Introduction
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Sakai-Sugimoto
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LFLT
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Hol. 0-sound
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Holographic zero sound at non-zero temperature
and baryon density in the Sakai-Sugimoto model
Matthias Ihl
Centro de Fı́sica do Porto
Faculdade de Ciências da Universidade do Porto
Vienna Theory Lunch Seminar, 11. 03. 2014
Holographic zero sound in the S-S model
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Results
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Introduction
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Sakai-Sugimoto
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LFLT
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Hol. 0-sound
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Outline of the talk
Based on recent paper: B. DiNunno, M. Ihl, N. Jokela, J. Pedraza, arXiv:1403.1827
Goal: Explaining the title.
Holographic zero sound at non-zero temperature and baryon
density in the Sakai-Sugimoto model
I
What is holography?
I
Introduction to the Sakai-Sugimoto model.
I
Landau-Fermi liquids, zero sound
I
Zero sound in Sakai-Sugimoto at non-zero T and µ
I
Results
Holographic zero sound in the S-S model
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Results
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Introduction
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Sakai-Sugimoto
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LFLT
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Hol. 0-sound
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What is holography?
I
I
I
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Holography refers to a duality between a string theory (in the
bulk) and a field theory (on the boundary) .
Compare to hologram: 3D information stored in 2D picture.
Original example: Maldacena duality (conjecture). N = 4
SYM in 3 + 1 dim. is dual to type IIB strings on AdS5 .
Strong-weak coupling duality.
Holographic zero sound in the S-S model
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Results
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Introduction
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Sakai-Sugimoto
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LFLT
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Hol. 0-sound
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What is holography? AdS/CFT and AdS/CMT
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AdS/CFT: Classical (weakly-coupled) theory of gravity e.g.
IIB SUGRA on AdS5 × S 5 ⇔ Strongly-coupled field theory
e.g. N = 4 SU(N) SYM.
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Allows us to compute strongly-coupled field theory properties
from gravity
AdS/CFT Dictionary: Field theory ⇔ Gravity
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Operators O, e.g. T µν , J µ (global sym.) ⇔
Fields φ, e.g. gµν , Aµ
Expectation values ⟨O⟩ ⇔ limr →∞ φ = J +
Equilibrium state ⇔ Solutions to EOMs
⟨O⟩
r
+ ...
AdS/CMT: Application to Condensed Matter Theory, such as
high-Tc superconductivity, strange metals, non-Fermi liquids,
graphene etc.
Holographic zero sound in the S-S model
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Results
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Introduction
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Sakai-Sugimoto
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LFLT
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Hol. 0-sound
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Towards (large Nc ) QCD
I
Original Maldacena duality: N = 4 SYM, highly symmetric.
I
Break some (or all) of SUSY and conformal sym. e.g.,
introduce non-zero temperature.
I
Add flavor degrees of freedom =⇒ Fundamental matter,
quarks, mesons.
Probe limit: Nf ≪ Nc flavor branes, no backreaction on
geometry, “quenched” approx. [Karch, Katz hep-th/0205236]
I
Semi-realistic models of chiral symmetry breaking,
deconfinement transition: e.g., Sakai-Sugimoto, [hep-th/0412141]
Holographic zero sound in the S-S model
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Results
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Introduction
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Sakai-Sugimoto
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LFLT
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Hol. 0-sound
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Sakai-Sugimoto model of holographic QCD
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Introduction [Aharony, Sonnenschein, Yankielowicz hep-th/0604161]
String theory dual to a field theory with chiral symmetry
breaking at T = 0.
IIA string theory on circle x 4 , Nc D4-branes filling
01234-directions, Nf D8-branes at x4 = 0, Nf D8-branes at
du
x4 = L, dx
= 0 at u = u0 , U-shaped embedding x4 (u).
4
Global chiral symmetry visible as gauge theory on
D8 − D8-branes.
Massless spectrum of 3+1 U(Nc ) gauge theory coupled to Nf
massless fermions.
Holographic zero sound in the S-S model
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Results
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Introduction
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Sakai-Sugimoto
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LFLT
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Hol. 0-sound
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Sakai-Sugimoto model of holographic QCD
Finite temp.: confinement/deconfinement transition
Holographic zero sound in the S-S model
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Results
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Introduction
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Sakai-Sugimoto
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LFLT
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Hol. 0-sound
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Sakai-Sugimoto model of holographic QCD
Phase diagram
Holographic zero sound in the S-S model
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Results
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Introduction
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Sakai-Sugimoto
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LFLT
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Hol. 0-sound
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Landau’s Fermi liquid theory and zero sound
I
(non-interacting fermions: ground state is a filled Fermi
sphere.)
