CONDENSED MATTER HOLOGRAPHIC DUALS OF

CHARGED ADS BLACK HOLES M. Cadoni

University of CagliariI will outline the use of holographic  methods for reproducing condensed matter phenomena out of  AdS gravity. Holographic superconductors and  exotic metals  dual to charged dilatonic AdS � �black holes are discussed in detail.

Mainly based on M.C, G. D’Appollonio, P. Pani, JHEP 1003,100 (2010), [arxiv:0912.3520]

Summary

Motivations for using holographic methods for Motivations for using holographic methods for investigating condensed matter systemsinvestigating condensed matter systems

The basics of the holographic AdS/CFT The basics of the holographic AdS/CFT correspondencecorrespondence

Holographic superconductorsHolographic superconductors

Charged dilatonic ADS black hole and holographic Charged dilatonic ADS black hole and holographic “exotic metals“exotic metals””

Holography and condensed matter physics

The holographic principle The holographic principle :: QFT in d-dimensions equivalent to QFT in d-dimensions equivalent to gravitational theory in d+1-dimensions. Main realization : the AdS/CFT gravitational theory in d+1-dimensions. Main realization : the AdS/CFT correspondence ( correspondence ( t’Hooft, Susskind , Maldacenat’Hooft, Susskind , Maldacena……..)..). .

Why use the holographic correspondence to study condensed matter Why use the holographic correspondence to study condensed matter physics (CMF) ?physics (CMF) ?

1.1. Most of the physical intuition we have on CM systems is based on Most of the physical intuition we have on CM systems is based on the standard paradigms, the standard paradigms, theory of symmetry breaking theory of symmetry breaking and and the notion the notion of weakly coupled quasiparticle of weakly coupled quasiparticle e.g bosonic quasiparticles are e.g bosonic quasiparticles are expected to condensate while fermionic to build a Fexpected to condensate while fermionic to build a Feermi surface. But rmi surface. But in CMF there are many strongly coupled systems for which this in CMF there are many strongly coupled systems for which this paradigms paradigms typically fail (typically fail (non Fermi liquids, high temperature non Fermi liquids, high temperature superconductorssuperconductors……)). The AdS/CFT correspondence allows us to . The AdS/CFT correspondence allows us to deal with strongly coupled QFTs in d-dimensions by investigating deal with strongly coupled QFTs in d-dimensions by investigating gravitational physics gravitational physics in d+1 dimensionsin d+1 dimensions

2.2. Dual condensed matter systems in principle could be used as holographic Dual condensed matter systems in principle could be used as holographic “ANALOGS” “ANALOGS” of gravitational systems (e.g. black holes). They could of gravitational systems (e.g. black holes). They could represent a way for testing experimentally concepts of high energy and represent a way for testing experimentally concepts of high energy and gravitational physics. gravitational physics.

3.3. Holographic description allows a quantitative description of universality Holographic description allows a quantitative description of universality classes of low temperature behaviour in terms of classes of low temperature behaviour in terms of thermodynamicsthermodynamics, , phase phase transitions and transport coefficients transitions and transport coefficients

Basic dictionary of the AdS/CFT correspondenceA) DUAL THEORIES

• Yang-Mills theory in d-dim String theory in AdSd+1 • Relevant string theories live in nontrivial curved background and are

not computationally under control • In the large N limit string theory can be approximated by the

effective classical gravity theory but the gauge theory remains strongly coupled , λ=gYM N ≈ L4 >>1

• Large N gauge theory in d-dim (Semi)classical AdSd+1 gravity

• The universal sector of the classical gravitational theory is the ST metric gμν

• Other relevant bulk fields are U(1) gauge field Aμ controlling charge density on the boundary and scalar ψ controlling the coupling constant and its running in the IR

• This is a consistent truncation: there is a mass gap in the spectrum of anomalous dimension separating BPS states from string excitation

• The universal sector is described by Einstein-Hilbert AdSd+1 gravity.

€

S =1

2κ 2dd +1x∫ −g R +

d d −1( )L2

⎛

⎝ ⎜

⎞

⎠ ⎟

• The most symmetric solution of the theory is AdS spacetime

• The isometry group of the ST is the conformal group in d-dim: SO(d,2) which contains a scale symmetry

• Thus the QFT living in the x=0 boundary of the AdS ST is conformally invariant.

