RELATIONS BETWEEN
TRANSPORT & CHAOS IN
HOLOGRAPHIC THEORIESRichard 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
Tstrangemetal

NON-QUASIPARTICLE STATES• Cartoon of a normal metal:✴ electron-like excitations with charge e,mass m, speed vF , lifetime 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-quasiparticlestates.
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 nearhorizon 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 ofT μν and J μat small (ω, k)• Examples: electrical resistivity, thermal resistivity, shear viscosity, diﬀusivityof energy,….

• Transport properties are important experimental observables✴

They are relatively easy to measure✴

They exhibit universality across diﬀerent systems

TRANSPORT PROPERTIES• There are also two theoretical reasons that transport properties are privileged.

(1) The dynamics of T μν and J μ are constrained by symmetries
governed by a simple eﬀective theory over long distances andtimescales: hydrodynamics
For a given QFT, we just need to determine the parameters of theeﬀective theory.

(2) Transport is directly related to the dynamics of the basic gravitationalvariables:

Tμν

⟷ gμν

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:• Equations of motion:ε ≡ T 00(t, x)∂tε + ∇ ⋅ j = 02∂ε ≪ 1j = − D ∇ε − Γ ∇3 ε + O( ∇5 )46

∂tε = D ∇ ε + Γ ∇ ε + O( ∇ )orω = − iDk 2 − iΓk 4 + O(k 6)• Hydrodynamics: energy diﬀuses over long distances.

What sets the values of the transport parameters D, Γ, etc ?

CHAOTIC PROPERTIES• Chaotic dynamics are seemingly something very diﬀerent from transport.

C(t, x) = − ⟨[V(t, x), W(0,0)] ⟩2T• In theories with a classical gravity dual, these correlations have the formC(t, x) ∼ e

τL−1(t − | x | /vB)✴

The timescale is always τL = (2πT )−1✴

But the “butterfly velocity” vB depends on the particular theory.

Shenker, Stanford (1306.0622)

Roberts, Stanford, Susskind (1409.8180)

MAIN RESULTS• In QFTs with a gravity dual, the transport properties are constrained by vB, τL• The collective modes that transport energy are characterized by theirdispersion relations ω(k).

ω(k*) = iτL

There is always a mode withwhere

k*2 = − (vBτL)−2 .

• Under appropriate conditions, the diﬀusivity of energy is set by2
D ∼ vBτL( In a normal metal, D ∼ vF2 τ )THE GRAVITATIONAL THEORIES• I will discuss asymptotically AdSd+2 black branes supported by matter fields:

2drds 2 = − f(r)dt 2 ++ h(r)dx2df(r)In ingoing co-ordinates

ds 2 = − f(r)dv 2 + 2dvdr + h(r)dx2d• For definiteness:

S= d∫

d+2

1x −g R − Z(ϕ)F − (∂ϕ)2 + V(ϕ)()
22• Matter fields induce an RG flow from the UV CFT : Fvr(r) ≠ 0

Numerical solution of equations of motion yield f(r), h(r) etc.

&ϕ(r) ≠ 0

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 = r0✴

normalizability near the AdS boundary r → ∞• e.g.

probe scalar field∂a( −g∂aδφ) − m 2 −gδφ = 0✴

2 independent solutions: δφnorm(r, ω, k)✴

If δφnorm is regular at the horizonand δφnon−norm(r, ω, k)quasi-normal mode.

• 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
ω(k) of a fieldpoles ω(k) of retarded Green’s
function of dual operator
Horowitz, Hubeny (hep-th/9909056)
Son, Starinets (hep-th/0205051)• The spectrum depends indetail on the particular theory, spacetime, field, etc

Numerical computation is required even in very simple cases.
e.g. massless scalar field in
Schwarzschild-AdS5Im(ω)Re(ω)

Plots from hep-th/0207133
by A. Starinets

kk

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

δφ(r) =∞

∑n=0✴

Solve iteratively for φn>0 :dh′(r0)2h(r0)(2πT − iω)φ1 = k + m h(r0) + iω
φ0(2 )
2✴

φn(r − r0)n2

At (ω1, k1) both solutions are regular at the horizon !

ω1 = − i2πT,

k12 = − (m 2h(r0) + dπTh′(r0))

etc.

