holographic zero sound from spacetime filling branesjuettner/slides/ronnie_rodgers.pdf ·...

Holographic zero sound fromspacetime filling branes

Ronnie RodgersWith Nikola Gushterov and Andy O’Bannon

Based on arXiv:1807.11327

Outline

Background and motivation- Fermi liquids- Holographic zero sound

The modelResultsSummary and outlook

AdS/CMTGauge/gravity duality:

Strongly coupled QFTs⇔Weakly coupled gravity theories

Playground for strongly coupled physics without a quasiparticledescriptionNo quantitative predictions, but one can try to identify universalqualitative phenomena

2

Fermi liquidsSystem of fermions: adiabatically turn on repulsive interactionsLandau theory: effective description of low-energy excitations interms of quasiparticlesFermi liquids in nature:• Helium-3

• Electron sea in metalsUseful reference point for understanding non-Fermi liquids(strange metals)

3

Zero sound in Fermi liquidsδnp(t,x) quasiparticles per unit momentum p

Boltzmann equation:

∂δnp∂t

+ vp · ∇δnp + interactions = collisions

4

Zero sound in Fermi liquidsδnp(t,x) quasiparticles per unit momentum p

Boltzmann equation:

∂δnp∂t

+ vp · ∇δnp + interactions = collisions

Low temperature: neglect collisionsSolution: “zero sound”

ω = ±vk − iΓk2 +O(k3)

Non-isotropic deformation of Fermi surface

4

Properties of zero soundSpeed v ≥ speed of sound vs

Zero sound

First sound

0 5 10 15 20 250.0

0.5

1.0

1.5

2.0

2.5

3.0

5

Properties of zero soundSpeed v ≥ speed of sound vs

Quasiparticle scattering rate: ν ∼ π2T 2 + ω2

µ(1− e−ω/T )

Dial up temperature, attenuation:• Quantum collisionless, T � ω, Γ ∼ T 0

• Thermal collisionless, T 2/µ� ω � T , Γ ∼ T 2

Hydrodynamic sound, ω � T 2/µ,Γ ∼ T−2

Zero sound→ hydrodynamic sound as temperature increases

5

(Zero) sound attenuation

Maximum defines collisionless-to-hydrodynamic crossover 6

(Zero) sound attenuationZero sound attenuation in Helium-3

[Abel, Anderson, Wheatley, Phys. Rev. Lett. 17 (Jul, 1966) 74-78] 7

Holographic zero soundHolographic models with bulk gauge field. Dual field theory:• U(1) global symmetry

• Non-zero chemical potential µ, charge density 〈Jt〉

• Compressible, d〈Jt〉 /dµ 6= 0

Spectrum of collective excitations (quasinormal modes)includes low-temperature longitudinal modes with sound-likedispersion

ω = ±vk − iΓk2 +O(k3)

“Holographic zero sound” (HZS)

Poles in two-point functions of Tµν and Jµ

8

HZS from probe branesProbe Dq-branes with worldvolume ⊃ AdSp+1 factor[Karch, Son, Starinets, 0806.3796; Davison, Starinets, 1109.6343]

ActionS = SEH − Tq

∫dp+2ξ

√−det(g + 2πα′F )

Probe limit GNL2Tq � 1 – no back-reactionNon-zero electric field A0 = A0(z)⇒ chemical potential µ

At T = 0, QNMs

ω = ± k√p− ik2

2pµ+O(k3)

Pole in 〈JJ〉 correlators

9


Attenuation, e.g. p = 2:

-8 -7 -6 -5 -4 -3-11.0

-10.5

-10.0

-9.5

-9.0

-8.5

-8.0

-7.5

10


T > 0

⨯

⨯

⨯

⨯⨯⨯⨯⨯ ⨯ ⨯ ⨯ ⨯ ⨯

⨯⨯⨯⨯

⨯

⨯

⨯

■

■

■

■

■

■

■

■

■

0

Crossover to hydrodynamics when poles collide11

HZS in Einstein-MaxellU(1) gauge field minimally coupled to gravity[Edalati, Jottar, Leigh, 1005.4075; Davison, Kaplis, 1111.0660]

S =1

16πGN

∫dd+1x

√−det g

(R+

d(d− 1)

L2− L2F 2

)AdS-Reissner-Nordstrom solution:Non-zero electric field A0 = A0(z)⇒ chemical potential µLow temperature pole in 〈JJ〉 and 〈TT 〉 of form

ω = ±vk − iΓk2 +O(k3)

Continuously becomes hydrodynamic sound at highertemperatures

12


Attenuation, d = 3

⨯ ⨯ ⨯⨯⨯⨯⨯⨯

⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯

-7 -6 -5 -4 -3 -2 -1 0

-12.6

-12.4

-12.2

-12.0

-11.8

-11.6

Small maximum – crossover?13


⨯⨯

■■

⨯⨯

0

No pole collision14

What is the HZS mode?These systems are not Fermi liquids:Einstein-Maxwell models can have Fermi surface[Liu, McGreevy, Vegh, 0903.2477; Cubrovic, Zaanen, Schalm, 0904.1993]But at T = 0: near horizon AdS2 ⇒ emergent scaling symmetry[Faulkner, Liu, McGreevy, Vegh, 0907.2694]

Probe branes:• No evidence for Fermi surface

• C ∼ T 2p

No symmetry breaking⇒ not (superfluid) phonon

Properties of HZS show significant qualitative differencesbetween the two models – why?What effective theories support zero sound?

