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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, 1302.6586
See: www.simonsfoundation.org
Gauge/gravity duality can reproduce many properties of condensed matter systems, even in the limit where the bulk is described by classical general relativity: 1) Fermi surfaces 2) Non-Fermi liquids 3) Superconducting phase transitions 4) … It is not clear why it is working so well.
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, Siopsis et al, Flauger et al) Main result: We will find surprising similarities to the optical conductivity of the cuprates.
Simple model of a conductor
Suppose electrons in a metal satisfy If there are n electrons per unit volume, the current density is J = nev. Letting E(t) = Ee-iωt, find J = σ E, with where K=ne2/m. This is the Drude model.
mdv
dt= eE �m
v
⌧
�(!) =K⌧
1� i!⌧
Re(�) =K⌧
1 + (!⌧)2, Im(�) =
K!⌧2
1 + (!⌧)2
Note:
(1) For (2) In the limit : This can be derived more generally from Kramers-Kronig relation.
⌧ ! 1
Re(�) / �(!), Im(�) = K/!
!⌧ � 1, |�| ⇡ K/!
Our gravity model
We start with just Einstein-Maxwell theory: This is the simplest context to describe a conductor. We require the metric to be asymptotically anti-de Sitter (AdS)
S =
Zd
4x
p�g
R+
6
L
2� 1
2Fµ⌫F
µ⌫
�
ds
2 =�dt
2 + dx
2 + dy
2 + dz
2
z
2
Want finite temperature: Add black hole Want finite density: Add charge to the black hole. The asymptotic form of At is
µ is the chemical potential and ρ is the charge density in the dual theory.
At = µ� ⇢z +O(z2)
Introduce the lattice by making the chemical potential be a periodic function: We numerically find solutions with smooth horizons that are static and translationally invariant in one direction.
µ(x) = µ [1 +A0 cos(k0x)]
0 �2
� 3 �2
2�
0
1
2
3
4
x
⇥⇤ �x⇥Charge density for A0 = ½, k0 = 2, T/µ = .055
Solutions are rippled charged black holes.
!
!
z
To compute the optical conductivity using linear response, we perturb the solution
Boundary conditions:
ingoing waves at the horizon
δgµν normalizable at infinity
δAt ~ O(z), δAx = e-iωt [E/iω + J z + …]
induced current
Conductivity
gµ⌫ = gµ⌫ + �gµ⌫ , Aµ = Aµ + �Aµ
Using Ohm’s law, J = σE, the optical conductivity is given by Since we impose a homogeneous electric field, we are interested in the homogeneous part of the conductivity σ(ω).
�(!, x) = limz!0
�F
zx
(x, z)
�F
xt
(x, z)
0 5 10 15 20 25
0.2
0.4
0.6
0.8
1.0
⇥⇤T
Re��⇥
0 5 10 15 20 25
0
1
2
3
4
5
6
⇥⇤TIm��⇥
Review: optical conductivity with no lattice (T/µ = .115)
With the lattice, the delta function is smeared out
0 10 20 30 400
5
10
15
20
25
30
⇥�T
Re�
0 10 20 30 40
0
5
10
15
20
⇥�T
Im�
The low frequency conductivity takes the simple Drude form:
�(!) =K⌧
1� i!⌧
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
10
15
20
25
30
⇥⇤T
Re��⇥
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.4514
15
16
17
18
⇥⇤T
Im��⇥
Intermediate frequency shows scaling regime:
⇥⇥⇥⇥⇥⇥⇥⇥
⇥⇥⇥⇥⇥⇥⇥
⇥⇥⇥⇥⇥⇥
⇥⇥⇥⇥⇥
⇥⇥⇥⇥
⇥⇥⇥⇥
⇥⇥⇥⇥
⇥⇥⇥
⇥⇥
⇥⇥
⇥⇥
⇥⇥
⇥⇥
��������
�������
������
�����
����
����
����
���
��
��
��
��
��
⇤⇤⇤⇤⇤⇤⇤⇤
⇤⇤⇤⇤⇤⇤⇤
⇤⇤⇤⇤⇤⇤
⇤⇤⇤⇤⇤
⇤⇤⇤⇤
⇤⇤⇤⇤
⇤⇤⇤⇤
⇤⇤
⇤⇤
⇤⇤
⇤⇤
⇤⇤
⇤⇤
⇤
⌅⌅⌅⌅⌅⌅⌅⌅
⌅⌅⌅⌅⌅⌅⌅
⌅⌅⌅⌅⌅⌅
⌅⌅⌅⌅⌅
⌅⌅⌅⌅
⌅⌅⌅⌅
⌅⌅⌅⌅
⌅⌅
⌅⌅
⌅⌅
⌅⌅
⌅⌅
⌅⌅
⌅
0.01 0.02 0.04 0.08 0.12
10
50
20
30
15
⌅�⇥
⇥⇤⇥�C|�| = B
!2/3+ C
Lines show 4 different temperatures: .033 < T/µ < .055
Comparison with the cuprates (van der Marel, et al 2003)
Bi2Sr2Ca0.92Y0.08Cu2O8+�
What happens in the superconducting regime?
We now add a charged scalar field to our action: 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).
