the ads/cft duality: an introduction. · the correspondence for those familiar with the real-space...

General Concepts A First Simple Example Holographic Superconductors Conclusions

The AdS/CFT Duality: an Introduction.

Marco Beria

SISSA, Trieste

December 14, 2010

M.Beria



Outline

1 General Concepts

2 A First Simple Example

3 Holographic Superconductors

4 Conclusions

M.Beria



The AdS Geometry

ds2 =

(L

z

)2 [dz2 + ηµνdxµdxν

], , z ∈ (0,∞)

IR UVz

d!1,1

z

RAdSd+1

minkowski

UVIR

...

Figure 1: The extra (‘radial’) dimension of the bulk is the resolution scale of the field theory.

The left figure indicates a series of block spin transformations labelled by a parameter z.

The right figure is a cartoon of AdS space, which organizes the field theory information

in the same way. In this sense, the bulk picture is a hologram: excitations with di!erent

wavelengths get put in di!erent places in the bulk image. The connection between these two

pictures is pursued further in [15]. This paper contains a useful discussion of many features of

the correspondence for those familiar with the real-space RG techniques developed recently

from quantum information theory.

of length. Although this is a dimensionful parameter, a scale transformation xµ ! !xµ can

be absorbed by rescaling the radial coordinate u! u/! (by design); we will see below more

explicitly how this is consistent with scale invariance of the dual theory. It is convenient to

do one more change of coordinates, to z " L2

u, in which the metric takes the form

ds2 =

!L

z

"2 #"µ!dxµdx! + dz2

$. (2.1)

These coordinates are better because fewer symbols are required to write the metric. z will

map to the length scale in the dual theory.

So it seems that a d-dimensional conformal field theory (CFT) should be related to a

theory of gravity on AdSd+1. This metric (2.1) solves the equations of motion of the following

action (and many others)4

Sbulk[g, . . . ] =1

16#GN

%dd+1x

#g ($2" + R+ . . . ) . (2.2)

Here,#

g "&| det g| makes the integral coordinate-invariant, and R is the Ricci scalar

but there is no proof for d > 1 + 1. Without Poincare invariance, scale invariance definitely does not implyconformal invariance; indeed there are scale-invariant metrics without Poincare symmetry, which do not havehave special conformal symmetry [16].

4For verifying statements like this, it can be helpful to use Mathematica or some such thing.

7

M.Beria



What is AdS/CFT duality?

Holographic Correspondence

QG on (d + 1)-asymp.AdS −→ d-CQFT on ∂AdS

IR UVz

d!1,1

z

RAdSd+1

minkowski

UVIR

...














ds2 =

!L

z


$. (2.1)






Sbulk[g, . . . ] =1

16#GN

%dd+1x

#g ($2" + R+ . . . ) . (2.2)

Here,#




7

QFT

QG

The conjecture:

ZQG = ZCQFT

M.Beria



How does it work?

CQFT AdS

{Oα} {Φα}∆ m2

g 1/g〈Oα1 . . .Oαn〉〈Φα1 . . .Φαn〉

Sources {φα0 }: ZCQFT [{φα0 }] = 〈e−R

ddxφα0 Oα〉CQFT

〈Oα1 . . .Oαn〉CQFT =δ

δφα10

. . .δ

δφαn0

ln (ZCQFT [{φα0 }])

Witten-Polyakov

φα0 ≡ Φα|∂AdS ⇒ ZQG [{φα0 }]

M.Beria



How does it work?

CQFT AdS

{Oα} {Φα}∆ m2

g 1/g〈Oα1 . . .Oαn〉〈Φα1 . . .Φαn〉

Sources {φα0 }: ZCQFT [{φα0 }] = 〈e−R

ddxφα0 Oα〉CQFT

〈Oα1 . . .Oαn〉CQFT =δ

δφα10

. . .δ

δφαn0

ln (ZQG [{φα0 }])

CQFT strong coupled ⇒ QG weak coupled

ZQG ≈ ZCG Classical Gravity

M.Beria



The Bulk Action

S =1

16πG

∫dd+1x

√g (R − 2Λ + fields)

CQFT AdS

vacuum geometryexcitations fields

The fields-Lagrangian takes care of

operator content {Oαi}, ∆;

symmetries.