15

What is the HZS mode?These systems are not Fermi liquids:Einstein-Maxwell models can have Fermi surface[Liu, McGreevy, Vegh, 0903.2477; Cubrovic, Zaanen, Schalm, 0904.1993]But at T = 0: near horizon AdS2 ⇒ emergent scaling symmetry[Faulkner, Liu, McGreevy, Vegh, 0907.2694]

Probe branes:• No evidence for Fermi surface

• C ∼ T 2p

No symmetry breaking⇒ not (superfluid) phonon

Properties of HZS show significant qualitative differencesbetween the two models – why?

What effective theories support zero sound? 15

Outline



ModelSpacetime filling brane with back-reaction

S =1

16πGN

∫d4x√−det g

(R+

d(d− 1)

L20

)− TD

∫d4x√−det(g + αF )

Admits charged black brane solutions:(2+1)-dimensional boundary CFT at T and µ

17

ModelSpacetime filling brane with back-reaction

S =1

16πGN

∫d4x√−det g

(R+

d(d− 1)

L20

)− TD

∫d4x√−det(g + αF )

Admits charged black brane solutions:(2+1)-dimensional boundary CFT at T and µDefine

L2 =3L2

0

3− 8πGNTDL20

, τ = 8πGNL2TD, α = α/L2

τ ∼ Nf/Nc number of flavours in CFTα measures non-linearity of interactionProbe DBI and Einstein-Maxwell appear as limits

17

PlanStudy the collective excitations in this setup• How does zero sound depend on parameters of the model?

• How do we recover previous regimes

For this talk: α = 1, vary τ

We have also computed spectral functions

18

Outline



Motion of polesτ = 0, α = 1, k/µ = 0.01

⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯⨯⨯⨯

■■

■

■

■

■

■

■

■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

⨯ 20

τ = 10−4

Motion of polesτ = 10−4, α = 1, k/µ = 0.01

⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯

⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯

■■■■■■■■■■■■

■

■

■

■

■

■

■

■

■

■

■

■

■

■

■

■

■

■

■

■

■

■

■■■■■■■■■■■■■■■■■■■■

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

22

τ = 10−3


⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯

■■■■■■■■■

■■▲▲▲▲

▲▲

▲

-1.0 -0.5 0.0 0.5 1.0-2.0

-1.5

-1.0

-0.5

0.0

24


⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯

■■■■

-1.0 -0.5 0.0 0.5 1.0-2.0

-1.5

-1.0

-0.5

0.0

24


⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯

⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯

■■■■■

■

■

■

■

■

■

■

■

■

■

■

■

■

■

■

■■■■■■■■■■■■■■■■■

-1.0 -0.5 0.0 0.5 1.0

-1.5

-1.0

-0.5

0.0

24

τ = 10−2


⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯

■■■■■■■■■■■■■■■■■■■■■■■■■■■■

****************************

▲▲

▲-4 -2 0 2 4

-10

-8

-6

-4

-2

0

26


⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯■■■■■■■■■■■■■■

▲

▲

▲

▲

▲

▲

▲

▲

▲

▲▲▲▲▲▲▲▲▲

-1.0 -0.5 0.0 0.5 1.0-10

-8

-6

-4

-2

0

Closest poles to real axis similar to Einstein-Maxwell26

HZS attenuationα = 1, k/µ = 0.01

◆ ◆ ◆ ◆ ◆◆

◆

◆

◆

◆

◆

◆◆

◆◆

◆

◆

◆

◆

◆

◆

◆

◆

◆

◆

◆

◆

◆

◆

◆

◆

⨯ ⨯

⨯

⨯

⨯

⨯⨯⨯ ⨯

⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯

+ + ++

+++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

■ ■■ ■ ■ ■■

■■■■■■■■

■■■■■■

■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■

* * * ** * *****************************************************************************

▲▲▲▲▲▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲

-8 -6 -4 -2 0

-12

-11

-10

-9

-8

◆

⨯

+

■

▲

* Reissner-Nordström

AdS-Schwarzschild sound

Probe zero sound

Qualitative resemblance to zero sound in Fermi liquids

Temperature scaling quantitatively different (closer for small τ )Maximum shrinks with increasing back-reaction

27

Outline



SummaryBack-reacted spacetime filling branes exhibit a holographiczero sound modeThis mode has qualitative similarities to zero sound in FermiliquidsThe back-reaction parameter τ ∼ Nf/Nc appears to control theappropriate effective theory

29

Outlook

How generic is this mode?• Is it universal in holographic models?• If not, what controls its appearance? Non-zero

spectral weight at zero frequency?• What does zero sound look like in holographic models

of Fermi liquids?

Outside of holography, do low temperaturesound modes exist in non-Fermi liquids?

Thank you!

30

Outlook

How generic is this mode?• Is it universal in holographic models?• If not, what controls its appearance? Non-zero

spectral weight at zero frequency?• What does zero sound look like in holographic models

of Fermi liquids?

Outside of holography, do low temperaturesound modes exist in non-Fermi liquids?

Thank you!30

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