1. Holographic Superconductors
Jiunn-Wei Chen (NTU)
w/ Ying-Jer Kao, Yu-Sheng Liu, Debaprasad Maity,
Wen-Yu Wen and Chen-Pin Yeh
(talk largely based on Wen’s slides)

2. Holography Holograms NMR 3d information encoded on 2d surfaces
Finite resolution helps.

3. Can this Universe a giant hologram?
Black hole entropy scales as the surface of the horizon.
Information upper bound scales as the surface of the system as well?

4. AdS/CFT Correspondence (Maldacena, 98)
4 dim gauge field theory (SYM) is equivalent to a 10 dim (AdS_5 x S_5) string theory--- a holography and a strong-weak interaction dual!

5. Some observations m2 = Δ(Δ-D) Operator O(x) of dimension Δ
Flat D-dimensional CFT
Conformal symmetry SO(D,2) z
(D+1)-dimensional anti-de Sitter
Isometry SO(D,2) IR
(□-m2)Φ(x,r)=0
m2 = Δ(Δ-D) x x’ UV
(Witten,98)
Operator O(x) of dimension Δ
<O(x)O(x’)> = |x-x’|-2Δ
Imagine a string stretching in between, we obtain
Coulomb potential for attractive force
V~1/|x-x’|
(Maldacena,98)
Lesson (AdS/CFT correspondence):
Interaction could be encoded into geometry

6. More surprise to come The proof? Top down vs. bottom up Gravity:
(Soft/hard) cut-off induces confinement z
(Karch-Katz-Son-Stephanov,06)
Linear potential for long string
Field Theory:
Modify InfraRed physics
Lesson 3 (AdS/ ? correspondence):
Interesting physics could appear while away from AdS/CFT
The proof? Top down vs. bottom up

7. Applied String Theory: strongly coupled system with approximate scaling symmetry
Quark Gluon Plasma (RHIC)
Drag force
Jet Quenching
η/s
QCD
Confinement/deconfinement
Gluon scattering
Baryon/Hadron
Quantum critical point
Superfluidity
High-Tc superconductivity (1911 discovered, 1950 GL, 1957 BCS, 1986 HTSC)

8. Today’s goals Goal #1 A minimum gravity model for HTSC
Goal #2 Fermionic spectral function of HTSC
Goal #3
From S-wave to D-wave SC’s

9. Superconductors
BCS theory: electron-electron pairing through phonon exchange; not enough for HTSC
Ginzburg-Landau theory: low energy effective theory; breaking the (local) U(1) symmetry spontaneously---massive EM fields (Higgs mechanism)

10. Holographic Superconductors
Minimum model:
Breaking the U(1) symmetry spontaneously [local U(1) in the “bulk”, global U(1) at the boundary]
Essential ingredients:
Finite temperature T
Chemical potential μ
Condensate φ (same quantum
number as a fermion pair)
(3+1) Gravity model
(2+1) HTSC

11. Finite temperature TH~ horizon size, large black hole is stable
HTSC is in thermal equilibrium with black hole at Hawking temperature TH
Hawking radiation
Small T
T=0
Large T

12. Finite chemical potential
Place electric field along radius direction, particles with opposite charges will accumulate on boundary and horizon, giving a charged balck hole
Voltage established between them can be interpretated as chemical potential (q)μ,which is the work done by moving a unit charge from horizon to boundary. ﹢ ﹣ Er

13. Condensate
φ field is in balance between two competing forces: gravitational attraction and electric repulsion.

14. When black hole is too heavy (high T), φ will fall into the horizon
When black hole is too heavy (high T), φ will fall into the horizon. (normal state)
When black hole is not so heavy (low T), φ safely stays outside the horizon and forms a condensate. (superconducting state)
N phase
SC phase
No hair
Hairy black hole =φ

16. Tc [Hartnoll,Herzog,Horowitz, 08]
Bosonic condensation
Fermionic condensation
strongly correlated?
usual BCS ~ 3.5

17. Hc [Nakano,Wen,Phys.Rev.D78 (08)]

18. Goal #2: Fermionic spectral function of HTSC---measurable experimentally

23. More story…

25. The gap we found in the s-wave superconductor is “soft”.
Summary
The gap we found in the s-wave superconductor is “soft”.
p-wave superconductor appears to have a hard gap at zero temperature

26. Towards a holographic model of D-wave superconductors (JWC, Kao, Maity, Wen, Yeh)
At the boundary (field theory side), we need a symmetric traceless 2nd rank tensor to form the condensate.
In the bulk, we higged a symmetric traceless 2nd rank tensor.
However, we have more components than we want and some of them are unstable---a remaining problem
Condensate vs T and DC, AC conductivitives worked out nicely.

27. Fermi arcs in d-wave superconductors (Benini, Herzog, and Yarom)

28. Fermi arcs in d+id superconductors (JWC, Liu, Maity)

29. Normal and Hall conductivity

30. Prospects Fermi arcs: in the pseugap phase not SC phase
D-wave: stability (a hard problem)
phase diagrams; quantum critical point (Sachdev, Liu, etc.) and insulator-superconductor phase transition (Takayanagi et al.)
microscopic mechanism

31. Thank You

32. A practical thing to do I should learn more condensed matter
BCS-BEC Graphene …

33. Abelian Higgs model in AdS black hole a.k.a hairy black hole solution
Ginzburg-Landau feels curvature from AdS-BH
AdS-BH metrics receives no back reaction from GL sector. (probe limit)
AdS-BH
T increases with BH mass GL
A: abelian gauge field U(1)
φ: Higgs
Mass term has no explicit T dependence
V has no other higher order term

34. State-Operator correspondence:
Scalar field (Higgs) with mass m
AdS bulk x
Boundary QFT
Operator of dimension Δ

35. Time component gauge potential encodes the message of chemical potential and charge density at the boundary
AdS Bulk
Boundary QFT