Entanglement, holography, and the quantum phases of...
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Entanglement,holography,
andthe quantum phases of matter
HARVARD
Yale University, September 10, 2012
Subir Sachdev
Lecture at the 100th anniversary Solvay conference,Theory of the Quantum World
arXiv:1203.4565Tuesday, September 11, 2012
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Liza Huijse Max Metlitski Brian Swingle
Tuesday, September 11, 2012
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Sommerfeld-Bloch theory of metals, insulators, and superconductors:many-electron quantum states are adiabatically
connected to independent electron states
Tuesday, September 11, 2012
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Sommerfeld-Bloch theory of metals, insulators, and superconductors:many-electron quantum states are adiabatically
connected to independent electron states
Band insulators
E
MetalMetal
carryinga current
InsulatorSuperconductor
kAn even number of electrons per unit cell
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E
MetalMetal
carryinga current
InsulatorSuperconductor
k
E
MetalMetal
carryinga current
InsulatorSuperconductor
k
Sommerfeld-Bloch theory of metals, insulators, and superconductors:many-electron quantum states are adiabatically
connected to independent electron states
Metals
An odd number of electrons per unit cell
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E
MetalMetal
carryinga current
InsulatorSuperconductor
k
Sommerfeld-Bloch theory of metals, insulators, and superconductors:many-electron quantum states are adiabatically
connected to independent electron states
Superconductors
Tuesday, September 11, 2012
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Modern phases of quantum matterNot adiabatically connected
to independent electron states:
Tuesday, September 11, 2012
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Modern phases of quantum matterNot adiabatically connected
to independent electron states:many-particle
quantum entanglement
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_
Quantum Entanglement: quantum superposition with more than one particleHydrogen molecule:
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Quantum Entanglement: quantum superposition with more than one particle
Tuesday, September 11, 2012
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Quantum Entanglement: quantum superposition with more than one particle
Tuesday, September 11, 2012
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Quantum Entanglement: quantum superposition with more than one particle
Einstein-Podolsky-Rosen “paradox”: Non-local correlations between observations arbitrarily far apart
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|Ψ� ⇒ Ground state of entire system,ρ = |Ψ��Ψ|
ρA = TrBρ = density matrix of region A
Entanglement entropy SE = −Tr (ρA ln ρA)
B
Entanglement entropy
A P
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Gapped quantum matter Spin liquids, quantum Hall states
Conformal quantum matter Quantum critical points in antiferromagnets, superconductors, and ultracold atoms; graphene
Compressible quantum matter Strange metals in high temperature superconductors, Bose metals
“Complex entangled” states of quantum matter,
not adiabatically connected to independent particle states
Tuesday, September 11, 2012
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“Complex entangled” states of quantum matter,
not adiabatically connected to independent particle states
Gapped quantum matter Spin liquids, quantum Hall states
Conformal quantum matter Quantum critical points in antiferromagnets, superconductors, and ultracold atoms; graphene
Compressible quantum matter Strange metals in high temperature superconductors, Bose metals
topological field theory
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Gapped quantum matter Spin liquids, quantum Hall states
Conformal quantum matter Quantum critical points in antiferromagnets, superconductors, and ultracold atoms; graphene
Compressible quantum matter Strange metals in high temperature superconductors, Bose metals
“Complex entangled” states of quantum matter,
not adiabatically connected to independent particle states
topological field theory
conformal field theory
Tuesday, September 11, 2012
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Gapped quantum matter Spin liquids, quantum Hall states
Conformal quantum matter Quantum critical points in antiferromagnets, superconductors, and ultracold atoms; graphene
Compressible quantum matter Strange metals in high temperature superconductors, Bose metals
“Complex entangled” states of quantum matter,
not adiabatically connected to independent particle states
topological field theory
conformal field theory
?Tuesday, September 11, 2012
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Gapped quantum matter Spin liquids, quantum Hall states
Conformal quantum matter Quantum critical points in antiferromagnets, superconductors, and ultracold atoms; graphene
Compressible quantum matter Strange metals in high temperature superconductors, Bose metals
“Complex entangled” states of quantum matter in d spatial dimensions
Tuesday, September 11, 2012
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SE = aP − b exp(−cP )where P is the surface area (perimeter)
of the boundary between A and B.
B
Entanglement entropy of a band insulator
A P
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H = J
�
�ij�
�Si · �Sj =
Mott insulator: Kagome antiferromagnet
P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).
