The fine grained energy spectrum of quantum chaotic systems is widely believed to be described by random matrix statistics. A basic scale in such a system is the energy range over which this behavior persists. We define the corresponding time scale by the time at which the linearly growing ramp region in the spectral form factor begins. We call this time $t_{\rm ramp}$. The purpose of this paper is to study this scale in many-body quantum systems that display strong chaos, sometimes called scrambling systems. We focus on randomly coupled qubit systems, both local and $k$-local (all-to-all interactions) and the Sachdev--Ye--Kitaev (SYK) model. Using numerical results for Hamiltonian systems and analytic estimates for random quantum circuits we find the following results. For geometrically local systems with a conservation law we find $t_{\rm ramp}$ is determined by the diffusion time across the system, order $N^2$ for a 1D chain of $N$ qubits. This is analogous to the behavior found for local one-body chaotic systems. For a $k$-local system with conservation law the time is order $\log N$ but with a different prefactor and a different mechanism than the scrambling time. In the absence of any conservation laws, as in a generic random quantum circuit, we find $t_{\rm ramp} \sim \log N$, independent of connectivity.
Out-of-time-order correlation functions provide a proxy for diagnosing chaos in quantum systems. We propose and analyze an interferometric scheme for their measurement, using only local quantum control and no reverse time evolution. Our approach utilizes a combination of Ramsey interferometry and the recently demonstrated ability to directly measure Renyi entropies. To implement our scheme, we present a pair of cold-atom-based experimental blueprints; moreover, we demonstrate that within these systems, one can naturally realize the transverse-field Sherrington-Kirkpatrick (TFSK) model, which e…
Scrambling, a process in which quantum information spreads over a complex quantum system becoming inaccessible to simple probes, happens in generic chaotic quantum many-body systems, ranging from spin chains, to metals, even to black holes. Scrambling can be measured using out-of-time-ordered correlators (OTOCs), which are closely tied to the growth of Heisenberg operators. In this work, we present a general method to calculate OTOCs of local operators in local one-dimensional systems based on approximating Heisenberg operators as matrix-product operators (MPOs). Contrary to the common belief …
Recent work has studied the growth of commutators as a probe of chaos and information scrambling in quantum many-body systems. In this work we study the effect of static disorder on the growth of commutators in a variety of contexts. We find generically that disorder slows the onset of scrambling, and, in the case of a many-body localized state, partially halts it. We access the many-body localized state using a standard fixed point Hamiltonian, and we show that operators exhibit slow logarithmic growth under time evolution. We compare the result with the expected growth of commutators in both…
A universal relation is established between the quantum work probability distribution of an isolated driven quantum system and the Loschmidt echo dynamics of a two-mode squeezed state. When the initial density matrix is canonical, the Loschmidt echo of the purified double thermofield state provides a direct measure of information scrambling and can be related to the analytic continuation of the partition function. Information scrambling is then described by the quantum work statistics associated with the time-reversal operation on a single copy, associated with the sudden negation of the syste…
Operator scrambling is a crucial ingredient of quantum chaos. Specifically, in the quantum chaotic system, a simple operator can become increasingly complicated under unitary time evolution. This can be diagnosed by various measures such as square of the commutator (out-of-time-ordered correlator), operator entanglement entropy etc. In this paper, we discuss operator scrambling in three representative models: a chaotic spin-$1/2$ chain with spatially local interactions, a 2-local spin model and the quantum linear map. In the first two examples, although the speeds of scrambling are quite diffe…
Thermalization of chaotic quantum many-body systems under unitary time evolution is related to the growth in complexity of initially simple Heisenberg operators. Operator growth is a manifestation of information scrambling and can be diagnosed by out-of-time-order correlators (OTOCs). However, the behavior of OTOCs of local operators in generic chaotic local Hamiltonians remains poorly understood, with some semiclassical and large N models exhibiting exponential growth of OTOCs and a sharp chaos wavefront and other random circuit models showing a diffusively broadened wavefront. In this paper …
If black holes are not to be dreaded syncs of information but be fully described by unitary evolution, they must scramble in-falling data and eventually leak it through Hawking radiation. Sekino and Susskind have conjectured that black holes are fast scramblers: they generate entanglement at a remarkably efficient rate, with characteristic time scaling logarithmically with the entropy. In this work, we focus on Matrix theory -- M theory in the light-cone frame -- and directly probe the conjecture. We develop a concrete test bed for quantum gravity using the fermionic variables of Matrix theory…
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…
BMN Matrix theory admits vacua in the shape of large spherical membranes. Per- turbing around such vacua, the setup provides for a controlled computational frame- work for testing information evolution in Matrix black holes. The theory realizes excitations in the supergravity multiplet as qubits. These qubits are coupled to matrix degrees of freedom that describe deformations of the spherical shape of the membrane. Arranging the ripples on the membrane into a heat bath, we use the qubit system as a probe and compute the associated Feynman-Vernon density matrix at one loop order. This allows us…
Characterizing the work statistics of driven complex quantum systems is generally challenging because of the exponential growth with the system size of the number of transitions involved between different energy levels. We consider the quantum work distribution associated with the driving of chaotic quantum systems described by random matrix Hamiltonians and characterize exactly the work statistics associated with a sudden quench for arbitrary temperature and system size. Knowledge of the work statistics yields the Loschmidt echo dynamics of an entangled state between two copies of the system …
