Abstract It has been proposed that the exponential decay and subsequent power law saturation of out-of-time-order correlation functions can be universally described by collective ‘scramblon’ modes. We develop this idea from a path integral perspective in several examples, thereby establishing a general formalism. After reformulating previous work on the Schwarzian theory and identity conformal blocks in two-dimensional CFTs relevant for systems in the infinite coupling limit with maximal quantum Lyapunov exponent, we focus on theories with sub-maximal chaos: we study the large-q limit of the SYK quantum dot and chain, both of which are amenable to analytical treatment at finite coupling. In both cases we identify the relevant scramblon modes, derive their effective action, and find bilocal vertex functions, thus constructing an effective description of chaos. The final results can be matched in detail to stringy corrections to the gravitational eikonal S-matrix in holographic CFTs, including a stringy Regge trajectory, bulk to boundary propagators, and multi-string effects that are unexplored holographically.
Abstract It has been proposed that the exponential decay and subsequent power law saturation of out-of-time-order correlation functions can be universally described by collective ‘scramblon’ modes. We develop this idea from a path integral perspective in several examples, thereby establishing a general formalism. After reformulating previous work on the Schwarzian theory and identity conformal blocks in two-dimensional CFTs relevant for systems in the infinite coupling limit with maximal quantum Lyapunov exponent, we focus on theories with sub-maximal chaos: we study the large-q limit of the SYK quantum dot and chain, both of which are amenable to analytical treatment at finite coupling. In both cases we identify the relevant scramblon modes, derive their effective action, and find bilocal vertex functions, thus constructing an effective description of chaos. The final results can be matched in detail to stringy corrections to the gravitational eikonal S-matrix in holographic CFTs, including a stringy Regge trajectory, bulk to boundary propagators, and multi-string effects that are unexplored holographically.
A bstract Fast scrambling is a distinctive feature of quantum gravity, which by means of holography is closely tied to the behaviour of large- c conformal field theories. We study this phenomenon in the context of semiclassical Liouville theory, providing both insights into the mechanism of scrambling in CFTs and into the structure of Liouville theory, finding that it exhibits a maximal Lyapunov exponent despite not featuring the identity in its spectrum. However, as we show, the states contributing to the relevant correlation function can be thought of as dressed scramblons. At a technical level we first use the path integral picture in order to derive the Euclidean four-point function in an explicit compact form. Next, we demonstrate its equivalence to a conformal block expansion, revealing an explicit but non-local map between path integral saddles and conformal blocks. By analytically continuing both expressions to Lorentzian times, we obtain two equivalent formulations of the OTOC, which we use to study the onset of chaos in Liouville theory. We take advantage of the compact form in order to extract a Lyapunov exponent and a scrambling time. From the conformal block expansion formulation of the OTOC we learn that scrambling shifts the dominance of conformal blocks from heavy primaries at early times to the lightest primary at late times. Finally, we discuss our results in the context of holography.
Abstract We consider a Hayden & Preskill like setup for both maximally chaotic and sub-maximally chaotic quantum field theories. We act on the vacuum with an operator in a Rindler like wedge R and transfer a small subregion I of R to the other wedge. The chaotic scrambling dynamics of the QFT Rindler time evolution reveals the information in the other wedge. The holographic dual of this process involves a particle excitation falling into the bulk and crossing into the entanglement wedge of the complement to r = R\I. With the goal of studying the locality of the emergent holographic theory we compute various quantum information measures on the boundary that tell us when the particle has entered this entanglement wedge. In a maximally chaotic theory, these measures indicate a sharp transition where the particle enters the wedge exactly when the insertion is null separated from the quantum extremal surface for r. For sub-maximally chaotic theories, we find a smoothed crossover at a delayed time given in terms of the smaller Lyapunov exponent and dependent on the time-smearing scale of the probe excitation. The information quantities that we consider include the full vacuum modular energy R\I as well as the fidelity between the state with the particle and the state without. Along the way, we find a new explicit formula for the modular Hamiltonian of two intervals in an arbitrary 1+1 dimensional CFT to leading order in the small cross ratio limit. We also give an explicit calculation of the Regge limit of the modular flowed chaos correlator and find examples which do not saturate the modular chaos bound. Finally, we discuss the extent to which our results reveal properties of the target of the probe excitation as a “stringy quantum extremal surface” or simply quantify the probe itself thus giving a new approach to studying the notion of longitudinal string spreading.