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H = J
�
�ij�
�Si · �Sj =
Mott insulator: Kagome antiferromagnet
Tuesday, September 11, 2012
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H = J
�
�ij�
�Si · �Sj =
Mott insulator: Kagome antiferromagnet
Tuesday, September 11, 2012
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H = J
�
�ij�
�Si · �Sj =
Mott insulator: Kagome antiferromagnet
Tuesday, September 11, 2012
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H = J
�
�ij�
�Si · �Sj =
Mott insulator: Kagome antiferromagnet
Tuesday, September 11, 2012
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H = J
�
�ij�
�Si · �Sj =
Mott insulator: Kagome antiferromagnet
Tuesday, September 11, 2012
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H = J
�
�ij�
�Si · �Sj =
Mott insulator: Kagome antiferromagnet
Tuesday, September 11, 2012
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Mott insulator: Kagome antiferromagnet
Alternative view Pick a reference configuration
Tuesday, September 11, 2012
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Mott insulator: Kagome antiferromagnet
Alternative view A nearby configuration
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Mott insulator: Kagome antiferromagnet
Alternative view Difference: a closed loop
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Mott insulator: Kagome antiferromagnet
Alternative view Ground state: sum over closed loops
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Mott insulator: Kagome antiferromagnet
Alternative view Ground state: sum over closed loops
Tuesday, September 11, 2012
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Mott insulator: Kagome antiferromagnet
Alternative view Ground state: sum over closed loops
Tuesday, September 11, 2012
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Mott insulator: Kagome antiferromagnet
Alternative view Ground state: sum over closed loops
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B
Entanglement in the Z2 spin liquid
A
The sum over closed loops is characteristic of the Z2 spin liquid, introduced in N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991), X.-G. Wen, Phys. Rev. B 44, 2664 (1991)
P
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B
Entanglement in the Z2 spin liquid
A
Sum over closed loops: only an even number of links cross the boundary between A and B
The sum over closed loops is characteristic of the Z2 spin liquid, introduced in N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991), X.-G. Wen, Phys. Rev. B 44, 2664 (1991)
P
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SE = aP − ln(2)where P is the surface area (perimeter)
of the boundary between A and B.
B
M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006); A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006);Y. Zhang, T. Grover, and A. Vishwanath, Phys. Rev. B 84, 075128 (2011).
Entanglement in the Z2 spin liquid
A P
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A
B
M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006); A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006);Y. Zhang, T. Grover, and A. Vishwanath, Phys. Rev. B 84, 075128 (2011).
B
Entanglement in the Z2 spin liquid
SE = aP − ln(2)where P is the surface area (perimeter)
of the boundary between A and B.
(4)
P
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SE = aP − ln(2)where P is the surface area (perimeter)
of the boundary between A and B.
B
M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006); A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006);Y. Zhang, T. Grover, and A. Vishwanath, Phys. Rev. B 84, 075128 (2011).
Entanglement in the Z2 spin liquid
A P
Tuesday, September 11, 2012
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Simeng Yan, D. A. Huse, and S. R. White, Science 332, 1173 (2011).
Mott insulator: Kagome antiferromagnet
S. Depenbrock, I. P. McCulloch, and U. Schollwoeck, arXiv:1205.4858
Hong-Chen Jiang, Z. Wang, and L. Balents, arXiv:1205.4289
Strong numerical evidence for a Z2 spin liquid
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Mott insulator: Kagome antiferromagnet
Evidence for spinonsYoung Lee,
APS meeting, March 2012
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Gapped quantum matter Spin liquids, quantum Hall states
Conformal quantum matter Quantum critical points in antiferromagnets, superconductors, and ultracold atoms; graphene
Compressible quantum matter Strange metals in high temperature superconductors, Bose metals
“Complex entangled” states of quantum matter in d spatial dimensions
Tuesday, September 11, 2012
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Gapped quantum matter Spin liquids, quantum Hall states
Conformal quantum matter Quantum critical points in antiferromagnets, superconductors, and ultracold atoms; graphene
Compressible quantum matter Strange metals in high temperature superconductors, Bose metals
“Complex entangled” states of quantum matter in d spatial dimensions
Tuesday, September 11, 2012
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M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
Ultracold 87Rbatoms - bosons
Superfluid-insulator transition
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g
T
gc
0
InsulatorSuperfluid
Quantumcritical
TKT
H = −t
�
�ij�
b†i bj +
U
2
�
i
ni(ni − 1) ; ni ≡ b†i bi
= U/t
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g
T
gc
0
InsulatorSuperfluid
Quantumcritical
TKT
Quantum critical point described by a CFT3
H = −t
�
�ij�
b†i bj +
U
2
�
i
ni(ni − 1) ; ni ≡ b†i bi
= U/t
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depth ofentanglement
D-dimensionalspace
Tensor network representation of entanglement at quantum critical point
M. Levin and C. P. Nave, Phys. Rev. Lett. 99, 120601 (2007)G. Vidal, Phys. Rev. Lett. 99, 220405 (2007)
F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac, Phys. Rev. Lett. 96, 220601 (2006)
d
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• Entanglement entropy obeys SE = aP − γ, whereγ is a shape-dependent universal number associatedwith the CFT3.
