Post on 09-Apr-2020
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
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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
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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
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
HZS from probe branesProbe Dq-branes with worldvolume ⊃ AdSp+1 factor[Karch, Son, Starinets, 0806.3796; Davison, Starinets, 1109.6343]
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
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HZS from probe branesProbe Dq-branes with worldvolume ⊃ AdSp+1 factor[Karch, Son, Starinets, 0806.3796; Davison, Starinets, 1109.6343]
T > 0
⨯
⨯
⨯
⨯⨯⨯⨯⨯ ⨯ ⨯ ⨯ ⨯ ⨯
⨯⨯⨯⨯
⨯
⨯
⨯
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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
HZS in Einstein-MaxellU(1) gauge field minimally coupled to gravity[Edalati, Jottar, Leigh, 1005.4075; Davison, Kaplis, 1111.0660]
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
HZS in Einstein-MaxellU(1) gauge field minimally coupled to gravity[Edalati, Jottar, Leigh, 1005.4075; Davison, Kaplis, 1111.0660]
⨯⨯
■■
⨯⨯
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
Background and motivation- Fermi liquids- Holographic zero sound
The modelResultsSummary and outlook
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
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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
Background and motivation- Fermi liquids- Holographic zero sound
The modelResultsSummary and outlook
Motion of polesτ = 0, α = 1, k/µ = 0.01
⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯⨯⨯⨯
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⨯ 20
τ = 10−4
Motion of polesτ = 10−4, α = 1, k/µ = 0.01
⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯
⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯
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-1.0 -0.5 0.0 0.5 1.0
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τ = 10−3
Motion of polesτ = 10−3, α = 1, k/µ = 0.01
⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯
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■■▲▲▲▲
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▲
-1.0 -0.5 0.0 0.5 1.0-2.0
-1.5
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-0.5
0.0
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Motion of polesτ = 10−3, α = 1, k/µ = 0.01
⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯
■■■■
-1.0 -0.5 0.0 0.5 1.0-2.0
-1.5
-1.0
-0.5
0.0
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Motion of polesτ = 10−3, α = 1, k/µ = 0.01
⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯
⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯
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-1.0 -0.5 0.0 0.5 1.0
-1.5
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0.0
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τ = 10−2
Motion of polesτ = 10−2, α = 1, k/µ = 0.01
⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯
■■■■■■■■■■■■■■■■■■■■■■■■■■■■
****************************
▲▲
▲-4 -2 0 2 4
-10
-8
-6
-4
-2
0
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Motion of polesτ = 10−2, α = 1, k/µ = 0.01
⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯⨯ ⨯■■■■■■■■■■■■■■
▲
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▲▲▲▲▲▲▲▲▲
-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
◆ ◆ ◆ ◆ ◆◆
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◆
⨯ ⨯
⨯
⨯
⨯
⨯⨯⨯ ⨯
⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯
+ + ++
+++++++
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
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* * * ** * *****************************************************************************
▲▲▲▲▲▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲
-8 -6 -4 -2 0
-12
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◆
⨯
+
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▲
* 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
Background and motivation- Fermi liquids- Holographic zero sound
The modelResultsSummary and outlook
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