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arXiv:2305.00298 · The European Physical Journal C

Chaos near to the critical point: butterfly effect and pole-skipping

B. Amrahi, M. Asadi, Farid Taghinavaz

Published May 14, 2024• 7 citations
arXivHTMLPDF

Abstract

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 $$(\omega _\star = i \lambda _L, \, k_\star = i\lambda _L/v_B),$$ (ω⋆=iλL,k⋆=iλL/vB), where $$\lambda _L$$ λL and $$v_B$$ 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 $$v_B^2 \ge c_s^2$$ vB2≥cs2 at each point of parameter space of the 1RCBH model where $$c_s$$ cs is the speed of sound wave propagation.

DOI
10.1140/epjc/s10052-024-12854-y

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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.
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.
A bstract The holographic system described by Einstein-Maxwell-Chern-Simons dynamics in the bulk of AdS exhibits a chiral magnetic effect and a quantum critical point. Through numerical calculations, we find that the butterfly velocity can serve as a new identifier for the quantum critical point in this system. We show that the critical point is the point at which the butterfly velocity is equal to the speed of light in the direction of the magnetic field, while in the opposite direction the butterfly propagation vanishes. Furthermore, by studying the pole-skipping points of the response function of the operator dual to the tensor part of the metric perturbation in the bulk, we discover a set of order parameters that distinguish the two states of the system near the quantum critical point. Each of these order parameters is the sum of the absolute values of the real parts of momentum at all pole-skipping points associated with a particular frequency. This quantity vanishes in the disordered state while taking a positive value in the ordered state. In addition, our results confirm the idea that the chiral magnetic effect can manifest macroscopically through quantum chaos.
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.
When the Lyapunov exponent λL in a quantum chaotic system saturates the bound λL ≤ 2πk B T , it is proposed that this system has a holographic dual described by a gravity theory. In particular, the butterfly effect as a prominent phenomenon of chaos can ubiquitously exist in a black hole system characterized by a shockwave solution near the horizon. In this paper we propose that the butterfly velocity can be used to diagnose quantum phase transition (QPT) in holographic theories. We provide evidences for this proposal with an anisotropic holographic model exhibiting metal-insulator transitions (MIT), in which the derivatives of the butterfly velocity with respect to system parameters characterizes quantum critical points (QCP) with local extremes in zero temperature limit. We also point out that this proposal can be tested by experiments in the light of recent progress on the measurement of out-of-time-order correlation function (OTOC).
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 In the current manuscript we perform a systematic investigation about the effects of nonlocal interaction to the spread of quantum information in many body system. In particular, we have studied how nonlocality influence the existing bound on the growth rate of the commutator involving two local operators, the butterfly velocity. For this purpose, we consider the nonlocal theory on the worldvolume of N ≫ 1, NS5 branes arising in the limit of vanishing string coupling, the ‘little string theory’. A direct evidence of nonlocality can be realized from the ‘volume law’ behavior for the most dominant part of holographic entanglement entropy. We obtain the butterfly velocity by studying the dynamics of the near horizon geometry backreacted by a high energy quanta in the form of a shockwave resulting from an early perturbation on the corresponding thermofield double state. We observe that the butterfly velocity increases with the nonlocal scale of little string theory, the inverse Hagedorn temperature βh, indicating a faster rate of information spread due to the nonlocal interaction. The same conclusion follows as the disruption of two sided mutual information is observed to occur at a faster rate for higher values of βh. Finally, we realize a direct connection between the parameters of quantum chaos and the quasinormal modes for collective excitations through the phenomenon of ‘pole skipping’.