Holographic Superconductors with Higher Curvature Corrections

awfulhihatΠολεοδομικά Έργα

15 Νοε 2013 (πριν από 3 χρόνια και 6 μήνες)

73 εμφανίσεις

Holographic Superconductors



with Higher Curvature Corrections

Sugumi

Kanno

(Durham)

work w/


Ruth Gregory (Durham)

Jiro

Soda (Kyoto)

arXiv:0907.3203, to appear in JHEP

Introduction

1

It would be very exciting if we could explain high temperature superconductivity

from black hole physics.

We verify this numerically and analytically.

According to holographic superconductors, scalar condensation in black hole system

exists. This deserves further study in relation to the “no
-
hair” theorem from gravity

perspective.

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.

c
T
What we are interested in is if


the
universal
relation between and :

is stable under stringy corrections.

8
g c
T

g

c
T
Horowitz & Roberts (2008)

g

: The gap in the frequency dependent conductivity

5
2
12


S d x g R
L
 
  
 
 

2
2 2
( )
r M
f r
L r
 
2
2
4
( ) 1 1
2
r
f r
L


 
  
 
 


2
4
2
R R R R R
 
 

  
2
4

1
M L
r
 

 
 
Gauss
-
Bonnet Black Hole

2

Action

BH solutions



2 2
2 2 2 2 2
2
( )
( )
dr r
ds f r dt dx dy dz
f r L
     
2
L
,0


2
2
L
2
,
4
L


Hawking temperature

1/4
2 3/2
1
( )
4
H
H
r r
r
M
T f r
L L
  


  
Gauss
-
Bonnet term

Coupling constant >0

Constant of integration related to the ADM mass of BH

2
eff
L
2
eff
1
L

Asymptotically vanishes.

( )
r

0

2 1/4
( )
H
r ML

Horizon is at

0


When

r
H

(=
M
) decreases, temperature decreases

(This is a nature of
AdS

spacetime
)


Chern
-
Simons limit


C
r




 
C
r



Gauss
-
Bonnet Superconductors


probe limit

3

Action (Maxwell field & charged complex scalar field)

2 2
5 2
1
4
S d x g F F iA m


  
 
      
 
 

EOMs

2
3 2
0
r f

  
 
  
2 2
2
3
0
f m
f r f f

  

 
 
 
    
 
 
 
 
Regularity at Horizon (2) :

( ) 0
H
r


4
( ) ( )
3
H H H
r r r
 

 
Asymtotic

behaviors

2
( )
r
r

 
 
2
eff
2 4 3
L
L


 
  
 
 
Boundary condition in the
asymptoric

AdS

region (2) :

0
C


is fixed

EOMs are nonlinear and coupled



( ) , ,
A r



( )
r
 

Static
ansatz
:

( )
i
A r
( )
r
A r
( )
i r
e

0
0
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 calculate

C


O
C

However…

We calculate this numerically first.

2
2
3
m
L
 
Numerical Results

4

1/3
0.198
c
T


0.0001


1/3
0.186
c
T


0.1


1/3
0.171
c
T


0.2


1/3
0.158
c
T


0.25


C


O
c
T
T
1
c
T


O
Critical Temperature

decrease

The effect of is to make condensation harder.


increase

Chern
-
Simons limit

1
L

Towards analytic understanding

5

E.g.)
The numerical solution for


r

Near horizon

4
( ) ( )
3
H H H
r r r
 

 
Near asymptotic
AdS

region

C
r









2
1
( ) ( ) ( ) ( )
2
H H H H H
r r r r r r r r
   
 
     
b.c
.

b.c
.

