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

Operator O(x) of dimension Δ<O(x)O(x’)> = |x-x’|-2Δ

x

x’

Flat D-dimensional CFTConformal symmetry SO(D,2)

(D+1)-dimensional anti-de SitterIsometry SO(D,2)

(□-m2)Φ(x,r)=0m2 = Δ(Δ-D)

z

Imagine a string stretching in between, we obtain Coulomb potential for attractive force

Lesson (AdS/CFT correspondence):Interaction could be encoded into geometry

(Witten,98)

(Maldacena,98)

IR

UV

V~1/|x-x’|

)( 2222

22 dzxddtz

Lds

More surprise to come

z

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)

Applied String Theory: strongly coupled system with approximate scaling symmetry

• Quark Gluon Plasma (RHIC)Drag forceJet Quenchingη/s

• QCDConfinement/deconfinementGluon scatteringBaryon/Hadron

• Quantum critical point• Superfluidity• High-Tc superconductivity

(1911 discovered, 1950 GL, 1957 BCS, 1986 HTSC)

Today’s goals

• Goal #1A minimum gravity model for HTSC

• Goal #2Fermionic 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 TChemical potential μCondensate 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

T=0 Small T Large T

Hawking radiation

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. (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 holeNo hair

SC phaseN phase

=φ

)0( ,0

0

1,22,1

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…

SummaryThe 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

BCS-BECGraphene…

I should learn more condensed matter

• 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 holea.k.a hairy black hole solution

Mass term has no explicit T dependenceV has no other higher order term

A: abelian gauge field U(1)φ: Higgs

• State-Operator correspondence:

x

AdS bulk

Boundary QFT

Operator of dimension Δ

Scalar field (Higgs) with mass m

21

21)(rr

r

22)3( Lm

• Time component gauge potential encodes the message of chemical potential and charge density at the boundary

rrr FE 0

AdS Bulk

Boundary QFT

r

r)(

),()( 0 rxAr