scholarly journals Phases of scrambling in eigenstates

2019 ◽  
Vol 7 (1) ◽  
Author(s):  
Tarek Anous ◽  
Julian Sonner
Keyword(s):  
Lyapunov Exponent ◽  
Effective Temperature ◽  
Arbitrary Number ◽  
Point Function ◽  
Expectation Values ◽  
Harmonic Oscillations ◽  
Time Order ◽  
Heavy State

We use the monodromy method to compute expectation values of an arbitrary number of light operators in finitely excited ("heavy") eigenstates of holographic 2D CFT. For eigenstates with scaling dimensions above the BTZ threshold, these behave thermally up to small corrections, with an effective temperature determined by the heavy state. Below the threshold we find oscillatory and not decaying behavior. As an application of these results we compute the expectation of the out-of-time order arrangement of four light operators in a heavy eigenstate, i.e. a six-point function. Above the threshold we find maximally scrambling behavior with Lyapunov exponent 2\pi T_{eff}2πTeff. Below threshold we find that the eigenstate OTOC shows persistent harmonic oscillations.

scholarly journals Holographic correlators at finite temperature

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Luis F. Alday ◽  
Murat Koloğlu ◽  
Alexander Zhiboedov
Keyword(s):  
Dispersion Relation ◽  
Finite Temperature ◽  
Arbitrary Number ◽  
Operator Product Expansion ◽  
Boundary Theory ◽  
Operator Product ◽  
Point Function ◽  
Consistency Conditions ◽  
Weakly Coupled ◽  
Analytic Expressions

Abstract We consider weakly-coupled QFT in AdS at finite temperature. We compute the holographic thermal two-point function of scalar operators in the boundary theory. We present analytic expressions for leading corrections due to local quartic interactions in the bulk, with an arbitrary number of derivatives and for any number of spacetime dimensions. The solutions are fixed by judiciously picking an ansatz and imposing consistency conditions. The conditions include analyticity properties, consistency with the operator product expansion, and the Kubo-Martin-Schwinger condition. For the case without any derivatives we show agreement with an explicit diagrammatic computation. The structure of the answer is suggestive of a thermal Mellin amplitude. Additionally, we derive a simple dispersion relation for thermal two-point functions which reconstructs the function from its discontinuity.

scholarly journals Pole skipping away from maximal chaos

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Changha Choi ◽  
Márk Mezei ◽  
Gábor Sárosi
Keyword(s):  
Lyapunov Exponent ◽  
Energy Density ◽  
Stress Tensor ◽  
Diffusion Constant ◽  
Special Point ◽  
Many Body ◽  
Point Function ◽  
Strongly Coupled ◽  
Universal Bound ◽  
Subtle Effect

Abstract Pole skipping is a recently discovered subtle effect in the thermal energy density retarded two point function at a special point in the complex (ω, p) planes. We propose that pole skipping is determined by the stress tensor contribution to many-body chaos, and the special point is at (ω, p)p.s. = $$ i{\lambda}^{(T)}\left(1,1/{u}_B^{(T)}\right) $$ i λ T 1 1 / u B T , where λ(T) = 2π/β and $$ {u}_B^{(T)} $$ u B T are the stress tensor contributions to the Lyapunov exponent and the butterfly velocity respectively. While this proposal is consistent with previous studies conducted for maximally chaotic theories, where the stress tensor dominates chaos, it clarifies that one cannot use pole skipping to extract the Lyapunov exponent of a theory, which obeys λ ≤ λ(T). On the other hand, in a large class of strongly coupled but non-maximally chaotic theories $$ {u}_B^{(T)} $$ u B T is the true butterfly velocity and we conjecture that uB ≤ $$ {u}_B^{(T)} $$ u B T is a universal bound. While it remains a challenge to explain pole skipping in a general framework, we provide a stringent test of our proposal in the large-q limit of the SYK chain, where we determine λ, uB, and the energy density two point function in closed form for all values of the coupling, interpolating between the free and maximally chaotic limits. Since such an explicit expression for a thermal correlator is one of a kind, we take the opportunity to analyze many of its properties: the coupling dependence of the diffusion constant, the dispersion relations of poles, and the convergence properties of all order hydrodynamics.

scholarly journals Exact four point function for large q SYK from Regge theory

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Changha Choi ◽  
Márk Mezei ◽  
Gábor Sárosi
Keyword(s):  
Lyapunov Exponent ◽  
Quantum Systems ◽  
Chaotic Regime ◽  
Closed Form Expression ◽  
Large Temperature ◽  
Complete Agreement ◽  
Leading Order ◽  
Point Function ◽  
Form Expression ◽  
Weakly Coupled

Abstract Motivated by the goal of understanding quantum systems away from maximal chaos, in this note we derive a simple closed form expression for the fermion four point function of the large q SYK model valid at arbitrary temperatures and to leading order in 1/N. The result captures both the large temperature, weakly coupled regime, and the low temperature, nearly conformal, maximally chaotic regime of the model. The derivation proceeds by the Sommerfeld-Watson resummation of an infinite series that recasts the four point function as a sum of three Regge poles. The location of these poles determines the Lyapunov exponent that interpolates between zero and the maximal value as the temperature is decreased. Our results are in complete agreement with the ones by Streicher [1] obtained using a different method.

