Chaos and Dynamics in Correlated Quantum Matter

We propose a unified description of the exponential growth and ballistic butterfly spreading of out-of-time ordered thermal correlation functions (OTOC) across different systems using a newly formulated "quantum hydrodynamics," which is valid at finite $\hbar$ and to all orders in derivatives. The scrambling of a generic few-body operator in a chaotic system is described as building up a "hydrodynamic cloud," and the exponential growth of the cloud arises from a shift symmetry of the hydrodynamic action. The shift symmetry also shields correlation functions of the energy density and flux, and time ordered correlation functions of generic operators from exponential growth, while leads to chaotic behavior in OTOCs. The theory also predicts an interesting phenomenon of the skipping of a pole at special values of complex frequency and momentum in two-point functions of energy density and flux. This pole-skipping phenomenon can be used to predict the Lyapunov exponent and butterfly velocity from energy density two-point function, and may be considered as a "smoking gun" for the hydrodynamic origin of the chaotic mode.

We consider isolated many-body quantum systems which may or may not thermalize, i.e., expectation values approach an (approximately) steady long-time limit which may or may not agree with the microcanonical prediction of equilibrium statistical mechanics. A general analytical theory is worked out for the typical temporal relaxation behavior in both cases and is compared with various experimental and numerical findings.

Certain wave functions of non-interacting quantum chaotic systems can exhibit "scars" in the fabric of their real-space density profile. Quantum scarred wave functions concentrate in the vicinity of unstable periodic classical trajectories. We introduce the notion of many-body quantum scars which reflect the existence of a subset of special many-body eigenstates concentrated in certain parts of the Hilbert space. We demonstrate the existence of scars in the Fibonacci chain – the onedimensional model with a constrained local Hilbert space realized in the 51 Rydberg atom quantum simulator [H. Bernien et al., Nature 2017]. The quantum scarred eigenstates are embedded throughout the thermalizing many-body spectrum, but surprisingly lead to direct experimental signatures such as robust oscillations following a quench from a charge-density wave state found in experiment. We develop a model based on a single particle hopping on the Hilbert space graph, which quantitatively captures the scarred wave functions. Our results suggest that scarred many-body bands give rise to a new universality class of quantum dynamics, which opens up opportunities for creating and manipulating novel states with long-lived coherence in systems that are now amenable to experimental study.

Using full random matrices, we derive analytical expressions that describe the entire evolution of the survival probability, density imbalance, and out-of-time-ordered correlator. These expressions provide bounds and serve as references for the studies of realistic many-body quantum systems. Following the same steps employed for full random matrices, we obtain expressions that match extremely well numerical results for disordered spin-1/2 models that are chaotic and quenched far from equilibrium. This comparison between the two models lead to the identification of dynamical features that are generic to many-body chaotic quantum systems; they include power-law decays and correlation holes.

The statistical mechanics of equilibrium systems is characterized by two fundamental ideas: that closed systems approach a late time thermal state and that of phase structure wherein such late time states exhibit singular changes as various parameters characterizing the system are changed. Recent progress has established generalizations of these ideas which apply to periodically driven, or Floquet, closed quantum systems. I will describe this progress, which centrally uses other recent advances in our understanding of many body localization. I will describe how it has resulted in the discovery of entirely new phases such as the Pi-spin glass/Floquet time crystal which exist only in driven quantum systems.

Entanglement and entropy are key concepts standing at the foundations of quantum and statistical mechanics, respectively. In the last decade the study of quantum quenches revealed that these two concepts are intricately intertwined. Although the unitary time evolution ensuing from a pure initial state maintains the system globally at zero entropy, at long time after the quench local properties are captured by an appropriate statistical ensemble with non zero thermodynamic entropy, which can be interpreted as the entanglement accumulated during the dynamics. Therefore, understanding the post-quench entanglement evolution unveils how thermodynamics emerges in isolated quantum systems. An exact computation of the entanglement dynamics has been provided only for non-interacting systems, and it was believed to be unfeasible for genuinely interacting models. Conversely, here we show that the standard quasiparticle picture of the entanglement evolution, complemented with integrability-based knowledge of the asymptotic state, leads to a complete analytical understanding of the entanglement dynamics in the space-time scaling limit. Our framework requires only knowledge about the steady state, and the velocities of the low-lying excitations around it. We provide a thorough check of our result focusing on the spin-1/2 Heisenberg XXZ chain, and considering quenches from several initial states.

