Butterfly Velocity Research Articles

Information scrambling and butterfly velocity in quantum spin glass chains

Information scrambling and butterfly velocity in quantum spin glass chains

Pole-skipping and chaos in D3-D7 brane systems

In this paper, we analyze the pole-skipping phenomena of finite temperature Yang-Mills theory with quark flavors, which is dual to D3-D7 brane systems in bulk. We also consider the external electric field in the boundary field theory, which is dual to the world volume electric field on the D7 brane. We will work in the probe limit where the D7 branes do not backreact to the D3 brane background. In this scenario, we decode the characteristic parameters of the chaos, namely, Lyapunov exponent λL and butterfly velocity vb from the pole-skipping points by performing the near effective horizon analysis of the linearized Einstein equations. Unlike pure Yang-Mills, once charged quarks with a background electric field are added into the system, the characteristic parameters of the chaos show nontrivial dependence on the quark mass and external electric field. We have observed that λL and vb decreases with increasing electric field. We further perform the pole-skipping analysis for the gauge invariant sound, shear, and tensor modes of the perturbation in the bulk and discuss their physical importance in the holographic context. Published by the American Physical Society 2024

Quantum chaos in the presence of nonconformality

The behavior of a chaotic system and its effect on existing quantum correlation has been holographically studied in the presence of nonconformality. Keeping in mind the gauge/gravity duality framework, the nonconformality in the dual field theory has been introduced by considering a Liouville type dilaton potential for the gravitational theory. The resulting black brane solution is associated with a parameter η which represents the deviation from conformality. The parameters of chaos, namely, the Lyapunov exponent and butterfly velocity are computed by following the well-known shock wave analysis. The obtained results reveal that the presence of nonconformality leads to suppression of the chaotic nature of a system. Further, for a particular value of the nonconformal parameter η, the system achieves Lyapunov stability resulting from the vanishing of both the Lyapunov exponent as well as butterfly velocity. Interestingly, this particular value of η matches with the previously given upper bound of η known as Gubser bound in the literature. The effects of chaos and nonconformality on the existing correlation of a thermofield doublet state have been quantified by holographically computing the thermomutual information in both the presence and absence of the shock wave. Furthermore, the entanglement velocity is also computed, and the effect of nonconformality on it has been observed. Finally, the obtained results for the Lyapunov exponent and the butterfly velocity have also been computed from the pole-skipping analysis. The results from the two approaches agree with each other. Published by the American Physical Society 2024

Shock waves, black hole interiors and holographic RG flows

We study holographic renormalization group (RG) flows perturbed by a shock wave in dimensions d ≥ 2. The flows are obtained by deforming a holographic conformal field theory with a relevant operator, altering the interior geometry from AdS-Schwarzschild to a more general Kasner universe near the spacelike singularity. We introduce null matter in the form of a shock wave into this geometry and scrutinize its impact on the near-horizon and interior dynamics of the black hole. Using out-of-time-order correlators, we find that the scrambling time increases as we increase the strength of the deformation, whereas the butterfly velocity displays a non-monotonic behavior. We examine other observables that are more sensitive to the black hole interior, such as the thermal a-function and the entanglement velocity. Notably, the a-function experiences a discontinuous jump across the shock wave, signaling an instantaneous loss of degrees of freedom due to the infalling matter. This jump is interpreted as a ‘cosmological time skip’ which arises from an infinitely boosted length contraction. The entanglement velocity exhibits similar dependence to the butterfly velocity as we vary the strength of the deformation. Lastly, we extend our analyses to a model where the interior geometry undergoes an infinite sequence of bouncing Kasner epochs.

Quantum Information Spreading in Generalized Dual-Unitary Circuits.

We study the spreading of quantum information in a recently introduced family of brickwork quantum circuits that generalizes the dual-unitary class. These circuits are unitary in time, while their spatial dynamics is unitary only in a restricted subspace. First, we show that local operators spread at the speed of light as in dual-unitary circuits, i.e., the butterfly velocity takes the maximal value allowed by the geometry of the circuit. Then, we prove that the entanglement spreading can still be characterized exactly for a family of compatible initial states (in fact, for an extension of the compatible family of dual-unitary circuits) and that the asymptotic entanglement slope is again independent on the Rényi index. Remarkably, however, we find that the entanglement velocity is generically smaller than 1. We use these properties to find a closed-form expression for the entanglement-membrane line tension.

