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

• In some QFTs, there are connections between transport properties and

underlying chaotic dynamics.

• First motivated by studying a particular transport process: diffusion of a U(1)

charge. 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.

OVERVIEW

Blake

MOTIVATION

• Transport properties characterize the dynamics of a system’s conserved

charges (e.g. energy, momentum, charge, etc.)

• 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?

U(1)

• Objects of interest: retarded Green’s functions of conserved charge densities

e.g. : change in energy density due to a small source

• Green’s function poles

dispersion relations

• f collective modes
• Can identify some general features, even in absence of quasiparticles

✴ When the system is in local equilibrium,

transport is governed by simple effective theories: hydrodynamics.

✴ In this regime, transport is dominated

by a handful of gapless modes

Gεε(ω, k) ⟨ε⟩ ⟷ ω(k) ωhydro(k)

TRANSPORT PROPERTIES

ω k

ωhydro(k) ωeq

keq

? ? ?

HYDRODYNAMICS

• Example: system whose only conserved charge is its total energy

✴ Local thermal equilibrium

state characterized by slowly-varying

✴ Dynamics of this variable are constrained by symmetries:

• r

Energy diffuses over long distances.

• The values of the parameters of the effective theory (

etc) depend on the details of the particular system. In a Fermi liquid, .

⟶ ε(x, t) ⟶ ∂tε = D∇2ε + Γ∇4ε + O(∇6) ωhydro(k) = − iDk2 − iΓk4 + O(k6) D, Γ D ∼ v2

Fτqp

CHAOTIC PROPERTIES

• Chaotic dynamics are seemingly something very different from transport.
• In theories with a classical gravity dual, these correlations have the form

✴ The timescale is always ✴ But the “butterfly velocity”

depends on the particular theory.

✴ In the gravity description, governed by near-horizon physics.

τL = (2πT)−1 vB

C(t, x) = − ⟨[V(t, x), W(0,0)]2⟩T C(t, x) ∼ eτ−1

L (t − |x|/vB)

Shenker, Stanford Roberts, Stanford, Susskind

SUMMARY OF RESULTS

• In QFTs with a gravity dual, the transport properties are constrained by
• There is a collective mode transporting

energy whose dispersion relation obeys where .

• In the limit of low temperatures, there is diffusive transport of energy with

vB, τL ω(k*) = iτL k2

* = − (vBτL)−2

Blake, RD, Grozdanov, Liu

D ∼ v2

BτL

Im(ω) Im(k)

(k*, ω*)

see also: Grozdanov, Schalm, Scopelliti

• Consistent with proposal that chaotic behavior has hydrodynamic origin
• If it is a hydro mode that satisfies

for And this mode is approximately diffusive up to

ωhydro(k*) = iτL k2

* = − (vBτL)−2

ωhydro(k) ∼ − iDk2 k = k*

INTERPRETATION

σ

V V

W W

σ

hydrodynamic mode of energy conservation :

Blake, Lee, Liu

D ∼ v2

BτL

• I will discuss asymptotically AdSd+2 black branes supported by matter fields:

Arising as classical solutions of

• Matter fields induce an RG flow from the UV CFT :
• Broad family of solutions with different symmetries and hydrodynamics.

ds2 = − f(r)dv2 + 2dvdr + h(r)dx2

d

S = ∫ dd+2x −g (R − Z(ϕ)F2 − 1 2(∂ϕ)2 + V(ϕ) − Y(ϕ)

d

∑

i=1

(∂χi)

2

)

ϕ(r) ≠ 0 Fvr(r) ≠ 0 χi = mxi

THE GRAVITATIONAL THEORIES

GREEN’S FUNCTIONS FROM GRAVITY

• Simplest case: scalar operator.
• Each Fourier mode has two independent solutions:

and

• Find the linear combination that is regular at the

horizon :

• The QFT retarded Green's function is

∂a( −g∂aφ) − m2 −gφ = 0 φnorm(r, ω, k) φnon−norm(r, ω, k) r = r0 φingoing = a(ω, k)φnorm + b(ω, k)φnon−norm G(ω, k) = b(ω, k) a(ω, k)

r0 r

• Green’s functions of conserved charges (e.g.

) depend in detail on the 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

✴

, limit where radial evolution is simple:

✴ Points in Fourier space

where the ingoing solution is not unique features of the Green’s functions that are insensitive to many details of the state

Gεε ω → 0 k → 0 d dr ( . . . φ′ (r)) = 0 (ω*, k*)

GREEN’S FUNCTIONS FROM GRAVITY

HORIZON CONSTRAINTS ON THE SPECTRUM

• Identifying points

where the ingoing solution is not unique exact constraints on the spectrum

• f collective modes
• Example: probe scalar field

✴ Ansatz: solution that is regular at the horizon ✴ Solve iteratively for

: etc.