S =
Zd
4x
p�g
R+
6
L
2� 1
2Fµ⌫F
µ⌫ � 2|(@ � ieA)�|2 + 4|�|2
L
2
�
The scalar field has mass m2 = -2/L2, since for this choice, its asymptotic behavior is simple: 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.
� = z�1 + z2�2 +O(z3)
We keep the same boundary conditions on At as before: 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.
µ(x) = µ [1 +A0 cos(k0x)]
0.05 0.10 0.15 0.20
1.0
1.5
2.0
2.5
3.0
3.5
4.0
T��
� L
Critical temperature as function of charge
A0 = 0 A0 = .8 A0 = 2
Lines correspond to different lattice amplitudes.
For fixed e, increasing A0 increases Tc (Ganguli et al, 2012)
Having found Tc, we now find solutions for T < Tc numerically. These are hairy, rippled, charged black holes.
!
!"0#
z
From the asymptotic behavior of Φ we read off the condensate as a function of temperature.
0.02 0.04 0.06 0.08 0.10 0.120.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Têm
HeXO\L1ê2êm
Condensate as a function of temperature
Lattice amplitude grows from 0 (inner line) to 2.4 (outer line).
In the homogeneous case, the zero temperature limit is known to take the form and near r = 0. With the lattice, the scalar field becomes more homogeneous on the horizon at low T, and S ~ T2.4 independent of the lattice amplitude.
ds
2= r
2(�dt
2+ dxidx
i) +
dr
2
g0r2(� log r)
� = 2(� log r)1/2
We again perturb these black holes as before and compute the conductivity as a function of frequency. Find that curves at small ω are well fit by adding a pole to the Drude formula The lattice does not destroy superconductivity (Siopsis et al, 2012; Iizuka and Maeda, 2012)
�(!) = i⇢s!
+⇢n⌧
1� i!⌧
Superfluid component
Normal component
Fit to: �(!) = i⇢s!
+⇢n⌧
1� i!⌧
superfluid density normal fluid density
0.0 0.2 0.4 0.6 0.8 1.00.0
0.1
0.2
0.3
0.4
0.5
0.6
TêTc
r sêm
0.0 0.2 0.4 0.6 0.8 1.00.0
0.1
0.2
0.3
0.4
0.5
0.6
TêTc
r nêm
The dashed red line through ρn is a fit to: with Δ = 4 Tc. This is like BCS with thermally excited quasiparticles but:
(1) The gap Δ is much larger, and comparable to what is seen in the cuprates.
(2) Some of the spectral weight remains uncondensed even at T = 0 (this is also true of the cuprates).
⇢n = a+ be��/T
The relaxation time rises quickly as the temperature drops:
0.0 0.2 0.4 0.6 0.8 1.020
40
60
80
100
120
T�Tc
⇥�
Line is a fit to with Δ1 = 4.3 Tc The scattering rate drops rapidly below Tc, another feature of the cuprates.
⌧ = ⌧1e�1/T
Intermediate frequency conductivity again shows the same power law: |�(!)| = B
!2/3+ C
⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥
⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥
�� � � � �������������������������������������������������������������������������������������������������
⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤
⌅⌅
⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅
0.01 0.02 0.05 0.10 0.20 0.50
10.0
5.0
2.0
20.0
3.0
30.0
15.0
7.0
⌅�⇥
⇥⇤⇧ ⇥�C
T/Tc = 1, .97, .86, .70
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.)
The Ferrell-Glover-Tinkham sum rule states:
Z 1
0+d!Re[�N (!)� �S(!)] =
⇡
2⇢s
normal phase
superconducting phase
Does this hold in our gravitational model?
0 1 2 3 40.0
0.5
1.0
1.5
2.0
wêm
ReHsL
0 1 2 3 40.0
0.1
0.2
0.3
0.4
0.5
wêm2 p‡ 0+w
êm ReHs
N-sSL
Yes
Blue: T = Tc Red: T = .7 Tc
2
⇡
Z !
0+d!Re[�N (!)� �S(!)]
ρs
For T < Tc, Re[σ] is reduced over a range of ω extending up to the chemical potential. This is also true for the cuprates, but not for conventional superconductors. In BCS, Re[σ] is reduced over a much smaller range of frequency: 2Δ = 3.5 Tc << µ.
Resonances
At larger frequencies, the optical conductivity has resonances. In the bulk, this is due to quasinormal modes of the charged black hole.
Quasinormal modes: modes that are ingoing at the horizon and normalizable at infinity. Only exist for a discrete set of complex frequencies.
They correspond to poles in retarded Green’s functions (Son and Starinets, 2002).
One can determine the quasinormal mode frequency by fitting One finds:
�(!) =GR(!)
i!=
1
i!
a+ b(! � !0)
! � !0
!0/µ = 1.48� 0.42i
Preliminary results on a full 2D lattice (T > Tc) show very similar results to 1D lattice.
!
!
z
The optical conductivity in each lattice direction is nearly identical to the 1D results.
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 • Sum rule holds only if one includes
frequencies of order chemical potential
But key differences remain
• Our superconductor is s-wave, not d-wave • Our power law has a constant off-set C • …