Gravity Dual of Spatially Modulated Phase

Shin Nakamura (Dept.Phys. Kyoto Univ.)

Reference:S.N. , Hirosi Ooguri (Caltech/IPMU), Chang-Soon Park (Caltech)

arXiv:0911.0679[hep-th] (to appear in Phys. Rev. D)

My talkI am going to talk about

Stability analysis of charged black holes in5d Einstein+Maxwell+Chern-Simons gravity.

Why am I talking in NFQCD?

Because, this can be re-phrased as

Stability analysis of finite density systems ofgauge-theory plasma in the presence of triangle anomaly,

by virtue of gauge/gravity correspondence.

What we will see:

A system with a triangle anomaly of a global symmetry may develop a spatially inhomogeneous instability,at sufficiently high densities of the global charge.

Black hole physics suggests that

A possibility to have a new (chiral) anomaly-induced inhomogeneous phase in QCD at high baryon density?(This part have not yet been considered in my work, andwould be a subject in the discussion.)

If we generalize the statementto the chiral anomaly in QCD,

AdS/CFT correspondenceOne intuitive explanation of necessity of • 5th direction• negatively curved spacetime

Black holeA solution to the Einstein’s equation.

Light can escape(un-trapped region)

Light cannot escape(trapped region)

radial direction

Horizon (Apparent horizon)

“strongly curved” “weakly curved”

Important quantities:• Area (A)• Surface gravity (κ)

Black hole thermodynamicsThermodynamics Black hole

0-th law T= const. at the equilibrium.

κ is constant in the static solution.

1st law dE=T dS dM=[κ/(8πGN)]dA2nd law Entropy never decreases. The area of horizon (A) never

decreases.3rd law We cannot reach T=0 in

any physical process.κ cannot reach zero in

any physical process.

κ: surface gravity (the gravitational acceleration at the horizon of the black hole)GN: Newton’s constant, M: mass of the black holeA: area of the horizon

κ and A mimic T and S, respectively.

Hawking found that the BH emits thermal radiation with T= κ /2π, if we quantize the fields around the BH.

NG

AST

4,

2

Black hole and holography

)(area4

1d

GTdM

NH

We need at least one more direction (the radial direction)to define the horizon : 5d gravity

Suppose that we want to find a correspondence betweenthe gravitational systems (black holes) and 3+1 dim. fieldtheories at finite temperature.

The entropy need to be proportional to the 3d volume:the horizon has to be 3+1 d surface.

radial direction

Horizon

3+1 d

Black hole and holography

However, black holes in a flat spacetime (BH’s in an asymptotically flat spacetime)have negative specific heat.

This problem is cured if the BH is embedded ina negatively curved spacetime.(BH in an asymptotically anti de Sitter (AdS) spacetime)

Black holes in 5d AdSis a good candidate for a gravitational descriptionof a 4d finite-temperature system.

Then,

T=0 limit

AdS/CFT correspondence

5d AdS-BH Some 4d systemat finite temp.

?

5d AdS Some 4d system(at zero temp.)

?

was promoted to a concrete equivalenceby using superstring theory. (Maldacena ‘97)

?

AdS/CFT correspondence

Type IIB supergravity on AdS5×S5 at the classical level

A typical example

4d SU(Nc) N=4 supersymmetric Yang-Mills (SYM) theory at the large-Nc and the large λ (‘t Hooft coupling) limitat the quantum level

conjectured to beequivalent

At the finite temperature, AdS5 becomes AdS-BH(if the SYM lives on Mnkowski)

Finite charge density

Going back to the black hole physics,

If we introduce a Maxwell field, the BH can carrythe U(1) charge Q.

dQAdG

TdMN

H 0)(area4

1

Is the charged AdS-BH equivalent to N=4 SYMwith some charge density?

Introduction of finite charge density intothe “thermodynamics of black hole.”

Yes.

