Gravity Dual of Spatially Modulated Phase

Instability of Black Holes Induced by Chern-Simons Terms

Shin Nakamura (Kyoto Univ.)Based on S.N.-Ooguri-Park, arXiv:0911.0679

MotivationChern-Simons terms in super-gravity play important roles. Especially from the viewpoint of holography,

• chiral current anomaly• baryons • ……….

We consider effects of the Chern-Simons termsin details in this talk.

in gauge theories are realized through the CS terms in their gravity duals.

Chern-Simons term in 5d SUGRAAn S5 reduction of 10d type IIB super-gravity :

LMJKIIJKLMMN

MN FFAFFRgL !34

112G16 5

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

• Any solution to this theory can be consistently embedded into the 10d type IIB super-gravity (namely, type IIB superstring).• This system is dual to 4d N=4 SYM theory with R-charge.• The CS coupling α is at some fixed value for the closure of SUSY.

321The bosonic part of the N=2 5d minimal gauged super-gravitywith a negative cosmological constant.

Günaydin, Sierra and Townsend, NPB242(1984)244;Cvetic et. al., NPB668(1999)96.

Equations of motion

.02

,41

2112

21 2

KLIJNIJKLMN

M

MNLNMLMNMN

FFFg

FgFFRgR

Let us consider only electrically charged solutions. Then 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.

Preparation: A flat-background example

Maxwell + CS on flat 5d (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 ,)(21 xkiti

NN exaA

.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

tachyonic?for the circular-polarized modes.

This does not necessarily mean the system is unstable, but it does in the following momentum range:

Ek 40

Dispersion relationThe dispersion relation is non-standard:

k

ω2

unstable range

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

Inhomogeneous condensation

The instability occurs at finite momenta.

The resulting condensation is inhomogeneous:the condensation is spatially modulated (and circularly polarized).

5d Maxwell theory + CS term + electric fieldis very interesting system.

Spontaneous breaking of translational and rotational symmetry.

What happens in the 5d SUGRA?The background spacetime is asymptotically AdS:

A slightly “tachyonic” mode is still stable:there is Breitenlohner-Freedman bound.

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

The U(1) gauge field couples with gravitons:

We need to diagonalize the possible fluctuations.

The background geometry is curved non-trivially.

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

Strategy• The electric field in the Reissner-Nordström black hole is maximum at the horizon and decreases towards the boundary.

Probably, the system is most unstable around the horizon since the “instability” was proportional to the electric field in 5d Maxell.

We take the near-horizon limit of the 5d extremal RN black hole.

AdS2×R3

Much simpler geometry than RN-BH.

The near-horizon analysis is at least a good starting point.

• The maximum electric field is realized in the extremal BH. (T=0, but SUSY is broken)

Breitenlohner-Freedman bound

2

222

zdzdxdx

Lds

In d+1-dimensional AdS spacetime,

a scalar fluctuation is stable if its mass m satisfies:

2

22

4Ldm

Breitenlohner-Freedman bound

Stable meansthe amplitude of the field fluctuation does not growalong the time evolution.

The near-horizon geometry

6222

406

24

21222

2

24111,

121,,

121)(

,)()(1

HH

HHH

z zqz

Tzqz

MzqFzqzMzf

dzzfxddtzfz

ds

RN black hole:

.0,212

,3,211)( 62

4222

HHH

zz

zz Tzq

zMzf

HH

Extremal limit choose the combination ofM and q so that T=0.

)(,10,1HH z

xzz Xyy Near-horizon limit

AdS2×R3

with elec. field of const. × vol. form.

.12162

121,

121

20

2222

2

F

ydxddtds

The Maxell + CS theory at the near-horizon

The contribution of the CS term exceeds the BF bound,if we do not include the gravitational fluctuations.

.12162,

121

20222

22

Fdxddtds

3)(4

12

121

22 mBF bound:

241

321

for SUGRA

222 )2(2 EEk

,396)2( 22 E,

241

condition for instability

What happens when the graviton switched on?

