Thermoelectric Transport and Black Hole Horizons

Jerome Gauntlett

Quantum gravity theory in (at least) one higher dimensionQuantum Field Theory

Gauge-Gravity Duality

Also known as holography or the AdS/CFT correspondence

Strongly coupled, large N Einstein gravity

Conformal field theory (CFT) living in d-dimensional spacetime boundary

Gravity theory living in d+1dimensional bulk anti-de-Sitter (AdS) spacetime

Simplest setup

Bulk field CFT operatorgµ⌫(r, x)

Aµ(r, x)

�(r, x)

Ja(x)

O(x)

T ab(x)

U(1)i.e CFT with symmetry

r

Black hole in the bulk

Thermal quantum field theory

Black holes in AdS spaces teach us about the different phases of some strongly coupled quantum field theories

AdS/CFT provides a beautiful realisation of black hole thermodynamics:

Also can learn much about black holes…

TH =

2⇡SBH =

A

4G

T, S

La2�xSrxCuO4

Plan

• Superconducting and spatially modulated phases

• Thermoelectric transport

• Holographic lattices - explicitly break transaltions

• Charge and energy diffusion modes for inhomogeneous media: Einstein relations from Einstein’s equations

• DC conductivity from solving fluid equations on the horizon

i.e. they spontaneously break U(1) and translations, respectively

µ 6= 0

L = L(gµ⌫ , Aµ, ...)

At(r, x) ! µ+⇢

r+ . . .

r ! 1

Consider the CFT on the a plane with

As

Add a source to CFT

LCFT ! LCFT + µJ t

Expectation value in CFT

hJ ti = ⇢

Superconducting phases

T > Tc

T Tc

µ 6= 0

L = L(gµ⌫ , Aµ,�)

CFT with

� = 0 ) hOi = 0

)

� ! 0r+

v

r2

hOi = v 6= 0

[Gubser] [Hartnoll,Herzog,Horowitz]

Spontaneously break U(1)

Spatially modulated phases

Striped order

Checkerboard order

Spontaneously break translations

Helical order

[Nakamura,Ooguri,Park] [JPG,Donos] [Withers]………

hOi / cos kx

µ 6= 0CFT with

More possibilities for CFTs with B 6= 0

Thermoelectric Conductivity

• Apply external AC electric field

• DC conductivity:

Eie�i!t

and thermal gradient ⇣ie�i!t

• Produces local electric currents and heat current

! ! 0

J̄ i Q̄i

J i(x)e�i!t

Qi(x)e�i!t

Focus on zero modes and

✓J̄ i

Q̄i

◆=

✓�ij(!) T↵ij(!)T ↵̄ij(!) T ̄ij(!)

◆✓Ej⇣j

◆

µ 6= 0Consider the CFT with

�DC = 1 ̄DC = 1

! ! 0 �(!) ⇠ ⇡�(!) +i

!

̄(!) ⇠ ⇡�(!) + i!

Black holes dual to CFT with periodic sources that explicitly break translation invariance (and time independent).

[Horowitz, Santos,Tong] [Donos,JPG][……]

Holographic Lattices

General framework for dissipating momentum:

�(r, x) ! �s(x)r

+ . . .

At(r, x) ! µ(x) + . . .

LCFT ! LCFT +

Zµ(x)J t(x) +

Zhab(x)T

ab +

Z�s(x)O(x)

gab(r, x) ! r2hab(x) + . . .

The lattice deformation can lead to black holes describing:

• Insulators and M-I transitions

• Novel ‘incoherent’ metals

[Donos,Hartnoll][Donos,JPG][……]

[Donos,JPG][Gouteraux][…..]

• ‘Coherent metals’ aka Drude physics - smoothed out [Hartnoll,Hoffman][Horowitz, Santos,Tong][…]

Arises when momentum is nearly conserved.

Holographic lattices are interesting

�(!)