A Combinatorial Method for Analyzing Sequential Firing Patterns Involving an Arbitrary Number of Neurons Based on Relative Time Order

2004 ◽  
Vol 92 (4) ◽  
pp. 2555-2573 ◽  
Author(s):  
Albert K. Lee ◽  
Matthew A. Wilson
Keyword(s):  
Arbitrary Number ◽  
Slow Wave Sleep ◽  
Statistical Significance ◽  
Reference Sequence ◽  
Sequence Matching ◽  
Combinatorial Method ◽  
Relative Time ◽  
Firing Patterns ◽  
Time Order ◽  
Sequential Firing

Information processing in the brain is believed to require coordinated activity across many neurons. With the recent development of techniques for simultaneously recording the spiking activity of large numbers of individual neurons, the search for complex multicell firing patterns that could help reveal this neural code has become possible. Here we develop a new approach for analyzing sequential firing patterns involving an arbitrary number of neurons based on relative firing order. Specifically, we develop a combinatorial method for quantifying the degree of matching between a “reference sequence” of N distinct “letters” (representing a particular target order of firing by N cells) and an arbitrarily long “word” composed of any subset of those letters including repeats (representing the relative time order of spikes in an arbitrary firing pattern). The method involves computing the probability that a random permutation of the word's letters would by chance alone match the reference sequence as well as or better than the actual word does, assuming all permutations were equally likely. Lower probabilities thus indicate better matching. The overall degree and statistical significance of sequence matching across a heterogeneous set of words (such as those produced during the course of an experiment) can be computed from the corresponding set of probabilities. This approach can reduce the sample size problem associated with analyzing complex firing patterns. The approach is general and thus applicable to other types of neural data beyond multiple spike trains, such as EEG events or imaging signals from multiple locations. We have recently applied this method to quantify memory traces of sequential experience in the rodent hippocampus during slow wave sleep.

scholarly journals Exponential growth of out-of-time-order correlator without chaos: inverted harmonic oscillator

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Koji Hashimoto ◽  
Kyoung-Bum Huh ◽  
Keun-Young Kim ◽  
Ryota Watanabe
Keyword(s):  
Quantum Mechanics ◽  
High Temperature ◽  
Lyapunov Exponent ◽  
Harmonic Oscillator ◽  
Exponential Growth ◽  
Local Maximum ◽  
Detailed Examination ◽  
One Dimension ◽  
One Dimensional ◽  
Time Order

Abstract We provide a detailed examination of a thermal out-of-time-order correlator (OTOC) growing exponentially in time in systems without chaos. The system is a one-dimensional quantum mechanics with a potential whose part is an inverted harmonic oscillator. We numerically observe the exponential growth of the OTOC when the temperature is higher than a certain threshold. The Lyapunov exponent is found to be of the order of the classical Lyapunov exponent generated at the hilltop, and it remains non-vanishing even at high temperature. We adopt various shape of the potential and find these features universal. The study confirms that the exponential growth of the thermal OTOC does not necessarily mean chaos when the potential includes a local maximum. We also provide a bound for the Lyapunov exponent of the thermal OTOC in generic quantum mechanics in one dimension, which is of the same form as the chaos bound obtained by Maldacena, Shenker and Stanford.

The isotope effect in electronic expectation values: an all-quantum study of C6H6 and C6D6

1998 ◽  
Vol 93 (5) ◽  
pp. 801-807
Author(s):  
JOACHIM SCHULTE ◽  
MICHAEL BOHM ◽  
RAFAEL RAMIREZ
Keyword(s):  
Isotope Effect ◽  
Expectation Values

Topological Characteristics of ECG Signal using Lyapunov Exponent and RBF Network

2011 ◽  
Vol 1 (9) ◽  
pp. 53-55
Author(s):  
Abinash Dahal ◽  
◽  
Deepashree Devaraj ◽  
Dr. N. Pradhan Dr. N. Pradhan
Keyword(s):  
Lyapunov Exponent ◽  
Ecg Signal ◽  
Rbf Network ◽  
Topological Characteristics

On the Harmonic Oscillations for the Motion of a Dynamical System

2017 ◽  
Vol 13 (2) ◽  
pp. 4657-4670
Author(s):  
W. S. Amer
Keyword(s):  
Dynamical System ◽  
Numerical Solutions ◽  
Equation Of Motion ◽  
Parameter Method ◽  
Kutta Method ◽  
Small Parameter Method ◽  
Harmonic Oscillations ◽  
Pivot Point ◽  
Runge Kutta Method ◽  
High Consistency

This work touches two important cases for the motion of a pendulum called Sub and Ultra-harmonic cases. The small parameter method is used to obtain the approximate analytic periodic solutions of the equation of motion when the pivot point of the pendulum moves in an elliptic path. Moreover, the fourth order Runge-Kutta method is used to investigate the numerical solutions of the considered model. The comparison between both the analytical solution and the numerical ones shows high consistency between them.

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