Source-linked AI summary
Stringy effects in scrambling
Stephen H. Shenker, Douglas Stanford
TL;DR
The paper asks how stringy and Planckian physics modifies holographic scrambling previously studied with Einstein gravity. It analyzes chaos-sensitive correlators as near-horizon high-energy scattering processes and finds that elastic string effects weaken, delay, and smear chaos, while inelastic effects are parametrically small for those correlators.
Problem
The paper addresses how stringy and Planckian corrections modify holographic scrambling and its boundary correlators.
Source: §1 · p. 2Method
The analysis computes chaos-sensitive correlators from near-horizon high-energy scattering, including elastic string corrections and estimates of inelastic effects.
Source: §1 · p. 2, §5 · p. 26, §6 · p. 32Results
Elastic string effects delay and smooth the development of chaos, while inelastic effects leave a parametrically small imprint on the studied correlators.
Source: §6 · pp. 30–32Takeaways & Limitations
Stringy chaos exhibits a novel diffusion-like spreading in the boundary theory, and specially chosen correlators can make inelastic effects dominant.
Source: §6 · pp. 31, 33The studied correlators do not expose truly Planckian physics because their important contributions remain at G_Ns ∼1, despite the state becoming largely inelastic at late times.
Source: §1.2 · p. 7
Abstract
from arXiv · showhide
In [1] we gave a precise holographic calculation of chaos at the scrambling time scale. We studied the influence of a small perturbation, long in the past, on a two-sided correlation function in the thermofield double state. A similar analysis applies to squared commutators and other out-of-time-order one-sided correlators [2-4]. The essential bulk physics is a high energy scattering problem near the horizon of an AdS black hole. The above papers used Einstein gravity to study this problem; in the present paper we consider stringy and Planckian corrections. Elastic stringy corrections play an important role, effectively weakening and smearing out the development of chaos. We discuss their signature in the boundary field theory, commenting on the extension to weak coupling. Inelastic effects, although important for the evolution of the state, leave a parametrically small imprint on the correlators that we study. We briefly discuss ways to diagnose these small corrections, and we propose another correlator where inelastic effects are order one.
1 Introduction
The paper studies holographic scrambling through high-energy scattering near black-hole horizons, extending Einstein-gravity analyses to stringy and inelastic effects. It finds that elastic string corrections weaken, delay, and spatially smear chaos, while inelastic effects remain parametrically small for the main correlators.
1 Introduction
1.1 Correlation functions that probe chaos
- The paper uses two-sided and one-sided out-of-time-order correlators to diagnose chaos, with their relations obtained by analytic continuation in imaginary time.The squared commutator becomes large when the corresponding chaos-sensitive correlator becomes small at late times.Source: §1.1 · pp. 5–6, Figure 2 · p. 6
1 Introduction
- The relevant bulk process is high-energy scattering near the bifurcation surface, with important modes satisfying G_Ns ∼1 in AdS units.The large energies arise from exponentially growing boosts of thermal-scale perturbations near the horizon.Source: §1 · pp. 2–3, §1.2 · pp. 6–7
1.2 Outline
- Stringy corrections delay the decrease of chaos-sensitive correlators and smear the region where they are affected.Transverse string spreading produces a parametrically large affected region, while Reggeization slows the growth of the scattering amplitude.Source: §1 · p. 4, §1.2 · p. 7
- Inelastic effects have little effect on the studied correlators because oscillatory elastic phases suppress contributions from large scattering momenta.Although most of the perturbed state becomes inelastic at late times, the correlator is controlled by the G_Ns ∼1 tails of the wavefunctions.Source: §1.2 · p. 7
- The authors propose additional observables to diagnose inelastic effects more directly, including a two-shock correlator where they can become dominant.The paper also discusses perturbative weak-coupling signatures and a connected spin-system comparison.Source: §1.2 · p. 7
2 Kinematics
The kinematic construction rewrites chaos-sensitive correlators as overlaps of bulk wavepacket states and a high-energy scattering amplitude near the black-hole horizons. Different operator orderings produce different in/out states and wavefunctions, while the dynamical input is the eikonal phase δ(s,b).