✴ At

both solutions are regular at the horizon

(ω*, k*) ω(k) φ(r) =

∞

∑

n=0

φn(r − r0)n φn>0 2h(r0)(2πT − iω)φ1 = (k2 + m2h(r0) + iω dh′ (r0) 2 ) φ0 (ω*, k*) ω* = − i2πT, k2

* = − (m2h(r0) + dπTh′

(r0))

Blake, RD, Vegh ; see also Grozdanov et al

• Moving infinitesimally away from

yields one regular solution: But this regular solution depends on the arbitrary slope .

• Can obtain an arbitrary combination of

and by tuning : there must be a pole with dispersion relation obeying

(ω*, k*) δk/δω φnorm φnon−norm δk/δω φingoing(ω* + iδω, k* + iδk) = C (1 − vz δk δω) φnorm + (1 − vp δk δω ) φnon−norm G(ω* + iδω, k* + iδk) = C δω − vzδk δω − vpδk ω(k*) = ω*

HORIZON CONSTRAINTS ON THE SPECTRUM

ω = ω* + iδω k = k* + iδk φ1 φ0 = 1 4h(r0) (4ik* δk δω − dh′ (r0))

• This constraint on

is obtained only from the near-horizon dynamics. 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.

• This argument can be generalized to any type of field.

ω(k)

HORIZON CONSTRAINTS ON THE SPECTRUM

e.g. Ceplak, Ramdial, Vegh a collective mode of the momentum density operator (Schwarzschild-AdS4)

• ()

exact (numerical) dispersion relation ω(k) near-horizon constraints

• For

, the equations are seemingly more complicated. couples to other metric perturbations and to matter field perturbations e.g.

• But the constraint is very simple, and independent of matter field profiles
• Robust to some further generalizations

e.g. higher-derivative gravity, magnetic fields & anomalies

Gεε(ω, k) δgvv

(−iωdh′ (r0) + 2k2) δgvv(r0) − 2 (2πT + iω) (ωδgxixi(r0) + 2kδgvx(r0)) = 4h(r0)(δTvv(r0) − Tvr(r0)δgvv(r0))

ω(k*) = + i2πT k2

* = − dπTh′

(r0) ω(k*) = + iτ−1

L

k2

* = − (vBτL)−2

CONSTRAINTS ON ENERGY DENSITY MODES

Grozdanov ; Abbasi, Tabatabaei

• It is reasonable to expect one of the hydro

modes to obey the universal constraint. In a few cases, this has been verified.

• In some cases,

is an excellent approximation up to

ωhydro(k) ≈ = − iDk2 k = k*

RELATION TO DIFFUSIVITY

Grozdanov, Schalm, Scopelliti ; Blake, RD, Grozdanov, Liu

D ≈ v2

BτL

(k*, ω*)

Im(ω) 2πT

• vB

2πT Im(k) ω = − iDk2 exact ω(k)

“linear axion” theory at low T

ω

non-hydro modes hydro mode

• dc conductivities are sensitive only to the near-horizon part of the spacetime

Thermal conductivity is independent of the matter field profiles

• At low (near IR fixed point), heat capacity is set by the horizon area.

Also independent of matter field profiles.

• In this limit, there is a collective mode of energy density with diffusivity

κ T c

THERMAL DIFFUSIVITY

D = κ c

κ ≡ κ − α2T σ = 4π f′ (r)h(r)d−2

d dr (f′

(r)h(r)d/2−1)

r=r0 Blake, RD, Sachdev

• Quantitative relations between diffusive transport and chaos

✴ For a large class of theories with AdS2xRd IR fixed points ✴ Generic IR fixed point has symmetry ✴ When

, diffusive approximation breaks down at .

t → Λzt, x → Λx z = 1 ω ≪ τ−1

L

LOW TEMPERATURE THERMAL DIFFUSIVITY

T → 0

as

Blake, RD, Sachdev

D = v2

BτL

T → 0

as

Blake, Donos

D = z 2(z − 1) v2

BτL

RD, Gentle, Goutéraux

• In some QFTs, there are connections between transport properties and

underlying chaotic dynamics.

• Near-horizon dynamics yield exact constraints on the dispersion relations of

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

• At low temperatures, there is a diffusive mode carrying energy with diffusivity

SUMMARY

D ∼ v2

BτL

ω(k*) = + iτ−1

L

k2

* = − (vBτL)−2

• Hydrodynamics and chaos

✴ Effective action of holographic theories ✴ Generalisations outside holography ✴ Regime of validity of (diffusive) hydro in holographic theories?

• Exact “pole-skipping” constraints from near-horizon dynamics

✴ Necessary conditions in a QFT? ✴ Robustness of universal constraint at ✴ Constraints on other transport coefficients?

ω = + i2πT

OPEN QUESTIONS

e.g. Gu, Qi, Stanford ; Patel, Sachdev ; Gu, Lucas, Qi ; Grozdanov, Schalm, Scopelliti ; … e.g. Hartman, Hartnoll, Mahajan ; Lucas ; Withers ; Grozdanov et al ; ….

THANK YOU!