Quantum chaos can be characterized by an exponential growth of the thermal out-of-time-order four-point function up to a scrambling time u[over ^]_{*}. We discuss generalizations of this statement for certain higher-point correlation functions. For concreteness, we study the Schwarzian theory of a one-dimensional time reparametrization mode, which describes two-dimensional anti-de Sitter space (AdS_{2}) gravity and the low-energy dynamics of the Sachdev-Ye-Kitaev model. We identify a particular set of 2k-point functions, characterized as being both "maximally braided" and "k-out of time order," which exhibit exponential growth until progressively longer time scales u[over ^]_{*}^{(k)}∼(k-1)u[over ^]_{*}. We suggest an interpretation as scrambling of increasingly fine grained measures of quantum information, which correspondingly take progressively longer time to reach their thermal values.
Quantum chaos can be characterized by an exponential growth of the thermal out-of-time-order four-point function up to a scrambling time u[over ^]{*}. We discuss generalizations of this statement for certain higher-point correlation functions. For concreteness, we study the Schwarzian theory of a one-dimensional time reparametrization mode, which describes two-dimensional anti-de Sitter space (AdS{2}) gravity and the low-energy dynamics of the Sachdev-Ye-Kitaev model. We identify a particular set of 2k-point functions, characterized as being both "maximally braided" and "k-out of time order," which exhibit exponential growth until progressively longer time scales u[over ^]{*}^{(k)}∼(k-1)u[over ^]{*}. We suggest an interpretation as scrambling of increasingly fine grained measures of quantum information, which correspondingly take progressively longer time to reach their thermal values.
Abstract Recent studies of out-of-time ordered thermal correlation functions (OTOC) in holographic systems and in solvable models such as the Sachdev-Ye-Kitaev (SYK) model have yielded new insights into manifestations of many-body chaos. So far the chaotic behavior has been obtained through explicit calculations in specific models. In this paper we propose a unified description of the exponential growth and ballistic butterfly spreading of OTOCs across different systems using a newly formulated “quantum hydrodynamics,” which is valid at finite ℏ 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 may be considered as a “smoking gun” for the hydrodynamic origin of the chaotic mode. We also discuss the possibility that such a hydrodynamic description could be a hallmark of maximally chaotic systems.
Fast scrambling and exponential growth of complexity are two distinctive features of quantum gravity, offering a unique window into the mechanisms that govern the semiclassical limit of the holographic duality. In this thesis, we explore these information-theoretic phenomena through two key diagnostics --- out-of-time ordered correlation functions (OTOCs) and spread complexity --- computed in the setting of two-dimensional holographic conformal field theories (CFTs). First, we focus on the large-c limit of Liouville theory, where the computation of OTOCs reveals distinctive insights into both the chaotic nature and the structural features of the theory. Remarkably, we find it exhibits a maximal Lyapunov exponent, despite lacking the identity operator in its spectrum, and a scrambling time logarithmic in the central charge. Using path integral techniques, we calculate heavy-light four-point functions, and uncover a conformal block expansion dominated by the exchange of an infinite tower of spinless double-trace operators. We attribute the chaotic behaviour of the OTOC to this exchange of operators, which can be thought of as dressed scramblons. Additionally, we observe that at the onset of scrambling, the dominance of conformal blocks shifts from heavy primaries at early times to light primaries at late times. Second, we turn to the diagnostic of spread complexity in locally excited holographic CFTs and argue that its time derivative is dual to the proper radial momentum of a massive point particle propagating in Anti-de Sitter spacetime (AdS). Exploiting the conformal symmetry and the emergence of an ``effective'' SL(2,R) group, we employ representation-theoretic techniques to compute return amplitudes, Lanczos coefficients, and spread complexities. We then verify our conjectured duality for three settings: the CFT on the line, dual to Poincaré AdS; the CFT on the circle, dual to global AdS; and the CFT at finite temperature, dual to a BTZ black hole. In the gravitational theory, the same SL(2,R) structure emerges as the isometry group of the infalling plane of the bulk particle. Third, we generalise the analysis of spread complexity to modular Hamiltonian evolution. Exploiting the rich algebraic structure of modular theory, we construct the modular Hamiltonians Kₘ and the half-sided modular translations G for the three example states above. Leveraging conformal symmetry, we once again compute modular return amplitudes, Lanczos coefficients, and modular spread complexities. In particular, we emphasise the role of ``conformal regularisation'' --- i.e. regulating with the operator that generates modular evolution - in preserving the effective SL(2,R) structure of the amplitudes. We find that evolution under Kₘ exhibits thermal spread complexity for β = 1 across all examples, while G-evolution universally leads to quadratic growth. Finally, we provide a formal explanation of these results, identifying the role of the Killing norm D of the evolution operator as the key factor governing the behaviour of complexity.
