Quantum Chaos and Holography

The idea of describing properties of complicated systems, such as complex atomic nuclei, using random numbers dates back to the 1950s. One application is in quantum mechanics, where random numbers are a tool for making accurate predictions about the statistics of discrete energy levels a system can assume. More precisely, the statistical distribution of the spacings between successive energy levels can be compared with distributions from random matrices, which provides a signature of whether the system behaves in a regular or a chaotic way. Random matrix theory (RMT) has since grown into an active branch between mathematics and physics, and has found applications in many branches of physics but also in biology or finance. Analyzing universal statistical properties of a spectrum requires unfolding. Unfortunately, this can lead to spurious results: for many-body systems, the density of states is generically far from being uniform, which makes the use of the unfolding procedure rather inaccurate. This is why in recent years the focus has shifted away from the statistics of energy spacings to the statistics of ratios between successive spacings. This ratio statistics is by now a widely used tool of quantum chaos, that allows to compare experimental or numerical observations with theoretical predictions. However, extra symmetries of the system, which may be hidden, can split the spectrum into independent random blocks, and thus modify these statistics. We show that it is possible to extend the theory of spacing ratio statistics to account for the presence of additional symmetries. Our results allow to probe for the existence of symmetries if they were unknown. We derive analytical surmises for random matrices with independent block structure, and illustrate our approach on a number of applications from many-body physics. This provides a tool not only to get a signature of chaos or regularity in systems with symmetries, but also to uncover these symmetries if they were previously unnoticed.

We develop a general non-perturbative characterisation of universal features of the density $\rho(\Gamma)$ of S-matrix poles (resonances) $E_n − i\Gamma_n$ describing waves incident and reflected from a disordered medium via a single M-channel waveguide/lead. Explicit expressions for $\rho(\Gamma)$ are derived for several instances of systems with broken time-reversal invariance, in particular for quasi-1D medium as well as for Random Regular Graph. In the case of perfectly coupled lead with $M \sim 1$ the most salient features are tails $\rho(\Gamma)\sim 1/\Gamma$ for narrow resonances reflecting exponential localization and $\rho(\Gamma)\sim 1/\Gamma^2$ for broad resonances reflecting states located in the vicinity of the attached wire. For multimode wires with $M\gg 1$ intermediate asymptotics $\rho(\Gamma)\sim 1/\Gamma^{3/2}$ is shown to emerge, reflecting diffusive nature of decay into wide enough contacts. The presentation will be based on a joint work with M. Skvortsov and K. Tikhonov.

The long-time correlation function is a well-known probe of information loss in black holes. I will show how the magnitude of the long-time correlator, averaged over a family of states, can be computed in gravity. This work extends recent progress in understanding corrections to semi-classical gravity to situations where an average over theories is not available, and to a wider class of observables. Based on https://arxiv.org/abs/2105.12771 and work in progress.

We study the power spectrum of eigen-angles of random matrices drawn from the circular unitary ensemble $({\rm CUE})$ and show that it can be evaluated in terms of either a Fredholm determinant, or a Toeplitz determinant, or a sixth Painlev\'e function. In the limit of infinite-dimensional matrices, we derive a {\it concise} parameter-free formula for the power spectrum which involves a fifth Painlev\'e transcendent. Further, we discuss a universality of the predicted power spectrum law (in random-matrix-theory context and beyond), and present a fair evidence that a universal Painlevé V curve is observed in the power spectrum of nontrivial zeros of the Riemann zeta function.

The tripartite information is an observable-independent measure for scrambling and delocalization of information. Therefore it is a good observable-independent indicator for distinguishing between many-body localized and delocalized regimes, which we confirm for the XXZ-chain in a random field. Specifically, we find that the tripartite information signal spreads inside a lightcone that only grows logarithmically in time in the many-body localized regime similar to the entanglement entropy. We also find that the tripartite information eventually reaches a plateau with an asymptotic value that is suppressed by strong disorder. N. Boelter and S. Kehrein, Phys. Rev. B 105 (2022) 104202

In this talk I will discuss manifestations of chaotic behavior in the solvable 2d JT gravity model. Boundary operators in this model are divided into two classes: generic operators that lead to chaotic behavior in out-of-time ordered correlators, and special "degenerate" operators that correspond to an integrable subsector of the model and that are related to the embedding of JT gravity within string theory (minimal string).