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

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),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\omega _\\star = i \\lambda _L, \\, k_\\star = i\\lambda _L/v_B),$$\\end{document} where λL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _L$$\\end{document} and vB\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$v_B$$\\end{document} 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≥cs2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$v_B^2 \\ge c_s^2$$\\end{document} at each point of parameter space of the 1RCBH model where cs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_s$$\\end{document} is the speed of sound wave propagation.

Probing pole-skipping through scalar Gauss-Bonnet coupling

The holographic phenomena of pole-skipping have been studied in the presence of scalar-Gauss-Bonnet interaction in a four-dimensional Anti-de Sitter-Schwarzchild black hole background. In this paper, we initiated a novel study on understanding the response of those pole-skipping points under the application of external sources. The source is identified with the holographic dual operator of the bulk scalar field with its non-normalizable solutions. We analyze in detail the dynamics of pole-skipping points in both sound and shear channels, considering linear perturbation in bulk. In the perturbative regime, characteristic parameters for chaos, namely Lyapunov exponent and butterfly velocity, remain unchanged. However, the momentum values of the pole-skipping points in various modes are found to be affected by the scalar source. Further, the diffusion coefficient has been observed to evolve non-trivially under the application of external sources.

Super-maximal chaos and instability

An upper bound on Lyapunov exponent of a thermal many body quantum system has been conjectured recently. It would be useful to develop a physical understanding of what prevents a system from violating this bound. In this article we suggest violation of this bound leads to instability. Our suggestion is based on careful scrutiny of the arguments that established the bound. We also present a physical system where this phenomenon is manifested. In the stable regime, along with incoherent metallic phase, the system exhibits another novel phase, where transport is controlled neither by quasi-particles nor by diffusion. At the phase transition, diffusion coefficient, butterfly velocity and Thouless time diverges.

Information trapping by topologically protected edge states: Scrambling and the butterfly velocity

Information trapping by topologically protected edge states: Scrambling and the butterfly velocity

Pole-skipping as order parameter to probe a quantum critical point

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.

Information scrambling and entanglement dynamics of complex Brownian Sachdev-Ye-Kitaev models

In this work, we study the information scrambling and the entanglement dynamics in the complex Brownian Sachdev-Ye-Kitaev (cBSYK) models, focusing on their dependence on the charge density n. We first derive the effective theory for scramblons in a single cBSYK model, which gives closed-form expressions for the late-time OTOC and operator size. In particular, the result for OTOC is consistent with numerical observations in [1]. We then study the entanglement dynamics in cBSYK chains. We derive the density dependence of the entanglement velocity for both Rényi entropies and the Von Neumann entropy, with a comparison to the butterfly velocity. We further consider adding repeated measurements and derive the effective theory of the measurement induced transition which shows U(2)L ⊗ U(2)R symmetry for non-interacting models.

From Dual Unitarity to Generic Quantum Operator Spreading.

Dual-unitary circuits are paradigmatic examples of exactly solvable yet chaotic quantum many-body systems, but solvability naturally goes along with a degree of nongeneric behavior. By investigating the effect of weakly broken dual unitarity on the spreading of local operators, we study whether, and how, small deviations from dual unitarity recover fully generic many-body dynamics. We present a discrete path-integral formula for the out-of-time-order correlator and recover a butterfly velocity smaller than the light-cone velocity, v_{B}<v_{LC}, and a diffusively broadening operator front, two generic features of ergodic quantum spin chains absent in dual-unitary circuit dynamics. The butterfly velocity and diffusion constant are determined by a small set of microscopic quantities, and the operator entanglement of the gates has a crucial role.