• Validity of the semiclassical description requires the AdS radius of curvature to be large in Planck units:

• c has to be interpreted has the number of DOF in the dual theory (central charge). c will scale as a power of N (e.g. N2).

€

ds2 =L2

x 2−dt 2 + dy idy i + dx 2

( )

€

c ≈Ld −1

κ 2>>1

B) CORRELATION FUNCTIONSB) CORRELATION FUNCTIONS Gauge inv. operator O in the QFT Gauge inv. operator O in the QFT dynamical field dynamical field ϕϕ in the bulk in the bulk; e.; e.g global g global

current Jcurrent Jμ μ in the QFT corresponds to Maxwell field A in the QFT corresponds to Maxwell field Aaa in the bulk, scalar operator O in the bulk, scalar operator OBB to scalar field to scalar field ϕϕ and so on. and so on.

• Generating function for the operator O is the partition function of the bulk gravitational Generating function for the operator O is the partition function of the bulk gravitational theory.theory.

€

Zbulk[Φ → Φ0] = ei d d xΦ 0∫ O

QFT

ϕϕ00 is the boundary value of is the boundary value of ϕ.ϕ. n- n- point functions for O point functions for O

are found by taking functional derivativesare found by taking functional derivatives of Zof Zbulkbulk with with

respect to respect to ϕϕ00..

• Scaling dimensions Δ for the QFT operator O is fixed by Scaling dimensions Δ for the QFT operator O is fixed by the masses m of bulk fields . For free bosons and free the masses m of bulk fields . For free bosons and free fermions in the bulk we havefermions in the bulk we have

• For instance for a bulk scalar we have near the boundary For instance for a bulk scalar we have near the boundary Φ= ΦΦ= Φ00xxd-Δd-Δ+Φ+Φ11xxΔΔ++…….……. AAnd nd the two-point function for the two-point function for

O is O is €

ΔB (ΔB − d) = (mL)2, ΔB =d

2+ Lm

€

O(y1)O(y2) =2Δ − d

L

Φ1

Φ0

• Thus, the holographic correspondence allows computation of correlators in certain strongly coupled quantum critical theories. The most universal deformation away from conformal invariance is placing the theory at finite temperature

• Processes at finite temperature are very difficult to compute also for a weak coupled QFT. This is not so for the holographic correspondence:

• Boundary QFT at temperature T AdS black hole at Hawking temperature T.

• Thermal states of the boundary QFT are identified with black hole solutions in the bulk (Schwarzschild-AdS black holes):

C) FINITE TEMPERATURE

€

ds2 =L2

x 2− f (x)dt 2 +

dr2

f (x)+ dy idy i

⎡

⎣ ⎢

⎤

⎦ ⎥; f (x) =1−

x

x+

⎛

⎝ ⎜

⎞

⎠ ⎟

d

; T =1

4πx+

• The simplest bulk quantities one can calculate at finite temperature are the free energy F=-Tlog Z and the entropy S

The dependence of S from the spatial volume V and the temperature T is fixed by scale invariance but the coefficient c (the central charge) counts the degrees of freedom of the dual QFT. Notice that from the bulk point of view c is an area measured in Planck units and that validity of the classical gravity description (large curvature radius of the AdS ST) requires c>>1 (large N approximation!!)

The black hole solution describes the theory at equilibrium. Perturbing the system we can compute response functions (correlators) using the same formulas used at T=0.

€

S =(4π )d Ld −1

2κ 2dd −1Vd −1T

d −1 ∝ cVd −1Td −1; c =

Ld −1

κ 2

This basic structure of the holographic correspondence has been used to compute spectral functions of the dual QFT, nonanalycities at complex frequencies (quasinormal modes), to investigate the long-wavelength dynamics (hydrodynamical limit) of the system etc.. For instance, a interesting proportionality relation between the entropy density of the black hole and the shear viscosity of the QFT. has been found. But the most interesting results come up when we switch on a U(1) gauge field on the bulk i.e. we consider a QFT with finite charge density on the boundary

HOLOGRAPHIC SUPERCONDUCTORSHOLOGRAPHIC SUPERCONDUCTORS ( Horowitz, Hartnoll, Herzog, Gubser( Horowitz, Hartnoll, Herzog, Gubser….)….)