Entanglement at the quantum critical point
B
M. A. Metlitski, C. A. Fuertes, and S. Sachdev, Physical Review B 80, 115122 (2009).H. Casini, M. Huerta, and R. Myers, JHEP 1105:036, (2011)
I. Klebanov, S. Pufu, and B. Safdi, arXiv:1105.4598
A P
Tuesday, September 11, 2012
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depth ofentanglement
D-dimensionalspace
Tensor network representation of entanglement at quantum critical point
M. Levin and C. P. Nave, Phys. Rev. Lett. 99, 120601 (2007)G. Vidal, Phys. Rev. Lett. 99, 220405 (2007)
F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac, Phys. Rev. Lett. 96, 220601 (2006)
d
Tuesday, September 11, 2012
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depth ofentanglement
D-dimensionalspaceA
d
Tensor network representation of entanglement at quantum critical point
Brian Swingle, arXiv:0905.1317Tuesday, September 11, 2012
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depth ofentanglement
D-dimensionalspaceA
Most links describe entanglement within A
d
Tensor network representation of entanglement at quantum critical point
Brian Swingle, arXiv:0905.1317Tuesday, September 11, 2012
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depth ofentanglement
D-dimensionalspaceA
Links overestimate entanglement
between A and B
d
Tensor network representation of entanglement at quantum critical point
Brian Swingle, arXiv:0905.1317Tuesday, September 11, 2012
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depth ofentanglement
D-dimensionalspaceA
Entanglement entropy = Number of links on
optimal surface intersecting minimal
number of links.
d
Tensor network representation of entanglement at quantum critical point
Brian Swingle, arXiv:0905.1317Tuesday, September 11, 2012
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Key idea: ⇒ Implement r as an extra dimen-sion, and map to a local theory in d + 2 spacetimedimensions.
r
Holography
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For a relativistic CFT in d spatial dimensions, themetric in the holographic space is uniquely fixedby demanding the following scale transformaion(i = 1 . . . d)
xi → ζxi , t → ζt , ds → ds
This gives the unique metric
ds2 =1
r2�−dt2 + dr2 + dx2
i
�
Reparametrization invariance in r has been usedto the prefactor of dx2
i equal to 1/r2. This fixesr → ζr under the scale transformation. This isthe metric of the space AdSd+2.
Tuesday, September 11, 2012
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For a relativistic CFT in d spatial dimensions, themetric in the holographic space is uniquely fixedby demanding the following scale transformaion(i = 1 . . . d)
xi → ζxi , t → ζt , ds → ds
This gives the unique metric
ds2 =1
r2�−dt2 + dr2 + dx2
i
�
Reparametrization invariance in r has been usedto the prefactor of dx2
i equal to 1/r2. This fixesr → ζr under the scale transformation. This isthe metric of the space AdSd+2.
Tuesday, September 11, 2012
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zr
AdS4 R2,1
MinkowskiCFT3
AdS/CFT correspondence
Tuesday, September 11, 2012
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zr
AdS4 R2,1
MinkowskiCFT3
AdS/CFT correspondence
A
Tuesday, September 11, 2012
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zr
AdS4 R2,1
MinkowskiCFT3
AdS/CFT correspondence
A
S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 18160 (2006).
Associate entanglement entropy with an observer in the enclosed spacetime region, who cannot observe “outside” : i.e. the region is surrounded by an imaginary horizon.
Tuesday, September 11, 2012
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zr
AdS4 R2,1
MinkowskiCFT3
AdS/CFT correspondence
A
S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 18160 (2006).
