Thermal states holographically dual to black holes in Einstein gravity display maximal Lyapunov growth as well as "butterfly effect cones". We study these effects in highly non-equilibrium states, obtained from an initial thermal state by the sudden injection of energy. We do this by computing out-of-time-order correlators (OTOCs) in BTZ-Vaidya spacetimes, which describe transitions between black holes at different temperatures. If both pairs of boundary operators appearing in the OTOC are inserted before the energy injection, we recover standard results, with butterfly effect cones displaying a light-cone structure. But when one pair of operators is inserted before and the other pair after the energy injection, the Lyapunov growth saturates the chaos bounds set by the local temperatures and the butterfly effect cones "open up", becoming superluminal, albeit in a way that does not violate causality. In the limiting case, in which the initial state is the vacuum, Lyapunov growth only starts after the energy injection. Our computations of the OTOCs are phrased in terms of gravitationally interacting particles, where fields are treated in a geodesic approximation and the eikonal phase shift is expressed in terms of stress tensors and shock waves associated to geodesics.
In this article, we explore dynamical aspects of Out-of-Time-Order correlators (OTOCs) for critical quenches, in which an initial non-trivial state evolves with a CFT-Hamiltonian. At sufficiently large time, global critical quenches exhibit a universal thermal-behaviour in terms of low-point correlators. We demonstrate that, under such a quench, OTOCs demarcate chaotic CFTs from integrable CFTs by exhibiting a characteristic exponential Lyapunov growth for the former. Upon perturbatively introducing inhomogeneity to the global quench, we further argue and demonstrate with an example that, such…
We study local quenches in 1+1 dimensional conformal field theories at large-c by operators carrying higher spin charge. Viewing such states as solutions in Chern-Simons theory, representing infalling massive particles with spin-three charge in the BTZ background, we use the Wilson line prescription to compute the single-interval entanglement entropy (EE) and scrambling time following the quench. We find that the change in EE is finite (and real) only if the spin-three charge q is bounded by the energy of the perturbation E, as |q|/c < E^2/c^2. We show that the Wilson line/EE correlator deep i…
We study non-equilibrium dynamics induced by a sudden quench of strongly correlated Hamiltonians with all-to-all interactions. By relying on a Sachdev-Ye-Kitaev (SYK) based quench protocol, we show that the time evolution of simple spin-spin correlation functions is highly sensitive to the degree of locality of the corresponding operators, once an appropriate set of fundamental fields is identified. By tracking the time-evolution of specific spin-spin correlation functions and their decay, we argue that it is possible to distinguish between operator hopping and operator growth dynamics; the la…
Various aspects of warped conformal field theories (WCFTs) are studied including entanglement entropy on excited states, the Renyi entropy after a local quench, and out-of-time-order four-point functions. Assuming a large central charge and dominance of the vacuum block in the conformal block expansion, (i) we calculate the single-interval entanglement entropy on an excited state, matching previous finite temperature results by changing the ensemble; and (ii) we show that WCFTs are maximally chaotic, a result that is compatible with the existence of black holes in the holographic duals. Finall…
We study the dynamics of (Rényi) mutual information, logarithmic negativity, and (Rényi) reflected entropy after exciting the ground state by a local operator. Together with recent results from Ref. [1], we are able to conjecture a close-knit structure between the three quantities that emerges in states excited above the vacuum, including both local and global quantum quenches. This structure intimately depends on the chaoticity of the theory i.e. there exist distinct sets of equivalences for integrable and chaotic theories. For rational conformal field theories (RCFT), we find all quantities …
The quantum dynamics of isolated systems under quench condition exhibits a variety of interesting features depending on the integrable/chaotic nature of system. We study the exact dynamics of trivially integrable system of harmonic chains under a multiple quench protocol. Out of time ordered correlator of two Hermitian operators at large time displays scrambling in the thermodynamic limit. In this limit, the entanglement entropy and the central component of momentum distribution both saturate to a steady state value. We also show that reduced density matrix assumes the diagonal form long after…
A global quench is an interesting setting where we can study thermalization of subsystems in a pure state. We investigate entanglement entropy (EE) growth in global quenches in holographic field theories and relate some of its aspects to quantities characterizing chaos. More specifically we obtain four key results: 1. We prove holographic bounds on the entanglement velocity $v_E$ and the butterfly effect speed $v_B$ that arises in the study of chaos. 2. We obtain the EE as a function of time for large spherical entangling surfaces analytically. We show that the EE is insensitive to the details…