In my talk, I intend to review some of the applications of a recently developed semi-semiclassical theory of non-equilibrium systems and quantum quenches [C.P Moca, M. Kormos, and G. Zaránd, Phys. Rev. Lett. 119, 100603 (2017)]. Our hybrid semiclassical method handles internal degrees of freedom completely quantum mechanically, and accounts efficiently for entanglement entropy generation by these. In non-equilibrium situations, we can follow time evolution up to timescales at which local thermalization occurs. As an application, we investigate the quench dynamics and phase fluctuations of a pair of tunnel coupled one dimensional Bose condensates described by the sine-Gordon model, where we have also obtained a full analytical semiclassical description of phase correlations in the so-called universal limit [M. Kormos and G. Zaránd, Phys. Rev. E 93, 062101 (2016)]. We validate our approach by non-Abelian TEBD calculations performed for the antiferromagnetic spin S=1 Heisenberg chain [M. Werner et al, unpublished]. We also apply our semiclassical approach to describe inhomogeneous charge (spin) relaxation and the formation of non-equilibrium steady states [M. Kormos, C.P Moca, and G. Zaránd, arXiv:1712.09466]. Again, in the universal limit, we obtain full analytical results, which we complete with Monte Carlo simulations for an antiferromagnetic spin 1 chain. Depending on the initial state, the spin transport is found to be ballistic or diffusive. In the ballistic case we identify a "second front" that moves more slowly than the maximal quasiparticle velocity, and spreads diffusively. We also observe local equilibration around the second front in terms of the densities of the particle species.

Classical many body interacting systems are typically chaotic and their microcanonical dynamics ensures that time averages and phase space averages are identical. In proximity to an integrable limit the properties of the network of nonintegrable action space perturbations will decide whether ergodicity will hold arbitrarily close to the limit (albeit with diverging relaxation times), or whether the system fragments into regular and chaotic parts and turns into a nonergodic conductor at a finite distance to the integrable limit.

Information scrambling and the butterfly effect in chaotic quantum systems can be diagnosed by out-of-time-ordered (OTO) commutators through an exponential growth and large late time value. We show that the latter feature shows up in a strongly correlated many-body system, a Luttinger liquid, whose density fluctuations we study at long and short wavelengths, both in equilibrium and after a quantum quench. We find rich behaviour combining robustly universal and non-universal features. The OTO commutators display temperature and initial state independent behaviour. For the short wavelength density operator, they reach a sizeable value after the light cone only in an interacting Luttinger liquid, where the bare excitations break up into collective modes. We benchmark our findings numerically on an interacting spinless fermion model in 1D, and find persistence of central features even in the non-integrable case. As a non-universal feature, the short time growth exhibits a distance dependent power.

Information in local ergodic systems spreads behind a quasi-causal light-cone with a constant velocity know as the Lieb-Robinson velocity. In this talk I will present two families of systems for which the Lieb-Robinson light-cones are deformed, and the causal regions are enhanced (suppressed), yielding asymptotically diverging (vanishing) Lieb-Robinson velocities. I will discuss the spatiotemporal shape of these anomalous light-cones.

Generic, clean quantum many-body systems approach a thermal equilibrium after a long time evolution. In order to reach a global equilibrium, conserved quantities have to be transported across the whole system which is a rather slow process governed by diffusion. By contrast, the scrambling of quantum information is ballistic and hence can be characterized by a "butterfly" velocity. One way of describing the propagation of quantum information is to study out-of-time ordered (OTO) correlation functions, which are unconventional correlation functions with time arguments that are not time ordered. Using matrix-product-state based numerical simulations, we compute such correlators at high temperatures in a one-dimensional Bose-Hubbard model, where well defined quasi-particles cease to exist. Finally, we will discuss ways of experimentally characterizing these unconventional OTO correlation functions in synthetic quantum matter.

