Lovelock holographic superconductors · Lovelock holographic superconductors Olivera Miıkovi·c and Ligeia ArÆnguiz Ponti–cia Universidad Católica de Valparaíso, Chile Universidad

Lovelock holographic superconductors

Olivera Miškovic and Ligeia Aránguiz

Pontificia Universidad Católica de Valparaíso, Chile

Universidad Técnica Federico Santa María, Chile

To appear soon

Thirteenth Marcel Grossmann Meeting - MG13, July 1-7, 2012, Stockholm

July 2, 2012Olivera Miškovic (PUCV) () Holographic superconductors July 2, 2012 1 / 28

Outline

Motivation and methodology

Action and equations of motion

Renormalization and boundary terms

Free energy of a holographic superconductor and phase transitions

Conclusions and open questions

Olivera Miškovic (PUCV) () Holographic superconductors July 2, 2012 2 / 28

Motivation

High-Tc superconductors: have a critical temperature above the upperlimit allowed by BCS theory (30K). There is no good theory that explainstheir properties.

AdS/CFT correspondence: powerful theorical method to investigatestrongly coupled field theories, such as high-Tc superconductors, analytically.

To explain various types of holographic high-Tc superconductors, one needsdifferent dual AdS gravities ⇒ Inclusion of higher-derivative correctionsin strongly coupled dual theory

Examples

Gauss-Bonnet term quadratic in the curvature changes the universal relationTc/energy gap and is able to make the condensation easier (higher Tc ).BI electrodynamics makes the condensation harder (lower Tc ).


Motivation

Field content necessary to describe superconductors in the context ofAdS/CFT

AdS gravity Holographic theory Comment

Metric fieldgµν(x)

TemperatureT = Const

In AdS gravity, T isthe Hawking Temperatureon the horizonof a non-extreme black hole

Gauge fieldAµ(x)

EM current J i Conductivity σ relatesJ i and the electricfield (Ohm Law)

Symm. breaking fieldΨ(x),Bµ(x), . . .

Order parameterO, V i , . . .

High-Tc superconductors,p-wave superconductors, . . .


Method

Fields in AdSd+1 gravity (bulk): X ={gµν,Aµ,Ψ

}Boundary conditions (r → ∞): gµν(x , r)→ g(0)ij (x)

Aµ(x , r)→ A(0)i (x)

Ψ(x , r)→ Ψ(0)(x)

Gravitational partition function : Z [X ] ' e−Iclass [X ]∣∣∣X→X(0)

AdS/CFT correspondence: ZGr[X(0)] = ZQFT [X(0)]

⇒ Iclass[X(0)] is the quantum effective action for QFT

Boundary fields in AdSd+1 gravity become the sources in QFTd :

〈O〉QFT =δIclass[X(0)]

δX(0)O =

{τij , J i ,OΨ

}(stress tensor, electric current, order parameter)


Motivation

In order to obtain information about a holographic superconductor, one has tosolve nonlinear field equations of matter fields coupled to gravity in AdS space,that is very diffi cult in general.

In recent work on the Holographic Superconductors (that deal with nonlineareffects), the authors study:

— [Franco et al 2009] They study the Stückelberg superconductor

— [Herzog 2011] Tc, superconductivity, critical exponent, free energy, inthe black hole background

— [Gubser 2008, Hartnoll et al 2008, Franco et al 2009,Jing et al 2010] Numerical calculations

— [Horowitz 2010, Herzog 2011, Cai et al 2011, Ge 2011, Kanno2011] Analytical calculations with/without backreaction


Motivation

Diffi culties in all these approaches:

Nonlinear equations cannot be solved exactly.

Study of very particular cases (no general analysis), for example:m2 = −3/`2, particular dimension d = 3 or 4, etc.Matching the expressions between three approximations: asymptoticsolutions, near-horizon solutions, and T ' Tc.

Our approach:

Based on the fact that all information about a superconductor is contained inthe free energy of the system.Knowing the free energy, the phase tansitions are analysed from itsdiscontinuities.Our approach has been made possible by the universal extrinsicregularization method easily applicable to higher-order gravities in anydimension.



