General Relativity
and the Cuprates
Gary Horowitz
UC Santa Barbara
G.H., J. Santos, D. Tong,
1204.0519, 1209.1098
G.H. and J. Santos, to appear
Can one do more than reproduce qualitative
features of condensed matter systems?
Can gauge/gravity duality provide a
quantitative explanation of some mysterious
property of real materials?
We will argue that the answer is yes!
Many previous applications have assumed
translational symmetry. But:
Momentum conservation + nonzero charge
density => Infinite DC conductivity
Can have effective momentum nonconservation
in a probe approximation (Karch, O’Bannon, 2007)
or by adding a lattice.
Plan: Calculate the optical conductivity of a
simple holographic conductor and
superconductor with lattice included.
A perfect lattice still has infinite conductivity.
So we work at nonzero T and include
dissipation. (Earlier work by: Kachru et al; Maeda et
al; Hartnoll and Hofman; Zaanen et al.)
Main result: We will find surprising similarities
to the optical conductivity of the cuprates.
Our gravity model
We work with just Einstein-Maxwell theory:

Z
6
1
4 p
S= d x
g R+ 2
Fµ⌫ F µ⌫
L
2
This is the simplest context to describe a
conductor. We require the metric to be
asymptotically AdS
ds2 =
dt2 + dx2 + dy 2 + dz 2
z2
Want finite temperature: Add black hole
Want finite density: Add charge to the black
hole. The asymptotic form of At is
At = µ
⇢z + O(z 2 )
µ is the chemical potential and ρ is the charge
density in the dual theory.
Introduce the lattice by making the chemical
potential be a periodic function:
µ(x) = µ̄ [1 + A0 cos(k0 x)]
We numerically find solutions with smooth
horizons that are static and translationally
invariant in one direction. (Have to solve 6
coupled nonlinear PDE’s in 2D.)
Solutions are rippled charged black holes.
Charge density for A0 = ½, k0 = 2,
T/µ = .055
4
⇥⇤ x⇥
3
2
1
0
0
3
2
2
x
2
Conductivity
To compute the optical conductivity using linear
response, we perturb the solution
gµ⌫ = ĝµ⌫ + gµ⌫ ,
Boundary conditions:
Aµ = µ + Aµ
ingoing waves at the horizon
δgµν normalizable at infinity
δAt ~ O(z),
δAx = e-iωt [E/iω + J z + …]
induced current
Using Ohm’s law, J = σE, the optical
conductivity is given by
Fzx (x, z)
˜ (!, x) = lim
z!0 Fxt (x, z)
Since we impose a homogeneous electric
field, we are interested in the homogeneous
part of the conductivity σ(ω).
Review: optical conductivity with no lattice
(T/µ = .115)
6
1.0
5
⇥
4
0.6
3
Im
Re ⇥
0.8
2
0.4
1
0.2
0
0
5
10
15
⇥⇤T
20
25
0
5
10
15
⇥⇤T
20
25
With the lattice, the delta function is
smeared out
20
30
25
15
10
Im
Re
20
15
10
5
5
0
0
0
10
20
⇥T
30
40
0
10
20
⇥T
30
40
The low frequency conductivity takes the simple
Drude form:
K⌧
(!) =
1 i!⌧
30
18
25
⇥
20
Im
Re ⇥
17
16
15
15
10
0.10
0.15
0.20
0.25
0.30
⇥⇤T
0.35
0.40
0.45
14
0.10
0.15
0.20
0.25
0.30
⇥⇤T
0.35
0.40
0.45
Intermediate frequency shows scaling
regime:
B
| | = 2/3 + C
!
