RELATIONS BETWEEN TRANSPORT & CHAOS IN HOLOGRAPHIC THEORIES - PowerPoint PPT Presentation

RELATIONS BETWEEN TRANSPORT & CHAOS IN HOLOGRAPHIC THEORIES Richard Davison Heriot-Watt University East Asian Strings Webinar, August 2020 based on 1809.01169 (with Mike Blake, Saso Grozdanov, Hong Liu) 1904.12883 (with Mike Blake, David Vegh)

MOTIVATION • Quantum field theories with strong interactions are important. Significant theoretical role in string theory / quantum gravity. • They are also relevant to some experimentally accessible systems: e.g. quark-gluon plasma ‘strange’ metals T strange metal

NON-QUASIPARTICLE STATES • Cartoon of a normal metal: ✴ electron-like excitations with charge , e mass , speed , lifetime m v F τ ✴ properties of these quasiparticles govern the properties of the metal • Strange metals have properties that seem inconsistent with a quasiparticle- based theory. • Strongly interacting QFT is a framework for describing non-quasiparticle states. But it is poorly understood.

INSIGHT FROM BLACK HOLES • Holographic duality gives us a handle on some strongly interacting QFTs Black holes have proven to be useful toy models of strange metals • Main reason: black holes exhibit some universal properties help to identify general features of strongly interacting QFTs • I will describe a new universal property of black holes, and its implications ✴ Certain features of black hole excitation spectrum depend only on near- horizon physics ✴ QFT transport properties are related to underlying chaotic dynamics

TRANSPORT PROPERTIES • Transport properties characterize the dynamics of a system’s conserved charges over long distances and timescales. i.e. the properties of and at small T μν J μ ( ω , k ) • Examples: electrical resistivity, thermal resistivity, shear viscosity, di ff usivity of energy,…. • Transport properties are important experimental observables ✴ They are relatively easy to measure ✴ They exhibit universality across di ff erent systems

TRANSPORT PROPERTIES • There are also two theoretical reasons that transport properties are privileged. (1) The dynamics of and are constrained by symmetries T μν J μ governed by a simple e ff ective theory over long distances and timescales: hydrodynamics For a given QFT, we just need to determine the parameters of the e ff ective theory. (2) Transport is directly related to the dynamics of the basic gravitational variables: T μν ⟷ g μν there is a degree of universality to transport in holographic theories

TRANSPORT PROPERTIES: AN EXAMPLE • Example : system whose only conserved charge is the total energy. • Local thermodynamic equilibrium state characterized by slowly- varying energy density: ε ≡ T 00 ( t , x ) ∂ ε ≪ 1 • Equations of motion: j = − D ∇ ε − Γ∇ 3 ε + O ( ∇ 5 ) ∂ t ε + ∇ ⋅ j = 0 ∂ t ε = D ∇ 2 ε + Γ∇ 4 ε + O ( ∇ 6 ) ω = − iDk 2 − i Γ k 4 + O ( k 6 ) or • Hydrodynamics: energy di ff uses over long distances. What sets the values of the transport parameters etc ? D , Γ ,

CHAOTIC PROPERTIES • Chaotic dynamics are seemingly something very di ff erent from transport. C ( t , x ) = − ⟨ [ V ( t , x ), W (0,0)] 2 ⟩ T • In theories with a classical gravity dual, these correlations have the form C ( t , x ) ∼ e τ − 1 L ( t − | x | / v B ) ✴ The timescale is always τ L = (2 π T ) − 1 Shenker, Stanford (1306.0622) ✴ But the “butterfly velocity” depends on the particular theory. v B Roberts, Stanford, Susskind (1409.8180)

MAIN RESULTS • In QFTs with a gravity dual, the transport properties are constrained by v B , τ L • The collective modes that transport energy are characterized by their dispersion relations . ω ( k ) There is always a mode with where . k 2 * = − ( v B τ L ) − 2 ω ( k * ) = i τ L • Under appropriate conditions, the di ff usivity of energy is set by D ∼ v 2 B τ L ( In a normal metal, ) D ∼ v 2 F τ

THE GRAVITATIONAL THEORIES • I will discuss asymptotically AdS d+2 black branes supported by matter fields: ds 2 = − f ( r ) dt 2 + dr 2 f ( r ) + h ( r ) dx 2 d In ingoing co-ordinates ds 2 = − f ( r ) dv 2 + 2 dvdr + h ( r ) dx 2 d − g ( R − Z ( ϕ ) F 2 − 1 S = ∫ d d +2 x 2( ∂ ϕ ) 2 + V ( ϕ ) ) • For definiteness: • Matter fields induce an RG flow from the UV CFT : F vr ( r ) ≠ 0 & ϕ ( r ) ≠ 0 Numerical solution of equations of motion yield , etc. f ( r ) h ( r )

QUASI-NORMAL MODES OF BLACK HOLES • Focus on one aspect of these spacetimes: quasi-normal modes . i.e. solutions to linearized perturbation equations, obeying appropriate BCs ✴ regularity (in ingoing coordinates) at the horizon r = r 0 ✴ normalizability near the AdS boundary r → ∞ • e.g. probe scalar field − g ∂ a δφ ) − m 2 ∂ a ( − g δφ = 0 ✴ 2 independent solutions: and δφ norm ( r , ω , k ) δφ non − norm ( r , ω , k ) ✴ If is regular at the horizon quasi-normal mode. δφ norm • Quasi-normal modes are characterized by their dispersion relations ω ( k )

