Holographic Superconductors with Higher Curvature Corrections

Sugumi Kanno (Durham)

work w/

Ruth Gregory (Durham)Jiro Soda (Kyoto)

arXiv:0907.3203, to appear in JHEP

2

Introduction

It would be very exciting if we could explain high temperature superconductivityfrom black hole physics.

We verify this numerically and analytically.

According to holographic superconductors, scalar condensation in black hole systemexists. This deserves further study in relation to the “no-hair” theorem from gravityperspective.

Holographic Superconductors

Since the stringy corrections in the bulk corresponds to the fluctuations from large N limit in holographic superconductors, it is expected the stringy corrections make holographic condensation harder.

Hartnoll, Herzog & Horowitz (2008)

・ the critical temperature is stable under stringy corrections.cT

What we are interested in is if

・ the universal relation between and : is stable under stringy corrections.8g cT g cT

Horowitz & Roberts (2008)g: The gap in the frequency dependent conductivity

3

52

12 S d x g R

L

2

2 2( )

r Mf r

L r

2

2

4( ) 1 1

2

rf r

L

242R R R R R

2

4

1M L

r

Gauss-Bonnet Black Hole

Action

BH solutions

2 2

2 2 2 2 22

( )( )

dr rds f r dt dx dy dz

f r L

2L , 0 2

2L 2

, 4L

Hawking temperature1/4

2 3/2

1( )

4H

H

r r

r MT f r

L L

Gauss-Bonnet term

Coupling constant >0

Constant of integration related to the ADM mass of BH

2effL

2eff

1L

Asymptotically vanishes.( )r

0

2 1/4( )Hr MLHorizon is at

0

When rH (=M ) decreases, temperature decreases(This is a nature of AdS spacetime)

… Chern-Simons limit

4

C

r

C

r

Gauss-Bonnet Superconductors – probe limit

Action (Maxwell field & charged complex scalar field)

2 25 21

4S d x g F F iA m

EOMs23 2

0r f

2 2

2

30

f m

f r f f

Regularity at Horizon (2) : ( ) 0Hr 4

( ) ( )3H H Hr r r

Asymtotic behaviors 2( )r

r

2

eff2 4 3L

L

Boundary condition in the asymptoric AdS region (2) : 0C is fixed

EOMs are nonlinear and coupled

( ) , , A r ( )r Static ansatz: ( )iA r( )rA r ( )i re 00

Mass of the scalar filed

Need 4 boundary conditions

Const. of Integrations

Solutions are completely determined

determined

According to AdS/CFT, we can interpret , so we want to calculateCO CHowever…We calculate this numerically first.

22

3m

L

5

Numerical Results

1/30.198cT

0.0001

1/30.186cT

0.1

1/30.171cT

0.2

1/30.158cT

0.25

CO

c

T

T

1

cT

O

Critical Temperature decrease

The effect of is to make condensation harder.

increase

Chern-Simons limit

1L

6

Towards analytic understanding

E.g.) The numerical solution for

r

Near horizon

4( ) ( )

3H H Hr r r N

ear a

sym

pto

tic AdS re

gio

n

C

r

21( ) ( ) ( ) ( )

2H H H H Hr r r r r r r r

b.c.

b.c.

Matching at somewhere

7

( ) (1) 1z z (1) 3

(1)4

Analytic approach

Change variable : Hrzr

EOMs2 2

4

1 20Hr

z z f

2 2

4 2 2

1 30Hrf

f z z f L f

2

2211 (1) (1) 1

2 2

Lz

4

222 2

15 3(1) (1) (1) 1

64 2 64 H

Lz

L r

0 1Hr r z

( ) (1) 1 z z

2( ) z z

( )z

Near horizon (z=1)

Near asymotoric AdS region (z=0)

(1)

2 2

411 1

1 2Hz

z z

r

z z f

21(1) 1

2z

(1) 0 3

(1) (1)4

Boundary Condition

Region :

2 2

4 2 21

1 1

1 3Hz

z z

rf

f z z f L f

21(1) 1

2z

2( )z qz ( )z D z D z

Solutions in the asymptotic region

(0) (1)z1

(0)2 q

21(0) (0) (0)

2z z D z

fixedHqr 0D

Boundary Condition

D z

Now, match these solutions smoothly at1

2z 1 1

, ,2 2

1

2z

1 1,

2 2

1

2z

'd

dz

8

3

3 24 1

2H

H

Br

Br L

Results of analytic calculation

Solutions

213(1)

8 2HrC

2 2

5 3 15 3(1) 8

2 2 64 2Hr

L L

Condensation is expressed by

: Critical temperature1/3

2 1cT BL L

11

213 3

3 3

13 22 1

8 2c

c c c

TT T

T T T T

O

2(1)

Go back to the original variable : Hr zr

HC r D 2

H

qr

2HBr

L 2

HrTL

Hawking temperature

3 3

2 3 3

21c

c

T T

L T T

COAdS/CFT dictionary gives a relation :

0 cT Tat0 cT Tfor

1/30.201 ( =0.0001)cT 1/30.196 ( =0.1)cT 1/30.191 ( =0.2)cT 1/30.188 ( =0.25)cT

1/30.198 ( =0.0001)cT 1/30.186 ( =0.1)cT 1/30.171 ( =0.2)cT 1/30.158 ( =0.25)cT

Numerical result

Good agreement!

1

2

1c

T

T

O Typical mean field theory result for the second order phase transition.

9

Conductivity of our boundary theory

Electromagnetic perturbations

If we see the asymptotic behavior of this solution,

2 22

2 2

1 k 20

fA A A

f r f r f f

4( ) ( ) H

irA r f r

A in the bulk AdS/CFT JGauge field Four-current on the CFT boundary

Consider perturbation of Aand its spatial components k x( , , ) ( )i i i ti iA t r x A r e e

B.c. near the horizon : ingoing wave function Hr The system is solvable.

(0) 2 2 2(2)eff(0)

2 2

k log

2

A LA rA A

r r

The conductivity is given by

(2)

(0)

k=0

2

2

A i

i A

: General solution

Arbitrary scale, which can be removed by an appropriate boundary counter term

Need to solve numerically with this b.c. toobtain , asymptotically.

( )A r(0)A (2)A

10

Conductivity and Universality

0.0001 0.152

c

TT

0.1 0.151

c

TT

0.2 0.152

c

TT

0.25 0.152

c

TT

realim

aginary

pole

exis

ts

0

0 0

0

pole

exis

ts

pole

exis

ts

pole

exis

ts

0.0001 0.1 0.2 0.25

The universal relation is unstable in the presence of GB correction.8g

cT

As increases, the gap frequency becomes large.

g g g g

11

Summary

The higher curvature corrections make the condensation harder.

The universal relation in conductivity is unstable under the higher curvature corrections.

We have found a crude but simple analytical explanation of condensation.

In the future, we will take into account the backreaction to the geometry.