The entropy of this region is bounded by its surface area (Bekenstein-Hawking-’t Hooft-Susskind)
Tuesday, September 11, 2012
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zr
AdS4 R2,1
MinkowskiCFT3
AdS/CFT correspondence
AMinimal
surface area measures
entanglemententropy
S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 18160 (2006).Tuesday, September 11, 2012
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depth ofentanglement
D-dimensionalspace
Entanglement entropy
A
Entanglement entropy = Number of links on
optimal surface intersecting minimal
number of links.
d
Brian Swingle, arXiv:0905.1317Tuesday, September 11, 2012
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depth ofentanglement
D-dimensionalspace
Entanglement entropy
Ad
Brian Swingle, arXiv:0905.1317
Entanglement entropy = Number of links on
optimal surface intersecting minimal
number of links.
Tuesday, September 11, 2012
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depth ofentanglement
D-dimensionalspace
Entanglement entropy
Ad
Emergent directionof AdSd+2 Brian Swingle, arXiv:0905.1317
Entanglement entropy = Number of links on
optimal surface intersecting minimal
number of links.
Tuesday, September 11, 2012
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zr
AdS4 R2,1
MinkowskiCFT3
AdS/CFT correspondence
A
S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 18160 (2006).
• Computation of minimal surface area yieldsSE = aP − γ,
where γ is a shape-dependent universal number.
Tuesday, September 11, 2012
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Many-particlequantum
entanglement
Quantum critical points of atoms and electrons
Holography
Tuesday, September 11, 2012
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Many-particlequantum
entanglement
Quantum critical points of atoms and electrons
Holography
CFT3
Tuesday, September 11, 2012
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Many-particlequantum
entanglement
Quantum critical points of atoms and electrons
Holographyand
string theory
Black holes
Tuesday, September 11, 2012
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Many-particlequantum
entanglement
Quantum critical points of atoms and electrons
Holographyand
string theory
Black holes
Tuesday, September 11, 2012
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• Allows unification of the standard model of particlephysics with gravity.
• Low-lying string modes correspond to gauge fields,gravitons, quarks . . .
String theory
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• A D-brane is a d-dimensional surface on which strings can end.
• The low-energy theory on a D-brane has no gravity, similar totheories of entangled electrons of interest to us.
• In d = 2, we obtain strongly-interacting CFT3s. These are“dual” to string theory on anti-de Sitter space: AdS4.
Tuesday, September 11, 2012
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• A D-brane is a d-dimensional surface on which strings can end.
• The low-energy theory on a D-brane has no gravity, similar totheories of entangled electrons of interest to us.
• In d = 2, we obtain strongly-interacting CFT3s. These are“dual” to string theory on anti-de Sitter space: AdS4.
Tuesday, September 11, 2012
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• A D-brane is a d-dimensional surface on which strings can end.
• The low-energy theory on a D-brane has no gravity, similar totheories of entangled electrons of interest to us.
• In d = 2, we obtain strongly-interacting CFT3s. These are“dual” to string theory on anti-de Sitter space: AdS4.
Tuesday, September 11, 2012
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depth ofentanglement
D-dimensionalspace
Tensor network representation of entanglement at quantum critical point
Emergent directionof AdS4 Brian Swingle, arXiv:0905.1317
d
Tuesday, September 11, 2012
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String theory near a D-brane
depth ofentanglement
D-dimensionalspace
Emergent directionof AdS4
d
Tuesday, September 11, 2012
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Many-particlequantum
entanglement
Quantum critical points of atoms and electrons
Holographyand
string theory
Black holes
Tuesday, September 11, 2012
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Many-particlequantum
entanglement
Quantum critical points of atoms and electrons
Holographyand
string theory
Black holes
Tuesday, September 11, 2012
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g
T
gc
0
InsulatorSuperfluid
Quantumcritical
TKT
Quantum critical point described by a CFT3
H = −t
�
�ij�
b†i bj +
U
2
�
i
ni(ni − 1) ; ni ≡ b†i bi
= U/t
Tuesday, September 11, 2012
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g
T
gc
0
InsulatorSuperfluid
Quantumcritical
TKT
Quantum critical point described by a CFT3
H = −t
�
�ij�
b†i bj +
U
2
�
i
ni(ni − 1) ; ni ≡ b†i bi
= U/t
Tuesday, September 11, 2012
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g
T
gc
0
InsulatorSuperfluid
Quantumcritical
TKT
CFT3 at T>0
Quantum critical point described by a CFT3
H = −t
�
�ij�
b†i bj +
U
2
�
i
ni(ni − 1) ; ni ≡ b†i bi
= U/t
Tuesday, September 11, 2012
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g
T
gc
0
InsulatorSuperfluid
Quantumcritical
TKT
CFT3 at T>0
Quantum critical point described by a CFT3
H = −t
�
�ij�
b†i bj +
U
2
�
i
ni(ni − 1) ; ni ≡ b†i bi
= U/t
Needed: Accurate theory of
quantum critical dynamics
Tuesday, September 11, 2012
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A 2+1 dimensional system at its
quantum critical point
String theory at non-zero temperatures
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A 2+1 dimensional system at its
quantum critical point
A “horizon”, similar to the surface of a black hole !