Matching at somewhere



( ) (1) 1
z z
 
  
(1)



3
(1)
4


Analytic approach

6

Change variable :

H
r
z
r

EOMs

2
2
4
1 2
0
H
r
z z f

  
 
  
2
2
4 2 2
1 3
0
H
r
f
f z z f L f

  

 
 
 
    
 
 
 
 


2
2
2
1
1 (1) (1) 1
2 2
L
z
 
 

  
 
 


4
2
2
2 2
15 3
(1) (1) (1) 1
64 2 64
H
L
z
L r

  
 

     
 
 
0 1
H
r r z
     


( ) (1) 1
z z
 

   
2
( )
z z

 
( )
z


Near horizon (z=1)

Near
asymotoric

AdS

region (z=0)

(1)

2
2
4
1
1
1
1 2
H
z
z
z
r
z z f

  



 
 


2
1
(1) 1
2
z


 
(1) 0


3
(1) (1)
4
 


Boundary Condition

Region :

2
2
4 2 2
1
1
1
1 3
H
z
z
z
r
f
f z z f L f

  




 
 
 
    
 
 
 
 


2
1
(1) 1
2
z


 
2
( )
z qz
 
 
( )
z D z D z
 

 
 
 
Solutions
in the asymptotic region

(0)

(1)
z



1
(0)
2




q

2
1
(0) (0) (0)
2
z z
  
 
  

D z



fixed
H
qr

0
D


Boundary Condition

D z




Now, match these solutions smoothly at

1
2
z

1 1
, ,
2 2
 
   

   
   
1
2
z

1 1
,
2 2
 
   

   
   
1
2
z

'
d
dz
 

 
 
3
3 2
4 1
2
H
H
Br
Br L


 

 
 
Results of analytic calculation

7

Solutions

2
13
(1)
8 2
H
r
C
 


 






2 2
5 3
15 3
(1) 8
2 2 64 2
H
r
L L








   

Condensation
is expressed by

: Critical
temperature

1/3
2 1
c
T
BL L


 

 
 
1
1
2
1
3
3
3 3
13 2
2 1
8 2
c
c c c
T
T T
T T T T









 
 
 
 
 
 
 
 

 
 
 
O
2
(1)


Go back to the original variable :

H
r
z
r

H
C r D


 

2
H
q
r


2
H
B
r
L

2
H
r
T
L


Hawking temperature

3
3
2 3 3
2
1
c
c
T
T
L T T
 

 
 
C

O
AdS
/CFT dictionary gives a relation :



0

c
T T

at

0

c
T T

for

1/3
0.201 ( =0.0001)
c
T
 

1/3
0.196 ( =0.1)
c
T
 

1/3
0.191 ( =0.2)
c
T
 

1/3
0.188 ( =0.25)
c
T
 

1/3
0.198 ( =0.0001)
c
T
 

1/3
0.186 ( =0.1)
c
T
 

1/3
0.171 ( =0.2)
c
T
 

1/3
0.158 ( =0.25)
c
T
 

Numerical result

Good agreement!

1
2
1
c
T
T
 
 
 
 
O
Typical mean field theory result for the second order phase transition.

Conductivity of our boundary theory

8

Electromagnetic perturbations

If we see the asymptotic behavior of this solution,

2 2
2
2 2
1 k 2
0
f
A A A
f r f r f f



 
 
 
     
 
 
 
 
4
( ) ( )
H
i
r
A r f r


A

in the bulk

AdS
/CFT

J

Gauge field

Four
-
current

on the CFT boundary

Consider perturbation of

A

and its spatial components

k x
(,,) ( )
i i i t
i i
A t r x A r e e

  

B.c
.
near the
horizon : ingoing wave function

H
r
The system is solvable.



(0) 2 2 2
(2)
eff
(0)
2 2
k
log
2
A L
A r
A 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
. to

obtain , asymptotically.

( )
A r
(0)
A
(2)
A
Conductivity and Universality

9

0.0001


0.152
c
T
T

0.1


0.151
c
T
T

0.2


0.152
c
T
T

0.25


0.152
c
T
T

pole exists

0


0


0


0


pole exists

pole exists

pole exists

0.0001


0.1


0.2


0.25


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

8
g
c
T


As increases, the gap frequency becomes large.


g

g

g

g

Summary

10

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.