Bounds on transport represent a way of understanding allowable regimes of quantum and classical dynamics. Numerous such bounds have been proposed either for classes of theories or universally for all theories. Few are inviolable. On the other hand, the inequalities recently derived for the growth rate of quantum chaos do appear to be exact and universal. In this talk, I will first review a few of the more influential bounds on transport from past decades and then discuss the ingredients that enter into proofs of bounds on quantum chaos: exponential (Lyapunov) and weak quantum chaos. I will then present a set of new methods for deriving exact, rigorous, and sharp bounds on all coefficients of hydrodynamic dispersion relations, including diffusivity and the speed of sound. These general techniques combine analytic properties of hydrodynamics and the theory of univalent (complex holomorphic and injective) functions. At least in systems with holographic duals, these methods allow to make a precise relation between rigorous bounds on transport and a property of quantum chaos known as pole-skipping.

The ergodic regime of many body quantum chaos defines the perhaps most universal phase of quantum mechanics. It is uniquely described by an effective zero-dimensional matrix theory, Efetov's supersymmetric nonlinear sigma model. To the best of our knowledge every chaotic quantum system maps onto said model in the long time limit. In this talk we apply this correspondence as a search principle in holography and two-dimensional quantum gravity. We will look at various effective models through this lens, among them the SYK model, JT gravity, and Kodaira-Spencer string field theory. Our main finding is that the latter is complete in that it is dual to the sigma model at large time scales. On this basis, it provides a full description of quantum chaos in the two dimensional bulk – from semiclassical gravitational fluctuations to deep microstate quantization at the largest time scales.

In this talk I will talk about recent as well as ongoing work which uses insights from quantum chaos to characterise non-perturbative and so-called doubly non-perturbative effects in quantum gravity. We will mostly be concerned with low dimensions, but our methods suggest ways forward in higher dimensions as well, as I shall outline.

We formulate a theory for resonances in the many-body localised phase in terms of local observables. These resonances constitute a mechanism by which integrability is ultimately broken, and so their investigation allows for an understanding of the emergence of quantum chaos. A key result is to show analytically that there are universal correlations between the matrix elements of local observables and the many-body level spectrum. Moreover, we reveal how the matrix elements encode the energy scales associated with resonance, and show that these energies are power-law distributed. Our theoretical approach is to consider the effect of varying a local field, which induces parametric dynamics of spectral properties.

A remarkable feature of chaos in many-body quantum systems is the existence of a bound on the quantum Lyapunov exponent. An important question is to understand what is special about maximally chaotic systems which saturate this bound. Here I will discuss a proposal for a `hydrodynamic' origin of chaos in such systems, and discuss hallmarks of maximally chaotic systems. In particular I will discuss how in maximally chaotic systems there is a suppression of exponential growth in commutator squares of generic few-body operators. This suppression appears to indicate that the nature of operator scrambling in maximally chaotic systems is fundamentally different to scrambling in non-maximally chaotic systems.

I will review recent progress concerning Krylov complexity

In quantum systems with a classical limit, advanced semiclassical methods provide the crucial link between classically chaotic dynamics and corresponding features at the quantum level. Recently, single-particle techniques dealing with ergodic wave interference in the usual semiclassical limit of small Planck constant have begun to be transformed into the field theoretical domain of large-N particle systems thereby accounting for genuine many-body quantum interference. This semiclassical many-body theory moreover lays the dynamical foundations for understanding random-matrix correlations of both single-particle and many-body quantum chaotic systems, also with implications for gravity models. After introducing this approach I will discuss two complementary phenomena where Ehrenfest time scales play a crucuial role: (i) Scarring, naturally requiring a classical limit, is intriguing because it indicates deviations from ergodicity for a fully chaotic system that globally shows eigenstate thermalization. I will demonstrate such scarring for many-body Bose-Hubbard systems with a high-dimensional associated classical phase space. (ii) Vice versa I will show scrambling of quantum information in an integrable Bose-Hubbard system.

I will review the recent results on the proof of random matrix spectral form factor and explicit computation of correlation functions of local observables in the so-called dual-unitary brickwork circuits (including integrable, non-ergodic, ergodic and chaotic cases). Further I will show how these results can be extended to another quantum-circuit platform, specifically to unitary interactions round-a-face (IRF). I will argue that correlation functions of these models are generally perturbatively stable with respect to breaking dual-unitarity, and describe a simple result within this framework.