String theory at non-zero temperatures
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Objects so massive that light is gravitationally bound to them.
Black Holes
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Horizon radius R =2GM
c2
Objects so massive that light is gravitationally bound to them.
Black Holes
In Einstein’s theory, the region inside the black hole horizon is disconnected from
the rest of the universe.
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Around 1974, Bekenstein and Hawking showed that the application of the
quantum theory across a black hole horizon led to many astonishing
conclusions
Black Holes + Quantum theory
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_
Quantum Entanglement across a black hole horizon
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_
Quantum Entanglement across a black hole horizon
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_
Quantum Entanglement across a black hole horizon
Black hole horizon
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_
Black hole horizon
Quantum Entanglement across a black hole horizon
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Black hole horizon
Quantum Entanglement across a black hole horizon
There is a non-local quantum entanglement between the inside
and outside of a black hole
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Black hole horizon
Quantum Entanglement across a black hole horizon
There is a non-local quantum entanglement between the inside
and outside of a black hole
Tuesday, September 11, 2012
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Quantum Entanglement across a black hole horizon
There is a non-local quantum entanglement between the inside
and outside of a black hole
This entanglement leads to ablack hole temperature
(the Hawking temperature)and a black hole entropy (the Bekenstein entropy)
Tuesday, September 11, 2012
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A “horizon”,whose temperature and entropy equal
those of the quantum critical point
String theory at non-zero temperatures
A 2+1 dimensional system at its
quantum critical point
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Friction of quantum criticality = waves
falling into black brane
A “horizon”,whose temperature and entropy equal
those of the quantum critical point
String theory at non-zero temperatures
A 2+1 dimensional system at its
quantum critical point
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A 2+1 dimensional system at its
quantum critical point
An (extended) Einstein-Maxwell provides successful description of
dynamics of quantum critical points at non-zero temperatures (where no other methods apply)
A “horizon”,whose temperature and entropy equal
those of the quantum critical point
String theory at non-zero temperatures
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R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, 066017 (2011)
h
Q2σ
0.0 0.5 1.0 1.5 2.00.0
0.5
1.0
1.5
ω
4πT
γ = 0
γ =1
12
γ = − 1
12
AdS4 theory of charge transport in a CFT3
σ(ω)
σ(∞)
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R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, 066017 (2011)
h
Q2σ
0.0 0.5 1.0 1.5 2.00.0
0.5
1.0
1.5
ω
4πT
γ = 0
γ =1
12
γ = − 1
12
AdS4 theory of charge transport in a CFT3
σ(ω)
σ(∞)
Effective theory on AdS4 obtained by a gradient expansion; all
parameters fixed by “OPE data” from CFT3.
SEM =
�d4x
√−g
�1
2κ2
�R+
6
L2
�− 1
4e2FabF
ab+
γL2
e2CabcdF
abF cd
�,
where Cabcd is the Weyl curvature tensor.
Stability and causality constraints restrict |γ| < 1/12.
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Gapped quantum matter Spin liquids, quantum Hall states
Conformal quantum matter Quantum critical points in antiferromagnets, superconductors, and ultracold atoms; graphene
Compressible quantum matter Strange metals in high temperature superconductors, Bose metals
“Complex entangled” states of quantum matter in d spatial dimensions
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Gapped quantum matter Spin liquids, quantum Hall states
Conformal quantum matter Quantum critical points in antiferromagnets, superconductors, and ultracold atoms; graphene
Compressible quantum matter Strange metals in high temperature superconductors, Bose metals
“Complex entangled” states of quantum matter in d spatial dimensions
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BaFe2(As1−x Px)2
K. Hashimoto, K. Cho, T. Shibauchi, S. Kasahara, Y. Mizukami, R. Katsumata, Y. Tsuruhara, T. Terashima, H. Ikeda, M. A. Tanatar, H. Kitano, N. Salovich, R. W. Giannetta, P. Walmsley, A. Carrington, R. Prozorov, and Y. Matsuda, Science 336, 1554 (2012).
