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

RELATIONS BETWEEN TRANSPORT & CHAOS IN HOLOGRAPHIC THEORIES Richard Davison Heriot-Watt University HoloTube, May 2020

OVERVIEW • In some QFTs, there are connections between transport properties and underlying chaotic dynamics. • First motivated by studying a particular transport process: di ff usion of a U(1) charge. Blake Its relation to underlying chaotic dynamics turns out to not be very robust. • I will describe subsequent work identifying more robust relations between transport and chaos. I will focus on QFTs with a gravity description.

MOTIVATION • Transport properties characterize the dynamics of a system’s conserved charges (e.g. energy, momentum, charge, etc.) U (1) • Transport properties are experimentally important ✴ Easy to measure ✴ Exhibit universality in interesting materials • Often governed by properties of underlying quasiparticle degrees of freedom. • What about in systems with no quasiparticle description?

TRANSPORT PROPERTIES • Objects of interest: retarded Green’s functions of conserved charge densities e.g. : change in energy density due to a small source ⟨ ε ⟩ G εε ( ω , k ) • Green’s function poles dispersion relations of collective modes ⟷ ω ( k ) • Can identify some general features, even in absence of quasiparticles ω ✴ When the system is in local equilibrium, ? ? transport is governed by simple e ff ective theories: hydrodynamics . ω eq ✴ In this regime, transport is dominated ? ω hydro ( k ) by a handful of gapless modes ω hydro ( k ) k k eq

HYDRODYNAMICS • Example : system whose only conserved charge is its total energy ✴ Local thermal equilibrium state characterized by slowly-varying ⟶ ε ( x , t ) ✴ Dynamics of this variable are constrained by symmetries: ∂ t ε = D ∇ 2 ε + Γ∇ 4 ε + O ( ∇ 6 ) ⟶ ω hydro ( k ) = − iDk 2 − i Γ k 4 + O ( k 6 ) or Energy di ff uses over long distances. • The values of the parameters of the e ff ective theory ( etc) depend on the D , Γ details of the particular system. In a Fermi liquid, . D ∼ v 2 F τ qp

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 ✴ But the “butterfly velocity” depends on the particular theory. v B Roberts, Stanford, Susskind ✴ In the gravity description, governed by near-horizon physics.

SUMMARY OF RESULTS • In QFTs with a gravity dual, the transport properties are constrained by v B , τ L Im ( ω ) • There is a collective mode transporting energy whose dispersion relation obeys where . k 2 * = − ( v B τ L ) − 2 ω ( k * ) = i τ L ( k * , ω * ) Blake, RD, Grozdanov, Liu see also: Grozdanov, Schalm, Scopelliti Im ( k ) • In the limit of low temperatures, there is di ff usive transport of energy with D ∼ v 2 B τ L

INTERPRETATION • Consistent with proposal that chaotic behavior has hydrodynamic origin W V Blake, Lee, Liu σ σ : hydrodynamic mode of energy conservation W V • If it is a hydro mode that satisfies for * = − ( v B τ L ) − 2 k 2 ω hydro ( k * ) = i τ L And this mode is approximately di ff usive up to ω hydro ( k ) ∼ − iDk 2 k = k * D ∼ v 2 B τ L

THE GRAVITATIONAL THEORIES • I will discuss asymptotically AdS d+2 black branes supported by matter fields: ds 2 = − f ( r ) dv 2 + 2 dvdr + h ( r ) dx 2 d Arising as classical solutions of − g ( R − Z ( ϕ ) F 2 − 1 S = ∫ d d +2 x ) d 2 2( ∂ ϕ ) 2 + V ( ϕ ) − Y ( ϕ ) ∑ ( ∂ χ i ) i =1 • Matter fields induce an RG flow from the UV CFT : χ i = mx i ϕ ( r ) ≠ 0 F vr ( r ) ≠ 0 • Broad family of solutions with di ff erent symmetries and hydrodynamics.

GREEN’S FUNCTIONS FROM GRAVITY • Simplest case: scalar operator. − g ∂ a φ ) − m 2 ∂ a ( − g φ = 0 • Each Fourier mode has two independent solutions: and φ norm ( r , ω , k ) φ non − norm ( r , ω , k ) • Find the linear combination that is regular at the r horizon : r = r 0 r 0 φ ingoing = a ( ω , k ) φ norm + b ( ω , k ) φ non − norm G ( ω , k ) = b ( ω , k ) • The QFT retarded Green's function is a ( ω , k )

GREEN’S FUNCTIONS FROM GRAVITY • Green’s functions of conserved charges (e.g. ) depend in detail on the G εε metric and matter field profiles throughout the spacetime. i.e. on many specific details of the particular QFT state. • But there are two situations when only near-horizon dynamics is important d , limit where radial evolution is simple: dr ( . . . φ ′ ( r ) ) = 0 ω → 0 k → 0 ✴ ✴ Points in Fourier space where the ingoing solution is not unique ( ω * , k * ) features of the Green’s functions that are insensitive to many details of the state