Lovelock gravity in d + 1 > 4 dimensions = General Relativity withhigher-order corrections, polynomial in the Riemann tensor Rµ

ναβ

IG [gµν] =1κ2

[d/2]∑p=0

αp2p+1

∫dd+1x

√g δ

ν1 ···ν2pµ1 ···µ2p R

µ1µ2ν1ν2 · · ·R

µ2p−1µ2pν2p−1ν2p

Einstein-Gauss-Bonnet (EGB) AdS action (first three terms )

IG [gµν] =12κ2

∫dd+1x

√g[R − 2Λ+ α

(R2 − 4RµνRµν + RµνλσRµνλσ

)]Standard notation: α0 = −2Λ = d (d−1)

`2α2 = α

α1 = 1 2κ2 = 16πGN

Various different effective AdS vacua. For being precise, we chose the one

that behaves as `2eff =2α(d−2)(d−3)

1−√1− 4α

`2(d−2)(d−3)

→ `2, when α→ 0.



Non-linear electrodynamics (NED) action for Abelian gauge field Aµ

INED =∫dd+1x

√−g L(F 2)L(F 2)= arbitrary function of the invariant F 2 = gµαg νβFµνFαβ

Complex Scalar Field, Stückelberg holographic superconductor

IS =12κ2

∫dd+1x

√−g

[−12(∂Ψ)2 − 1

2m2Ψ2 − 1

2F (Ψ) (∂p − A)2

]

Ψ = Ψ e ip complex scalar field (Ψ, p ∈ R)

F (Ψ) ≥ 0 U(1) invariant

U(1) transformations{

Ψ→ Ψ , p → p − αAµ → Aµ − ∂µα

Total bulk action I0(g ,A,Ψ, p) = IG(g) + INED(A, g) + IS(Ψ, p, g)



Equations of motion:

Rµν − 12 gµν R − gµν Λ+Hµν = Tµν + τµν gravitational field equation

∇ν

(F µν dL

dF 2

)= −F (Ψ) (∇µp − Aµ) generalized Gauss law(

�−m2)

Ψ =12dFdΨ

(∇p − A)2 scalar field equations

∇µ

[F (Ψ) (∇µp − Aµ)

]= 0

Hµν = Lanczoc tensor (quadratic in Rµναβ, correction due to the GB term)

Energy-momentum tensor of NED: Tµν =12 gµν L+ dL

dF 2 2FµλF λν

Energy-momentum tensor of the scalar field:τµν =

12 gµν

[− 12 (∂Ψ)2 − 1

2 m2Ψ2 − 1

2 F (Ψ) (∂p − A)2]+

+ 12 ∂µΨ∂νΨ+ 12 F (Ψ)

(∂µp − Aµ

)(∂νp − Aν)

U(1) gauge fixing: p = 0Olivera Miškovic (PUCV) () Holographic superconductors July 2, 2012 10 / 28

Charged black hole solutions

Static black hole in the coordinates xµ = (t, r , ym)

ds2 = gµν(x) dxµdxν = −f (r) dt2 + dr2

f (r)N(r)+ r2δmn dymdyn

Planar black hole: flat transversal section δmn dymdyn

Event horizon r+: f (r+) = 0 (the largest root)

Black hole hair: N(r) = eχ(r ) (no hair: χ = 0, Ψ = 0)

Hawking temperature on the horizon: T = 14π f

′(r+)√N(r+)

Gauge field : At = φ (r) , Ai = 0

Electric potential only, we do not look at the magnetic properties of thesuperconductor.