Resistivity∼ ρ0 +ATn
FL
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A Bose metal : a compressible phase of bosonswhich breaks no symmetries.
Bosons with correlated hopping
H = −t
�
�ij�
b†i bj +
U
2
�
i
ni(ni − 1) + w
�
ijk�∈�b†i b
†kbjb�
Tuesday, September 11, 2012
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A Bose metal : a compressible phase of bosonswhich breaks no symmetries.
Bosons with correlated hopping
H = −t
�
�ij�
b†i bj +
U
2
�
i
ni(ni − 1) + w
�
ijk�∈�b†i b
†kbjb�
Tuesday, September 11, 2012
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A Bose metal : a compressible phase of bosonswhich breaks no symmetries.
Bosons with correlated hopping
H = −t
�
�ij�
b†i bj +
U
2
�
i
ni(ni − 1) + w
�
ijk�∈�b†i b
†kbjb�
Tuesday, September 11, 2012
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AfAf = �Q�
• Bose metal: the boson, b, fractionalizes into (say) 2fermions, f1 and f2 (“quarks”), each of which formsa Fermi surface. Both fermions necessarily couple toan emergent gauge field, and so the Fermi surfacesare “hidden”.
O. I. Motrunich and M. P. A. Fisher, Physical Review B 75, 235116 (2007)
L. Huijse and S. Sachdev, Physical Review D 84, 026001 (2011)
S. Sachdev, to appear
Q = b†b
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AfAf = �Q�
O. I. Motrunich and M. P. A. Fisher, Physical Review B 75, 235116 (2007)
L. Huijse and S. Sachdev, Physical Review D 84, 026001 (2011)
S. Sachdev, to appear
Q = b†bb → f1f2
Gauge invariance:f1(x) → f1(x)eiθ(x),f2(x) → f2(x)e−iθ(x)
• Bose metal: the boson, b, fractionalizes into (say) 2fermions, f1 and f2 (“quarks”), each of which formsa Fermi surface. Both fermions necessarily couple toan emergent gauge field, and so the Fermi surfacesare “hidden”.
Tuesday, September 11, 2012
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A
• Area enclosed by the Fermi surface A = Q, thefermion density
• Particle and hole of excitations near the Fermi sur-face with energy ω ∼ |q|z; three-loop computationshows z = 3/2.
• The phase space density of fermions is effectively one-dimensional, so the entropy density S ∼ T deff/z withdeff = 1.
Bose metals
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A
• Area enclosed by the Fermi surface A = Q, thefermion density
• Particle and hole of excitations near the Fermi sur-face with energy ω ∼ |q|z; three-loop computationshows z = 3/2.
• The phase space density of fermions is effectively one-dimensional, so the entropy density S ∼ T deff/z withdeff = 1. M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075127 (2010)
→| q|←
Bose metals
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A
• Area enclosed by the Fermi surface A = Q, thefermion density
• Particle and hole of excitations near the Fermi sur-face with energy ω ∼ |q|z; three-loop computationshows z = 3/2.
• The phase space density of fermions is effectively one-dimensional, so the entropy density S ∼ T deff/z withdeff = 1.
→| q|←
Bose metals
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B
A
Y. Zhang, T. Grover, and A. Vishwanath, Physical Review Letters 107, 067202 (2011)
Entanglement entropy of Fermi surfaces
Logarithmic violation of “area law”: SE =1
12(kFP ) ln(kFP )
for a circular Fermi surface with Fermi momentum kF ,where P is the perimeter of region A with an arbitrary smooth shape.
Non-Fermi liquids have, at most, the “1/12” prefactor modified.
P
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r
Holography
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Consider the metric which transforms under rescaling as
xi → ζ xi
t → ζz t
ds → ζθ/d ds.
This identifies z as the dynamic critical exponent (z = 1 for“relativistic” quantum critical points).