• Finite charge density means a VEV for the time-component of a current J0 on the boundary but to build a superconductor we also need a scalar ψ describing the effective interaction of the charge carriers in the background of ion lattice, eventually this will result in a charged scalar condensate

€

S = d4 x∫ −g R +6

L2−

1

4Fμν F μν − ∂ψ − iqAψ

2− m2ψ 2 ⎛

⎝ ⎜

⎞

⎠ ⎟

In the bulk this corresponds to a U(1) gauge field Aμ and a covariantly coupled complex scalar ψ (from now on d=3 and 2κ2=1)

• Now we need to generate a second order phase transition at T=Tc between a phase with unbroken U(1) symmetry, <O>=0, for T>Tc and one with broken symmetry <O>≠0 for T<Tc.

• In the bulk this can be realised starting from the usual RN solution with a trivial scalar field at T>Tc:

If at low temperature the theory allows for a charged black hole solution with non trivial scalar hair and the RN solution becomes unstable we have a phase transition generating in the dual QFT a charged scalar condensate and a superconducting phase

€

ds2 = −g(r)e−χ dt 2 +dr2

g(r)+ r2(dx 2 + dy 2); g(r) =

r2

L2+

Q2

4r2−

2M

r;

A0 =Q

r−

Q

r+

; χ = 0; ψ = 0

This superconducting instability can be thought of as a polarisation of the spacetime. Above Tc the whole charge is inside the RN black hole. Below Tc the hairy black hole is energetically favourite and the charge is largely carried by the scalar outside the black hole horizon. Because of no-hair theorems it is difficult to find black hole solutions with non trivial scalar hairs. This theorems can be circumvented by considering charged scalars around a charged black hole (Gubser)

The field equations for the scalar are€

€

ψ' '+g'

g−

χ '

2+

2

r

⎛

⎝ ⎜

⎞

⎠ ⎟ψ '−meff

2 ψ = 0; meff2 = m2 −

q2A02eχ

g

Stability of the AdS ST requires m2 to be above the

BF bound – 9/4L2

• The instability of the RN solution is due to the negative contribution to the effective mass. At low temperature this will cause the scalar hair to form.

• The scalar and electric potential behaves asymptotically

€

ψ =ψ1

rλ++

ψ 2

rλ−+ .........; A0 = μ −

ρ

r; λ ± =

3 ± 9 + 4m2L2

2

• The holographic duality implies ψ 2 = <O2> an hairy black hole will correspond to the formation of a scalar charged condensate on the boundary.

Using appropriate boundary conditions one can integrate numerically the field equation of the theory and find the dependence of ψ 2 from the temperature

CONDUCTIVITY

According to the AdS/CFT correspondence transport phenomena in dual QFT are related to perturbations of bulk fields. Perturbations of gtx and Ax with harmonic time dependence and zero spatial momentum decouple and we have

• This curve is qualitatively similar to that obtained in BCS theory and observed in many superconducting materials. The condensate raises quickly when the material is cooled below Tc and goes to a constant as T goes to zero. Near Tc has the square root behavior (1-T/Tc)1/2 predicted by the Landau-Ginsburg theory

€

Ax ' '+g'

g−

χ '

2

⎛

⎝ ⎜

⎞

⎠ ⎟Ax '+

ω2

g2−

A0 '2

g

⎛

⎝ ⎜

⎞

⎠ ⎟e

χ −2q2ψ 2

g

⎡

⎣ ⎢

⎤

⎦ ⎥Ax = 0

• This has to be solved with purely ingoing boundary conditions at the horizon and asymptotically (r=∞)

From the AdS/CFT dictionary we have

for the electric field Ex , the current Jx and the conductivity σ on the boundary

€

Ax = Ax(0) +

Ax(1)

r+ ....

€

Ex = −Ax(0); Jx = Ax

(1), σ =Jx

Ex

= −iAx

(1)

ωAx(0)

• Upon numerical integration one finds the real and imaginary part of the conductivity

As we lower the temperature from Tc a gap opens as expected in BCS theory. There is a delta function at ω=0 (DC conductivity), but this cannot be seen from the numerical solution of the real part. The immaginari part has a pole Im(σ) ≈ 1/ω which from the Kramers-Kronig relations imply Im(σ) ≈ δ(ω).

In the limit T=0 from a BCS type description one would expect In the limit T=0 from a BCS type description one would expect an exponential suppression of the conductivity an exponential suppression of the conductivity ≈e≈e-Δ/T-Δ/T . .