M.Beria



Why is it useful?

Strongly Coupled Quantum Problems → Classical GravitationalProblems.

Build-up effective theories for difficult problems (HTc

Superconductors, Graphene, etc..) once provided:

{Oα}∆α

Symmetries.

Very suitable to compute linear response coefficient, χ.

Weakness

Few of control on the dual CQFT;

No exact correspondence between dual theories.

M.Beria



The Duality at Finite Temperature

Imaginary time, T 6= 0. Z = Tre−H/T = e−F/T on S1 × Σd−1

introduction of a scale;

Pure AdS is no longer the good dual vacuum.

The gravitational thermal vacuum is the Schwarzschild Black Hole.⇒ Schwarzschild-AdS space:

ds2 =

(L

z

)2 [fdτ2 +

1

fdz2 + dx idxi

], f = 1−

(z

zh

)d

M.Beria



The Duality at Finite Temperature

ρ2 = 4d

Lzh

(zh − z) and ψ = d2zhτ

ds2 ≈ ρ2dψ2 + dρ2 +

(L

zh

)2

dx idxi

2π-periodicity in ψ ⇒

T =d

4πzh

M.Beria



Outline

1 General Concepts



4 Conclusions

M.Beria



The Dual field-Action and Equation of Motion

S = −N2

∫dd+1x

√g(∂AΦ∂AΦ + m2Φ2

)

S = EOM − N2

∫

∂AdSddx√

gg zAΦ∂AΦ

(−� + m2

)Φ = 0 , Φ(x , z) = e ikµxµfk(z)

EOM ⇒ Bessel. fk(z) ∼ z∆ for z → 0

0 =(k2z2 + m2L2 −∆(∆− d)

)z∆

∆(∆− d) = m2L2

M.Beria



Two-point Correlator

IR UVz

d!1,1

z

RAdSd+1

minkowski

UVIR

...














ds2 =

!L

z


$. (2.1)






Sbulk[g, . . . ] =1

16#GN

%dd+1x

#g ($2" + R+ . . . ) . (2.2)

Here,#




7

QFT

QG �

Introduction A First Simple Example Holographic Superconductors Conclusions

Two-point Correlator

z ≈ �, fk(z) =zd/2K∆−d/2(kz)

�d/2K∆−d/2(k�)and F�(k) = z−d+1fk(z)∂z fk(z)

��z=�

S = −NLd−1

2

�

∂AdSddkφ0(−k, �)F�(k)φ0(k, �)

�O(k1)O(k2)�� = (2π)d δ(k1 + k2)F�(k1)

φ0(k, �) ≈ �d−∆φreno (x) and O ≈ �∆Oren

�O(x)O(0)� ∼ 1

|x |2∆

M.Beria


�, fk(z) =zd/2K∆−d/2(kz)

�d/2K∆−d/2(k�)

and F�(k) = z−d+1fk(z)∂z fk(z)��z=�

S = −NLd−1

2

∫

∂AdSε

ddkφ0(−k , ε)Fε(k)φ0(k , ε)

〈O(k1)O(k2)〉ε = (2π)d δ(k1 + k2)Fε(k1)

φ0(k , ε) ≈ εd−∆φreno (x) and O ≈ ε∆Oren

〈O(x)O(0)〉 ∼ 1

|x |2∆

M.Beria



Linear Response Theory (m=0)

S [x ] =∫ tfti

L (x(t), x ′(t)), δSδxi

= Πi = ∂L∂x ′

i. Π(x , z) = δL

δ∂zφ

〈O(x)〉CQFT =δW [φ0]

δφ0(x)= finite

{limz→0

(z

L

)d−∆Π(x , z)

}

Kubo formula: δ〈O(x)〉 −→k,ω→0

iωχφ0

χ = limk→0

limω→0

limz→0

Π(k , z)

iωΦ(k, z)

χ =1

4πG

√∣∣∣∣g

gzzgtt

∣∣∣∣

M.Beria



Outline

1 General Concepts



4 Conclusions

M.Beria



HTc-Superconductors

No well known theory, probably strong coupling,

Experimental data available,

Good candidate for Ads/CFT approach.