HORIZON CONSTRAINTS ON THE SPECTRUM• Moving infinitesimally away from (ω1, k1) yields one regular solution:ω = ω1 + iδωk = k1 + iδk

φ11δk=4ik1− dh′(r0))φ0 4h(r0) (δωBut this regular solution depends on the arbitrary slope δk/δω .
• Can obtain an arbitrary combination of φnorm and φnon−norm by tuning δk/δω :δkδkφingoing(ω1 + iδω, k1 + iδk) = 1 − vz
φnorm + C 1 − vp
φnon−norm((δω )δω )For an appropriate choice of slope (δω = vpδk), there is a quasi-normal mode.
there must be a dispersion relation obeying ω(k1) = ω1

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 = − i2πTn,k = knn = 1,2,3,…

for appropriate values kn .

• These points in complex Fourier space are called pole-skipping points.
Intersection of a line of poles with a line of zeroes
in the dual QFT 2-point function

G=Cδω − vzδkδω − vpδk

POLE-SKIPPING EXAMPLES• The argument can be generalized to other spacetimes
e.g. BTZ black hole / CFT2 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 kn are real✴

Fermionic fields: frequencies shifted to ω = − i2πT (n + 1/2)
Ceplak, Ramdial, Vegh (1910.02975)

CONSTRAINTS ON ENERGY DENSITY MODES• Usually very complicated to determine the collective modes of energy density

δgvv couples to other metric perturbations and to matter field perturbations• But in this case, near-horizon Einstein equations yield a simple constraint

ω(k*) = + i2πTIndependent of the matter field profiles.

k*2 = − dπTh′(r0)Blake, RD, Grozdanov, Liu (1809.01169)• Universal constraint on the collective modes of energy density:

ω(k*) = + iτL−1

k*2 = − (vBτL)−2First observed numerically in Schwarzschild-AdS5: Grozdanov, Schalm, Scopelliti (1710.00921)

HYDRODYNAMIC INTERPRETATION• An interpretation: chaotic behavior has hydrodynamic origin

Blake, Lee, Liu(1801.00010)

WVσσ : hydrodynamic mode ofenergy conservation

VW

See also: Gu, Qi, Stanford (1609.07832), Haehl, Rozali (1808.02898),…
• Conversely, the chaotic behavior constrains the hydrodynamic parameters oftheories with holographic duals.

IMPLICATIONS FOR HYDRODYNAMICS• There is typically a collective mode of energy density with dispersion relation

ωhydro(k) = − iDk 2 − iΓk 4 + O(k 6)
At long distances, this is the hydrodynamic diﬀusion of energy density.

−1• ωhydro(k*) = + iτLconstraint on hydrodynamic parameters.

• Make an additional assumption:

If diﬀusive approximation ωhydro(k) ≈ − iDk 2 is good up to ω = iτL−1, k = k*D ≈ − k*−2τL−1

2
D ≈ vBτL

LOW TEMPERATURE DIFFUSIVITIES• Consistent with the diﬀusivity of energy density at low temperatures✴

For a large class of theories with AdS2xRd IR fixed points
as✴

T→0Generic IR fixed point has symmetry t → Λzt,
as T → 0✴

2D = vBτLz
D=vB2τL2(z − 1)

Blake, Donos(1611.09380)

x → ΛxBlake, RD, Sachdev(1705.07896)

When z = 1, diﬀusive approximation breaks down at ω ≪ τL−1.

RD, Gentle, Goutéraux
(1808.05659)

SUMMARY• Near-horizon dynamics yield exact constraints on the dispersion relations ofcollective modes.

• There is a universal constraint for collective modes of energy density

ω(k*) = + iτL−1

k*2 = − (vBτL)−2• Under appropriate conditions, this constrains the diﬀusivity of energy density2
D ∼ vBτLi.e. transport is related to underlying chaotic properties.

OPEN QUESTIONS• How robust is the universal constraint on the energy density collective modes?

More direct evidence of the chaos/hydrodynamics link?
Grozdanov (1811.09641), Ahn et al (1907.08030, 2006.00974), Natsuume, Okamura(1909.09168), Abbasi & Tabatabaei (1910.13696), Liu, Raju (2005.08508), …
• Regime of validity of (diﬀusive) hydrodynamics in holographic theories?

Withers (1803.08058), Kovtun et al (1904.01018), …
• What generalizes to other (non-holographic) strongly interacting systems?

Gu, Qi, Stanford (1609.07832), Patel, Sachdev (1611.00003), …
• Precise restrictions on transport parameters from near-horizon constraints.

Grozdanov (2008.00888)• Pole-skipping points in more general spacetimes / field theories ?

Ahn et al (2006.00974)

THANK YOU!