2 Kinematics
- Each correlator separates into wavefunction kinematics and a two-to-two scattering amplitude depending on impact parameter b and energy variable s.The wavefunctions are obtained from bulk-to-boundary propagators Fourier transformed along horizon coordinates.Source: §2 · pp. 7–8
2.1 Derivation
- The one-sided correlator is represented as an overlap of an ‘in’ state and an ‘out’ state in the thermofield-double purification.The construction places W and V states on opposite horizon slices and uses null-momentum and transverse-position bases.Source: §2.1 · pp. 9–10
- At large relative boost, momentum conservation makes the scattering amplitude approximately diagonal in the relevant null momenta and transverse positions.The high-energy collision preserves the dominant momenta and impact-parameter data up to curvature-scale corrections.Source: §2.1 · p. 11
- Within the two-particle sector, scattering multiplies the state by e^{iδ(s,b)} plus an orthogonal inelastic component.The complex phase δ(s,b) is the dynamical quantity that controls the correlator calculation.Source: §2.1 · pp. 11–12
2.2 A two-sided case
- The two-sided configuration is likewise an overlap of ‘in’ and ‘out’ states, but its operators are already spacelike related, avoiding intermediate time evolution.Its overlap can be interpreted as propagation in a shock-wave geometry sourced by a superposition of the other pair of quanta.Source: §2.2 · p. 12
- The derivation neglects V–W annihilation contributions, which are expected to be small after a few thermal timescales.The operators are also assumed to be temporally smeared over a thermal scale so the correlators remain finite.Source: §2.1 · p. 11, §2.2 · p. 12
3 Elastic eikonal gravity approximation
The elastic eikonal approximation reduces near-horizon scrambling to high-energy shock-wave scattering and wave-function overlaps. Its correlators show universal early exponential behavior, while late-time decay depends on operator wave-function tails and geometry.
3.1 The basic amplitude
- The elastic eikonal amplitude is computed by linearizing gravity and evaluating the classical action sourced by fixed high-energy trajectories near the two horizons.The relevant fields take a shock-wave form, with finite-width wave functions justifying the transverse delta-function approximation when separations exceed their width.Source: §3.1 · pp. 13–14
3.2 Relation to free propagation on a shock background
- The shock geometry acts as a transverse-position-dependent shift v → v − h(x), so the correlator becomes a wave-function overlap weighted by the shock strength.The overlap can equivalently be viewed as either V propagation through a W-generated shock or W propagation through a V-generated shock.Source: §3.2 · pp. 15–16
3.3 The integral over momenta
- The correlator is an integral weighted by e^{iδ}, with the important region set by δ ≲ 1; in an asymmetric setup, one momentum integral can be evaluated exactly before a saddle-point treatment.Without the phase, the integral gives the disconnected overlap ⟨VV⟩⟨WW⟩.Source: §3.3 · p. 17, AdS3 example · p. 18
AdS3 example
- In the localized AdS3 example, the late-time decay is e^{-∆t}, slower than the e^{-2∆t} decay for spherical shocks because transverse wave-function tails dominate.The result reflects a compromise between tail suppression and shock strength; homogeneous shocks or compact spaces do not permit the same compromise.Source: AdS3 example · pp. 18–19
General background
- For general backgrounds, the initial correlator change is universal and exponential, while late-time compact-space decay is governed by the lowest operator-dependent quasinormal frequency.A precise analysis is unavailable without bulk-to-boundary propagators; the compact-space asymptotic form involves antipodal wave-function support and the butterfly velocity vB of [1] [3].Source: General background · pp. 19–21
4 Tree level stringy corrections
The paper derives tree-level string corrections by replacing the flat-space Regge operator with a curvature-corrected Pomeron operator in the black-hole background. These corrections slow the growth of the amplitude and smooth its transverse profile, while longitudinal spreading remains provisionally controlled.
4 Tree level stringy corrections
- The tree-level string calculation uses highly boosted bulk-to-boundary vertex operators and a Regge-limit worldsheet OPE retaining all powers of ws.The resulting Pomeron operator reproduces standard flat-space Regge behavior before curvature corrections are included.Source: §4 · p. 22, Flat space Pomeron operator · pp. 22–23
First curvature correction
- Only shifts in worldsheet conformal dimensions produce energy-growing curvature corrections; vertex operators and OPE coefficients remain at Gaussian order.The correction is organized through worldsheet L0 and L̄0 operators and matched to Einstein’s equations.Source: First curvature correction · pp. 23–24
The amplitude
- String corrections make the amplitude grow more slowly with s than gravity and replace the gravitational exponential transverse profile with a smoother Gaussian profile.The smoothing reflects transverse string spreading over a scale √log s.Source: The amplitude · p. 25
- The corrected amplitude has a small imaginary part associated with inelastic long-string production, while eikonal exponentiation is expected to remain accurate at small GN with GNs and b fixed.The exponential correction is a power-series expansion in ℓs^2/ℓAdS^2, whose full series may matter at large time.Source: The amplitude · p. 25
Longitudinal spreading
- The estimated longitudinal spreading is too small to substantially affect the weakly curved analysis, but its uncertain nature leaves the control of the calculation provisional.A substantially different estimate could make near-singularity curvatures important.Source: Longitudinal spreading · p. 26
5 Inelastic effects
Inelastic string and gravitational processes are estimated using flat-space scattering intuition and modeled as smooth high-energy cutoffs. They are parametrically subleading in the correlators studied, despite becoming important for the underlying state at sufficiently high energies.
