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:

⇥⇥⇥⇥⇥⇥⇥⇥

⇥⇥⇥⇥⇥⇥⇥

⇥⇥⇥⇥⇥⇥

⇥⇥⇥⇥⇥

⇥⇥⇥⇥

⇥⇥⇥⇥

⇥⇥⇥⇥

⇥⇥⇥

⇥⇥

⇥⇥

⇥⇥

⇥⇥

⇥⇥

��������

�������

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�����

����

����

����

���

��

��

��

��

��

⇤⇤⇤⇤⇤⇤⇤⇤

⇤⇤⇤⇤⇤⇤⇤

⇤⇤⇤⇤⇤⇤

⇤⇤⇤⇤⇤

⇤⇤⇤⇤

⇤⇤⇤⇤

⇤⇤⇤⇤

⇤⇤

⇤⇤

⇤⇤

⇤⇤

⇤⇤

⇤⇤

⇤

⌅⌅⌅⌅⌅⌅⌅⌅

⌅⌅⌅⌅⌅⌅⌅

⌅⌅⌅⌅⌅⌅

⌅⌅⌅⌅⌅

⌅⌅⌅⌅

⌅⌅⌅⌅

⌅⌅⌅⌅

⌅⌅

⌅⌅

⌅⌅

⌅⌅

⌅⌅

⌅⌅

⌅

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

⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥

⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥⇥

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⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤⇤

⌅⌅

⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅⌅

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 •  …