The out-of-time-ordered correlator (OTOC) has been proposed as an indicator of chaos in quantum systems due to its simple interpretation in the semiclassical limit. In particular, its rate of possible exponential growth at $\hbar \to 0$ is closely related to the classical Lyapunov exponent. Here we explore how this approach to "quantum chaos" relates to the random-matrix theoretical description. To do so, we introduce and study the level statistics of the logarithm of the out-of-time-ordered operator, $\hat{\Lambda}(t) = \ln \left( - \left[\hat{x}(t),\hat{p}(0) \right]^2 \right)/(2t)$, that we dub the "Lyapunovian" or "Lyapunov operator" for brevity. The Lyapunovian's level statistics is calculated explicitly for the quantum stadium billiard. It is shown that in the bulk of the filtered spectrum, this statistics perfectly aligns with the Wigner-Dyson distribution. One of the advantages of looking at the spectral statistics of this operator is that it has a well defined semiclassical limit where it reduces to the matrix of uncorrelated classical Lyapunov exponents in a partitioned phase space. We provide heuristic picture interpolating these two limits using Moyal quantum mechanics. Our results show that the Lyapunov operator may serve as a useful tool to characterize "quantum chaos" and in particular quantum-to-classical correspondence in chaotic systems, by connecting the semiclassical Lyapunov growth that we demonstrate at early times, when the quantum effects are weak, to universal level repulsion that hinges on strong quantum interference effects.

Lessons from Anderson localization highlight the importance of dimensionality of real space for localization due to disorder. More recently, studies of many-body localization have focussed on the phenomena in one dimension using techniques of exact diagonalization and tensor networks. On the other hand experiments in two dimensions have provided concrete results going beyond the previously numerically accessible limits while posing several challenging questions. I will present the first large-scale numerical examination of a disordered Bose-Hubbard model in two dimensions realized in cold atoms, which shows entanglement based signatures of many-body localization. By generalizing a low-depth quantum circuit to two dimensions we approximate eigenstates in the experimental parameter regimes for large systems, which is beyond the scope of exact diagonalization. A careful analysis of the eigenstate entanglement structure provides an indication of the putative phase transition marked by the peak in the fluctuations of entanglement entropy in a parameter range potentially consistent with experiments.

In the first part [1] I will show that electron-magnon interaction delocalizes the particle in a system with strong charge disorder. The analysis is based on results obtained for a single hole in the one–dimensional $t$-$J$ model. Unless there exists a mechanism that localizes spin excitations, the charge carrier remains delocalized even for a very strong charge disorder and shows subdiffusive motion up to the longest accessible times [1]. In the second part [2] I will present a study of dynamics of a single hole subject to a random magnetic field. Strong disorder that couples only to the spin sector localizes both spin and charge degrees of freedom. While we cannot precisely pinpoint the threshold disorder, we conjecture that there are two distinct transitions. Weaker disorder first causes localization in the spin sector. Carriers become localized for somewhat stronger disorder when the spin localization length is of the order of a single lattice spacing. I will also discuss finite doping. [1] J. Bonca and M. Mierzejewski, PRB \bf{95}, 214201 (2017). [2] G. Lemut, M. Mierzejewski, and J.Bonca, PRL \bf{119} 246601 (2017).

Glassy dynamics has long been recognized as having a time-reparametrization (almost) invariance, physically related to the diverging susceptibility of the relaxation timescale of these systems with respect to perturbations (such as shear). Zero temperature entropy also appears naturally, once one considers the generator of the dynamics, as a consequence of the multiplicity of metastable states. The developments around the SYK model offer the hope of a fruitful exchange of ideas.

Out of equilibrium phases of matter exhibiting order in individual eigenstates, such as many-body localised spin glasses and discrete time crystals, can be characterised by inherently dynamical quantities such as spatiotemporal correlation functions. In this work, we introduce dynamical potentials which act as generating functions for such correlations and capture eigenstate phases and order even at infinite temperatures. These potentials show formal similarities to their equilibrium counterparts, namely thermodynamic potentials. We provide three representative examples: a disordered XXZ chain showing many-body localisation, a disordered Ising chain exhibiting spin-glass order and its periodically-driven cousin exhibiting time-crystalline order.