I
with interactions: ground state still filled Fermi sphere of
degenerate fermionic quasiparticles, provided T ≪ µ.
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Quasiparticles interact over short distances ⇒ excitations
more complicated.
I
Also: ω, k ≪ µ, excitations remain close to Fermi surface.
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T=0: If quasiparticle interaction repulsive, get bosonic,
longitudinal, gapless collective excitation corresponding to
shape oscillations of the Fermi surface. This is the zero sound
mode.
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Can be characterised by dispersion relation:
ωT =0 (k) = ωTR =0 (k) − iΓT =0 (k).
Holographic zero sound in the S-S model
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Results
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Introduction
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Sakai-Sugimoto
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LFLT
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Hol. 0-sound
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Landau’s Fermi liquid theory and zero sound
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What happens at T ̸= 0?
Thermal excitations become important for ω ∼ T ,
2
thermal collisions dominate for ω ∼ Tµ .
⇒ LFLT predicts 3 different regimes: collisionless quantum
(A), collisionless thermal (B) and hydrodynamic (C).
Behaviour of damping rate ΓT (k) different in each regime:
I
(A): ΓT (k) = Γ0 (k) ∼
I
(B): ΓT (k) = Γ0 (k) +
( )2
(C): ΓT (k) ∼ µ Tω .
I
I
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k2
µ , independent
2
c Tµ .
of T .
Has been experimentally observed in liquid Helium-3.
Question: What are the low energy properties of holographic
quantum liquids (field th. states at large density of matter).
How does our holographic quantum liquid at strong coupling
behave compared to the (weakly-coupled) LFLT?
Similarities/differences?
Holographic zero sound in the S-S model
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Results
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Introduction
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Sakai-Sugimoto
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LFLT
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Hol. 0-sound
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What do we want to compute?
I
Bosonic excitations are encoded in retarded Green’s functions
of bosonic operators, GR (ω, k).
Dominant poles yield dispersion relation ω(k), and correspond
to field theory excitations. (Smaller imaginary part means
longer-lived modes).
I
Can compute these from gravity by fluctuating the relevant
fields around their background value and solving their
equations of motion.
I
We will be mostly interested in (longitudinal) massless
excitations coupling to the (charge) density operator which
correspond to zero sound (and diffusion).
Holographic zero sound in the S-S model
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Results
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Introduction
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Sakai-Sugimoto
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LFLT
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Hol. 0-sound
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Results
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Holographic zero sound in the Sakai-Sugimoto model.
I
Introduce baryon density/chemical potential by turning on a
flux Ftu ̸= 0 on the world volume of the D8-branes (Au = 0):
)
(
∫ ∞
d
1 3 13
d2
du ∂u At =
µ = At (u → ∞) =
F
, , ,− 5 .
3/2 2 1 2 10 10
uH
uH
3πuH
I
Want to study the (longitudinal) massless excitation coupled
to the density operator. This requires analyzing the linearized
e.o.m.s that follow from the quadratic DBI action describing
the fluctuations of the gauge fields living on the
D8/D8-branes.
Expand (kµ = (−ω, k, 0, 0))
∫
d 4 k ikµ x µ
µ
e
at (kµ , u),
At (x , u) = At (u) +
(2π)4
∫
d 4 k ikµ x µ
Ax (x µ , u) =
e
ax (kµ , u),
(2π)4
I
Holographic zero sound in the S-S model
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Introduction
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Sakai-Sugimoto
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LFLT
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Hol. 0-sound
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Results
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Holographic zero sound in the Sakai-Sugimoto model.
I
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DBI action yields 2 equations for at and ax and one constraint
equation.
Introduce gauge invariant quantity
E (kµ , u) := kat (kµ , u) + ωax (kµ , u)
Resulting equation for E (u):
)
(
15u 4
ω 2 G ′ (u)
5
+ 2
E ′ (u)
E ′′ (u) + − + 2
u 2g (u) k − ω 2 G (u)
(
(
))
R 3 u 2 k 2 − ω 2 G (u)
−
E (u) = 0,
g 2 (u)f (u)
where f (u) = 1 −
with TH =
3uH 1/2
4πR 3/2
Holographic zero sound in the S-S model
13/24
( uH )3
u
, g (u) :=
and µ
b=
d 1/5
.
