1. Toward a Holographic Model of d-wave Superconductors
Chen-Pin Yeh
National Taiwan University
With Jiunn-Wei Chen, Ying-Jer Kao, Debaprasad Maity and Wen-Yu Wen.

2. Outline Introduction to high temperature superconductor
How does AdS/CFT work for s-wave superconductor
A model for the holographic d-wave superconductor

3. Introduction (normal SuperConductor)
Many metals are superconducting when lowered to certain temperature, Tc (~few Kelvin)
Properties:
Zero D.C resistance,
Perfect diamagnetic,
Critical frequency

4. Introduction (normal SC)
Explained by the BCS theory (almost fifty years after the discovery)
Fermi surface
Why Tc small? k -k
BCS works because of
this hierarchy
Normal state: Fermi Liquid
Paring SC state

5. Introduction (High Tc SC)
BCS theory is so successful that it is embarrassed to discover the high Tc material, Cuprate in 1986, for
Tc~35K
Why Tc high?

6. High Tc physics Cuprate
Two dimension layer made of copper-oxygen plane
Un-doped parent compound: Mott Insulator and anti-ferromagnetic.

7. High Tc physics t-j model xt>>j Fermi-liquid metal ? xt<<j
t: hopping energy
U: on-site energy
J=4(t^2)/U:exchange energy
t-j model
xt>>j
Fermi-liquid metal ?
xt<<j
Pseudo-gap (spin liquid?) SC

8. d-wave symmetry High Tc ground state has d-wave symmetry
Quasi particle spectrum in BCS state

9. Experimental evidence of d-wave symmetry
s-wave
d-wave
Tunneling from
Normal metal to SC

10. Why AdS/CFT can help
In general sense, AdS/CFT is a duality between gravity theory with isometry
identified as conformal
symmetry in gauge theory
Quantum critical point is
a good starting place to
apply AdS/CFT
Study of gauge theory in strong
coupled region (like high Tc)
is usually difficult. AdS/CFT provides a alternative

11. holographic model for non-Fermi liquid
Non-Fermi liquid behavior is found in charged AdS black hole
Liu, McGreevy and Vegh 09’
One thing people can get
holographically is the
correlation functions of
the dual gauge theory
in large N, large coupling.

12. holographic model for s-wave SC
Charged scalar in (3+1) AdS black hole
Hartnoll, Herzog and Horowitz 08’
Linear response to electric perturbations
Order parameter is dual to ψ(r)

13. Simple model for d-wave SC
Landau-Ginsberg type model
N. Mermin 73’
Order parameter is the mix of all d-wave states
Particular choice of potential will give symmetry
with

14. Holographic model for d-wave SC
Put the following
In the (3+1) AdS Black hole background (probe limit)
Order parameter in field theory
is dual to Bij (i,j=x,y)

15. Holographic model for d-wave SC
Background solution ansatz
The equations of motion
The asymptotic behavior

16. Holographic model for d-wave SC
Condensate f1
Near the critical point
T/Tc
Signal the second order phase transition

17. Electrical perturbation
Need to turn on perturbations of B accordingly
Linearized equations of motion

18. Conductivity Ax equation of motion can be solved independently.
Equations of motion are the same for x y
Conductivity is isotropic
gap
Re[σ]
In-falling at horizon.
Near the boundary:
ω/Tc

19. Perturbation of B
For m^2<2 we have the following asymptotic behavior
After imposing in-falling condition for b at horizon, it
is hard to make b normalizable. However configuration of
finite energy can be found.

20. Issues of negative energy
When B is uncharged, the negative kinetic energy can be
Removed by imposing
When B is charged, the constrain is unknown to us yet

21. Two possibility
The d-wave background solution in this set up doesn’t exist
after including the constrains.
The background ansatz automatically satisfies the constrains
(this is true if B is uncharged). And the perturbation of A is
still decoupled from B and consistent solutions for B exit.
So the calculation of conductivity still holds.

22. Conclusion Currently done in AdS/CFT Properties of the strange metal
Strange metal-SC transition
Insulator-SC transition at zero temperature…
Our attempt
Strange metal-SC transition with the d-wave symmetry
order parameter.

23. Conclusion Future works
Magnetic interaction is important in the high Tc materials;
gravity theory should account for it.
How to understand the doping of Mott insulator
and pseudo-gap region holographically.
It is still hard to identify the boundary degrees of freedom.
Thus less can be said about quasi-particle spectrum, how
and why the paring formed.