Scalar field: Ψ = Ψ(r)



Exact solution without condensate: Ψ = 0, for suffi ciently high temperaturesMetric function:

f 2(r) =r2

2α (d − 2) (d − 3)

{[1−

(1− 4α (d − 2) (d − 3)

`2− µ

rd

+1

d (d − 1) rd

(rdL− 4qrE + (d − 1) 4qφ

)∣∣∣∣rr+

)1/2

Electric field: E dLdF 2 = −

qr d−1 ⇒ E (r)

Electric potential: dφdr = −E ⇒ φ(r)

Integration constants: q, µ —related to the electric charge and BH mass



Solution with scalar condensate: Ψ 6= 0, for temperatures T ≤ TcElectric potential: φ(r) = −

r∫r+dr E (r) (measured with respect to the horizon)

Chemical potential: Φ = φ(∞)− φ(r+)

Integral field equations, convenient for obtaining iterative solutions:

rd−1√Nf Ψ′ =

∫dr r

d−1√N

(m2Ψ− dF

dΨφ2

2f

)4rd−1

√NE

dLdF 2

= −q +∫dr r

d−1Fφ√Nf

rd−2Nf − α (d − 2) (d − 3) rd−4N2f 2 − r d`2= −µ+

∫dr2rd−1

d − 1 (Ttt + τtt )

If the free energy of this solution is lower than the one without condensate, aphase transition will occur.


Euclidean action

Free energy in QFTd on finite T from the Euclidean action: F = T IEclassTo avoid a conical singularity in the Euclidean spacetime at r+, theEuclidean time τ has to be periodic: τ ' τ + T−1

Flat horizon γmn = δmn gives an infinite volume Vd−1 =∫dd−1y

√γ

⇒ we work with densities (energy density, entropy density, charge density)

The bulk action evaluated on-shell

IE0 = −Vd−12κ2T

∞∫r+dr r

d−1√N(LEGB + LNED + LS)

We do not need an explicit form of the solution to integrate out this action:

IE0 =Vd−12κ2T

√Nrd−1

[f ′(1− 2α (d − 1) (d − 2) Nfr 2

)− 4φE

dLdF 2

]∞

r+

This expression is IR divergent (at r = ∞) ⇒ we have to renormalize itby adding the boundary terms


Counterterm regularization in AdS gravity

We define the total action that contains the boundary term: I = I0 + B

The total action satisfies:— Its Euclidean action evaluated on-shell is finite.— Corresponding Noether charges are finite.— The action principle determines the thermodynamic ensemble of thecorresponding QFT.

The boundary terms are known for Lovelock AdS gravity in any dimension[Kofinas, Olea 2006]

(Note that this is not the case for holographic renolmalization, where the series isapproximative and written explicitly only up to a certain dimension.)

These counterterms depend explicitly on the extrinsic curvature Kij = ∂r gijof the boundary, they are called “Kounterterms”.

They are background independent and consistent with standardholographic renormalization procedure.


Kounterterms

Pure gravitational boundary terms

Even dimensions, d + 1 = 2nThe boundary term B2n−1 is the n-th Chern form:

B2n−1 = c2n−1∫dd x√h1∫0dt δ

[j1 ···j2n−1 ][i1 ···i2n−1 ]K

i1j1

(12 R

i2 i3j2 j3− t2K i2j2 K

i3j3

)· · · ×

(12 R

i2n−2 i2n−1j2n−2 j2n−1

− t2K i2n−2j2n−2K i2n−1j2n−1

)Odd dimensions, d + 1 = 2n+ 1The boundary term B2n comes from the transgression form:

B2n = c2n∫dd x√h1∫0dt

t∫0ds δ

[j1 ···j2n ][i1 ···i2n ] K

i1j1

δi2j2

(12 R

i3 i4j3 j4− t2K i3j3 K

i4j4+ s2`2eff

δi3j3δi4j4

)· · · ×

(12R

i2n−1 i2nj2n−1 j2n

− t2K i2n−1j2n−1K i2nj2n +

s2`2eff

δi2n−1j2n−1

δi2nj2n

)


Kounterterms

For simplicity, we look at even dimensions only

c = − 1κ2(1− α∗)(2n− 2)! (−`

2eff)

n−1 (constant)

α∗ ≡ 2α

`2eff(d − 1) (d − 2) (convenient definition)

Planar black holes: Ri1 i2j1 j2 = 0 (boundary curvature)

Euclidean evaluation of the gravitational boundary term:

BEG = −V2n−22κ2T

`2n−2eff (1− α∗) limr→∞

(Nn−

12 f n−1f ′

).