A remarkable result in holography

DC conductivity matrix is universally obtained exactlyby solving generalised, linearised Navier-Stokes equations for an incompressible charged fluid on the curved black hole horizon

[Donos,JPG]

[Damour][McDonald,Price,Thorne][…]

• Not a special case of fluid-gravity correspondence

• Can view as an exact version of old membrane paradigm

Recall S =A

4G

Calculating conductivities in holography (in general)

• Start with equilibrium holographic lattice black hole

• Linear perturbation with specific boundary conditions:

Ingoing boundary conditions at the black hole horizon

Only sources are and

• Solve PDEs: the subleading behaviour of the perturbation at encodes the response of the currents (and other operators)

Ei ⇣i

Simplification for DC conductivity

• Use Hamiltonian-like version of the equations of motion with respect to a radial foliation of spacetime. Gives constraint equations and radial equations

• Constraint equations on the horizon:

Give closed set of equations for a subset of the perturbation

Solving them gives , as a function of ,

• Radial equations imply

J iH(x) Qi

H(x) Ei ⇣i

J̄ iH

= J̄ i1 Q̄i

H= Q̄i1

and hence have , as a function of , Q̄i1J̄i1 Ei ⇣i

and hence DC conductivity.

Define

Find system of Navier-Stokes equations on the horizon

(�g(0)it , �g(0)rt , �a

(0)t )$

�2rir(ivj) = (4⇡T ⇣j �rj p) + a(0)t (Ej +rjw)

(vi, p, w)

[Donos,JPG]E.g. for Einstein-Maxwell theory

and static holographic lattices (preserve T in CFT)

QiH

= 4⇡Tphvi

J iH

=ph[a(0)

tvi + hij(Ej + @j!)]

@iJi

H= 0

@iQi

H= 0

Comments I

• Gravity theory can have arbitrary matter content

Formalism can be generalised:

[Banks,Donos,JPG]

• Black holes that break time reversal invariance

Find: extra Lorentz and Coriolis terms in Stokes equations on the horizon [Donos,JPG,Griffin,Melgar]

[Cooper,Halperin,Ruzin]Identify transport currents

• Can generalise to higher derivative theories of gravityGives higher derivative Stokes equations

[Donos,JPG,Griffin,Melgar]

Comments II

There are some special holographic lattices for which we can say more about the Stokes equations

• a) One dimensional lattices - can solve equations in closed form

[Banks,Donos,JPG,Griffin,Melgar,…]

• b) Perturbative lattice (Drude physics) - expand about flat horizon and solve Stokes equations perturbatively

̄ = L�14⇡sT↵ = ↵̄ = L�14⇡⇢� = L�14⇡⇢2

s

) ̄(�T )�1 = s2

⇢2Generalised Wiedemann-Franz Law

Comments III

•Can show as (NOT just for holography)

Return to the case of spontaneous breaking of translations

Generalises result for translationally invariant systems[Davison,Gouteraux,Hartnoll]

[Donos,JPG,Griffin,Ziogas]

T ↵̄ij = T↵ij(!) !✓⇡�(!) +

i

!

◆⇢Ts

"+ p�ij � µ�ij0

T ̄ij(!) !✓⇡�(!) +

i

!

◆(Ts)2

"+ p�ij + µ2�ij0

�ij(!) !✓⇡�(!) +

i

!

◆⇢2

"+ p�ij + �ij0

• Can also calculate in holography from horizon…

! ! 0

�ij0

Einstein’s relation from Einstein equations?

What else can we get from the horizon?

Diffusion modes

Holographic lattices

Hydrodynamic modes that should exist for any !, ki ! 0 kL

• Charge is conserved quasinormal modes associated with diffusion of charge

)

• Satisfy Einstein relation? D = ���1

• Subtlety: charge and energy diffusion modes mix [Hartnoll]

kL

Result:

• Susceptibilities c⇢ ⌘ T (@s/@T )⇢

⇠ ⌘ (@s/@µ)T = (@⇢/@T )µ

� ⌘ (@⇢/@µ)T

• Define ̄(k) = ̄ijkikj , ↵(k) = ↵ijkikj , �(k) = �ijkikj

(k) ⌘ ̄(k)� ↵2(k)T

�(k)

Dispersion relation for coupled diffusion modes

i!+ + i!� =(k)

c⇢+

�(k)

�+

T [�↵(k)� ⇠ �(k)]2

c⇢�2�(k)

i!+ i!� =(k)

c⇢

�(k)

�

[Donos,JPG,Ziogas]

Final Comments

• DC thermoelectric conductivity can be obtained by solving Stokes equations on black hole horizons

• Dispersion relations for diffusion modes can also be calculated

• What else can be found at the horizon?

Corollary: thermodynamic instability

Extend to superfluids? Spontaneous breaking of translations?

) dynamic instability c.f. old conjecture of [Gubser]

Can be used for explicit and spontaneous breaking of translations

for case of explicit breaking: gives generalised Einstein relations

Bounds on diffusion constants? [Hartnoll][Blake][…]