Anti-De Sitter spaceA C o m m o n R o o m f o r S t r i n g s a n d A t o m s ?

Outline

Quark Gluon Plasma and Cold Atoms

The AdS/CFT Correspondence

The viscosity calculation

A holographic superfluid

Holography with Schrödinger Symmetry?

Elliptic flow in non-central collisions

Elliptic flow in non-central collisions

dN

d2pT dY=

dN

2πpT dpT dY[1 + 2v2(pT ) cos(2φ) + . . . ]

can be measured and calculated from hydrodynamic simulations

23

0 100 200 300 400N

Part

0

0.02

0.04

0.06

0.08

0.1

v2

PHOBOS

Glauber

!/s=10-4

!/s=0.08

!/s=0.16

0 1 2 3 4p

T [GeV]

0

5

10

15

20

25v

2 (p

erce

nt)

STAR non-flow corrected (est.)STAR event-plane

Glauber

!/s=10-4

!/s=0.08

!/s=0.16

0 100 200 300 400N

Part

0

0.02

0.04

0.06

0.08

0.1

v2

PHOBOS

CGC

!/s=10-4

!/s=0.08

!/s=0.16

!/s=0.24

0 1 2 3 4p

T [GeV]

0

5

10

15

20

25

v2 (p

erce

nt)

STAR non-flow corrected (est).STAR event-plane

CGC!/s=10

-4

!/s=0.08

!/s=0.16

!/s=0.24

FIG. 8: (Color online) Comparison of hydrodynamic models to experimental data on chargedhadron integrated (left) and minimum bias (right) elliptic flow by PHOBOS [85] and STAR [87],respectively. STAR event plane data has been reduced by 20 percent to estimate the removal

of non-flow contributions [87, 88]. The line thickness for the hydrodynamic model curves is anestimate of the accumulated numerical error (due to, e.g., finite grid spacing). The integrated v2

coefficient from the hydrodynamic models (full lines) is well reproduced by 12ep (dots); indeed, the

difference between the full lines and dots gives an estimate of the systematic uncertainty of thefreeze-out prescription.

experimental data from STAR with the hydrodynamic model is shown in Fig. 8.For Glauber-type initial conditions, the data on minimum-bias v2 for charged hadrons

is consistent with the hydrodynamic model for viscosities in the range η/s ∈ [0, 0.1], whilefor the CGC case the respective range is η/s ∈ [0.08, 0.2]. It is interesting to note thatfor Glauber-type initial conditions, experimental data for both the integrated as well as theminimum-bias elliptic flow coefficient (corrected for non-flow effects) seem to be reproducedbest7 by a hydrodynamic model with η/s = 0.08 " 1

4π . This number has first appeared in the

7 In Ref. [22] a lower value of η/s for the Glauber model was reported. The results for viscous hydrodynamics

shown in Fig. 8 are identical to Ref. [22], but the new STAR data with non-flow corrections became

[Romatschke, Luzum, Phys.Rev.C78:034915,2008]

η

s=

14π ?

From Quark Matter conference 2009, Session 12: AdS/CFT, Cold Atoms, Flow

John Thomas (Duke University)

1 10 100 1000T, K

0

50

100

150

200

Helium 0.1MPaNitrogen 10MPa

Water 100MPa

Viscosity bound

4! "

sh

Figure 2: The viscosity-entropy ratio for some common substances: helium, nitrogen and

water. The ratio is always substantially larger than its value in theories with gravity duals,

represented by the horizontal line marked “viscosity bound.”

experimentally whether the shear viscosity of these gases satisfies the conjectured bound.

This work was supported by DOE grant DE-FG02-00ER41132, the National Science

Foundation and the Alfred P. Sloan Foundation.

References

[1] S.W. Hawking, “Particle creation by black holes,” Commun. Math. Phys. 43, 199 (1975).

[2] J.D. Bekenstein, “Black holes and entropy,” Phys. Rev. D 7, 2333 (1973).

[3] G.T. Horowitz and A. Strominger, “Black strings and p-branes,” Nucl. Phys. B 360,

197 (1991).

[4] L.D. Landau and E.M. Lifshitz, Fluid Mechanics (Pergamon Press, New York, 1987).

[5] G. ’t Hooft, “Dimensional reduction in quantum gravity,” gr-qc/9310026.

7

[Kovtun, Son, Starinets, Phys.Rev.Lett.94:111601,2005] (KSS bound)

AdS/CFT

String Theory: Quantum Gravity

Elementary Particles Strings

Closed Strings: Graviton

Open Strings: Gauge Bosons, Matter

Supersymmetry

Higher Dimensions (10, 11)

D-Branes

D-brane : place where open strings end

arbitrary dimension (up to 10)

Non-abelian Gauge theories

non-perturbative D-brane geometry

Decoupling

IIb String theory with N D3-branes

N=4 SUSY + Gravity + massive Stringstates

decouple by taking

leaves N=4 SUSY gauge theory

Geometry = Anti-deSitter space

α′ → 0

AdS/CFT

Maldacena 1998:

metric

IIb strings in AdS5 x S5 = SYM

strong/weak coupling duality

ds2 =r2

L2(−dt2 + d!x2) +

L2dr2

r2+ L2dΩ2

5

L4

α′2 = g2Y MN gs =

1N

L =1

g2Y M

tr

(−1

4FµνFµν + iψiσ

µDµψi +12DµΦijD

µΦij . . .