ButBut this is not the case, in this limit there is still a small this is not the case, in this limit there is still a small conductivity even at small frequencies. conductivity even at small frequencies.

CHARGED DILATONIC ADS BLACK HOLE AND HOLOGRAPHIC “EXOTIC METALS”

(work done in collaboration with P. Pani and G. D’Appollonio, JHEP

3(2010)100) • The idea is to consider bulk AdS Einstein-Maxwell

dilaton gravity in which a REAL scalar is not covariantly but NONMINIMALLY coupled to the U(1) field

€

S = d4 x∫ −g R −1

4f (ψ )Fμν F μν −

1

2(∂ψ )2 −V (ψ )

⎛

⎝ ⎜

⎞

⎠ ⎟

• f(ψ) is a coupling function. To allow the RN-AdS black f(ψ) is a coupling function. To allow the RN-AdS black hole solution at ψ=0 ( corresponding to a UV fixed point) hole solution at ψ=0 ( corresponding to a UV fixed point) we must havewe must have

€

f (ψ ) =1+α

2ψ 2 + O(ψ 3); V = −

6

L2+

β

2L2ψ 2 + O(ψ 3)

• SUGRA actions stemming from string theory indicate the SUGRA actions stemming from string theory indicate the ψ=∞ IR behaviour f(ψ)= eψ=∞ IR behaviour f(ψ)= ea ψa ψ

• Motivations for considering Motivations for considering nonminimal couplings between nonminimal couplings between the scalar and the U(1) field:the scalar and the U(1) field:

SUGRA and Low-energy effective actions for string SUGRA and Low-energy effective actions for string theorytheory If hairy dilatonic black hole solution exist they should If hairy dilatonic black hole solution exist they should be dual to NEUTRAL CONDENSATE in the boundarybe dual to NEUTRAL CONDENSATE in the boundaryDiscovery of phase transitions of the dual QFT not Discovery of phase transitions of the dual QFT not generated by an U(1) symmetry breaking, possibly generated by an U(1) symmetry breaking, possibly describing some exotic metallic phasedescribing some exotic metallic phase

Stability of the RN-ADS solution against small scalar Stability of the RN-ADS solution against small scalar perturbations is investigated by expanding ψ in Fourier modesperturbations is investigated by expanding ψ in Fourier modes

€

ψωk =R(r)

re i(k1x +k2y−ωt ), f 2R' '+ ff 'R'+[ω2 −V ]R = 0,

V = fk 2

r2+

f '

r+ meff

2 ⎡

⎣ ⎢

⎤

⎦ ⎥, meff

2 = m2 −α

2A0 '2

€

Tc =QL

16πL2

12 − γ

γ1/ 4

⎛

⎝ ⎜

⎞

⎠ ⎟; γ =

2

α

9

4+ β

⎛

⎝ ⎜

⎞

⎠ ⎟

• The presence of this instability is confirmed by explicit numerical The presence of this instability is confirmed by explicit numerical solution of the field equations: below Tsolution of the field equations: below Tc c appears an hairy solution of appears an hairy solution of the field equations whose free energy Fthe field equations whose free energy Fhairyhairy < F < FRNRN

If the nonminimal coupling α is large enough it can lower the mass If the nonminimal coupling α is large enough it can lower the mass of the excitation below the BF bound generating a tachyonic mode of the excitation below the BF bound generating a tachyonic mode destabilizing the RN background. Approximate criteria for destabilizing the RN background. Approximate criteria for instability give the critical temperatureinstability give the critical temperature

• Below Tc the black hole solution develops a neutral scalar hair in the dual theory we have a second order phase transition and the formation of a neutral condensate. We have studied this transition for the following models

• Tc ≈ (ρ)1/2 and the behaviour of the condensate at constant charge density ρ as a function of T is

Free energy (left) and specific heat (right) of the hairy (Red) and RN black hole below Tc

Near the critical temperature we have the universal Landau-Ginsburg scaling behaviour (1-T/Tc)1/2

• The equation for the perturbations of the gauge potential Ax is now

€

Ax ' '+g'

g−

χ '

2+

1

f

df

dψψ '