Chaos and pole-skipping in a simply spinning plasma

We study the relationship between many-body quantum chaos and energy dynamics in holographic quantum field theory states dual to the simply-spinning Myers-Perry-AdS5 black hole. The enhanced symmetry of such black holes allows us to provide a thorough examination of the phenomenon of pole-skipping, that is significantly simpler than a previous analysis of quantum field theory states dual to the Kerr-AdS4 solution. In particular we give a general proof of pole-skipping in the retarded energy density Green’s function of the dual quantum field theory whenever the spatial profile of energy fluctuations satisfies the shockwave equation governing the form of the OTOC. Furthermore, in the large black hole limit we are able to obtain a simple analytic expression for the OTOC for operator configurations on Hopf circles, and demonstrate that the associated Lyapunov exponent and butterfly velocity are robustly related to the locations of a family of pole-skipping points in the energy response. Finally, we note that in contrast to previous studies, our results are valid for any value of rotation and we are able to numerically demonstrate that the dispersion relations of sound modes in the energy response explicitly pass through our pole-skipping locations.

Information scrambling of the dilute Bose gas at low temperature

We calculate the quantum Lyapunov exponent $\lambda_L$ and butterfly velocity $v_B$ in the dilute Bose gas at temperature $T$ deep in the Bose-Einstein condensation phase. The generalized Boltzmann equation approach is used for calculating out-of-time ordered correlators, from which $\lambda_L$ and $v_B$ are extracted. At very low temperature where elementary excitations are phonon-like, we find $\lambda_L\propto T^5$ and $v_B\sim c$, the sound velocity. At relatively high temperature, we have $\lambda_L\propto T$ and $v_B\sim c(T/T_*)^{0.23}$. We find $\lambda_L$ is always comparable to the damping rate of a quasiparticle, whose energy depends suitably on $T$. The chaos diffusion constant $D_L=v_B^2/\lambda_L$, on the other hand, differs from the energy diffusion constant $D_E$. We find $D_E\ll D_L$ at very low temperature and $D_E\gg D_L$ otherwise.

A holographic study of the characteristics of chaos and correlation in the presence of backreaction

In this work, we perform a holographic study to estimate the effect of backreaction on the correlation between two subsystems forming the thermofield double (TFD) state. Each of these subsystems is described as a strongly coupled large-Nc thermal field theory, and the backreaction imparted to it is sourced by the presence of a uniform distribution of heavy static quarks. The TFD state we consider here holographically corresponds to an entangled state of two AdS blackholes, each of which is deformed by a uniform distribution of static strings. In order to make a holographic estimation of correlation between two entangled boundary field theories in presence of backreaction we compute the holographic mutual information in the backreacted eternal blackhole. The late time exponential growth of an early perturbation is a signature of chaos in the boundary thermal field theory. Using the shock wave analysis in the dual bulk theory, we characterize this chaotic behaviour by computing the holographic butterfly velocity. We find that there is a reduction in the butterfly velocity due to a correction term that depends on the backreaction parameter. The late time exponential growth of an early perturbation destroys the two-sided correlation, whereas the backreaction always acts in favour of it. Finally we compute the entanglement velocity that essentially encodes the rate of disruption of correlation between two boundary theories.

Real-time correlators in chaotic quantum many-body systems

We study real-time correlators $\ensuremath{\langle}\mathcal{O}(\mathbf{x},t)\mathcal{O}(0,0)\ensuremath{\rangle}$ of local operators in chaotic quantum many-body systems. These correlators show universal structure at late times, determined by the geometry of the dominant operator-space Feynman trajectories for the evolving operator $\mathcal{O}(\mathbit{x},t)$. These trajectories, which involve the operator contracting to a point at both the initial and final time, are qualitatively different from those that dominate the out-of-time-order correlation function. In the absence of conservation laws, local correlations decay exponentially: $\ensuremath{\langle}\mathcal{O}(\mathbf{x},t)\mathcal{O}(0,0)\ensuremath{\rangle}\ensuremath{\sim}exp(\ensuremath{-}{s}_{\text{eq}}\phantom{\rule{0.16em}{0ex}}r(\mathbit{v})\phantom{\rule{0.16em}{0ex}}t)$, where $\mathbit{v}=\mathbit{x}/t$ defines a ray in spacetime, and $r(\mathbit{v})$ is a rate function associated with this ray. We express $r(\mathbit{v})$ in terms of cost functions for various spacetime structures. In $1+1\mathrm{D}$ the operator histories can exhibit a phase transition at a critical value ${v}_{c}$ of the ray velocity, which leads to a singular behavior in $r(\mathbit{v})$. At low velocities, the dominant Feynman histories are ``fat'': The operator grows to a size of order ${t}^{\ensuremath{\alpha}}$ (with $\ensuremath{\alpha}=1/2$ in the simplest case) before contracting to a point again. At high velocities the trajectories are ``thin'': The operator always remains of order-one size. In a Haar-random quantum circuit, this transition maps to a simple binding transition for a pair of random walks, which represent the left and right spatial boundaries of the operator. In higher-dimensional systems, thin trajectories always dominate. We discuss the circumstances in which the butterfly velocity ${v}_{B}$ can be deduced from a time-ordered two-point function, rather than the out-of-time ordered correlator. In the random circuit, correlators may also be computed in the framework of an effective Ising-like statistical mechanics model: we describe this calculation, as well as a special feature of the weights in the case of a $1+1\mathrm{D}$ Haar-random brickwork circuit. The present paper addresses lattice models, but also suggests the possibility of morphological phase transitions for real-time Feynman diagrams in continuum quantum field theories.