Within the AdS/CFT correspondence, the entanglement properties of the CFT are related to wormholes in the dual gravity theory. This gives rise to questions about the factorisation properties of the Hilbert spaces on both sides of the correspondence. We review these issues and show how the Berry phase, a geometrical phase encoding information about topology, may be used to reveal similarities between the Hilbert space structure on both sides of the correspondence. Mathematical concepts such as coadjoint orbits play an important role. In addition to its relevance for quantum gravity, this analysis also suggests how to experimentally realise the Berry phase and its relation to entanglement in table-top experiments involving photons or electrons.

In an isolated system, the time evolution of a given observable in the Heisenberg picture can be efficiently represented in Krylov space. In this representation, an initial operator becomes increasingly complex as time goes by, a feature that can be quantified by the Krylov complexity. We introduce a fundamental and universal limit to the growth of the Krylov complexity by formulating a Robertson uncertainty relation, involving the Krylov complexity operator and the Liouvillian, as generator of time evolution. We further show the conditions for this bound to be saturated and illustrate its validity in the paradigmatic models of quantum chaos. Ref. Niklas Hörnedal, Nicoletta Carabba, Apollonas S. Matsoukas-Roubeas, Adolfo del Campo, Ultimate Physical Limits to the Growth of Operator Complexity, arXiv:2202.05006

Generic perturbations of an integrable system lead to chaos. How they lead to chaos in the classical setting has been understood for decades, and is captured by the Kolmogorov-Arnold-Moser theorem. No such result exists for quantum systems; moreover, the scaling of the threshold to chaos with system size is not well understood even classically. We propose some answers to these questions for quantum many-body systems, arguing that the onset of chaos can be understood as a Fock-space delocalization process. We corroborate our theoretical predictions against large-scale numerical simulations in a variety of model systems.

The typical entanglement entropy of subsystems of random pure states is known to be (nearly) maximal, while the typical entanglement entropy of random Gaussian pure states has been recently shown to exhibit a qualitatively different behavior, with a coefficient of the volume law that depends on the fraction of the system that is traced out [1]. In this talk we show evidence that the typical entanglement entropy of eigenstates of quantum-chaotic Hamiltonians mirrors the behavior in random pure states [2], while that of integrable Hamiltonians mirrors the behavior in random Gaussian pure states [3]. Based on these results, we conjecture that the typical entanglement entropy of Hamiltonian eigenstates can be used as a diagnostic of quantum chaos and integrability [3]. We discuss subtleties that emerge as a result of conservation laws, such as particle number conservation [1,2], as well as of lattice translational invariance [4]. References: [1] E. Bianchi, L. Hackl, M. Kieburg, MR, and L. Vidmar, arXiv:2112.06959. [2] L. Vidmar and MR, PRL 119 220603 (2017). [3] T. LeBlond, K. Mallayya, L. Vidmar, and MR, PRE 100 062134 (2019). [4] L. Vidmar, L. Hackl, E. Bianchi, and MR, PRL 119 020601 (2017).

Dual-unitary circuits have recently gained attention as a class of many-body models for which the dynamics is both chaotic and tractable. After a short introduction to dual-unitary circuit dynamics and the underlying space-time duality I will show how dual-unitary dynamics can be combined with specific projective measurements without losing this underlying duality. This will be illustrated using a new kind of emergent random matrix behaviour following a quantum quench: starting from a time-evolved state, an ensemble of pure states supported on a small subsystem can be generated by performing projective measurements on the remainder of the system, leading to a projected ensemble. In chaotic quantum systems it was conjectured that such projected ensembles become indistinguishable from the uniform Haar-random ensemble and lead to a quantum state design, and I will argue that this holds exactly in dual-unitary circuit dynamics.

We investigate the charged many many-body interacting Sachdev-Ye-Kitaev (SYK) model in the grand-canonical setting. By varying the chemical potential or temperature, we find that, in the maximally chaotic regime, the system undergoes a phase transition between high and low entropies. A similar transition in entropy is seen in charged AdS black holes transitioning between a large and small event horizon. We also find a first-order chaotic-to-integrable (quantum) phase transition, where the finite extensive entropy drops to zero, reminiscent of the Hawking-Page (HP) transition. Analytically calculating the critical exponents associated with the continuous phase transition, we find that our model in fact shares a universality class with large number of charged black hole phase-transitions. https://arxiv.org/abs/2204.09629

