quantum critical transport, duality, and m-theory hep-th/0701036 christopher herzog (washington)...
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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
eT
h
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
j
UnH t
† 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 ~
Holes ~
Excitations of the insulator:
†Particles ~
Holes ~
scs
Insulator
0
0
Superfluid
0
scs
Insulator
0
0
Superfluid
0
Wil
Con
son
formal
-Fishe
field theor
r fixed p
y:
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 dimensionsJ
22
= p p
J p J p K pp
K: a universal number analogous to the level number of the Kac-Moody algebra in 1+1 dimensions
2
Application of Kubo formula shows that
4 2
eK
h
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
However: computation is at 0, with 0,
while experimental measurements are for
B
T
k T
Does this matter ?
CFT correlator of U 1 current in 1+1 dimensionsJ
2
2
2 2
Charge density correlation at 0 :
1 , 0 ~
, , ~
R R
t t
T
J x Jix
kJ k J k
k
CFT correlator of U 1 current in 1+1 dimensionsJ
2 2
2
2
2 2
Charge density correlation at 0 :
, 0 ~ sin
, , ~
R R
t n t nn
T
TJ x J
T ix
kJ k i J k i
k
Conformal mapping of plane to cylinder with circumference 1/T
CFT correlator of U 1 current in 1+1 dimensionsJ
2 2
2
2
2 2
2
2 2
Charge density correlation at 0 :
, 0 ~ sin
, , ~
, , ~
R R
t n t nn
t t
T
TJ x J
T ix
kJ k i J k i
k
kJ k J k
k
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
Does this matter ?
However: computation is at 0, with 0,
while experimental measurements are for
B
T
k T
Does this matter ?
Correlators of conserved charges are
independent of the ratio
For CFTs in 1+1 dimensions: NO
Bk T
No diffusion of charge, and no hydrodynamics
However: computation is at 0, with 0,
while experimental measurements are for
B
T
k T
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
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
physi
on-dominated transport
c or
s
t me
c
i
B
B
B
dec collisio
k T
oherk nceT n
T
e
k
K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).
CFT correlator of U 1 current in 2+1 dimensionsJ
22
= p p
J p J p K pp
K: a universal number analogous to the level number of the Kac-Moody algebra in 1+1 dimensions
2
Application of Kubo formula shows that
4 2
eK
h
CFT correlator of U 1 current at 0J T
22
= p p
J p J p K pp
K: a universal number analogous to the level number of the Kac-Moody algebra in 1+1 dimensions
2
Application of Kubo formula shows that
4 2
eKT h
CFT correlator of U 1 current at 0J T
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
2 2
Application of Kubo formula shows that
4 4 2 0, 2 0,T Le e
K KT h h
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 ~
Holes ~
Approaching the transition from the superfluid Excitations of the superfluid: (A) Spin waves
Approaching the transition from the superfluid Excitations of the superfluid: (B) Vortices
vortex
Approaching the transition from the superfluid Excitations of the superfluid: (B) Vortices
vortex
E
Approaching the transition from the superfluid Excitations of the superfluid: Spin wave and vortices
scs
Insulator
0
0
Superfluid
0
Wil
Con
son
formal
-Fishe
field theor
r fixed p
y:
oint
C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981)
s, ccs s
Insulator
0
0
0
Superfluid
0
0
Wil
Con
son
formal
-Fishe
field theor
r fixed p
y:
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 that
4 4 2 0, 2 0,T Le e
K KT h h
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
3
4
k
T
diffusion peakImCtt/k2
3
4
k
T
34 T
ab 2 2, , = , ,a b T T L LJ k J k k P K k P K k
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 k
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 k
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/N expansion to control strongly-coupled gravity theory).