)

AdS CFT

Finite temperature

Black hole metric

Horizon

Hawking temperature = field theory temperature

dual to N=4 plasma

ds2 = r2(−f(r)dt2 + d!x2) +dr2

f(r)r2

f(r) = 1− π4T 4

r4

Finite Temperature

AdS CFT: Green Functions

field in AdS with fixed boundary value

acts as source for gauge invariant operatore.g. metric

allows calculation of n-point functions

in practice: large N and strong coupling: classical gravity (5D Einstein + cosmological constant)

ZAdS [Φ0] = 〈eR

Φ0O〉CFT

gµν → Tµν

retarded Gf’s

conformal weight

boundary

at horizon: infalling

retarded Green function

1√−g

∂r

(√−ggrr∂rφ

)− (k2 + m2)φ = 0

limr→∞

r∆−φ = φ0

∆(∆− d) = m2

φ(r) = [Ar−∆−(1 + . . . ) + Br−∆+(1 + . . . )]φ0

GR =B

A

retarded Gf’s

Linear response

Response to an external source j(t,x):

〈Φ(t,x)〉 = −∫

dτ d3ξ GR(t − τ,x − ξ) j(τ, ξ) , (1)

Im ν

Re ν

Hydro and beyond – p.12/28

GR has poles at A=0 : quasinormal frequencies

ωn = ±Ωn − iΓn Dissipation

Shear ViscosityThe theorists way of measuring viscosity:

gravitational waves!

Hydrodynamics predicts

D =η

sT

Compute from gravitational shear wave in AdS!

Either compute pole or set k=0 and use Kubo formula

〈TxyTxy〉 =iηω2

ω + iDk2

universal result: valid for all holographic field theories to leading order (even for anisotropic systems, Miesowicz coeff [K.L., J. Mas, JHEP 0707:088,2007] )

Conjectured lower bound (recent counter examples for sub-leading N contributions)

Consistent with low value at RHIC (and cold atoms?)

Shear Viscosityη

s=

14π

[G. Policastro, D.T. Son, A. Starinets, PRL 87:081601]

Quantum Criticality[C. Herzog, P. Kovtun, S. Sachdev, D.T. Son; Phys.Rev.D75:085020,2007]

• AdS4 (2+1) dim CFT

• study conductivity

• hydrodynamic-to-collisionless ω ∝ q2 ω ∝ q

0.2 0.4 0.6 0.8 1.0

!0.6

!0.5

!0.4

!0.3

!0.2

!0.1

σ(ω) = σ0

[J. Mas, M. Kaminski, K.L., J. Mas, J. Tarrio, to appear]

A Holographic Superfluid

abelian Higgs model in AdS4 Black Hole

charge scalar condenses for low T

charge manages to hoover over the horizon

chem. potential At -component

[S. Gubser, Phys.Rev.Lett.101:191601,2008][S. Hartnoll, C. Herzog, G. Horowitz, Phys.Rev.Lett.101:031601,2008]

L = −14F 2

µν −m2ΨΨ−DµΨDµΨ

At = µ − n

r+ O(1/r2)

Ψ =J

r+

〈O〉r2

+ O(1/r3)

A Holographic SuperfluidQuasinormal frequencies for Ψ unbroken phase 〈O〉 = 0

T < Tc

[I. Amado, M. Kaminski, K.L., JHEP 0905:021,2009]

〈O〉 ∝(

1− T

Tc

)1/2

A Holographic SuperfluidConductivity:

σn = limω→0

![σ(ω)] = exp(−∆/T ) ∆ =2 ωg ∆ ∼ 8Tc

A Holographic Superfluid2nd Sound: speed and attenuation

Further Results

p-wave superconductors (S.Gubser)

explicit string models (J. Erdmenger et al.)

models with pseudogap regions (J. Erdmenger et al.)

dynamical universality class: Model A with z=2 (K. Maeda et al.)

Meissner effect (1/2 of it: generation of currents)

(optimistic) hope: relevant for high Tc superconductors

Schrödinger Symmetry

Quantum critical point with z=2

Symmetry group of Schrödinger equation

Fermions at unitarity

Generators are

central extension

t→ λzt , "x→ λ"x

Translations Pi , Rotations Mij

Galilean boosts Ki , time translations H

Dilatations D , special conformal C

[Ki, Pj ] = −iδijN

Particle Number

Schrödinger Symmetry

Schrödinger algebra on the lightcone: x± = (x0 ± x3)

E2 − !p 2 = 0 −→ p− =p2⊥

2p+

Hamiltonian H = p−

Mass N = p+

Boosts Ki = M i+

Dilatations D = Dr + M+−

Special conformal C = K+r

Schrödinger Symmetry

ds2 = −β2r4dt2 + r2(−2dξdt + dx2⊥) +

dr2

r2

[D.T. Son, Phys. Rev. D 78 (2008) 046003][K. Balasubramanian,

J. McGreevy, Phys.Rev.Lett.

101:061601,2008]

• Schrödinger = isometries

• d+1+2 dimensions

(space , time, holographic and extra)

• particle number ~

compactify to get discrete spectrum

• non-relativistic causal structure

• singular free energy

• superfluid at low T?

• DLCQ of non-commutative theory?

∂ξ

F = −cN2V MT 2

(T

µ

)2

SummaryHolographic Quantum Field Theories

strongly coupled transport theory

useful for QGP

maybe useful for QCPs

relativistic superfluids, (Non-)Fermi liquids, ...

non-relativistic ?

holographic condensed matter physics ?

Additional Reading

Gauge-Gravity Duality: S. Gubser, A. Karch, arXiv:0901.0953 [hep-th]

Lectures on Holography and Condensed Matter Physics: S. Hartnoll, arXiv:0903.3246 [hep-th]

Lectures on holographic Superfluidity and Superconductivity, C. Herzog, arXiv:0904.1975 [hep-th]

Nearly Perfect Fluidity: QGP and Cold Atoms,T. Schaefer, D. Teaney, arXiv:0904.3107 [hep-ph]