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arXiv:2010.08558 · Journal of High Energy Physics

Pole skipping away from maximal chaos

Changha Choi, Márk Mezei, Gábor Sárosi

Published Feb 24, 2021• 37 citations
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Abstract

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.= $$ i{\lambda}^{(T)}\left(1,1/{u}_B^{(T)}\right) $$ iλT11/uBT , whereλ(T)= 2π/βand $$ {u}_B^{(T)} $$ uBT 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 obeysλ≤λ(T). On the other hand, in a large class of strongly coupled but non-maximally chaotic theories $$ {u}_B^{(T)} $$ uBT is the true butterfly velocity and we conjecture thatuB≤ $$ {u}_B^{(T)} $$ uBT 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-qlimit of the SYK chain, where we determineλ, 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.

DOI
10.1007/jhep02(2021)207

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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.= iλ(T)(1,1/uB(T))i{\lambda}^{(T)}\left(1,1/{u}_B^{(T)}\right)iλ(T)(1,1/uB(T)​) iλT11/uBT , whereλ(T)= 2π/βand uB(T){u}_B^{(T)}uB(T)​ uBT 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 obeysλ≤λ(T). On the other hand, in a large class of strongly coupled but non-maximally chaotic theories uB(T){u}_B^{(T)}uB(T)​ uBT is the true butterfly velocity and we conjecture thatuB≤ uB(T){u}_B^{(T)}uB(T)​ uBT 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-qlimit of the SYK chain, where we determineλ, 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.
Abstract We study the butterfly effect and pole-skipping phenomenon for the 1RCBH model which enjoys a critical point in its phase diagram. Using the holographic idea, we compute the butterfly velocity and interestingly find that this velocity can probe the critical behavior of this model. We calculate the dynamical exponent of this quantity near the critical point and find a perfect agreement with the value of the other quantity’s dynamical exponent near this critical point. We also find that at special points, namely the (ω⋆=iλL, k⋆=iλL/vB),(\omega _\star = i \lambda _L, \, k_\star = i\lambda _L/v_B),(ω⋆​=iλL​,k⋆​=iλL​/vB​), (ω⋆=iλL,k⋆=iλL/vB), where λL\lambda _LλL​ λL and vBv_BvB​ vB are Lyapunov exponent and butterfly velocity respectively, the phenomenon of pole-skipping appears which is a sign of a multivalued retarded Green’s function. Furthermore, we observe that vB2≥cs2v_B^2 \ge c_s^2vB2​≥cs2​ vB2≥cs2 at each point of parameter space of the 1RCBH model where csc_scs​ cs is the speed of sound wave propagation.
A bstract We analyze pole skipping of stress tensor two-point functions in two-dimensional quantum field theories perturbed away from conformality by a relevant deformation. The retarded two-point Green’s function can be formally computed in conformal perturbation theory, though it results in singular expressions. We propose a natural interpretation of these expressions and compute the resulting Green’s function to leading nontrivial order in the deformation. As a check of our results, we compare the Lyapunov exponents and butterfly velocities we find from our computed skipped poles to those obtained from both a leading-order conformal field theory analysis using Ward identities, as well as to a holographic gravitational dual perturbed by a massive scalar field; we find precise agreement. We comment on extensions to sub-leading order, where agreement with holographic expectations would no longer be expected.
In this paper, we study pole-skipping in non-maximally quantum chaotic systems. Using Rindler conformal field theories and the large-qqq SYK chain as illustrative examples, we argue that the pole-skipping points of few-body operators organize into trajectories in the complex frequency-momentum plane, with the leading trajectory encoding the quantum Lyapunov exponent. We further propose that these trajectories admit a natural interpretation as Regge trajectories of stringy excitations in a dual stringy black hole geometry. From this perspective, pole-skipping for an individual operator can be viewed as tracking the stringy horizon through the response of a single excitation. Our results suggest that pole-skipping reflects intrinsic properties of quantum chaotic systems and may be deeply connected to the structure of horizons in the stringy regime.
Pole-skipping offers compelling evidence for the hydrodynamic origin of chaotic behavior in strongly coupled quantum systems. We demonstrate that the cumulative effect of higher-order corrections to the hydrodynamic diffusive mode, captured by the parameter Ω, determines the thermal diffusivity through chaos parameters, providing new insights into the interplay between chaos and hydrodynamics. In the incoherent limit, where momentum relaxation is pronounced, we show that the thermodynamically stable phase corresponds to a semi-locally critical IR fixed point. This finding extends previous analyses of simple IR fixed points and may offer a gravity dual description for stable phases beyond the Sachdev-Ye-Kitaev model in the incoherent regime. Additionally, we connect our findings to a universal bound on the thermal diffusion constant in holography , establishing a direct link with the dynamical critical exponent z . We derive a new relation, Ω = ( 2 − z ) / ( 2 z − 2 ) , and propose that it characterizes the thermodynamically stable phase for generic values of z .
Abstract A simple probe of chaos and operator growth in many-body quantum systems is the out of time ordered four point function. In a large class of local systems, the effects of chaos in this correlator build up exponentially fast inside the so called butterfly cone. It has been previously observed that the growth of these effects is organized along rays and can be characterized by a velocity dependent Lyapunov exponent, λ(v). We show that this exponent is bounded inside the butterfly cone as λ(v) ≤ 2πT (1 − | v |/v B ), where T is the temperature and v B is the butterfly speed. This result generalizes the chaos bound of Maldacena, Shenker and Stanford. We study λ(v) in some examples such as two dimensional SYK models and holographic gauge theories, and observe that in these systems the bound gets saturated at some critical velocity v * < v B . In this sense, boosting a system enhances chaos. We discuss the connection to conformal Regge theory, where λ(v) is related to the spin of the leading large N Regge trajectory, and controls the four point function in an interpolating regime between the Regge and the light cone limit. Finally, we comment on the generalization of the chaos bound to boosted and rotating ensembles and clarify some recent results on this in the literature.
Abstract Recent studies of out-of-time ordered thermal correlation functions (OTOC) in holographic systems and in solvable models such as the Sachdev-Ye-Kitaev (SYK) model have yielded new insights into manifestations of many-body chaos. So far the chaotic behavior has been obtained through explicit calculations in specific models. In this paper we propose a unified description of the exponential growth and ballistic butterfly spreading of OTOCs across different systems using a newly formulated “quantum hydrodynamics,” which is valid at finite ℏ and to all orders in derivatives. The scrambling of a generic few-body operator in a chaotic system is described as building up a “hydrodynamic cloud,” and the exponential growth of the cloud arises from a shift symmetry of the hydrodynamic action. The shift symmetry also shields correlation functions of the energy density and flux, and time ordered correlation functions of generic operators from exponential growth, while leads to chaotic behavior in OTOCs. The theory also predicts an interesting phenomenon of the skipping of a pole at special values of complex frequency and momentum in two-point functions of energy density and flux. This pole-skipping phenomenon may be considered as a “smoking gun” for the hydrodynamic origin of the chaotic mode. We also discuss the possibility that such a hydrodynamic description could be a hallmark of maximally chaotic systems.