50
⌅
⇤
⇥
⌅
⇤
⇥
⌅
⇤
⇥
⇥⇤⇥ C
30
20
15
10
0.01
⌅
⇤
⇥
⌅
⇤
⇥
⌅
⇤
⇥
⌅
⇤
⇥
⌅
⇤
⇥ ⌅
⇤
⇥ ⌅
⇤
⇥ ⌅
⇤
⇥ ⇥
⌅
⇤
⌅
⇥
⇤
⌅
⇤
⇥⇤
⌅
⌅⌅
⇥⇤
⇥⇤
⌅⌅
⇥⇤
⇥⇤
⌅⌅
⇥⇤
⇥⇤
⌅⌅
⇥⇤
⇥⇤
⌅⌅
⇥⇤
⇥⇤
⌅⌅
⇥⇤
⇥⇤
⌅⌅
⇥⇤
⌅⌅
⇥⇤
⇥⇤
⌅⌅⌅
⇥⇤
⇥⇤
⇥⇤
⌅⌅⌅
⇥⇤
⇥⇤
⇥⇤
⌅⌅⌅
⇥⇤
⇥⇤
⌅⌅⌅
⇥⇤
⇥⇤
⇥⇤
⌅⌅⌅
⇥⇤
⇥⇤
⇥⇤
⌅⌅⌅
⇥⇤
⇥⇤
⇥⇤
⌅⌅
⇥⇤
⇥⇤
⇥⇤
⇥⇤
⇥
0.02
⌅⇥
0.04
0.08 0.12
Lines show 4 different temperatures:
.033 < T/µ < .055
Comparison with the cuprates
(van der Marel, et al 2003)
Bi2 Sr2 Ca0.92 Y0.08 Cu2 O8+
What happens in the
superconducting regime?
We now add a charged scalar field to our action:
S=
Z
d4 x
p

6
g R+ 2
L
1
Fµ⌫ F µ⌫
2
2|(@
2
4|
|
ieA) |2 +
L2
Gubser (2008) argued that at low temperatures,
charged black holes would have nonzero Φ.
Hartnoll, Herzog, G.H. (2008) showed this was
dual to a superconductor (in homogeneous case).
The scalar field has mass m2 = -2/L2, since for
this choice, its asymptotic behavior is simple:
=z
2
+
z
1
3
+
O(z
)
2
This is dual to a dimension 2 charged scalar
operator O with source ϕ1 and <O> = ϕ2.
We set ϕ1 = 0.
For electrically charged solutions with only At
nonzero, the phase of Φ must be constant.
We keep the same boundary conditions on At
as before:
µ(x) = µ̄ [1 + A0 cos(k0 x)]
Start with previous rippled charged black holes
with Φ = 0 and lower T. When do they become
unstable?
Onset of instability corresponds to a static
normalizable mode of the scalar field.
Tc depends on the charge e of Φ. Larger e
makes it easier to condense Φ giving higher Tc.
Critical temperature as function of charge
4.0
3.5
A0 = 0
Lines correspond
to different lattice
amplitudes.
A0 = .8
3.0
A0 = 2
L 2.5
2.0
1.5
1.0
0.05
0.10
T
0.15
0.20
For fixed e,
increasing A0
increases Tc
(Ganguli et al,
2012)
Having found Tc, we now find solutions for T < Tc.
These are hairy, rippled, charged black holes.
From the asymptotic behavior of Φ we read off
the condensate as a function of temperature.
Condensate as a function of temperature
1.4
1.2
HeXO\L1ê2 êm
1.0
Lattice amplitude
grows from
0 (inner line) to
2.4 (outer line).
0.8
0.6
0.4
0.2
0.0
0.02
0.04
0.06
0.08
Têm
0.10
0.12
We again perturb these black holes as before
and compute the conductivity as a function of
frequency.