Ingredients

CQFT AdS

Jµ AµΨ†Ψ Φ

S = N∫

d4x√

g

(R − 2Λ− 1

4FµνFµν − DµΦDµΦ−m2Φ2

)

g : Reissner-Nordstrom-AdS Black Hole;

under Tc other hairy-charged Black Hole metric required.

M.Beria



The Condensate

29

To be consistent with ref. [42] I choose to work in the canonical ensemble where the charge density n

is fixed. The chemical potential µ is then determined dynamically through the differential equationsand the other boundary conditions.

For the scalar, normally the leading behavior would correspond to an external control parameterfor the field theory, i.e. a source, and the subleading behavior would be an expectation value, justas occurred for Aµ and the grand canonical ensemble. However, here we have a choice of interpre-tations and a corresponding choice of boundary condition [43]. In the AdS/CFT correspondence, abulk scalar with mass m corresponds to a scalar operator in the field theory with conformal dimen-sion ∆ through the relation m2L2 = ∆(∆ − 3) (for AdS4). Thus a value of the mass m2L2 = −2corresponds to ∆ = 1 or 2. For a scalar with ∆ = 2, a is interpreted as a source for the operator inthe field theory while b ∼ �O2� is an expectation value. To study phase transitions, we should lookfor solutions with no source for the scalar, a = 0, but where b becomes nonzero at some criticaltemperature. On the other hand, for a scalar with ∆ = 1, things are switched, a ∼ �O1�, and oneshould set b = 0. With a single choice of mass parameter, I get two models for the price of one.

The other boundary conditions for this field theory I set at the horizon of the black hole, z = zh.For the scalar, the condition ψ < ∞ at the horizon eliminates one of the integration constants inthe differential equation. On the other hand, for the gauge field, I need to choose At = 0 at thehorizon in order to ensure the gttA2

t < ∞.Given the boundary conditions and the set of differential equations for At and ψ, I have a well

posed problem which unfortunately does not appear to have an analytic solution. However, thedifferential equations are relatively straightforward to solve numerically, and one can look for aphase transition as a function of the dimensionless ratio n/T 2. In practice we fix the temperatureby setting the horizon radius to zh = 1, and tune n. For 0 < n < nc, the black hole solution withψ = 0 appears to be stable. However, for n > nc, there is a phase transition to a black hole withscalar hair. We can equally well think of this phase transition from an ordinary black hole to ahairy black hole as occurring as we lower the temperature at fixed n. Plots of the behavior of �Oi�as a function of temperature are given in Figure 7. The phase transition is second order (see for

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

TTc

!O1"Tc

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

TTc

!O2"Tc

FIG. 7: The condensate as a function of temperature for the two operators O1 and O2. The condensategoes to zero at T = Tc ∝ n1/2.

M.Beria



Outline

1 General Concepts



4 Conclusions

M.Beria



Conclusions

CQFT AdS

d d+1{Oα} {Φα}

∆ m2

g 1/g〈Oα1 . . .Oαn〉〈Φα1 . . .Φαn〉

Vacuum GeometryExcitations Fields

Compute quantum correlators in a classical gravitation framework(linear response);

thermal theories live in a Schwarzschild-AdS space;

{Oα}, {∆α} and Symmetries → effective theories for difficultproblems (HTc Superconductors).

M.Beria



References

E.Witten: Anti De Sitter space and holography

J.McGreevy: Holographic duality with a view towards many-bodyphysics

E.Hubeny,M.Rangamani: A holographic view on physics out ofequilibrium

G.T.Horowitz: Surprising connection between general relativity andcondensed matter

A.S.T.Pires: AdS/CFT correspondence in condensed matter

S.A.Hartnoll Lectures on holographic methods for condensed matterphysics

C.P.Herzog Lectures on holographic superfluidity andsuperconductivity

Papadimitriou,K.Skenbderis: AdS/CFT correspondence andgeometry

M.Beria


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