The coupling between the gauge field and the gravitoncomes from the kinetic term:

....41

iiMN

MN FhFgFFg

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

Our modes couple 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

jjd

jkiijkj

jij

ij

iiii

hFEg

FAEFFFF

ikhhKKgL

)2(

2)()(2

)()(

)2d(

eff

6!32

141

21

41

graviton

Maxwell+CS

coupling

1141662464142

2/32424622

22min

Em

The equations of motion reduce to

.0

,042AdS

2AdS

2AdS

22

2

kjijkj

ji

kjijkkjijkij

j

KfE

KEfEf

)(

01212,

21 i

ijkijki KKFf

,04

det222

22

kmEmEEkkm The mass eigenvalue

should satisfy:321,62 E

= - 2.96804…Oops! slightly stable!

Result at the near horizon

(The SUSY is broken even at the extremal case.)

Einstein +Maxwell+CS theory near the horizon of the RN-BH becomes unstable if the CS coupling α is α> αc=0.2896…. while the 5d SUGRA has α=3-1/2/2=0.2887..which is barely outside the unstable region.

AdS2 (xμ=0,1)

Ai scalar

hμi KK gauge field

h2i Stückelberg field

(mass)2=k2

(mass)2=k2

decouples

The CS makes the “scalars” tachyonic but they couples withthe “massive KK gauge fields.”

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….

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.

CommentAlthough αc is not modified in the full-geometry analysis,the momentum-range for instability is corrected.

The instability range goes until some upper limit of k like

The red curve: the BF bound from the NH analysis.Curve I: the lower bound for the isntability region.Curve II: on-set of higher-excited unstable modes.

but only the lowercurves are shown.

unstable

Unstable region

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

Decoupling modesThere are unstable modes which are not detectedin the near-horizon analysis.

They become non-normalizable mode in the near-horizongeometry.

Intuitive but not precise cartoon:amplitude

rhorizon boundary

near-horizonregion

mode for curve I: no node

mode for curve II: 1 node This goes down faster.

More precisely:

......)( 1232

21

m

rr

An appropriate combination of the gauge field and the gravitonnear the horizon (r=0) :

“Normalizable” means finite.*

0

dr

rrrer mmrm

log2cosconst.log2cos 123

1231

2log1 2212

32

Full-geometry analysis: the normalizability is checked numerically. Near-horizon analysis: take the above profile as if it is valid until r=∞.

The modes outside the curve II satisfy the BF bound andare non-normalizable in the near-horizon analysis.

implies divergence

but, if the BF bound is violated (m2< - 3),

integration oscillating but finite

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 unknown reason?

A Phenomenological Approach

CS coupling at arbitrary values Presence of CS term is quite general for gravity theories coming from superstring theory through various com-pactifications.

Cf. Holographic superconductor

Investigation of general nature of the gravity theory with Maxwell+CS is important, being motivated by the possible correspondence to yet to known CFT.

Let us go a little bit further to investigate the natureof Einstein+Maxwell+CS theory at arbitrary CS coupling.

A possible phase transition

Then we have unstable RN-AdS BH’s at sufficientlylarge CS couplings.

αc is a function of the temperature.

We have a phase transition into a modulated phasewhen we cool down the temperature towards zero.

Suppose that the CS coupling is a parameterwhich we can choose phenomenologically.

In other words, let us start with 5d Einstein + Maxwell + CStheory with electric field.

A holographic interpretation

2

22 )(~

zdzdxdxzg

ds

......)( )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

Circularly-polarized current

momentum k

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

“Phase diagram”

The boundary of the two regions: onset of the instability.This may not be the real phase boundary which is determined by thecompetition of the free energies of the two phases, since we do not knowthe exact free energy for the modulated phase.

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 modes is non-standard.

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

Van Hove singularityExample: Plasmino in QCD

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

Even for N=4 SYM at finite R-charge, the anomaly induced Van Hove singularity exists.

Summary

• Einstein(Λ<0)+Maxwell+CS theory is an interesting theory, which has a potential possibility to be a gravity dual of phase transitions to inhomogeneous phases.

• 5d gravity theories reduced from type IIB SUGRA have a CS term.•The circularly-polarized modes of gauge fields in Einstein+Maxwell+CS theory with electric flux have non-standard dispersion relations.•If the CS interaction is strong enough, the circularly- polarized modes become tachyonic.• As far as we have analyzed, the gravity theories coming from SUGRA are barely stable although SUSY is broken.