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arXiv:2311.12121 · Journal of High Energy Physics

Scramblon loops

Douglas Stanford, Shreya Vardhan, Shunyu Yao

Published Oct 9, 2024• 4 citations
arXivHTMLPDF

Abstract

A bstract In large N chaotic quantum systems, the butterfly effect is mediated by a collective field mode known as the “scramblon.” We study self-interactions of the scramblon in variants of the Sachdev-Ye-Kitaev model. In spatially extended versions of the model and for large spatial separation, fluctuations described by loop diagrams can invalidate the single-scramblon approximation well before its contribution to out-of-time-order correlators becomes of order one. We find a qualitative difference between an incoherent regime at high temperaure (or in a Brownian version of the model) and a coherent regime at low temperature.

DOI
10.1007/jhep10(2024)073

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A bstract In large N chaotic quantum systems, the butterfly effect is mediated by a collective field mode known as the “scramblon.” We study self-interactions of the scramblon in variants of the Sachdev-Ye-Kitaev model. In spatially extended versions of the model and for large spatial separation, fluctuations described by loop diagrams can invalidate the single-scramblon approximation well before its contribution to out-of-time-order correlators becomes of order one. We find a qualitative difference between an incoherent regime at high temperaure (or in a Brownian version of the model) and a coherent regime at low temperature.
A method has been proposed to calculate the out-of-time time ordered correlator in the generalization of the Sachdev–Ye–Kitaev model with a nonzero spatial dimension. The result is applicable not only at small times, when the chaotic properties of the system are developed weakly, but also at large times of about the Erenfest time. It has been shown that information on the applied perturbation, which is described by this correlator, propagates ballistically in the form of a front. The velocity of the front for models of this type has been calculated for the first time.
Information scrambling refers to the propagation of information throughout a quantum system. Its study not only contributes to our understanding of thermalization but also has wide implications in quantum information and black hole physics. Recent studies suggest that information scrambling in large-N systems with all-to-all interactions is mediated by collective modes called scramblons. However, a criterion for the validity of scramblon theory in a specific model is still missing. In this work, we address this issue by investigating the signature of the scramblon effective theory in random spin models with all-to-all interactions. We demonstrate that, in scenarios where the scramblon description holds, the late-time operator size distribution can be predicted from its early-time value, requiring no free parameters. As an illustration, we examine whether Brownian circuits exhibit a scramblon description and obtain a positive confirmation both analytically and numerically. Our findings provide a concrete experimental framework for unraveling the scramblon field theory in random spin models using quantum simulators.
A bstract Understanding the emergence of complex correlations in strongly interacting systems remains a fundamental challenge in quantum many-body physics. One fruitful approach is to develop solvable toy models that encapsulate universal properties shared by realistic systems. In this work, we introduce the Brownian SYK-Hubbard model, which combines the all-to-all random interactions of the Sachdev-Ye-Kitaev (SYK) model with on-site Hubbard-type interactions. This hybrid construction enables the study of the interplay between nonlocal random dynamics and local correlation effects: (1) As the interaction strength increases, the single-particle spectrum exhibits a transition from a single peak to a two-peak structure, signaling the onset of Mottness. (2) The spectral form factor undergoes a sequence of dynamical transitions as the evolution time increases before reaching the plateau in the long-time limit under strong Hubbard interactions. (3) The out-of-time-order correlator is computed by summing a series of modified ladder diagrams, which determines the quantum Lyapunov exponent and reveals a violation of the bound on branching time. Our results establish a new analytically tractable platform for exploring the effects of Hubbard interactions in chaotic many-body systems.
We describe the dynamics of many-body quantum chaotic systems at all time scales by studying the Green's and out-of-time order correlation (OTOC) functions of the four-body, NNN-Majorana Sachdev-Ye-Kitaev model. By combining the scramblon formalism and random-matrix-theory techniques, we obtain analytical expressions for these functions at all times. The early exponential growth of the OTOC is followed by an exponential decay at a rate governed by that of the Green's function (the real part of the leading complex Ruelle-Pollicott resonances). For late times that scale exponentially with NNN, both functions have a dip-ramp-plateau pattern for Nmod8=2,6N \mathrm{mod}8 = 2, 6Nmod8=2,6 that deviates substantially from the ergodic prediction due to local-in-energy correlations of matrix elements and eigenvalues, even after the Heisenberg time.
The Sachdev-Ye-Kitaev (SYK) model describes Majorana fermions with random interaction, which displays many interesting properties such as non-Fermi liquid behavior, quantum chaos, emergent conformal symmetry and holographic duality. Here we consider a SYK model or a chain of SYK models with N Majorana fermion modes coupled to another SYK model with N 2 Majorana fermion modes, in which the latter has many more degrees of freedom and plays the role as a thermal bath. For a single SYK model coupled to the thermal bath, we show that although the Lyapunov exponent is still proportional to temperature, it monotonically decreases from 2π/β (β = 1/(k B T), T is temperature) to zero as the coupling strength to the thermal bath increases. For a chain of SYK models, when they are uniformly coupled to the thermal bath, we show that the butterfly velocity displays a crossover from a T \sqrt{T} T​ -dependence at relatively high temperature to a linear T-dependence at low temperature, with the crossover temperature also controlled by the coupling strength to the thermal bath. If only the end of the SYK chain is coupled to the thermal bath, the model can introduce a spatial dependence of both the Lyapunov exponent and the butterfly velocity. Our models provide canonical examples for the study of thermalization within chaotic models.
The existence of a quantum butterfly effect in the form of exponential sensitivity to small perturbations has been under debate for a long time. Lately, this question has gained increased interest due to the proposal to probe chaotic dynamics and scrambling using out-of-time-order correlators. In this work we study echo dynamics in the Sachdev-Ye-Kitaev model under effective time reversal in a semiclassical approach using the truncated Wigner approximation, which accounts for nonvanishing quantum fluctuations that are essential for the dynamics. We demonstrate that small imperfections introduced in the time-reversal procedure result in an exponential divergence from the perfect echo, which allows us to identify a Lyapunov exponent ${\ensuremath{\lambda}}_{L}$. In particular, we find that ${\ensuremath{\lambda}}_{L}$ is twice the Lyapunov exponent of the semiclassical equations of motion. This behavior is attributed to the growth of an out-of-time-order double commutator that resembles an out-of-time-order correlator.