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)

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

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?

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!

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

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

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)

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

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)

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

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

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

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

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

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

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

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

More story…

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

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.

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

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

Normal and Hall conductivity

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

Thank You

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

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

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

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