1. Holographic Models for High-Tc superconductors Jiunn-Wei Chen (NTU) w/ Ying-Jer Kao, Debaprasad Maity, Wen-Yu Wen and Chen-Pin Yeh (talk largely based on Wen’s slides)

2. R=8πGT Why bother to know about gravity?

3. 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!

4. Some physical observations Dilute Gas S ~ V Black hole Bekenstein-Hawking: S ~A Fluid/Solid Horizon Lesson 1 (Holographic Principle): In the strong coupling regime, field theory degrees of freedom could have been repacked into gravity degrees of freedom weakly interact strongly interact (`tHooft,98)

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

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

7. Applied String Theory (3 yrs old) for strongly coupled system 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 T H ~ horizon size, large black hole is stable HTSC is in thermal equilibrium with black hole at Hawking temperature T H T=0 Small T Large T Hawking radiation

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. ﹢ ﹢ ﹢ ﹢ ﹢ ﹢ ﹣ ﹣ ﹣ ﹣ ﹣ ﹣ ErEr

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

16. Tc [Hartnoll,Herzog,Horowitz, 08] Bosonic condensationFermionic 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. 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 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. Prospects Gap D-wave phase diagrams; quantum critical point (Sachdev, Liu, etc.) and insulator-superconductor phase transition (Takayanagi et al.) microscopic mechanism

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

29. Thank You

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

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

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