Global charge in N=4 SYMN=4 SYM has SU(4) global symmetrythat is called as R-symmetry.

“Re-arrangement” of 4 super-symmetries.

If we consider only the U(1) sector of the R-symmetry,we can introduce the global U(1) R-charge to the system.

String theory tells

This can be achieved in the 5d gravity by introducinga Maxwell field and the U(1) charge of the BH.

5 d asymptotically AdS charged black hole (AdS-Reissner-Nordström black hole)

AdS/CFT correspondence

4d N=4 SYM theory with U(1) R-charge.

But, 5d Einstein+Maxwell is not very enough.

So far, we have reached

5d Einstein+Maxwell

AnomalyIt is known that there is U(1)R anomaly in N=4 SYM.

Recall that,

Quantum SYM Classical gravityAdS/CFT

The effect of the anomaly has to already be includedin the gravity-theory side. Is it the case? Yes.

If we derive the 5d gravity from superstring theory, we obtain Einstein+Maxwell+Chern-Simons theory. The CS part describes the anomaly.

Let us consider 5d Einstein+Maxwell+Chern-simonsin details,

as a finite density system with anomaly.

The details of my analysis

The gravity side

LMJKIIJKLMMN

MN FFAFFRgL !34

112G16 5

5-dim. Einstein (Λ<0)+ Maxwell theory with a CS term.

Gauge-theory side:The CS term holographicaly represents the R-sym. anomaly of SYM.

The U(1) truncation of the S5 reduction of 10d type IIB supergravity has a CS term with . 321

Origin of the CS term

What does the CS term?

.02

,4

1

2

112

2

1 2

KLIJNIJKLMN

M

MNL

NMLMNMN

FFFg

FgFFRgR

If we consider only electrically charged solutions,the CS term does not affect the solutions.

The solution is just an ordinary AdS-Reissner-Nordström black hole.

However, if consider the fluctuations around the solution,the CS term plays an important role.

Equations of motion:

A flat-space excercise

Maxwell + CS on the 5d Minkowski (without gravity)

.02

KLIJNIJKLMN

M FFF

If we have a background electric field F01=E, the e.o.m.under the Lorenz gauge *is

.04 2012 AOAEA LKNKL

N

For linear perturbations of ,2xkiti

NN eaA

.0)(4

,0)(4

342

432

AikEA

AikEA

(* we can also formulate in a gauge invariant way.)

)2(.0)(4

)1(,0)(4

342

432

AikEA

AikEA

0)(4)( 434322 iAAEkiAAk

222 )2(2 EEk

)2()1( i

The dispersion relation is

tachyonicfor the circular-polarized modes.

This does not always mean the instability, but the systemis unstable in the following momentum range:

Ek 40

Dispersion relationThe dispersion relation is non-standard: “roton-like” dispersion

k

ω2

unstable range

The minimum of the spectrum is located at k=2αE ≠ 0.

The “instability” is proportional to α and E.

Inhomogeneous condensation

The instability occurs at finite momenta.

The resulting condensation of the gauge field is spatially modulated (and circularly polarized).

Spontaneous breaking of translational and rotational symmetry.

Cartoon of the unstable mode

momentum k

• Helical strudture• Spontaneous breaking of the rotational and the translational symmetries

What happens in the black-hole geometry?

The background spacetime is asymptotically AdS:

A slightly “tachyonic” mode is still stableif it satisfies the Breitenlohner-Freedman bound.

Breitenlohner-Freedman, Ann. Phys. (1982) 144; Phys. Lett. (1982) 197.

We need to check whether the effect of the CSterm breaks the BF bound or not.

Breitenlohner-Freedman bound

2

222

z

dzdxdxLds

In d+1-dimensional AdS spacetime,

a scalar fluctuation is stable if its mass m satisfies:

2

22

4L

dm

Breitenlohner-Freedman bound

The question is whether our potentially unstablemode really breaks the BF bound or not.