2R 3/2
√
d 2 + u 5 , G (u) =
g 2 (u)
,
u 5 f (u)
Introduction
....
Sakai-Sugimoto
...
LFLT
..
Hol. 0-sound
....
Holographic zero sound in the √
Sakai-Sugimoto model.
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Change of coordinates: y = 2
rescaling y 7→ yH ye,
eµ
k
yH
kµ 7→
(dimensionless).
Asymptotic behavior: near boundary, 2 local solutions:
y ) + BZ II (e
y ), with exponents 0 and 3.
E (e
y ) = AZ I (e
at horizon, choose incoming-wave solution:
E (e
y ) = (1 − ye)−i ω̃/6 R(e
y ), where R(e
y ) regular at horizon.
General prescription to find retarded Green’s functions: Derive
boundary action twice w.r.t. source E (±k, 0) = A(±k).
9/2
2
2
B
Result: GRxx (ω, k) = 6NTy R3 k 2ω−ω2 A
= ωk 2 GRtt (ω, k).
H
I
R3
u ,
Spectral functions: χ(ω, k) = −2Im (GR (ω, k)) .
Need to numerically study quasi normal modes of the E (u)
equation above; precisely those solutions satisfying
incoming-wave bdy. cond. at horizon and vanish at boundary
=⇒ poles of GR (ω, k).
Holographic zero sound in the S-S model
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Results
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Introduction
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Sakai-Sugimoto
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LFLT
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Hol. 0-sound
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Quasinormal modes and dispersion relation.
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Obtain dispersion relation ω(k) from the dominant poles of
the correlation functions (for fixed parameters T and µ,and a
given k).
I
Zero temperature result: [Kulaxizi, Parnachev arXiv:0811.2262]
√
2
( )
−1+2γ+ln k
π
ω(k)
= ± 25 k + 103/2 µ̂2 10 k 3 − i 103/2
k3 + O k5 .
µ̂2
I
Instability of the whole system (from transveral mode sector
coupling to Chern-Simons term): for high densities, i.e.
µ
e ≈ 3.714 and above, the model becomes unstable towards
formation of baryon charge density waves (spatially modulated
phases).
TH =0
Holographic zero sound in the S-S model
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Results
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Introduction
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Sakai-Sugimoto
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LFLT
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Hol. 0-sound
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Results
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Quasinormal modes and dispersion relation.
Ž
Ω
Non-zero temperatures (increasing)
0.01
0.00
- 0.01
0.05
0.10
0.15
Ž
k
0.20
- 0.02
- 0.03
- 0.04
- 0.05
- 0.06
Figure: Real (red dashed curve) and imaginary (blue and black solid
curve) parts of the dispersion relation ω̃(k̃) for µ̃ = 5. The black solid
curve indicates the more stable, longer-lived solution, with less negative
imaginary part.
Holographic zero sound in the S-S model
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Introduction
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Sakai-Sugimoto
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LFLT
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Hol. 0-sound
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Results
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Quasinormal modes and dispersion relation.
Ž
Ω
Non-zero temperatures (increasing)
0.5
0.0
- 0.5
0.5
1.0
1.5
2.0
2.5
3.0
Ž
k
- 1.0
- 1.5
- 2.0
- 2.5
- 3.0
Figure: Real (red dashed curve) and imaginary (blue and black solid
curve) parts of the dispersion relation ω̃(k̃) for µ̃ = 1.2. The black solid
curve indicates the more stable, longer-lived solution, with less negative
imaginary part.
Holographic zero sound in the S-S model
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Introduction
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Sakai-Sugimoto
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LFLT
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Hol. 0-sound
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Results
.......
Quasinormal modes and dispersion relation.
Ž
Ω
Non-zero temperatures (increasing)
0.5
0.0
- 0.5
0.5
1.0
1.5
2.0
2.5
3.0
Ž
k
- 1.0
- 1.5
- 2.0
- 2.5
- 3.0
Figure: Real (red dashed curve) and imaginary (blue and black solid
curve) parts of the dispersion relation ω̃(k̃) for µ̃ = 0.8. At large k̃, the
zero sound mode (blue solid curve) is the most stable one, but for lower
k̃ another quasi-normal mode (black solid curve) becomes more stable.
Holographic zero sound in the S-S model
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Introduction
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Sakai-Sugimoto
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LFLT
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Hol. 0-sound
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Behavior of ΓT
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Four different regimes
region I (collisionless quantum): ΓT is a constant (independent of TH ) and essentially the same as for zero
temperature, Γ0 .
region II (collisionless thermal): ΓT ∼ TH3 .
region IIIa (crossover between collisionless thermal and hydrodynamical): ΓT ∼ TH−3 .
region IIIb (hydrodynamical): ΓT ∼ TH−1 , in complete agreement with high temperature diffusion result
˛
˛
k2 .
