays within a time-scale, T_3≈T_2/R with R= (0.15±0.01), confirming the emergence of a perturbation independent regime. Thus, the central hypothesis of irreversibility is solidly supported once more.

The relaxation of out-of-time-ordered correlators (OTOCs) has been studied as a means to characterize the scrambling properties of a quantum system. We show that the presence of local conserved quantities typically results in, at the fastest, an algebraic relaxation of the OTOC provided (i) the dynamics is local and (ii) the system follows the eigenstate thermalization hypothesis. Our result relies on the algebraic scaling of the infinite-time value of OTOCs with system size, which is typical in thermalizing systems with local conserved quantities, and on the existence of finite speed of propagation of correlations for finite-range-interaction systems. We show that time independence of the Hamiltonian is not necessary as the above conditions (i) and (ii) can occur in time-dependent systems, both periodic or aperiodic. We also remark that our result can be extended to systems with power-law interactions. Published in Physical Review B 104, 104306 (2021)

Far out-of-equilibrium many-body quantum dynamics in isolated systems necessarily generate interferences beyond an Ehrenfest time scale, where quantum and classical expectation values diverge. Ultracold atomic gases provide a promising setting to explore these phenomena. Theoretically speaking, the heavily-relied-upon truncated Wigner approximation leaves out these interferences. We develop a semiclassical theory of coherent state propagation for many-body bosonic systems, which properly incorporates such missing quantum effects. For mesoscopically populated Bose-Hubbard systems, it is shown that this theory captures post-Ehrenfest quantum interference phenomena very accurately, and contains relevant phase information to perform many-body spectroscopy with high precision.

We demonstrate analytically and verify numerically that the out-of-time order correlator is given by the thermal average of Loschmidt echo signals. This provides a direct link between the out-of- time-order correlator – a recently suggested measure of information scrambling in quantum chaotic systems – and the Loschmidt echo, a well-appreciated familiar diagnostic that captures the dynam- ical aspect of chaotic behavior in the time domain, and is accessible to experimental studies. (Bin Yan, Lukasz Cincio, Wojciech H. Zurek, Information Scrambling and Loschmidt Echo, Phys. Rev. Lett. 124, 160603 (2020), arXiv:1903.02651)

The bound on the growth rate of the out-of-time-order four-point correlator in chaotic quantum many-body quantum systems, conjectured by J. Maldacena, S. Shenker and D. Stanford, can be derived from the general structure of operator matrix elements that follows from the eigenstate thermalization hypothesis (ETH). This talk is based on joint work with C. Murthy.

I will show that Hawking radiation-like phenomena may be observed in systems that exhibit butterfly effects. Suppose that a classical dynamical system has a Lyapunov exponent $\lambda$, and is deterministic and non-thermal ($T=0$). If we quantize this system, the quantum fluctuations may imitate thermal fluctuations with temperature $T \sim \lambda/2\pi$ in a semi-classical regime, and it may cause analogous Hawking radiation. I will also discuss that this proposal may provide an intuitive explanation of the existence of the bound of chaos proposed by Maldacena, Shenker and Stanford.

Trotterization of quantum many-body dynamics is a central tool for both computational methods and digital quantum simulation on quantum computers. It is the goal of this talk to show that localization and quantum chaos play a key role in estimating the accuracy of such Trotterized dynamics. Specifically, it will be argued that a localization phenomenon strongly bounds these errors for local observables, leading to an error independent of system size and simulation time. Trotterized time evolution is thus intrinsically much more robust than suggested by known error bounds on the global many-body wave function. This robustness is characterized by a sharp threshold as a function of the Trotter step size, which separates a localized region with controllable Trotter errors from a quantum chaotic regime where errors grow rapidly with time. These different dynamical regimes will be characterized by standard quantities such as out-of-time-ordered correlators.

We study the Loschmidt echo for quenches in open one-dimensional lattice models with symmetry protected topological phases. For quenches where dynamical quantum phase transitions do occur we find that cusps in the bulk return rate at critical times $t_c$ are associated with sudden changes in the boundary contribution. For our main example, the Su-Schrieffer-Heeger model, we show that these sudden changes are related to the periodical appearance of two eigenvalues close to zero in the dynamical Loschmidt matrix. We demonstrate, furthermore, that the structure of the Loschmidt spectrum is linked to the periodic creation of long-range entanglement between the edges of the system.

Local measurements provide a mechanism to counteract and suppress the entanglement generation in complex interacting many-body systems, and when sufficiently strong can lead to a transition with an non extensive behaviour of the steady-state entanglement entropy. I discuss how this steady state behaviour relates to the dynamical microscopic processes participating in the creation and propagation of entanglement. Diagnosing entanglement dynamics in noisy and disordered spin chains via the measurement-induced steady-state entanglement transition T. Boorman, M. Szyniszewski, H. Schomerus, A. Romito, arXiv:2107.11354 (2021) Universality of Entanglement Transitions from Stroboscopic to Continuous Measurements M. Szyniszewski, A. Romito, and H. Schomerus, Phys. Rev. Lett. 125, 210602 (2020). Entanglement transition from variable-strength weak measurements M. Szyniszewski, A. Romito, and H. Schomerus, Phys. Rev. B 100, 064204 (2019).