AdS2 ×R3

• The near-horizon geometry is much simpler.

5d AdS-Reissner-Nordström black hole

BH

horizon boundary5th diresction

• Consider near the horizon where the electric field is biggest.

• The charge at a given mass has an upper limit. Consider the maximum charge: extremal, T=0

= -8

The near-horizon geometryIn my convention,

.62,12

1)(0

222

2

2AdSz gFdzdtz

ds

3)(4

1

121

22 mBF bound:

Breaks the BF bound.

22 96)2( E

62E32

1 (for SUGRA)

Another point in the BH analysis

We need to diagonalize the possible fluctuations.

The Maxwell field couples with gravitons:

....4

1 i

iMNMN fhFgFFg

)1,0(),(,, AAfhgg iiiiii

The Maxwell field couples to the off-diagonal part of the metric.

From the viewpoint of AdS2

The Maxwell field couples to the off-diagonal part of the metric.

KK-gauge field from the2d (AdS2) point of view.

AdS2 ×R3 (xμ=0,1, x2, xi=3,4) AdS2 (xμ=0,1)

Ai Maxwell scalar

hμi off-diagonal graviton KK gauge field

h2i off-diagonal graviton Stückelberg field

Diagonalize these modes and find the lowest mass eigenvalue.

jjd

jkiijkj

jij

ij

iiii

hFEg

FAEFFFF

ikhhKKg

L

)2(

2)()(2

)()(

)2d(

eff

6!32

1

4

1

2

1

4

1graviton

Maxwell+CS

coupling

1141662464142

2/32424622

22min

E

m

The equations of motion reduce to

.0

,042AdS

2AdS

2AdS

22

2

kjijkj

ji

kjijkkjijkij

j

KfE

KEfEf

)2(

)(01,

2

1

d

i

ijkijkig

KKFf

,04

det222

22

kmEm

EEkkm The mass eigenvalueshould satisfy:

= - 2.96804…Oops! slightly stable! > -3 (BF bound)

32

1,62 E

For SUGRA

Result at the near horizon

(Notice: the SUSY is broken even at the extremal case.)

Einstein +Maxwell+CS theory near the horizon of the extremal RN-BH becomes unstable if the CS coupling α is

α > αc=0.2896…. The 5d SUGRA has α=1/2√3=0.2887…. and barely stable.

Full-geometry analysisThe near-horizon analysis provides a sufficient conditionfor instability but not the necessary condition.

We found that the numerical analysis on the full RN-BHgeometry yields the same critical value αc=0.2896….

Numerical analysis on the full AdS-RN-BH geometry:

• consider normalizable modes ,• impose “in-going” boundary condition at the horizon,• see whether its amplitude decays or grows along time.

The behavior depends on k and α and T.

Other geometriesWe have also analyzed the following cases at the near-horizon limit.

• 5d extremal AdS-RN-BH with the boundary geometry S3.• 3-charge solutions

As far as we have analyzed, the BF bound is alwaysbarely satisfied (within 0.4%) if we employ the CScoupling coming from type IIB super-gravity.

Are there any reason? (We do not know so far.) Can we find some unstable example from SUGRA?

CS coupling at arbitrary values

We have found that the U(1)R anomaly is not largeenough to achieve the instability in N=4 SYM. (But, it is very close to the critical value.)

What happens if the anomaly were large enough?

To pose this question would be still important to motivate further detailed analysis in moreQCD-like models.

Let us go to the large α region.

Unstable regionThe unstable range in k is broader than the result in the near-horizon analysis, although the critical coupling is unchanged.

The upper limit can be drawn like this.