ω(k)˛
∼ −i 2πT
TH →∞
H
Log
GT
G0
10
8
6
4
2
-8
Holographic zero sound in the S-S model
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-6
-4
-2
2
Ž
-Log Μ
Results
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Introduction
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Sakai-Sugimoto
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LFLT
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Hol. 0-sound
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Results
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Behavior of the spectral function
Ž
אΩ2
Ž
k =5
70 000
Ž
אΩ2
Ž
k =5
60 000
60 000
50 000
50 000
40 000
40 000
30 000
30 000
20 000
20 000
10 000
10 000
0
0
0
2
4
Ž
Ω
6
8
10
3.00
3.05
3.10
3.15
Ž
Ω
3.20
3.25
3.30
Figure: Spectral function χxx (ω̃) (normalized by ω̃ 2 ) in region I,
collisionless quantum regime, for ke = 5 and µ̃ = 10 (red), µ̃ = 15 (blue),
µ̃ = 20 (orange). The sharp, distinct peaks correspond to the zero sound
mode. Note that decreasing µ̃ corresponds to raising the temperature, so
that the highest peak occurs for the lowest temperature and largest µ̃.
Holographic zero sound in the S-S model
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Introduction
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Sakai-Sugimoto
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LFLT
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Hol. 0-sound
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Results
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Behavior of the spectral function
Ž
אΩ2
Ž
k =0.5
10
Ž
אΩ2
Ž
k =0.5
1.5´106
8
1.0´106
6
500 000
0
0.300
4
2
0.305
0.310
0.315
Ž
Ω
0.320
0.325
0.330
0.335
2
4
Ž
Ω
6
8
10
Figure: Spectral function χxx (ω̃) (normalized by ω̃ 2 ) in region II,
collisionless thermal regime, for ke = 0.5 and µ̃ = 11 (red), µ̃ = 13 (blue),
µ̃ = 15 (orange).
Holographic zero sound in the S-S model
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Introduction
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Sakai-Sugimoto
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LFLT
..
Hol. 0-sound
....
Results
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Behavior of the spectral function
Ž
אΩ2
Ž
k =0.01
25
Ž
אΩ2
Ž
k =0.01
8´107
20
6´107
15
4´107
10
2´107
5
0
0
0.000
0.002
0.004
0.006
0.008
Ž
Ω
0.010
0.012
0.014
5
10
Ž
Ω
15
20
Figure: Spectral function χxx (ω̃) (normalized by ω̃ 2 ) in region III,
hydrodynamic regime, for ke = 0.01 and µ̃ = 5 (red), µ̃ = 10 (blue),
µ̃ = 15 (orange). The hydrodynamic diffusive contribution becomes
dominant compared to the remnants of the zero sound mode.
Holographic zero sound in the S-S model
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25
Introduction
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Sakai-Sugimoto
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LFLT
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Hol. 0-sound
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Absence of a Fermi surface
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Discontinuity in distribution fct. shows up as singularity at
k = 2kF in the retarded
Green’s
(
) ( function
) at small frequency ω:
GR (ω → 0, k) ∼ 2kkF − 1 ln 2kkF − 1 .
Can evaluate GR numerically in this limit, but no sign of
discontinuity/Fermi surface.
40
30
Re@GR D
Ž
Ω = 0 20
10
0
0
Holographic zero sound in the S-S model
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1
2
Ž
k
3
4
5
Results
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Introduction
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Sakai-Sugimoto
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LFLT
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Hol. 0-sound
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Conclusions and Outlook
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Some properties of our model compatible with LFLT; spec.
5/3
heat Cv ∼ T at low T , eq. of state ∼ nB as expected for
non-relativistic fermions.
Stable zero sound mode with non-standard k 3 -dependence of
ΓT at low and moderate T .
Found interesting crossover regime between collisionless
thermal and hydro regime where ΓT ∼ T −3 .
No sharp Fermi surface: Either flavour back reaction effect, or
no Fermi surface at all.
Further research necessary to decide whether holographic QCD
has a true low energy description in terms of quasiparticle
excitations of a Fermi surface, or it is a novel type of
strongly-coupled quantum liquid phase without Fermi surface.
Thank you!
Holographic zero sound in the S-S model
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Results
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