The entanglement evolution after a quantum quench became one of the tools to distinguish integrable versus chaotic (non-integrable) quantum many-body dynamics. Following this line of thoughts, here we propose that the revivals in the entanglement entropy provide a finite-size diagnostic benchmark for the purpose. Indeed, integrable models display periodic revivals manifested in a dip in the block entanglement entropy in a finite system. On the other hand, in chaotic systems, initial correlations get dispersed in the global degrees of freedom (information scrambling) and such a dip is suppresse…
We compute the disruption of mutual information in a TFD state dual to a Kerr black hole with equal angular momenta in $AdS_5$ due to an equatorial shockwave. The shockwave respects the axi-symmetry of the Kerr geometry with specific angular momenta $\mathcal{L}_{ϕ_1}$ & $\mathcal{L}_{ϕ_2}$. The sub-systems considered are hemispheres in the $left$ and the $right$ dual CFTs with the equator of the $S^3$ as their boundary. We compute the change in the mutual information by determining the growth of the HRT surface at late times. We find that at late times leading upto the scrambling time the min…
The onset of quantum chaos in quantum field theory may be studied using out-of-time-order correlators at finite temperature. Recent work argued that a timescale logarithmic in the central charge emerged in the context of two-dimensional conformal field theories, provided the intermediate channel was dominated by the Virasoro identity block. This suggests a wide class of conformal field theories exhibit a version of fast scrambling. In the present work we study this idea in more detail. We begin by clarifying to what extent correlators of wavepackets built out of superpositions of primary opera…
In this paper we investigate measures of chaos and entanglement in rational conformal field theories in 1+1 dimensions. First, we derive a universal formula for the late time value of the out-of-time-ordered correlators for this class of theories. Our universal result can be expressed as a particular combination of the modular S-matrix elements known as the anyon monodromy scalar. Next, in the explicit setup of a $SU(N)_k$ Wess-Zumino-Witten model, we compare the late time behavior of the out-of-time-ordered correlators and the purity. Interestingly, in the large-c limit, the purity grows loga…
Out-of-time ordered (OTO) correlation functions describe scrambling of information in correlated quantum matter. They are of particular interest in incoherent quantum systems lacking well defined quasi-particles. Thus far, it is largely elusive how OTO correlators spread in incoherent systems with diffusive transport governed by a few globally conserved quantities. Here, we study the dynamical response of such a system using high-performance matrix-product-operator techniques. Specifically, we consider the non-integrable, one-dimensional Bose-Hubbard model in the incoherent high-temperature re…
We consider the logarithmic negativity and related quantities of time evolution operators. We study free fermion, compact boson, and holographic conformal field theories (CFTs) as well as numerical simulations of random unitary circuits and integrable and chaotic spin chains. The holographic behavior strongly deviates from known non-holographic CFT results and displays clear signatures of maximal scrambling. Intriguingly, the random unitary circuits display nearly identical behavior to the holographic channels. Generically, we find the "line-tension picture" to effectively capture the entangle…
We study a precise and computationally tractable notion of operator complexity in holographic quantum theories, including the ensemble dual of Jackiw-Teitelboim gravity and two-dimensional holographic conformal field theories. This is a refined, "microcanonical" version of K-complexity that applies to theories with infinite or continuous spectra (including quantum field theories), and in the holographic theories we study exhibits exponential growth for a scrambling time, followed by linear growth until saturation at a time exponential in the entropy $\unicode{x2014}$a behavior that is characte…
We study the finite-temperature scrambling behavior of a quantum system described by a Hamiltonian chosen from a random matrix ensemble. This effectively (0+1)-dimensional model admits an exact calculation of various ensemble-averaged out-of-time-ordered correlation functions in the large-$N$ limit, where $N$ is the Hilbert space dimension. For a Hamiltonian drawn from the Gaussian unitary ensemble, we calculate the ensemble averaged OTOC at all temperatures. In addition to an early time quadratic growth of the averaged out-of-time-ordered commutator (OTOC), we determine that the OTOC saturate…
We analyze the out-of-time-order correlation functions of a solvable model of a large number, $N$, of $M$-component quantum rotors coupled by Gaussian-distributed random, infinite-range exchange interactions. We focus on the growth of commutators of operators at a temperature $T$ above the zero temperature quantum critical point separating the spin-glass and paramagnetic phases. In the large $N,~M$ limit, the squared commutators of the rotor fields do not display any exponential growth of commutators, in spite of the absence of any sharp quasiparticle-like excitations in the disorder-averaged …
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 first propose and study a quantum toy model of black hole dynamics. The model is unitary, displays quantum thermalization, and the Hamiltonian couples every oscillator with every other, a feature intended to emulate the color sector physics of large-$\mathcal{N}$ matrix models. Considering out of equilibrium initial states, we analytically compute the time evolution of every correlator of the theory and of the entanglement entropies, allowing a proper discussion of global thermalization/scrambling of information through the entire system. Microscopic non-locality causes factorization of red…
We study chaos in the classical limit of the matrix quantum mechanical system describing D0-brane dynamics. We determine a precise value of the largest Lyapunov exponent, and, with less precision, calculate the entire spectrum of Lyapunov exponents. We verify that these approach a smooth limit as $N \rightarrow \infty$. We show that a classical analog of scrambling occurs with fast scrambling scaling, $t_* \sim \log S$. These results confirm the k-locality property of matrix mechanics discussed by Sekino and Susskind.
