Download A Gravity Model for Superconductors & (Non

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)
R=8πGT
Why bother to know about gravity?
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 physical observations
Lesson 1 (Holographic Principle):
In the strong coupling regime, field theory degrees of freedom
could have been repacked into gravity degrees of freedom
Horizon
Black hole
Dilute Gas
S~V
weakly interact
Fluid/Solid
Bekenstein-Hawking: S
(`tHooft,98)
strongly interact
~A
Some observations

L2
ds  2 ( dt 2  dx 2  dr 2 )
r
2
Flat D-dimensional CFT
Conformal symmetry SO(D,2)
x’
r
IR
UV
(D+1)-dimensional anti-de Sitter
Isometry SO(D,2)
(□-m2)Φ(x,r)=0
m2 = Δ(Δ-D)
(Witten,98)
x
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 2 (AdS/CFT correspondence):
Interaction could be encoded into geometry
More surprise to come
r
Gravity:
(Soft/hard) cut-off induces confinement
(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 (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)
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)
(2+1) HTSC
(3+1) Gravity model
Finite temperature
• TH~ horizon size, large black hole is stable
• HTSC is in thermal equilibrium with black hole at
Hawking temperature TH
Hawking radiation
T=0
Small T
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. (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
Hairy black hole
No hair
=φ
0
 1, 2  0, ( 2,1  0)
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…
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
• 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.
Prospects
• Gap
• D-wave
• phase diagrams; quantum critical point (Sachdev, Liu,
etc.) and insulator-superconductor phase transition
(Takayanagi et al.)
• microscopic mechanism
A practical thing to do
BCS-BEC
Graphene
…
I should learn more condensed matter
Thank You
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
(   3)  m2 L2
( r ) 
1
r
1

2
r
2
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
( r )  A0 ( x, r )
Er  F0 r   r 
AdS Bulk
Boundary QFT
(r)   

r
