quantum critical transport, duality, and m-theory
DESCRIPTION
Quantum critical transport, duality, and M-theory. hep-th/0701036. Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh Son (Washington). Talk online at http://sachdev.physics.harvard.edu. - PowerPoint PPT PresentationTRANSCRIPT
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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)
Talk online at http://sachdev.physics.harvard.edu
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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
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Trap for ultracold 87Rb atoms
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M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
Velocity distribution of 87Rb atoms
Superfliud
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M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
Velocity distribution of 87Rb atoms
Insulator
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Outline
1. Boson Hubbard model – conformal field theory (CFT)
2. Hydrodynamics of CFTs
3. Duality
4. SYM3 with N=8 supersymmetry
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Outline
1. Boson Hubbard model – conformal field theory (CFT)
2. Hydrodynamics of CFTs
3. Duality
4. SYM3 with N=8 supersymmetry
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†
†
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
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The insulator:
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Excitations of the insulator:
†Particles ~
Holes ~
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Excitations of the insulator:
†Particles ~
Holes ~
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scs
Insulator
0
0
Superfluid
0
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scs
Insulator
0
0
Superfluid
0
Wil
Con
son
formal
-Fishe
field theor
r fixed p
y:
oint
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Outline
1. Boson Hubbard model – conformal field theory (CFT)
2. Hydrodynamics of CFTs
3. Duality
4. SYM3 with N=8 supersymmetry
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CFT correlator of U 1 current J
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)
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However: computation is at 0, with 0,
while experimental measurements are for
B
T
k T
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However: computation is at 0, with 0,
while experimental measurements are for
B
T
k T
Does this matter ?
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However: computation is at 0, with 0,
while experimental measurements are for
B
T
k T
2
2 2ret
For
Cha
CFTs in 1
rge densi
+1 dimensions, NO.
ty correlation at
Charge density correlation at 0 :
0 : conformally map plane to a
cyl
inder of
1circumf
erence , and
,
t
,t t
T
T
kJ k J
k
T
k K
2
2 2ret
hen analytically continue to real frequencies:
, ,t t
kJ k J k K
k
No diffusion of charge, and no hydrodynamics
Does this matter ?
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However: computation is at 0, with 0,
while experimental measurements are for
B
T
k T
(almost always)
: Hydrodynamic, collision-dom
For CFTs in 2+1 dimensions, YES .
is a characteristic or time
: Collisionless physics
inaB
B
B
coll
k
decoherence isiok T
T
n
T
k
ted transport
K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).
Does this matter ?
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However: computation is at 0, with 0,
while experimental measurements are for
B
T
k T
Does this matter ?
(almost always)For CFTs in 2+1 dimensions, YES .
K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).
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CFT correlator of U 1 current J
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
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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
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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
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Outline
1. Boson Hubbard model – conformal field theory (CFT)
2. Hydrodynamics of CFTs
3. Duality
4. SYM3 with N=8 supersymmetry
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Excitations of the insulator:
†Particles ~
Holes ~
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Approaching the transition from the superfluid Excitations of the superfluid: (A) Spin waves
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Approaching the transition from the superfluid Excitations of the superfluid: (B) Vortices
vortex
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Approaching the transition from the superfluid Excitations of the superfluid: (B) Vortices
vortex
E
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Approaching the transition from the superfluid Excitations of the superfluid: Spin wave and vortices
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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)
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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)
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Consequences of duality on CFT correlators of U 1 currents
2 2, , = , ,T T L LJ 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
C. Herzog, P. Kovtun, S. Sachdev, and D.T. Son, hep-th/0701036
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Outline
1. Boson Hubbard model – conformal field theory (CFT)
2. Hydrodynamics of CFTs
3. Duality
4. SYM3 with N=8 supersymmetry
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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
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
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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
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
Holographic self-duality
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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).