⎛

⎝ ⎜

⎞

⎠ ⎟Ax '+

ω2

g2−

A0 '2 f

g

⎛

⎝ ⎜

⎞

⎠ ⎟e

χ ⎡

⎣ ⎢

⎤

⎦ ⎥Ax = 0

• The equation with the usual boundary conditions is solved by numerical integration

• The new phase although not superconductive shows interesting electric transport properties presumably caused by the interaction of the charge carriers with the condensate. The optical conductivity behaves qualitatively similar for the four class of models

• The conductivity approaches a constant value at large ω (determined by a UV relativistic dispersion relation) , has a minimum at low frequencies then reaches a constant value at ω=0 which can be considerably larger then the constant value at high ω . This is enhanced for large values of the non-minimal coupling. The effect is reminiscent of a DRUDE PEAK in ordinary metals.

€

Re[σ ] =kτ

1+ ω2τ 2; k =

ne2

m; τ ⇒ relaxation time due to scattering

€

• The resistivity does not increase monotonically with the temperature as for usual conductors but displays a minimum

• This is reminiscent of the KONDO effect caused in real metals with magnetic impurities by the interaction of the magnetic moment of the conduction electrons with the magnetic moment of the impurity

ZERO TEMPERATURE LIMITZERO TEMPERATURE LIMIT• Phase transition occurs also at T=0 Phase transition occurs also at T=0 Quantum phase Quantum phase

transitiontransition • AT T=0 the near-horizon geometry of the RN BH is AT T=0 the near-horizon geometry of the RN BH is

AdSAdS22 × R × R22 . . • At T=0 the instability of the RN solution is related to At T=0 the instability of the RN solution is related to

the fact that the BF bound for the fact that the BF bound for AdSAdS2 2 is more stringent is more stringent than that for AdSthan that for AdS44

• Generically, the near-horizon behaviour of the hairy Generically, the near-horizon behaviour of the hairy solution is of the Lifshitz type: solution is of the Lifshitz type:

• In our case we have the near-extremal solutionIn our case we have the near-extremal solution€

ds2 = −dt 2

r2υ+ l2 dr2

r2+

dx idx i

r2

€

ds2 = −λdt 2 + λ−1dr2 + r2−2hdx idx i,

λ = λ 0rw 1−

m

rη

⎛

⎝ ⎜

⎞

⎠ ⎟, ψ =ψ 0 −ξ ln r

• The frequency dependence of the conductivity in the extremal limit can be studied by rewriting the equation for the perturbation of Ax as a Schrödinger equation

€

d2Ψ

dz2+ ω2 −V (z)( )Ψ = 0, Ψ = f Ax ,

dr

dz= ge−χ , V (z) = gf (A0

' )2 −1

f

d2 f

dz2,

• One has just to solve a 1D Schrödinger equation for a particle incident on a potential barrier from the right, with purely ingoing conditions on the horizon (z=-∞)

•The conductivity is determined by the reflection coefficient R of the barrier €

Ψ=e ikz + R e−ikz,

€

σ(ω) =1− R

1+ R−

i

2ω

1

f

df

dz

⎡

⎣ ⎢

⎤

⎦ ⎥z= 0

• Using the analytic form of the near horizon solution one finds the small ω scaling behaviour for the T=0 solution

• Similar scaling behaviour have been found by Goldstein et al . [arxiv: 0911.3586] for extremal BH in models with exponential coupling function eaψ . Typical Schrodinger potential (left) and electrical conductivity (right) [black: a=1,b=0; red: a=b=(3)-

-1/2]

€

σ(ω) ≈ ω2+

bξ

1−bξ

• In the “Schrödinger description” the presence of a peak in the real part of σ at small values of ω when T>0 is due to the fact that V(z) is not positive definite and can support a resonance near ω=0. The potential receives a negative contribution from the nonminimal coupling term between the scalar and the gauge field:

LAST DEVELLOPMENTS: Introducing in the Lagrangian a Stückelberg field θ nonminimally coupled with the scalar

one can see θ as a phase and ψ as a modulus of a complex field. The phase with broken U(1) symmetry becomes superconductive, whereas the exotic metal phase is recovered for large values of α/q [Liu, Sun, arxiv:1006.2726]WORK IN PROGRESS: Presence of background magnetic field, i.e hairy dyonic black hole solutions in the bulk

AND MANY OPEN QUESTIONS……………

€

J(ψ ,q)(∂μθ − Aμ )2