QUASI-NORMAL MODES OF BLACK HOLES • Collective excitations of the dual QFT are encoded in the quasi-normal modes. quasi-normal modes poles of retarded Green’s ω ( k ) of a field function of dual operator ω ( k ) Horowitz, Hubeny (hep-th/9909056) Son, Starinets (hep-th/0205051) • The spectrum depends in detail on the particular theory, spacetime, field, etc Numerical computation is required even in very simple cases. Im ( ω ) Re ( ω ) e.g. massless scalar field in Schwarzschild-AdS 5 Plots from hep-th/0207133 by A. Starinets k k

HORIZON CONSTRAINTS ON THE SPECTRUM • Certain features of the spectrum depend only on the near-horizon dynamics. Blake, RD, Vegh (1904.12883) see also Kovtun et al (1904.12862) • Example : probe scalar field ∞ ∑ ✴ Ansatz: solution that is regular at the horizon φ n ( r − r 0 ) n δφ ( r ) = n =0 ✴ Solve iteratively for : φ n >0 2 h ( r 0 )(2 π T − i ω ) φ 1 = ( k 2 + m 2 h ( r 0 ) + i ω dh ′ ) φ 0 ( r 0 ) etc. 2 ✴ At both solutions are regular at the horizon ! ( ω 1 , k 1 ) 1 = − ( m 2 h ( r 0 ) + d π Th ′ k 2 ( r 0 ) ) ω 1 = − i 2 π T ,

HORIZON CONSTRAINTS ON THE SPECTRUM • Moving infinitesimally away from yields one regular solution: ( ω 1 , k 1 ) 4 h ( r 0 ) ( 4 ik 1 ( r 0 ) ) φ 1 1 δ k ω = ω 1 + i δω = δω − dh ′ k = k 1 + i δ k φ 0 But this regular solution depends on the arbitrary slope . δ k / δω • Can obtain an arbitrary combination of and by tuning : δ k / δω φ norm φ non − norm φ ingoing ( ω 1 + i δω , k 1 + i δ k ) = ( 1 − v z δω ) φ norm + C ( 1 − v p δω ) φ non − norm δ k δ k For an appropriate choice of slope ( ), there is a quasi-normal mode. δω = v p δ k there must be a dispersion relation obeying ω ( k 1 ) = ω 1

HORIZON CONSTRAINTS ON THE SPECTRUM • This feature of the spectrum is independent of the rest of the spacetime. Near-horizon dynamics yield exact constraints on the dispersion relations ω ( k ) • A more complete analysis of this type yields infinitely many constraints ω = − i 2 π Tn , k = k n n = 1,2,3,… for appropriate values . k n • These points in complex Fourier space are called pole-skipping points . Intersection of a line of poles with a line of zeroes G = C δω − v z δ k in the dual QFT 2-point function δω − v p δ k

POLE-SKIPPING EXAMPLES • The argument can be generalized to other spacetimes e.g. BTZ black hole / CFT 2 at non-zero T Δ = 5/2 pole-skipping point dispersion relation of pole dispersion relation of zero • And it can be generalized to non-scalar fields/operators, e.g. ✴ U(1) Maxwell field: some are real k n ✴ Fermionic fields: frequencies shifted to ω = − i 2 π T ( n + 1/2) Ceplak, Ramdial, Vegh (1910.02975)

CONSTRAINTS ON ENERGY DENSITY MODES • Usually very complicated to determine the collective modes of energy density couples to other metric perturbations and to matter field perturbations δ g vv • But in this case, near-horizon Einstein equations yield a simple constraint k 2 ω ( k * ) = + i 2 π T * = − d π Th ′ ( r 0 ) Independent of the matter field profiles. Blake, RD, Grozdanov, Liu (1809.01169) • Universal constraint on the collective modes of energy density: ω ( k * ) = + i τ − 1 k 2 * = − ( v B τ L ) − 2 L First observed numerically in Schwarzschild-AdS 5 : Grozdanov, Schalm, Scopelliti (1710.00921)

HYDRODYNAMIC INTERPRETATION • An interpretation: chaotic behavior has hydrodynamic origin Blake, Lee, Liu (1801.00010) W V σ σ : hydrodynamic mode of energy conservation V W See also: Gu, Qi, Stanford (1609.07832), Haehl, Rozali (1808.02898),… • Conversely, the chaotic behavior constrains the hydrodynamic parameters of theories with holographic duals.

IMPLICATIONS FOR HYDRODYNAMICS • There is typically a collective mode of energy density with dispersion relation ω hydro ( k ) = − iDk 2 − i Γ k 4 + O ( k 6 ) At long distances, this is the hydrodynamic di ff usion of energy density. constraint on hydrodynamic parameters. ω hydro ( k * ) = + i τ − 1 • L • Make an additional assumption: If di ff usive approximation is good up to ω hydro ( k ) ≈ − iDk 2 ω = i τ − 1 L , k = k * D ≈ v 2 B τ L D ≈ − k − 2 * τ − 1 L