Kounterterms

Conserved quantities obtained from the Noether theorem give finite charges forany solution

Electric charge for an arbitrary NED Lagrangian— from the Noether current for a U(1) gauge transformation

Q = −2Vd−1κ2

limr→∞

(rd−1

√NE

dLdF 2

)=Vd−12κ2

q (finite)

Black hole mass and vacuum energy— associated to the Killing vector ∂t

M =Vd−12κ2

limr→∞

Md (r) =(d − 1)Vd−1

2κ2µ (finite)

Evac = 0 (present only in odd dimensions)

Meven(r) = rd−1√Nf ′

1− α∗`2effNf

r2− (1− α∗)

(`2effNf

r2

) d−12


Thermodynamic ensemble choice

Adition of a boundary term in the AdS gravity action IE

⇔Legendre transformation of energy in a holographic theory

Action principle determines thermodynamic ensemble:

Canonical ensemble Grand canonical ensembleFree energy F (T ,Q) = M − TS G (T ,Φ) = M − TS −QΦFirst Law δF = −SδT +ΦδQ δG = −SδT −QδΦQSR IEcan = T

−1F IEgrand = T−1G

QSR=Quantum Statistical Relation

δIEcan = 0 when δT , δQ = 0δIEgrand = 0 when δT , δΦ = 0


Electromagnetic boundary term

We work in canonical ensemble:

BNED =2κ2

∫dd+1x ∂µ

(√−g AνF νµ dL

dF 2

)

because the on-shell variation of the total NED action is

δ (INED + BNED) = − 2κ2

∫dd x Aµ δ

(√−h nνF νµ dL

dF 2

)∼

∫dd x Φ δQ

Euclidean continuation of the NED boundary term:

BENED =2Vd−1κ2T

limr→∞

(rd−1φ

√NE

dLdF 2

)= −ΦQ

T.

Scalar field boundary term: BES = 0 (no divergences ).


Free energy

Summing up all contributions, the Euclidean action is:

IE =Vd−12κ2T

limr→∞

rd−1√Nf ′

1− α∗`2effNf

r2− (1− α∗)

(`2effNf

r2

) d−12+

−2πrd−1+ Vd−1κ2

Note that:

At the infinity, all divergences cancel out.

At the infinity, the finite terms give exactly M and Q.

On the horizon, using f (r+) = 0 and φ(r+) = 0, the only expression thatdoes not vanish is the entropy:

S =2πVd−1

κ2rd−1+ =

Area4G


Free energy

The final result for the free energy is the Quantum Statistical Relation

TIE = M − TS

This is a general formula for static, hairy black holes in any d , for arbitraryfunctions L(F 2) and F (Ψ), and arbitrary GB coupling constant α, where

M =Vd−12κ2

limr→∞

rd−1√Nf ′

1− α∗`2effNf

r2− (1− α∗)

(`2effNf

r2

) d−12

T =14π

f ′(r+)√N(r+)

General solution is known if Ψ = 0 [Miškovic & Olea, 2011]


Phase transitions

In order to study the phase transitions, first remember:

—Helmholtz free energy in the canonical ensemble is F (T ,Q) = M − TS .—The system tends to the state where F is minimal.

—Local extremum of F at fixed temperature means

∂F∂Q

= 0⇒ Q = Qextr(T )

—States with minimal F :

Fmin(T ) = F (T ,Qextr(T ))

—There is a second order phase transition if d2Fmin(T )dT 2 has a discontinuity at Tc

when Fmin(T ) reaches the absolute minimum (for any T ).


Phase transitions

In our case:

It is enough to look at the asymptotic solutions. Similarly to theholographic reconstruction of spacetime, one can solve the equations alongradius from the asymptotic region r >> èff to the horizon r = r+.

We study a class of gravity theories where the matter couplings and unknownfunctions behave as power series in èff/r .Boundary conditions are:

φ(r+) = 0 f (r+) = 0 N(r+) > 0 Ψ(r+) = 0φ(∞) = Φ f (∞) = fAdS N(∞) = NAdS Ψ(∞) < ∞

Comment: Physical quantity is chemical potential Φ = φ(∞)− φ(r+), butone cannot choose the referent point with φ(r+) 6= 0 when the scalar field Ψis present because its effective mass becomes infinite,m2eff = m

2 + g tt (r+)φ2(r+).