Stable

Unstable region

Next slide T=0

full

near horizon

SUGRA

Unstable region

b: “unstable region” from the near-horizon analysis

A holographic interpretation of the instability

2

22 )(~

z

dzdxdxzgds

......)( )2(2)0( AzAzAnon-normalizable mode

source(A0

(0) =μ)

normalizable mode

<current>

Condensation of the gauge field:development of the non-zero expectation valueof the current without the external source.

spatially modulated current(circularly polarized helical current)

Finite momentum

dQAdG

TdMN

H 0)(area4

1

Spontaneous generation of helical current

momentum k

• The current has a helical structure• Spontaneous breaking of translational and rotational symmetries.

Brazovskii modelBrazovskii model: a model for phase transitions to inhomogeneous phases.

Brazovskii, Sov. Phys. JETP 41 (1975) 85.

The non-standard dispersion relation in which the spectrum has the minimum at a finite momentum is postulated there.

• weakly anisotropic antiferromagnets• cholesteric liquid crystals• pion condensates in neutron stars• Rayleigh-Bénard convection• symmetric diblock copolymers

Applied to various physics, such as

In our model, the non-standard dispersion relation is realized by the CS term.

Cholesteric liquid crystal

“A cholesteric liquid crystal is a type of liquid crystal with a helical structure and which is therefore chiral. Cholesteric liquid crystals are also known as chiral nematic liquid crystals……..”

Taken from Wikipedia

Van Hove singularityω

kEven if we do not have instability, the dispersion relationof the circularly-polarized mode is non-standard.

At the minimum of the spectrum, the density of statesper unite energy diverges: Van Hove singularity.

Density of state

0 002

2

0 02

2020

))((1

4

))((4))((

3

33

3

kkk

kkk)g(

dkd

dk

dkkd

If dω/dk=0 at non-zero k, the density of state per unitenergy diverges.

For the standard dispersion relation, dω/dk=0 at k=0 but k2(dω/dk)-1 is not divergent.

Example of Van Hove singularityExample: Plasmino in QCD

Di-lepton production rate diverges at the Van Hove singularity.

Quark dispersion relation in a medium (left)and the di-lepton production rate (right).(Barteen-Pisarski-Yuan, PRL64(1990)2242.)

For N=4 SYM at finite R-charge, the anomaly induced Van Hove singularity exists (althoughthe CS-induced instability does not exist)in the gluonic sector.

We found the non-standard dispersion relation for the normalizable mode of the circularly polarizedbulk U(1) gauge field.

Summary

• Einstein+Maxwell+CS theory is an interesting theory, which has a potential possibility to be a gravity dual of phase transitions to inhomogeneous phases.

• We found a new instability of the AdS-RN black holes at sufficiently large CS interaction.

• The circularly-polarized mode has non-standard dispersion relation even for stable cases.

• The CS terms from SUGRA are barely small to achieve the instability, as far as we have analyzed.

• The instability occurs at finite momenta: the conden- sation is spatially modulated and helical.

Discussions

• Are all the SUGRA-origin models stable?. If so, can we prove it?

• Other generalization? introduction of angular momentum, consideration of non-abelian gauge field,….

To superstring theorists:

To the specialists of BH physics:

Discussions for nuclear physicists

Domokos-Harvey, PRL99(2007)141602

Actually, Domokos and Harvey have already proposeda similar mechanism in the meson systems within theframework of a phenomenological AdS/QCD approach.

• The effective theory of mesons can be given as a 5d gauge theory in a negatively curved background.• CS coupling exists as a result of chiral anomaly. (holographic counter-part of the WZW term in ChPT)

In holographic QCD:

They have pointed out that:

• the presence of vector ー axial-vector mixing at the finite baryon density,

• the possibility of the inhomogeneous condensation of these mesons at the sufficiently high density.

But,It is true that there is a similar mathematical structurein the effective theory of mesons as what I have donein my talk.

• Do you have any idea related to the anomaly?• Do you have any idea related to the non-standard dispersion relation? • Do you have any idea related to the anomaly-induced mixing of modes?

I am sure that you know much better than me.

Hope to be a good starting point for discussion.