It has been recently shown that small subsystems of finite quantum systems generically equilibrate. We extend these results to infinite-dimensional Hilbert spaces of field theories and matrix models. We consider a quench setup, where initial states are chosen from a microcanonical ensemble of finite energy in free theory, and then evolve with an arbitrary non-perturbative Hamiltonian. Given a dynamical assumption on the expectation value of particle number density, we prove that small subsystems reach equilibrium at the level of quantum wave-function, and with respect to all observables. The p…
Chaos and complexity entail an entropic and computational obstruction to describing a system, and thus are intrinsically difficult to characterize. In this paper, we consider time evolution by Gaussian Unitary Ensemble (GUE) Hamiltonians and analytically compute out-of-time-ordered correlation functions (OTOCs) and frame potentials to quantify scrambling, Haar-randomness, and circuit complexity. While our random matrix analysis gives a qualitatively correct prediction of the late-time behavior of chaotic systems, we find unphysical behavior at early times including an $\mathcal{O}(1)$ scrambli…
Two topics, evolving rapidly in separate fields, were combined recently: The out-of-time-ordered correlator (OTOC) signals quantum-information scrambling in many-body systems. The Kirkwood-Dirac (KD) quasiprobability represents operators in quantum optics. The OTOC has been shown to equal a moment of a summed quasiprobability. That quasiprobability, we argue, is an extension of the KD distribution. We explore the quasiprobability's structure from experimental, numerical, and theoretical perspectives. First, we simplify and analyze the weak-measurement and interference protocols for measuring t…
Recent work by Shenker, Stanford, and Kitaev has related the black hole horizon geometry to chaotic behavior. We extend this from eternal black holes to black holes that form and then evaporate. This leads to an identity for the change in the black hole S-matrix (over times shorter than the scrambling time) due an addition infalling particle, elaborating an idea of 't Hooft.
Recent gravity discussions of a traversable wormhole indicate that in holographic systems signals generated by a source could reappear long after they have dissipated, with the need of only performing some simple operations. In this paper we argue the phenomenon, to which we refer as "regenesis", is universal in general quantum chaotic many-body systems, and elucidate its underlying physics. The essential elements behind the phenomenon are: (i) scrambling which in a chaotic system makes out-of-time-ordered correlation functions (OTOCs) vanish at large times; (ii) the entanglement structure of …
We construct a one-parametric family of the double-scaling limits in the hermitian matrix model $Φ^6$ for 2D quantum gravity. The known limit of Bresin, Marinari and Parisi belongs to this family. The family is represented by the Gurevich-Pitaevskii solution of the Korteveg-de Vries equation which describes the onset of nondissipative shock waves in media with small dispersion. Numerical simulation of the universal Gurevich-Pitaevskii solution is made.
We argue that in a large class of disordered quantum many-body systems, the late time dynamics of time-dependent correlation functions is captured by random matrix theory, specifically the energy eigenvalue statistics of the corresponding ensemble of disordered Hamiltonians. We find that late time correlation functions approximately factorize into a time-dependent piece, which only depends on spectral statistics of the Hamiltonian ensemble, and a time-independent piece, which only depends on the data of the constituent operators of the correlation function. We call this phenomenon "spectral de…
High energy hadron interactions are commonly described by using a probabilistic parton model that ignores quantum entanglement present in the light-cone wave functions. Here we argue that since a high energy interaction samples an instant snapshot of the hadron wave function, the phases of different Fock state wave functions cannot be measured - therefore the light-cone density matrix has to be traced over these unobservable phases. Performing this trace with the corresponding $U(1)$ Haar integration measure leads to "Haar scrambling" of the density matrix, and to the emergence of entanglement…
We study numerically the onset of chaos and thermalization in the Banks-Fischler-Shenker-Susskind (BFSS) matrix model with and without fermions, considering Lyapunov exponents, entanglement generation, and quasinormal ringing. We approximate the real-time dynamics in terms of the most general Gaussian density matrices with parameters which obey self-consistent equations of motion, thus extending the applicability of real-time simulations beyond the classical limit. Initial values of these Gaussian density matrices are optimized to be as close as possible to the thermal equilibrium state of the…
Scrambling in interacting quantum systems out of equilibrium is particularly effective in the chaotic regime. Under time evolution, initially localized information is said to be scrambled as it spreads throughout the entire system. This spreading can be analyzed with the spectral form factor, which is defined in terms of the analytic continuation of the partition function. The latter is equivalent to the survival probability of a thermofield double state under unitary dynamics. Using random matrices from the Gaussian unitary ensemble (GUE) as Hamiltonians for the time evolution, we obtain exac…
