Quantum critical transport, duality, and M-theory

hep-th/0701036

Christopher Herzog (Washington) Pavel Kovtun (UCSB)

Subir Sachdev (Harvard) Dam Thanh Son (Washington)

Talks online at http://sachdev.physics.harvard.edu

D. B. Haviland, Y. Liu, and A. M. Goldman, Phys. Rev. Lett. 62, 2180 (1989)

( )( )

( )

Superconductor

Insulator

2

Quantum critical point

0

Conductivity

0 0

40

T

T

eTh

σ

σ

σσ → = ∞

→ =

→ ≈

Trap for ultracold 87Rb atoms

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

Velocity distribution of 87Rb atoms

Superfliud

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

Velocity distribution of 87Rb atoms

Insulator

Outline

1. Boson Hubbard model – conformal field theory (CFT)

2. Hydrodynamics of CFTs

3. Duality

4. SYM3 with N=8 supersymmetry

Outline

1. Boson Hubbard model – conformal field theory (CFT)

2. Hydrodynamics of CFTs

3. Duality

4. SYM3 with N=8 supersymmetry

†

†

Degrees of freedom: Bosons, , hopping between the

sites, , of a lattice, with short-range repulsive interactions.

- ( 1)2

j

i j jj j

ji j

j

b

b b n n

jU nH t μ= − + − +∑ ∑ ∑

† j j jn b b≡

Boson Hubbard model

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

For small , superfluid

For large , insulator

Ut

Ut

The insulator:

Excitations of the insulator:

†Particles ~ ψ

Holes ~ ψ

Excitations of the insulator:

†Particles ~ ψ

Holes ~ ψ

scs

Insulator0

0ψσ

=

=

Superfluid0

ψσ

≠

= ∞

scs

Insulator0

0ψσ

=

=

Superfluid0

ψσ

≠

= ∞

WilCon

sonformal

-Fishe field theor

r fixed py:oint

Outline

1. Boson Hubbard model – conformal field theory (CFT)

2. Hydrodynamics of CFTs

3. Duality

4. SYM3 with N=8 supersymmetry

( )CFT correlator of U 1 current in 2+1 dimensionsJμ

( ) ( ) 22 =

p pJ p J p K p

pμ ν

μ ν μνη⎛ ⎞

− −⎜ ⎟⎝ ⎠

K: a universal number analogous to the level number of the Kac-Moody algebra in 1+1 dimensions

2

Application of Kubo formula shows that4 2e Kh

σ π=

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

However: computation is at 0, with 0, while experimental measurements are for B

T

k T

ω

ω

= →

However: computation is at 0, with 0, while experimental measurements are for B

T

k T

ω

ω

= →

Does this matter ?

( )CFT correlator of U 1 current in 1+1 dimensionsJμ

( ) ( )( )

( ) ( )

2

2

2 2

Charge density correlation at 0 :1 , 0 ~

, , ~

R R

t t

T

J x Jix

kJ k J kk

ττ

ω ωω

=

+

− −−

( )CFT correlator of U 1 current in 1+1 dimensionsJμ

( ) ( ) ( )( )

( ) ( )

2 2

2

2

2 2

Charge density correlation at 0 :

, 0 ~ sin

, , ~

R R

t n t nn

TTJ x J

T ix

kJ k i J k ik

πτπ τ

ω ωω

≥

+

− −+

Conformal mapping of plane to cylinder with circumference 1/T

( )CFT correlator of U 1 current in 1+1 dimensionsJμ

( ) ( ) ( )( )

( ) ( )

( ) ( )

2 2

2

2

2 2

2

2 2

Charge density correlation at 0 :

, 0 ~ sin

, , ~

, , ~

R R

t n t nn

t t

TTJ x J

T ix

kJ k i J k ik

kJ k J kk

πτπ τ

ω ωω

ω ωω

≥

+

− −+

− −−

Conformal mapping of plane to cylinder with circumference 1/T

However: computation is at 0, with 0, while experimental measurements are for B

T

k T

ω

ω

= →

Does this matter ?

However: computation is at 0, with 0, while experimental measurements are for B

T

k T

ω

ω

= →

Does this matter ?