The eigenstate thermalisation hypothesis, when stated naively, harbours a paradox. The faithful description of quantum evolution from an initially weakly entangled state requires an exponentially growing number of variables. If the system thermalises, however, ultimately only the energy density can be used to distinguish states of the system. The resolution is connected with the implicit focus upon a minor partition of the system so that the major part acts as an effective bath in which information is lost. Unlike, the related paradox for black holes, this information horizon does not evaporate. The question remains, however, how does the exponentially increasing number of variables eventually reduce to the one required to describe the thermal state. We connect this question to ideas of classical thermalisation. In that context, thermalisation and ergodicity are associated with dynamical chaos, characterized by the Lyapunov spectrum. Quantum Thermalisation can be recast in a similar way by projecting the dynamics onto a variational manifold of states that can capture the observables on the minor partition. Evolving on this manifold using the time-dependent variational principle projects the quantum dynamics onto a classical Hamiltonian dynamics. We carry out this procedure for an infinite spin chain and extract the full Lyapunov spectrum of the resulting chaotic dynamics. This spectrum provides an alternative perspective upon eigenstate thermalisation, pre-thermalisation and integrability as well as providing a resolution of the information paradox of quantum thermalisation.

The average entanglement entropy of subsystems of random pure states is (nearly) maximal [1]. In this talk, we discuss the average entanglement entropy of subsystems of highly excited eigenstates of integrable and nonintegrable many-body lattice Hamiltonians. For translationally invariant quadratic models we prove that, when the subsystem size is not a vanishing fraction of the entire system, the average eigenstate entanglement entropy departs from that of random pure states. We also prove that typical eigenstates of such Hamiltonians exhibit eigenstate thermalization for local observables [2]. For random pure states with a fixed particle number (away from half filling) and normally distributed real coefficients, we prove that the deviation from the maximal value grows with the square root of the system's volume. The behavior of the entanglement entropy of highly excited eigenstates of a particle number conserving quantum chaotic model is shown to be consistent with the analytical results for the random canonical states [3]. References: [1] D. N. Page, Phys. Rev. Lett. 71, 1291 (1993). [2] L. Vidmar, L. Hackl, E. Bianchi, and M. Rigol, Phys. Rev. Lett. 119, 020601 (2017). [3] L. Vidmar and M. Rigol, Phys. Rev. Lett. 119, 220603 (2017).

We will argue that sampling complexity, that is the question of how hard it is to classically produce a sample from a given probability distribution, lies at the heart of understanding quantum many-body systems in and out of equilibrium. For example, a single measurement of a pure quantum state of n qubits in the computational basis produces a single sample from the probability distribution defined by the squared norms of the 2^n amplitudes of the state. If this state is a ground state of a Hamiltonian, or if this state was produced via unitary time evolution following a quantum quench, one can use the complexity of sampling from the resulting distribution as an effective order parameter to draw phase boundaries (including dynamical phase boundaries) in parameter space and in time. We have made the first steps along this research direction in [Deshpande et al, arXiv:1703.05332].

Out-of-time-order correlation (OTOC) functions have recently received a lot of interest as tools to analyze the dynamics of interacting many-body systems and to quantify the scambling of information in these systems. They exhibit a characteristic exponential growth as a function of time until the scrambling time, respectively, Ehrenfest time, where they abruptly saturate. We will show for quantum chaotic many-body (Bose-Hubbard) systems that this saturation arises from quantum interference effects and derive corresponding analytical expressions for the OTOC. This is non-perturbatively achieved by expressing the OTOC in terms of coherent sums over interfering solutions of the corresponding classical mean-field equations, thereby including correlation effects. Moreover, on the numerical side we devise a semiclassical propagator for large-N Bose-Hubbard systems far-out-of equilibrium that allows to calculate with high precision many-body quantum interference in Fock space on time scales far beyond the scrambling/Ehrenfest time.

A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT). For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry (1985) within the so-called diagonal approximation of semiclassical periodic-orbit sums. Derivation of the full RMT spectral form factor K(t) from semiclassics has been completed only much later in a tour de force by Mueller et al (2004). In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming at the forefront even for simple many-body quantum systems, such as locally interacting spin chains. Such systems display two universal types of behaviour which are termed as `many-body localized phase' and `ergodic phase'. In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder. In my talk I will outline the first theoretical explanation for these observations. I will show how to compute K(t) explicitly in the leading two orders in t and show its agreement with RMT for non-integrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin 1/2 models in a periodically kicking transverse field.