Pole Skipping in Holographic Theories with Bosonic Fields.

We study pole skipping in holographic conformal field theories dual to diffeomorphism invariant theories containing an arbitrary number of bosonic fields in the large N limit. Defining a weight to organize the bulk equations of motion, a set of general pole skipping conditions are derived. In particular, the frequencies simply follow from general covariance and weight matching. In the presence of higher-spin fields, we find that the imaginary frequency for the highest-weight pole skipping point equals the higher-spin Lyapunov exponent which lies outside of the chaos bound. Without higher-spin fields, we show that the energy density Green's function has its highest-weight pole skipping happening at a location related to the out-of-time-order correlator for arbitrary higher-derivative gravity, with a Lyapunov exponent saturating the chaos bound and a butterfly velocity matching that extracted from a shockwave calculation. We also suggest an explanation for this matching at the metric level by obtaining the on-shell shockwave solution from a regularized limit of the metric perturbation at the skipped pole.

A tale of two butterflies: an exact equivalence in higher-derivative gravity

We prove the equivalence of two holographic computations of the butterfly velocity in higher-derivative theories with Lagrangian built from arbitrary contractions of curvature tensors. The butterfly velocity characterizes the speed at which local perturbations grow in chaotic many-body systems and can be extracted from the out-of-time-order correlator. This leads to a holographic computation in which the butterfly velocity is determined from a localized shockwave on the horizon of a dual black hole. A second holographic computation uses entanglement wedge reconstruction to define a notion of operator size and determines the butterfly velocity from certain extremal surfaces. By direct computation, we show that these two butterfly velocities match precisely in the aforementioned class of gravitational theories. We also present evidence showing that this equivalence holds in all gravitational theories. Along the way, we prove a number of general results on shockwave spacetimes.

Out-of-Time-Order correlators in driven conformal field theories

We compute Out-of-Time-Order correlators (OTOCs) for conformal field theories (CFTs) subjected to either continuous or discrete periodic drive protocols. This is achieved by an appropriate analytic continuation of the stroboscopic time. After detailing the general structure, we perform explicit calculations in large-c CFTs where we find that OTOCs display an exponential, an oscillatory and a power-law behaviour in the heating phase, the non-heating phase and on the phase boundary, respectively. In contrast to this, for the Ising CFT representing an integrable model, OTOCs never display such exponential growth. This observation hints towards how OTOCs can demarcate between integrable and chaotic CFT models subjected to a periodic drive. We further explore properties of the light-cone which is characterized by the corresponding butterfly velocity as well as the Lyapunov exponent. Interestingly, as a consequence of the spatial inhomogeneity introduced by the drive, the butterfly velocity, in these systems, has an explicit dependence on the initial location of the operators. We chart out the dependence of the Lyapunov exponent and the butterfly velocities on the frequency and amplitude of the drive for both protocols and discuss the fixed point structure which differentiates such driven CFTs from their undriven counterparts.

Quantum information scrambling and quantum chaos in little string theory

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’.