For definiteness, we fix A0 = 2, k0 = 2, µ̄ = 1
Optical conductivity for T/Tc = 1, .97, .86, .70
⌅
⇤
⇤⇥
50
⇤ ⇥⇥
⌅
⇥
⇤
⇤ ⇥⇥
⇥⇥
⌅ ⇤
⇤
⇥⇥
⌅
⇤
⇥⇥
40
⇤
⇤
⇥⇥
⌅
⌅
⇤
⇥⇥
⇤
⇥⇥
⌅
⇤
⇥⇥
⌅
⇤
⌅
⇥⇥
30
⇤
⇥⇥
⇤
⌅
⇤
⇥⇥
⇤
⌅
⇤
⌅
⇥
⇥⇥
⇤
⇤
⌅
⇥⇥
⇤
⌅
⇥⇥
⇤
⇤
⌅
⌅
⇥⇥
⇤⇤
⇤
⌅
⇥
20
⇤
⇥⇥⇥
⌅
⌅⌅
⇤
⇤
⇤⇤
⇥
⇤
⇥
⇤⇤
⌅
⌅⌅
⇥⇥⇥
⇤
⇤
⇤
⇥⇥⇥
⇤⇤
⌅⌅
⌅
⇤⇤
⇥
⇤
⌅⌅
⇤⇤⇤ ⌅
⇤⇤
⌅⌅
⌅
⌅⌅
⇥
⇤
⌅⌅
⇤
⇤
⌅⌅
⇤
⇥
⇤
⌅
⇤
⇤
⌅⌅⌅⌅⌅⌅
⇤
⇤
⇤
⇤⇤⇤
⇤
⇥
⌅⌅
⌅⌅⌅⌅⌅⌅⌅⌅
⇤
⇤
⇤
⇥
⇤
⇤
10
⇤
⇤
⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅
⇤
⇤⇤⇤⇤⇤
⇤
⇥
⇤⇥
⌅⌅
⇤⇤⇤⇤⇤
⇥
⇥ ⇤
⇤⇤⇤⇤⇤
⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅
⇤
⇤
⇤⇤⇤⇤⇤⌅
⌅⌅
⇤⇤
⌅⌅
⌅⌅⌅
⇤
⌅⌅
⇤
⌅⌅
⇤⇤⇤
⌅⌅
⇤
⌅⌅⌅⌅⌅⌅
⇤
⇤
⇤
⌅⌅
⇤
⇤
⇤⇤
⇤ ⇤ ⇤ ⇤
⌅⌅
⇤⌅
⇤⇤
⇥
⇤
⌅⌅⌅⌅⌅⌅⌅⌅
⇤
⇤⇤⇤⇤ ⇥
⇥
⇥⇤
⇤ ⇤ ⇤ ⇤ ⇤ ⇤⇥⇥⇥
⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥⇥⇥⇥ ⇤⇥⇤
⇤
⇤
⇤
⇤
⇥
⇥
⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅
⇥
⇥
⇥
⇤
⇤
⇤
⇥
⇤
⇤
⇤ ⇤
⇥
⇥
⌅⌅
⌅⌅
⌅⌅
⌅⌅
⌅⌅⌅
⇥
8
Im ⇥⇥
Re ⇥⇥
6
4
2
0.00
0.02
0.04 0.06
⇤⇤
0.08
0.10
0.00 0.02 0.04 0.06 0.08 0.10
⇤⇤
Curves at small ω are well fit by adding a pole to
the Drude formula
Fit to:
⇢s
⇢n ⌧
(!) = i +
!
1 i!⌧
0.6
0.6
0.5
0.5
0.4
rn êm
rs êm
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
0.2
0.4
0.6
TêTc
0.8
superfluid density
1.0
0.0
0.0
0.2
0.4
0.6
TêTc
0.8
normal fluid density
1.0
The dashed red line through ρn is a fit to:
⇢n = a + be
with
/T
Δ = 4 Tc.
This suggests that some of the spectral weight
remains uncondensed even at T = 0.
The relaxation time rises quickly as the
temperature drops:
120
100
⇥
80
60
40
20
0.0 0.2 0.4 0.6 0.8 1.0
T Tc
Line is a fit to
⌧ = ⌧1 e
1 /T
with Δ1 = 4.3 Tc
Intermediate frequency conductivity again
shows the same power law:
B
| (!)| = 2/3 + C
30.0
!