We propose a characterization of quantum many-body chaos: given a collection of simple operators, the set of all possible pair correlations between these operators can be organized into a matrix with a random-matrix-like spectrum. This approach is particularly useful for locally interacting systems, which do not generically show exponential Lyapunov growth of out-of-time-ordered correlators. We demonstrate the validity of this characterization by numerically studying the Sachdev-Ye-Kitaev model and a one-dimensional spin chain with random magnetic field (XXZ model).
Recent years have witnessed tremendous progress in developing a fine-grained low-dimensional holographic correspondence, specifically the construction of quantum mechanical boundary theories as holographic duals of two-dimensional gravity. In these developments, quantum chaos played a crucial role, both as source of universality and as a guiding principle for the matching of bulk and boundary signatures of gravity. In this article we review the construction of the chaos-assisted low-dimensional holographic correspondence for non-experts. We open with an introductory discussion of the two main protagonists of the theory, the SYK model and two-dimensional Jackiw-Teitelboim gravity. Within this framework we will discuss two independent 'bridges' between bulk and boundary physics, one pertaining to early time chaotic instabilities, the other to late time quantum chaos up to and including time scales of the order of the gravitational quantum level spacing. We will demonstrate that the resolution of these fine-grained quantum scales requires the extension of semiclassical gravity by elements of string theory. We conclude with an outlook towards higher dimensional generalizations of the chaotic holographic correspondence.
We develop a finite-dimensional formulation of the recently introduced notion of ``timelike entanglement'', defined in terms of two-point functions between operators supported on different Cauchy slices. Using a local orthonormal operator basis, we recast this construction in terms of a generalized response tensor. Building on this, we introduce a generalized spacetime density kernel (GSDK) corresponding to higher-point correlation functions, including time-ordered as well as out-of-time-ordered correlators. We show that the Haar-averaged (2N)-point function yields the (2N)-th moment of the spectral form factor (SFF), evaluated at an N-enhanced effective temperature. The correlation functions of the GSDK operators also yield the SFF, with an effective (1/N)-reduction of the physical time-scales. The GSDK places both scrambling diagnostics and spectral statistics on a similar footing and clarifies how higher-point correlators and non-trivial time ordering capture fine-grained dynamical information of a quantum system.
The exact role of the internal degrees of freedom (a.k.a. d.o.f.) in holography is not well-understood. Thus, in this thesis, we study a toy model of holography without space: the Sachdev-Ye-Kitaev (SYK) Model. This 0+1 theory of all possible 4-body interactions of N fermion "flavors"/"colors" features a low energy limit reproducing aspects of 1+1 Jackiw-Teitelboim gravity. First, we show that the inherent discreteness of the quantum spectrum results in universal late-time behavior due to eigenvalue repulsion. We then note that the theory's four-point functions probe the phenomenon of operator growth, where an internal d.o.f. goes on to epidemically evolve into larger products of internal d.o.f.s. In this manner where small operators smoothly grow into superpositions of increasing products of operators, we observe a sort of "size" locality, which is intimately tied with the notion of a conformal primary "descending" along its descendants. In fact, we find that the underlying structure of the SYK epidemic limits to that of a probe particle falling into a AdS2 black hole. In other words, similar to how nearest neighbor interactions lead to dynamics on a flat space background, we demonstrate that many internal interactions lead to dynamics on a higher dimensional geometry.