Correlators of conserved charges are

independent of the ratio

For CFTs in 1+1 dimensions: NO

Bk Tω

No diffusion of charge, and no hydrodynamics

However: computation is at 0, with 0, while experimental measurements are for B

T

k T

ω

ω

= →

Does this matter ?For CFTs in 2+1 dimensions: YES

However: computation is at 0, with 0, while experimental measurements are for B

T

k T

ω

ω

= →

Does this matter ?For CFTs in 2+1 dimensions: YES

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

: Collisionless: Hydrodynamic, collisi

is a characteristi

physion-dominated transport

c or

s

t me

c

i

B

B

B

dec collisio

k T

oherk nceT n

T

e

kωω

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

( )CFT correlator of U 1 current in 2+1 dimensionsJμ

( ) ( ) 22 =

p pJ p J p K p

pμ ν

μ ν μνη⎛ ⎞

− −⎜ ⎟⎝ ⎠

K: a universal number analogous to the level number of the Kac-Moody algebra in 1+1 dimensions

2

Application of Kubo formula shows that4 2e Kh

σ π=

( )CFT correlator of U 1 current at 0J Tμ =

( ) ( ) 22 =

p pJ p J p K p

pμ ν

μ ν μνη⎛ ⎞

− −⎜ ⎟⎝ ⎠

K: a universal number analogous to the level number of the Kac-Moody algebra in 1+1 dimensions

( )2

Application of Kubo formula shows that4 2e KT h

ωσ π= ∞ =

( )CFT correlator of U 1 current at 0J Tμ >

( ) ( ) ( ) ( )( )

( )

2 2

,

2 2

The projectors are defined by

while , are universal functions o

and ; ( ,

f and

)

, , = , ,

i jT L Tij ij

L T

T T L L

k k p pP P P p

kK k T T

kk p

J k J k k P K k P K k

μ νμν μν μν

μ ν μν μν

ωω

δ η ω

ω ω ω ω ω

= − = − − =

− − − +

( ) ( ) ( )2 2

Application of Kubo formula shows that4 4 2 0, 2 0,T Le eK KT h h

ωσ π ω π ω= =

Superfluid Insulator

Superfluid Insulator

CFT at T > 0

Outline

1. Boson Hubbard model – conformal field theory (CFT)

2. Hydrodynamics of CFTs

3. Duality

4. SYM3 with N=8 supersymmetry

Excitations of the insulator:

†Particles ~ ψ

Holes ~ ψ

Approaching the transition from the superfluidExcitations of the superfluid: (A) Spin waves

Approaching the transition from the superfluidExcitations of the superfluid: (B) Vortices

vortex

Approaching the transition from the superfluidExcitations of the superfluid: (B) Vortices

vortex

E

Approaching the transition from the superfluidExcitations of the superfluid: Spin wave and vortices

scs

Insulator0

0ψσ

=

=

Superfluid0

ψσ

≠

= ∞

WilCon

sonformal

-Fishe field theor

r fixed py:oint

C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981)

s, ccs s

Insulator0

00

ψ

ϕσ

=

≠

=

Superfluid0

0

ψ

ϕσ

≠

=

= ∞

WilCon

sonformal

-Fishe field theor

r fixed py:oint

s

C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981)

( )Consequences of duality on CFT correlators of U 1 currents

( ) ( ) ( )2 2

Application of Kubo formula shows that4 4 2 0, 2 0,T Le eK KT h h

ωσ π ω π ω= =

C. Herzog, P. Kovtun, S. Sachdev, and D.T. Son, hep-th/0701036

Outline

1. Boson Hubbard model – conformal field theory (CFT)

2. Hydrodynamics of CFTs

3. Duality

4. SYM3 with N=8 supersymmetry

ImCtt/k2 CFT at T=0

34 T

ωπ

34

kTπ

diffusion peakImCtt/k2

34

kTπ

34 T

ωπ

( ) ( ) ( ) ( )( )ab 2 2, , = , ,a b T T L LJ k J k k P K k P K kμ ν μν μνω ω δ ω ω ω− − − +

The self-duality of the 4D SO(8) gauge fields leads to

C. Herzog, P. Kovtun, S. Sachdev, and D.T. Son, hep-th/0701036

( ) ( ) ( ) ( )( )ab 2 2, , = , ,a b T T L LJ k J k k P K k P K kμ ν μν μνω ω δ ω ω ω− − − +

The self-duality of the 4D SO(8) gauge fields leads to

C. Herzog, P. Kovtun, S. Sachdev, and D.T. Son, hep-th/0701036

( ) ( ) ( ) ( )( )ab 2 2, , = , ,a b T T L LJ k J k k P K k P K kμ ν μν μνω ω δ ω ω ω− − − +

The self-duality of the 4D SO(8) gauge fields leads to

C. Herzog, P. Kovtun, S. Sachdev, and D.T. Son, hep-th/0701036

Holographic self-duality

Open questions

1. Does KLKT = constant (i.e. holographic self-duality) hold for SYM3 SCFT at finite N ?

2. Is there any CFT3 with an Abelian U(1) current whose conductivity can be determined by self-duality ? (unlikely, because global and topological U(1) currents are interchanged under duality).

3. Is there any CFT3 solvable by AdS/CFT which is not (holographically) self-dual ?

4. Is there an AdS4 description of the hydrodynamics of the O(N) Wilson-Fisher CFT3 ? (can use 1/Nexpansion to control strongly-coupled gravity theory).