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![Page 1: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/1.jpg)
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
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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
![Page 7: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/7.jpg)
Outline
1. Boson Hubbard model – conformal field theory (CFT)
2. Hydrodynamics of CFTs
3. Duality
4. SYM3 with N=8 supersymmetry
![Page 8: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/8.jpg)
Outline
1. Boson Hubbard model – conformal field theory (CFT)
2. Hydrodynamics of CFTs
3. Duality
4. SYM3 with N=8 supersymmetry
![Page 9: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/9.jpg)
†
†
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 ~
![Page 12: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/12.jpg)
Excitations of the insulator:
†Particles ~
Holes ~
![Page 13: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/13.jpg)
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
![Page 16: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/16.jpg)
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)
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However: computation is at 0, with 0,
while experimental measurements are for
B
T
k T
![Page 18: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/18.jpg)
However: computation is at 0, with 0,
while experimental measurements are for
B
T
k T
Does this matter ?
![Page 19: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/19.jpg)
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
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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
![Page 21: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/21.jpg)
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
![Page 22: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/22.jpg)
However: computation is at 0, with 0,
while experimental measurements are for
B
T
k T
Does this matter ?
![Page 23: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/23.jpg)
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
![Page 24: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/24.jpg)
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
![Page 25: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/25.jpg)
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
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K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).
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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
![Page 28: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/28.jpg)
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
![Page 30: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/30.jpg)
Superfluid Insulator
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Superfluid Insulator
CFT at T > 0
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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
![Page 33: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/33.jpg)
Excitations of the insulator:
†Particles ~
Holes ~
![Page 34: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/34.jpg)
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)
![Page 39: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/39.jpg)
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)
![Page 40: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/40.jpg)
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
![Page 41: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/41.jpg)
Outline
1. Boson Hubbard model – conformal field theory (CFT)
2. Hydrodynamics of CFTs
3. Duality
4. SYM3 with N=8 supersymmetry
![Page 42: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/42.jpg)
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ImCtt/k2 CFT at T=0
34 T
3
4
k
T
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diffusion peakImCtt/k2
3
4
k
T
34 T
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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
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
C. Herzog, P. Kovtun, S. Sachdev, and D.T. Son, hep-th/0701036
![Page 53: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/53.jpg)
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
![Page 54: Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh.](https://reader037.fdocuments.in/reader037/viewer/2022110207/56649d385503460f94a10dee/html5/thumbnails/54.jpg)
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).