Theory of the Nernst effect near the superfluid-insulator ...qpt.physics.harvard.edu/talks/aspen_2007.pdf · Theory of the Nernst effect near the superfluid-insulator transition Sean

Theory of the Nernst effect

near the superfluid-insulator transition

Sean Hartnoll (KITP), Christopher Herzog (Washington), Pavel Kovtun (KITP), Marcus Mueller (Harvard), Subir Sachdev (Harvard), Dam Son (Washington)

Outline

1. Superfluid/supersolid/insulator quantum transitions Insulators at integer and commensurate densities

2. Theory of quantum-critical transport Collisionless-t0-hydrodynamic crossover of

conformal field theories

3. Hydrodynamics at incommensurate densities withimpurities and a magnetic field

Exact relations between thermoelectric co-efficients

4. Nernst effect in the cuprate superconductors

Outline







Trap for ultracold 87Rb atoms

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

Boson Hubbard model

M.P.A. Fisher, P.B. Weichmann, G. Grinstein, and D.S. Fisher Phys. Rev. B 40, 546 (1989).

g = ring exchange (Sandvik)

La2CuO4

Phase diagram of doped antiferromagnets

g

La2CuO4

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).


T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

g

La2CuO4


S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991).

Hole density

g

La2CuO4



Hole density

g

La2CuO4



Hole density

VBS insulator

VBS supersolid

d-wave superconductor

0 Doping

p

4

d-wave

A1

A2

q

X

broken

g

VBS supersolid

M. Vojta and S. Sachdev, Phys. Rev. Lett. 83, 3916 (1999)

0 Doping

p

4

d-wave

A1

A2

q

X

broken

g

VBS supersolid


“pair density wave”

0 0.5 1 1.5 2 2.5

t / J

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Vnn

/ t

d-wave

Stripes p x 1, p 8

VBS supersolids

Insulators


Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma,

M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, Science 315, 1380 (2007)











“Glassy” Valence Bond Supersolid

Outline







Outline







The insulator:

Excitations of the insulator:

Excitations of the insulator:

Superfluid Insulator

Non-zero temperature phase diagram

Depth of periodic potential




Dynamics of the classical Gross-Pitaevski equation




Dilute Boltzmann gas of particle and holes




No wave or quasiparticle description

D. B. Haviland, Y. Liu, and A. M. Goldman,

Phys. Rev. Lett. 62, 2180 (1989)

Resistivity of Bi films

M. P. A. Fisher, Phys. Rev. Lett. 65, 923 (1990)

σSuperconductor(T → 0) = ∞σInsulator(T → 0) = 0

σQuantum critical point(T → 0) ≈ 4e2

h







Collisionless-to

hydrodynamic

crossover of a

conformal field

theory (CFT)

K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

Collisionless-to-hydrodynamic crossover of a CFT in 2+1 dimensions


Collisionless-to-hydrodynamic crossover of a CFT in 2+1 dimensions


Hydrodynamics of a conformal field theory (CFT)

The scattering cross-section of the thermal

excitations is universal and so transport co-

efficients are universally determined by kBT

Charge diffusion constant

Conductivity



The AdS/CFT correspondence (Maldacena, Polyakov) relates

the hydrodynamics of CFTs to the quantum gravity theory of

the horizon of a black hole in Anti-de Sitter space.

Holographic

representation of black

hole physics in a 2+1

dimensional CFT at a

temperature equal to the

Hawking temperature of

the black hole.

Black hole

3+1 dimensional AdS space


The AdS/CFT correspondence (Maldacena, Polyakov) relates

the hydrodynamics of CFTs to the quantum gravity theory of

the horizon of a black hole in Anti-de Sitter space.


Hydrodynamics of

a CFT

Waves of gauge

fields in a curved

background


P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, 085020 (2007)

Spin diffusion constant

Spin conductivity

For the (unique) CFT with a SU(N) gauge field and

16 supercharges, we know the exact diffusion

constant associated with a global SO(8) symmetry:

Collisionless-to-hydrodynamic crossover of solvable SYM3


ImC/k2 CFT at T=0


Collisionless-to-hydrodynamic crossover of solvable SYM3


diffusion peakImC/k2


Outline







Outline







For experimental applications, we must move away from the ideal CFT

e.g.


e.g.

• A chemical potential μ


e.g.


CFT


e.g.


CFT

Supersolid


e.g.


CFT


e.g.



e.g.


CFT


e.g.


• A magnetic field BCFT

S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, arXiv:0706.3215


Conservation laws/equations of motion


Constitutive relations which follow from Lorentz transformation to moving frame


Single dissipative term allowed by requirement of positive entropy production. There is only one

independent transport co-efficient


e.g.

CFT


• A magnetic field B


e.g.



• An impurity scattering rate 1/ imp (its T dependence

follows from scaling arguments)

CFT


From these relations, we obtained results for the transport co-efficients, expressed in terms of a “cyclotron” frequency and damping:












Outline







Outline











Transverse thermoelectric co-efficient(

h

2ekB

)αxy = ΦsB (kBT )2

(2πτimp

�

)2ρ2 + ΦσΦε+P (kBT )3 �/2πτimp

Φ2ε+P (kBT )6 + B

2ρ2(2πτimp/�)2

,

whereB = Bφ0/(�v)2 ; ρ = ρ/(�v)2.

LSCO - Theory


LSCO - Experiments

N. P. Ong et al.

Only input parameters Output

LSCO - Theory


�v = 47 meV Aτimp ≈ 10−12 s

ωc = 6.2GHz · B

1T

(35KT

)3

Only input parameters Output

LSCO - Theory


�v = 47 meV Aτimp ≈ 10−12 s

ωc = 6.2GHz · B

1T

(35KT

)3

Similar to velocity estimates by A.V. Balatsky and Z-X. Shen, Science 284, 1137 (1999).

To the solvable supersymmetric, Yang-Mills theory CFT, we add



After the AdS/CFT mapping, we obtain the Einstein-Maxwell theory of a black hole with

• An electric charge

• A magnetic charge

The exact results are found to be in precise accord with

all hydrodynamic results presented earlier


• General theory of transport in a weakly disordered

``vortex liquid’’ state.

• “Relativistic” magnetohydrodynamics offers an efficient

approach to disentangling momentum and charge

transport

• Exact solutions via black hole mapping have yielded

first exact results for transport co-efficients in interacting

many-body systems, and were valuable in determining

general structure of hydrodynamics.

• Simple model reproduces many trends of the Nernst

measurements in cuprates.

Conclusions

Documents

Theory of the Nernst effect near the superfluid-insulator ...qpt.physics.harvard.edu/talks/aspen_2007.pdf · Theory of the Nernst effect near the superfluid-insulator transition Sean