θ is the violation of hyperscaling exponent.The most general choice of such a metric is
ds2 =1
r2
�− dt2
r2d(z−1)/(d−θ)+ r2θ/(d−θ)dr2 + dx2
i
�
We have used reparametrization invariance in r to choose sothat it scales as r → ζ(d−θ)/dr.
ds2 =1
r2
�− dt2
r2d(z−1)/(d−θ)+ r2θ/(d−θ)dr2 + dx2
i
�
L. Huijse, S. Sachdev, B. Swingle, Physical Review B 85, 035121 (2012)Tuesday, September 11, 2012
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Consider the metric which transforms under rescaling as
xi → ζ xi
t → ζz t
ds → ζθ/d ds.
This identifies z as the dynamic critical exponent (z = 1 for“relativistic” quantum critical points).
θ is the violation of hyperscaling exponent.The most general choice of such a metric is
ds2 =1
r2
�− dt2
r2d(z−1)/(d−θ)+ r2θ/(d−θ)dr2 + dx2
i
�
We have used reparametrization invariance in r to choose sothat it scales as r → ζ(d−θ)/dr.
ds2 =1
r2
�− dt2
r2d(z−1)/(d−θ)+ r2θ/(d−θ)dr2 + dx2
i
�
L. Huijse, S. Sachdev, B. Swingle, Physical Review B 85, 035121 (2012)Tuesday, September 11, 2012
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ds2 =1
r2
�− dt2
r2d(z−1)/(d−θ)+ r2θ/(d−θ)dr2 + dx2
i
�
At T > 0, there is a horizon, and computation of its
Bekenstein-Hawking entropy shows
S ∼ T (d−θ)/z.
So θ is indeed the violation of hyperscaling exponent as
claimed. For a compressible quantum state we should
therefore choose θ = d− 1.
No additional choices will be made, and all subsequent re-
sults are consequences of the assumption of the existence
of a holographic dual.
L. Huijse, S. Sachdev, B. Swingle, Physical Review B 85, 035121 (2012)Tuesday, September 11, 2012
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θ = d− 1
L. Huijse, S. Sachdev, B. Swingle, Physical Review B 85, 035121 (2012)
Holography of Bose metals
The null energy condition (stability condition for gravity)yields a new inequality
z ≥ 1 +θ
d
In d = 2, this implies z ≥ 3/2. So the lower bound isprecisely the value obtained from the field theory.
N. Ogawa, T. Takayanagi, and T. Ugajin, JHEP 1201, 125 (2012).
ds2 =1
r2
�− dt2
r2d(z−1)/(d−θ)+ r2θ/(d−θ)dr2 + dx2
i
�
Tuesday, September 11, 2012
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θ = d− 1
L. Huijse, S. Sachdev, B. Swingle, Physical Review B 85, 035121 (2012)N. Ogawa, T. Takayanagi, and T. Ugajin, JHEP 1201, 125 (2012).
ds2 =1
r2
�− dt2
r2d(z−1)/(d−θ)+ r2θ/(d−θ)dr2 + dx2
i
�
Application of the Ryu-Takayanagi minimal area formula toa dual Einstein-Maxwell-dilaton theory yields
SE ∼ Q(d−1)/dP lnP
with a co-efficient independent of UV details and of the shapeof the entangling region. These properties are just as ex-pected for a circular Fermi surface with Q ∼ kdF .
Holography of Bose metals
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Holographic theory of a Bose metal
Hidden Fermi
surfacesof “quarks”
Fully fractionalized state has all the electricflux exiting to the horizon at r = ∞
Electric flux
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• Electric flux exiting the horizon corresponds tofractionalized component of the conserved den-sity Q, which is proposed to be associated with“hidden” Fermi surfaces of gauge-charged par-ticles.
• Gauss Law and the “attractor” mechanism inthe bulk⇔ Luttinger theorem on the boundary theory.
Holography, fractionalization,and hidden Fermi surfaces
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Conclusions
Realizations of many-particle entanglement:
Z2 spin liquids and conformal quantum critical points
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Conclusions
More complex examples in metallic states are experimentally
ubiquitous, but pose difficult strong-coupling problems to conventional methods of field
theory
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Conclusions
String theory and gravity in emergent dimensions
offer a remarkable new approach to describing states with many-particle quantum entanglement.
Much recent progress offers hope of a holographic description of “strange metals”
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r
anti-de Sitter space
Emergent holographic direction
Rd,1
Minkowski
AdSd+2
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anti-de Sitter space
Tuesday, September 11, 2012