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.
Hydrodynamics as an effective field theory (EFT) has been immensely successful for predicting the behaviour of time-ordered correlation functions in the long-time, large-distance regime. In this paper, we analyse the predictions of hydrodynamics for scrambling as probed by out-of-time order correlators (OTOCs). To do so, we consider an all-derivative order generalisation of the Schwinger-Keldysh EFT of hydrodynamics extended to a four-fold time contour required to capture OTOCs. A crucial ingredient of the construction is a local SL(2,R) symmetry, which has previously been suggested as a mechanism for scrambling. We also derive this exact EFT from the microscopic formulation of the maximally chaotic low-temperature limit of the SYK lattice, thereby determining all transport coefficients for this model. As a novel application, we compute the OTOC of energy currents, find that its exponential growth saturates the chaos bound, and match it to expectations based on a shockwave (or scramblon) effective description. Our work shows that the universality of hydrodynamics degrees of freedom extends beyond traditional time-ordered correlators to more complex observables such as OTOCs.
Abstract We study six-point correlation functions in two dimensional conformal field theory, where the six operators are grouped in pairs with equal conformal dimension. Assuming large central charge c and a sparse spectrum, the leading contribution to this correlation function is the six-point Virasoro identity block — corresponding to each distinct pair of operators fusing into the identity and its descendants. We call this the star channel. One particular term in the star channel identity block is the stress tensor SL(2, ℝ) (global) block, for which we derive an explicit expression. In the holographic context, this object corresponds to a direct measure of nonlinear effects in pure gravity. We calculate additional terms in the star channel identity block that contribute at the same order at large c as the global block using the novel theory of reparametrizations, which extends the shadow operator formalism in a natural way. We investigate these blocks’ relevance to quantum chaos in the form of six-point scrambling in an out-of time ordered correlator. Interestingly, the global block does not contribute to the scrambling mode of this correlator, implying that, to leading order, six-point scrambling is insensitive to the three-point graviton coupling in the bulk dual. Finally, we compare our findings with a different OPE channel, called the comb channel, and find the same result for the chaos exponent in this decomposition.
The complex Sachdev-Ye-Kitaev (SYK) model, featuring fermions with all-to-all interactions, serves as a dual paradigm for understanding non-Fermi liquid behavior and the holographic nature of charged black holes. Two defining characteristics of the standard SYK model are its maximal chaos (Lyapunov exponent λL=2πT at temperature T), and its finite zero-temperature residual entropy. While previous studies have largely focused on couplings drawn from a zero-mean Gaussian distribution, we investigate a generalized model with a finite mean-to-standard-deviation ratio, g≡J0/δJ of the coupling distribution in order to get deeper insight into the evolution of chaos. We find that increasing g yields the following effects: (i) The system remains a fast scrambler with λL=AT, but with a suppressed coefficient A<2π. (ii) In the limit g→∞, out-of-time-ordered correlators (OTOCs) no longer exhibit exponential growth with λL≃0. (iii) The spectral correlations indicative of late-time chaos maintain Wigner-Dyson level spacing statistics for all values of g. (iv) The system preserves a finite residual entropy, albeit with reduced magnitude, for all g values. We conclude that in this generalized SYK model, there is a chaotic to non-chaotic crossover. Moreover different measures of chaos decouple, demonstrating that the presence of finite residual entropy does not strictly imply maximal chaos.