5 Inelastic effects
- The relevant inelastic channels include oscillator excitation, long-string production, multigraviton production, and ultimately macroscopic black-hole production.Their impact is assessed using impact parameter b and center-of-mass energy √s, with transverse string spreading setting the relevant scale.Source: §5 · p. 26, Oscillator excitation · p. 27, Long string creation · p. 27
Oscillator excitation
- Diffractive oscillator excitation is suppressed relative to elastic scattering by (ℓs/ℓAdS)^2 ∼ 1/λ or, at smaller impact parameter, parametrically by 1/log N^2.The smaller suppression at b < bI follows from the reduced gradient of the string-corrected shock profile.Source: Oscillator excitation · p. 27
Long string creation
- Long-string production is exponentially suppressed at b > bI and suppressed by ℓs^2/ℓAdS^2 relative to the real phase at b < bI.The latter regime is also where bulk curvature is significant.Source: Long string creation · p. 27
Multiparticle production
- At the scrambling scale GNs ∼ 1, multigraviton production remains suppressed by gs^2, while black-hole production occurs only at higher energies where the elastic amplitude is exponentially small.Macroscopic black-hole production becomes relevant when GN^2s ∼ 1 for b ∼ 1.Source: Multiparticle production · p. 27, Black hole production · p. 27
Effect on correlators
- Because the correlator integrals are dominated where the elastic phase is order one, inelastic effects produce only parametrically small corrections to the studied correlators.The paper estimates oscillator and long-string corrections as multiplicative perturbations and proposes other observables where inelastic effects can be order one.Source: Effect on correlators · pp. 28–29
6 Discussion
The discussion finds that elastic string effects smear and diffuse chaotic growth, while inelastic effects usually leave simple correlators nearly unchanged but can dominate specially designed two-shock observables.
Stringy corrections
- String spreading smooths ballistic chaos into an initially diffusive disk, after which a ballistic region moves outward and leaves slower growth behind.The diffusive disk is parametrically large at scrambling time, while its later growth depends on wave-function convolution.Source: §6 · p. 30, Figure 7 · p. 30
- The resulting stringy diffusion is a novel boundary signature related in the bulk to branched diffusion on the horizon.It smooths the ballistic growth associated with precursor size and tensor-network growth, while the field-theoretic explanation remains open.Source: §6 · pp. 30–31
Weak coupling
- At weak coupling, the suggested scrambling time scales as t∗ ∼ β/(2πc2λ) log N^2, parametrically longer than in the gravitational limit.The discussion relates this behavior to the small-λ Regge intercept and resummed gluon-ladder effects.Source: §6 · pp. 31–32
Inelastic effects
- Inelastic effects are parametrically small in the studied simple correlators because their wave-function integrals select scattering tails with G_Ns ∼ 1.Although most of the evolved state becomes inelastic at late times, the correlators do not directly expose truly Planckian physics.Source: §6 · p. 32
- A two-shock correlator can instead become sensitive to strong inelasticity when G_N^2s ∼ 1, where geodesics traverse a nonlinear post-collision region.For G_N^2s ≪ 1 nonlinear effects are suppressed, while for G_N^2s ≫ 1 the post-collision region is isolated near the singularity.Source: §6 · p. 33
A Brownian circuit
A two-body nonlocal Brownian circuit models scrambling with random time-dependent couplings, reducing the correlator calculation to 2^n linear ODEs. The ensemble-averaged correlator remains nearly unchanged until t∼log n, then decays exponentially, with scrambling time proportional to log n.
Method
- Randomness reduces the correlator calculation to a closed system of 2^n linear ODEs for subsystem-purity variables.Permutation symmetry groups these variables into functions indexed by subsystem size, from which D(t) is recovered.Source: A Brownian circuit · pp. 34–36
Scrambling time
- t∗∝log n is the time for the correlator to become O(1) affected, matching the information-theoretic scrambling time.This follows by relating the correlator to the slowest-relaxing purity, that of the full system minus one site.Source: A Brownian circuit · pp. 34, 37
Model
- The two-body nonlocal Brownian circuit uses Gaussian-random couplings between pairs of n spins and Pauli indices.Its infinitesimal time evolution is driven by random variables with zero mean and variance proportional to dt.Source: A Brownian circuit · pp. 34–35
Correlator dynamics
- After t∼log n, the ensemble-averaged correlator exponentially decays, with late-time behavior set by the ODE system’s eigenvalue gap.Individual Brownian-circuit realizations fluctuate around the mean by O(2^-n).Source: A Brownian circuit · pp. 34, 36
Interpretation
- The logarithmic scrambling behavior is quantitatively consistent with ballistic wave propagation in a logarithmic coordinate.Near the endpoint controlling the correlator, the evolution becomes a translating wave, yielding the log-n timescale.Source: A Brownian circuit · pp. 36–37
Correlator dynamics
- The ensemble-averaged correlator D(t) is plotted for different values of n in Figure 9.Source: Figure 9 · p. 37