⌅
T/Tc = 1, .97, .86, .70
⌅
⌅
⇤
20.0
⇤
15.0
⇥⇤⇧⇥ C
10.0
7.0
5.0
3.0
2.0
0.01
⇤
⌅
⌅
⇤⇤ ⌅ ⌅
⇤
⇤ ⌅⌅
⇤⇤
⇤⇤⌅
⌅
⇤⇤
⌅⌅
⇤⇤
⌅⌅
⇤
⇤
⇤⇤
⌅⌅
⇤⇤
⌅
⌅⇤
⌅
⌅⇤
⇥ ⇥ ⇥ ⇥⇥
⌅
⌅⇤
⌅
⌅⌅
⇤
⇥⇥⇥⇥
⌅⇤
⌅
⌅⌅
⇥⇥⇥⇥ ⇤
⌅
⇤
⌅
⌅
⇥⇥⇥⇥ ⇤
⌅
⇤
⌅⌅
⌅
⌅
⇤
⌅⌅
⇥⇥⇥⇥ ⇤
⌅
⌅
⇤
⌅
⌅
⇥⇥⇥⇥⇤
⇤
⌅
⇤
⌅
⇤
⌅
⌅
⇤
⌅
⌅
⇥⇥⇥
⇤
⌅
⌅
⇤
⌅
⇤
⇥⇥
⌅
⇤
⌅
⇥
⇤
⌅
⇥
⇤
⇥
⌅
⇤
⇥
⌅
⇥
⇤
⌅
⇤
⇥
⌅
⇥
⇤
⇥
⌅
⇤
⇥
⌅
⇤
⇥
⌅
⇤
⇥
⇥
⇤
⌅
⇥
⇤
⌅
⇥
⇤
⌅
⇥
⇤
⌅
⇥
⇤
⇥
⌅
⇤
⇥
⌅⇤
⇤⇤
⇥
⇥⇤
⇥
⇤⇤
⇥
⇤⇤
⇤⇤
⇤⇤⇤⇤
⇤⇤⇥
⇤⇤
⇥
⇤
⇤
⇥
⇤
⇥
⇤
⇤
⇥
⇤
⇤
⇥
⇤
⇤
⇥
⇤
⇥
⇤
⇤
⇥
⇤
⇤
⇥
⇤
⇥
⇤
⇤
⇥⇥
⇥⇥⇥
⇥⇥⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
⇥
0.02
0.05
0.10
⌅ ⇥
0.20
0.50
Coefficient B and
exponent 2/3 are
independent of T
and identical to
normal phase.
8 samples of
BSCCO with
different doping.
Each plot includes
T < Tc as well as
T > Tc.
No change in the
power law.
(Data from Timusk
et al, 2007.)
There are two ways to compute the superfluid
density: coefficient of pole in Im σ, or
Ns,sum ⌘
Z
!0
d! Re[
n (!)
s (!)]
T > Tc
T < Tc
0+
The integral over σs by can be approximated
by the Drude peak, and just gives ρn.
So if the two definitions agree:
Ns,sum ⌘
Z
!0
d! Re[
n (!)
s (!)]
0+
1.0
then ρn + ρs must be
constant.
Ns,sum underestimates
ρs by about 15%.
0.8
⇤n ⇤s ⇥⇤⇥
But its not.
= ρs
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
T⇤Tc
0.8
1.0
A similar discrepancy is seen in the
cuprates.
Letting Ns = ρs be coefficient of pole:
(data from Timusk et al, 2007)
Our simple gravity model reproduces
many properties of cuprates:
• 
• 
• 
• 
• 
• 
Drude peak at low frequency
Power law fall-off ω-2/3 at intermediate ω
Rapid decrease in scattering rate below Tc
Gap 2Δ = 8 Tc
Normal component doesn’t vanish at T = 0
Calculation of superfluid density from Re σ
underestimates value from Im σ.
But key differences remain
•  Our superconductor is s-wave, not d-wave
•  Our power law has a constant off-set C
•  …