Asymptotic behavior of all functions can be classified depending on matterinteractions.

We find that the scalar field can propagate to the boundary and producespontanous symmetry breaking.


Phase transitions

- For example, in the range of the parameters a < a1,b < b1 < {2∆+ b− d + δ} and 0 < δ < min {δ1, d}, the solution is:

F (Ψ) =F0ra+F1ra1

+ · · ·

L(F 2) =L0rb+L1rb1

+ · · ·

f (r) =r2

`2eff

[1− r

d+

rd− f1

(1rd− r

b−d+

rb

)+ · · ·

]

N(r) = 1+N1r2∆ +

N2r2∆+δ

+ · · ·

Ψ(r) =Ψ0r∆ +

Ψ1r∆+δ

+ · · ·

φ(r) = φ0

(1

rb−d− 1

rb−d+

)+ φ1

(1

rb1−d− 1

rb1−d+

)+ · · ·

- The parameters a, b define the interaction of scalar and EM field, respectivelly.- Minimal scalar coupling (a = 2∆) and Born—Infeld-like fields (b = 2d − 2)belong to this class of solutions.


Phase transitions

Results:Conformal dimension ∆ of the scalar field is related to its mass as∆ (∆− d) = m2`2eff. We choose a positive branch ∆ ≥ d

2 .Arbitrary constants f0 and φ0 are related to the conserved charges

M = − (d−1)Vd−12κ2`2eff

(1− α∗) f0 and Q =Vd−1(λ+d )L02κ2λ(λ+1)φ0

.

The scalar field constant Ψ0 ≡ 〈O〉 becomes an order parameter in theholographic QFT.All other constants are determined in terms of the power parameters, chargesand the order parameter. For example, the scalar field produces nonvanishinghair N(r),

N1 =∆Ψ20

2 (1− α∗) (d − 1) , N2 =2∆ (a− ∆+ 2)Ψ0Ψ1(d − 1) (1− α∗) (a+ 2)

.

We also find the temperature as function T (r+, φ0,Ψ0).Free energy is expressed in terms of (r+, φ0,Ψ0) as

F = β1(r+, φ0) + β2(r+, φ0)Ψ20 + β3(r+, φ0)Ψ

δ0 + · · ·

where βn are known functions.Olivera Miškovic (PUCV) () Holographic superconductors July 2, 2012 26 / 28

Phase transitions

Note that we deal with the planar black holes (r+ >> èff) consistent withthe high-Tc limit, where F is also large.

Our expression for the free energy F (T ,Q,O) has the form

F (r+,Ψ, φ0) = F (T (r+, φ0,Ψ0),Q(φ0),Ψ0)

Dependent variables are T = T (r+, φ0,Ψ0) and Q = Q(φ0).The extremum for fixed T and Q is(

∂F∂r+

)Ψ,φ

(∂r+∂Ψ0

)T ,Q

+

(∂F

∂Ψ0

)r+,φ

= 0

There are at least two solutions Ψ0 = 0 and Ψ0 6= 0 for critical values Tcand Qc , that depend on the parameters in the theory. Second order phase

transition is characterized by discontinued d 2FmindΨ20

∣∣∣Tc ,Qc

and

Fmin(Ψ0 6= 0) < Fmin(Ψ = 0).


Conclusions and open questions

We found exact (Quantum Statistical Relation) and approximate (large r+)expressions for the Helmholtz free energy.

We have to complete the parameters analysis to identify all matterinteractions with gravity whose holographic duals correspond to thesuperconductors.

Then we have to leave thermal equilibrium and compute a conductivity σ forobtained high-Tc superconductors.

We have to analyse the consequences of our expressions when T ' Tc andfind the critical exponent.

Also, we have to study particular cases of L(F 2), F (Ψ) and compare ourresults with the known numerical and analytic solutions.


Documents

Lovelock holographic superconductors · Lovelock holographic superconductors Olivera Miıkovi·c and Ligeia ArÆnguiz Ponti–cia Universidad Católica de Valparaíso, Chile Universidad