HORIZON CONSTRAINTS ON THE SPECTRUM • Identifying points where the ingoing solution is not unique ( ω * , k * ) exact constraints on the spectrum of collective modes ω ( k ) • Example : probe scalar field Blake, RD, Vegh ; see also Grozdanov et al ∞ ∑ ✴ Ansatz: solution that is regular at the horizon φ n ( r − r 0 ) n φ ( 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 * , k * ) * = − ( m 2 h ( r 0 ) + d π Th ′ ( r 0 ) ) k 2 ω * = − i 2 π T ,

HORIZON CONSTRAINTS ON THE SPECTRUM • Moving infinitesimally away from yields one regular solution: ( ω * , k * ) 4 h ( r 0 ) ( 4 ik * ( r 0 ) ) φ 1 δ k ω = ω * + i δω 1 δω − dh ′ = k = k * + 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 : φ norm φ non − norm δ k / δω φ ingoing ( ω * + i δω , k * + i δ k ) = C ( 1 − v z δω ) φ norm + ( 1 − v p δω ) φ non − norm δ k δ k G ( ω * + i δω , k * + i δ k ) = C δω − v z δ k δω − v p δ k there must be a pole with dispersion relation obeying ω ( k * ) = ω *

HORIZON CONSTRAINTS ON THE SPECTRUM • This constraint on is obtained only from the near-horizon dynamics. ω ( k ) a part of the spectrum that is independent of the rest of the spacetime • A more thorough analysis yields infinitely many constraints of this kind e.g. �� ( � ) a collective mode of the momentum � � � � density operator (Schwarzschild-AdS 4 ) ��� ��� ��� ��� ��� ��� ��� � � - ��� � - ��� exact (numerical) dispersion relation ω ( k ) - ��� near-horizon constraints � - ��� - ��� � - ��� • This argument can be generalized to any type of field. e.g. Ceplak, Ramdial, Vegh

CONSTRAINTS ON ENERGY DENSITY MODES • For , the equations are seemingly more complicated. G εε ( ω , k ) couples to other metric perturbations and to matter field perturbations δ g vv e.g. ( r 0 ) + 2 k 2 ) δ g vv ( r 0 ) − 2 (2 π T + i ω ) ( ωδ g x i x i ( r 0 ) + 2 k δ g vx ( r 0 ) ) = 4 h ( r 0 ) ( δ T vv ( r 0 ) − T vr ( r 0 ) δ g vv ( r 0 ) ) ( − i ω dh ′ • But the constraint is very simple, and independent of matter field profiles k 2 * = − d π Th ′ ω ( k * ) = + i 2 π T ( r 0 ) ω ( k * ) = + i τ − 1 * = − ( v B τ L ) − 2 k 2 L • Robust to some further generalizations e.g. higher-derivative gravity, magnetic fields & anomalies Grozdanov ; Abbasi, Tabatabaei

RELATION TO DIFFUSIVITY • It is reasonable to expect one of the hydro ω modes to obey the universal constraint. hydro mode In a few cases, this has been verified. Grozdanov, Schalm, Scopelliti ; Blake, RD, Grozdanov, Liu non-hydro modes “linear axion” theory at low T Im ( ω ) • In some cases, is ω hydro ( k ) ≈ = − iDk 2 ω = − iDk 2 2 π T ��� an excellent approximation up to k = k * exact ω ( k ) ��� D ≈ v 2 B τ L ( k * , ω * ) ��� ��� ��� ��� ��� ��� ��� v B 2 π T Im ( k )

THERMAL DIFFUSIVITY • dc conductivities are sensitive only to the near-horizon part of the spacetime Thermal conductivity is independent of the matter field profiles κ ( r ) h ( r ) d − 2 κ ≡ κ − α 2 T f ′ = 4 π Blake, RD, Sachdev d σ dr ( f ′ ( r ) h ( r ) d /2 − 1 ) r = r 0 • At low (near IR fixed point), heat capacity is set by the horizon area. T c Also independent of matter field profiles. • In this limit, there is a collective mode of energy density with di ff usivity D = κ c

LOW TEMPERATURE THERMAL DIFFUSIVITY • Quantitative relations between di ff usive transport and chaos ✴ For a large class of theories with AdS 2 xR d IR fixed points D = v 2 as B τ L Blake, Donos T → 0 ✴ Generic IR fixed point has symmetry t → Λ z t , x → Λ x z 2( z − 1) v 2 as D = B τ L T → 0 Blake, RD, Sachdev ✴ When , di ff usive approximation breaks down at . ω ≪ τ − 1 z = 1 L RD, Gentle, Goutéraux