Using a recent proposal of circuit complexity in quantum field theories introduced by Jefferson and Myers, we compute the time evolution of the complexity following a smooth mass quench characterized by a time scale $δt$ in a free scalar field theory. We show that the dynamics has two distinct phases, namely an early regime of approximately linear evolution followed by a saturation phase characterized by oscillations around a mean value. The behavior is similar to previous conjectures for the complexity growth in chaotic and holographic systems, although here we have found that the complexity …
A class of tensor models were recently outlined as potentially calculable examples of holography: their perturbative large-$N$ behavior is similar to the Sachdev-Ye-Kitaev (SYK) model, but they are fully quantum mechanical (in the sense that there is no quenched disorder averaging). These facts make them intriguing tentative models for quantum black holes. In this note, we explicitly diagonalize the simplest non-trivial Gurau-Witten tensor model and study its spectral and late-time properties. We find parallels to (a single sample of) SYK where some of these features were recently attributed t…
Complete eigenvalue spectra of the staggered Dirac operator in quenched $4d$ compact QED are studied on $8^3 \times 4$ and $8^3 \times 6$ lattices. We investigate the behavior of the nearest-neighbor spacing distribution $P(s)$ as a measure of the fluctuation properties of the eigenvalues in the strong coupling and the Coulomb phase. In both phases we find agreement with the Wigner surmise of the unitary ensemble of random-matrix theory indicating quantum chaos. Combining this with previous results on QCD, we conjecture that quite generally the non-linear couplings of quantum field theories le…
We employ holographic techniques to investigate the dynamics of the order parameter of a strongly coupled superconductor after a perturbation that drives the system out of equilibrium. The gravity dual that we employ is the ${\rm AdS}_5$ Soliton background at zero temperature. We first analyze the normal modes associated to the superconducting order parameter which are purely real since the background has no horizon. We then study the full time evolution of the order parameter after a quench. For sufficiently a weak and slow perturbation we show that the order parameter undergoes simple undamp…
Operator scrambling denotes the evolution of a simple operator into a complicated one (in the Heisenberg picture), which characterizes quantum chaos in many-body systems. More specifically, a simple operator evolves into a linear superposition of many operators, most of which are many-body operators supported on a region of size much larger than $1$. In general, an operator does not have a definite size but is characterized by a probability distribution of size. The operator size is related to out-of-time-order correlation functions, but these are generically difficult to obtain from experimen…
We investigate the eigenvalue spectrum of the staggered Dirac matrix in SU(3) gauge theory and in full QCD as well as in quenched U(1) theory. As a measure of the fluctuation properties of the eigenvalues, we consider the nearest-neighbor spacing distribution. We find that in all regions of their phase diagrams, compact lattice gauge theories have bulk spectral correlations given by random matrix theory, which is an indication for quantum chaos. In the confinement phase, the low-lying Dirac spectrum of these quantum field theories is well described by random matrix theory, exhibiting universal…
We investigate the eigenvalue spectrum of the staggered Dirac matrix in SU(3) gauge theory and in full QCD as well as in quenched U(1) theory on various lattice sizes. As a measure of the fluctuation properties of the eigenvalues, we consider the nearest-neighbor spacing distribution, $P(s)$. We further study two-color QCD at nonzero chemical potential, $μ$, by constructing the spacing distribution of adjacent eigenvalues in the complex plane. We find that in all regions of their phase diagrams, compact lattice gauge theories have bulk spectral correlations given by random matrix theory, which…
We consider the Brownian SYK model of $N$ interacting Majorana fermions, with random couplings that are taken to vary independently at each time. We study the out-of-time-ordered correlators (OTOCs) of arbitrary observables and the Rényi-$2$ tripartite information of the unitary evolution operator, which were proposed as diagnostic tools for quantum chaos and scrambling, respectively. We show that their averaged dynamics can be studied as a quench problem at imaginary times in a model of $N$ qudits, where the Hamiltonian displays site-permutational symmetry. By exploiting a description in term…
We consider the time evolution of mixed state correlation measures in two-dimensional conformal field theories, such as logarithmic negativity, odd entropy, and reflected entropy, after quantum quenches of various kinds. These correlation measures, in the holographic context, are all associated to the entanglement wedge cross section. We contrast various classes of conformal field theories, both rational and irrational (pure) conformal field theories. First, for rational conformal field theories, whose dynamics can be well described by the quasi-particle picture, we find all four quantities fo…
National Tsing-Hua University（国立清华大学）；Center for Theory-Computation-Data Science Research, National Tsing-Hua University（国立清华大学理论计算数据科学研究中⼼）；National Center for Theoretical Sciences（国家理论科学中⼼）
Probing quantum chaos near a wormhole throat with a circular string
School of Science, Guangxi University of Science and Technology（广西科技大学理学院）；Department of Physics, Institute of Fundamental Physics and Quantum Technology, Ningbo University（宁波大学物理系基础物理与量子技术研究所）；Department of Applied Physics, College of Science, China Agricultural University（中国农业大学理学院应用物理系）