We study thermalization and spectral properties of extended systems connected, through their boundaries, to a thermalizing Markovian bath. Specifically, we consider periodically driven systems modeled by brickwork quantum circuits where a finite section (block) of the circuit is constituted by arbitrary local unitary gates while its complement, which plays the role of the bath, is dual unitary. We show that the evolution of local observables and the spectral form factor are determined by the same quantum channel, which we use to characterize the system's dynamics and spectral properties. In particular, we identify a family of fine-tuned quantum circuits—which we call strongly nonergodic—that fails to thermalize even in this controlled setting, and, accordingly, their spectral form factor does not follow the random matrix theory prediction. We provide a set of necessary conditions on the local quantum gates that lead to strong nonergodicity, and in the case of qubits, we provide a complete classification of strongly nonergodic circuits. We also study the opposite extreme case of circuits that are almost dual unitary, i.e., where thermalization occurs with the fastest possible rate. We show that, in these systems, local observables and spectral form factor approach, respectively, thermal values and random matrix theory prediction exponentially fast. We provide a perturbative characterization of the dynamics and, in particular, of the timescale for thermalization.

The general form of the Eigenstate Thermalization Hypothesis (ETH), describing all the relevant correlations of matrix elements, may be derived on the basis of a `typicality' argument of invariance with respect to local rotations involving nearby energy levels. In this talk, I will discuss the close relation between ETH and Free Probability theory, as applied to a thermal ensemble or an energy shell. This mathematical framework allows one to express in an unambiguous way high order connected correlation functions (here identified as free cumulants) in terms of standard correlation functions. This perspective naturally incorporates the consistency property that local functions of ETH operators also satisfy ETH. The present results open a direct connection between quantum thermalization and the mathematical structure of Free Probability, thus offering the basis for insightful analogies and new developments.

The paradigm of black holes as displaying quantum-chaos signatures got a strong push with the celebrated work of Saad, Shenker and Stanford where the characteristic genus expansion of spectral correlations within quantized Jackiw-Teitelboim (JT) gravity was shown to be dual to a special type of Random Matrix Theory (RMT) ensemble. Periodic Orbit Theory (POT), being a far reaching elaboration of the semiclassical program describing quantum systems in the asymptotic regime of large actions, provides an unique framework where the quantum signatures of classical (or mean-field) chaos are systematically addressed and it constitutes a fundamental pillar of the quantum chaos field. In fact, starting from the celebrated Gutzwiller trace formula expressing partition functions in terms of coherent sums over periodic orbits, the careful analysis of orbit bunching successfully explains the emergence of universal spectral correlations in chaotic systems (the Bohigas-Giannoni-Schmidt conjecture). In this talk I will present a progress report on the efforts to understand (and interpret) the structure of JT spectral correlations by means of POT. For this, I first introduce the relevant semiclassical tools that, starting with the asymptotic analysis of the path integral, allow us to understand spectral universality and the emergence of RMT fluctuations in single systems, as opposed to ensembles where disorder techniques have been used recently. Armed with this machinery I will show how the existence of a finite classical action scale leads naturally to an effective description of spectral functions in terms of a subset of periodic orbits that, in turn, demands them to acquire a specific structure with the form of a genus-like expansion. This is the starting point for a formal identification of POT structures with quantum-gravitational objects, and opens the possibility of identifying a single classical chaotic system that, once quantized, describes generic structural aspects of JT spectral functions.

Spatiotemporal correlation functions provide the key diagnostic tool for studying spatially extended complex quantum many-body systems. In ergodic systems scrambling causes initially local observables to spread uniformly over the whole available Hilbert space and causes exponential suppression of correlation functions with the spatial size of the system. In this talk, we present a perturbed free quantum circuit model, in which ergodicity is induced by a unitary impurity placed on the system's boundary and that allows for demonstrating the underlying mechanism governing the asymptotic scaling of correlations with system size. This is achieved by mapping dynamical correlation functions of local operators in a system of linear size $L$ at time $t$ to a partition function with complex weights defined on a two-dimensional lattice of smaller size $t/L \times L$ with a helix topology. We evaluate this partition function in terms of suitable transfer matrices. As this drastically reduces the complexity of the computation of correlation functions, we are able to treat system sizes far beyond what is accessible by exact diagonalization. By studying the spectra of transfer matrices numerically and combining our findings with analytical arguments we determine the asymptotic scaling of correlation functions with system size. For impurities that remain unitary under partial transpose, we demonstrate that correlation functions at times proportional to system size $L$ are generically exponentially suppressed with $L$. In contrast, for generic unitary impurities correlations show persistent revivals with a period given by the system size.