theory of quantum matter: from quantum fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfsalam...
TRANSCRIPT
![Page 1: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/1.jpg)
Theory of Quantum Matter: from Quantum Fields
to Strings
HARVARD
Salam Distinguished Lectures The Abdus Salam International Center for Theoretical Physics
Trieste, Italy January 27-30, 2014
!Subir Sachdev
Talk online: sachdev.physics.harvard.edu
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1. The simplest models without quasiparticles
A. Superfluid-insulator transition
of ultracold bosons in an optical lattice
B. Conformal field theories in 2+1 dimensions and
the AdS/CFT correspondence
2. Metals without quasiparticles
A. Review of Fermi liquid theory
B. A “non-Fermi” liquid: the Ising-nematic
quantum critical point
C. Holography, entanglement, and strange metals
Outline
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1. The simplest models without quasiparticles
A. Superfluid-insulator transition
of ultracold bosons in an optical lattice
B. Conformal field theories in 2+1 dimensions and
the AdS/CFT correspondence
2. Metals without quasiparticles
A. Review of Fermi liquid theory
B. A “non-Fermi” liquid: the Ising-nematic
quantum critical point
C. Holography, entanglement, and strange metals
Outline
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(a) Entanglement
(b) Holography, entanglement, and CFTs
(c) Generalized holography beyond CFTs
(d) Holography of strange metals
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(a) Entanglement
(b) Holography, entanglement, and CFTs
(c) Generalized holography beyond CFTs
(d) Holography of strange metals
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| i ) Ground state of entire system,
⇢ = | ih |
⇢A = TrB⇢ = density matrix of region A
Entanglement entropy SE = �Tr (⇢A ln ⇢A)
B
Entanglement entropy
A P
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| i ) Ground state of entire system,
⇢ = | ih |
Take | i = 1p2(|"iA |#iB � |#iA |"iB)
Then ⇢A = TrB⇢ = density matrix of region A=
12 (|"iA h"|A + |#iA h#|A)
Entanglement entropy SE = �Tr (⇢A ln ⇢A)= ln 2
Entanglement entropy
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Band insulators
E
MetalMetal
carryinga current
InsulatorSuperconductor
kAn even number of electrons per unit cell
Entanglement entropy of a band insulator
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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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Logarithmic violation of “area law”: SE = CE 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.
The prefactor CE is independent of the shape of the entangling
region, and dependent only on IR features of the theory.
B
A P
Entanglement entropy of Fermi surfaces
B. Swingle, Physical Review Letters 105, 050502 (2010)Y. Zhang, T. Grover, and A. Vishwanath, Physical Review Letters 107, 067202 (2011)
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Logarithmic violation of “area law”: SE = CE 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.
The prefactor CE is independent of the shape of the entangling
region, and dependent only on IR features of the theory.
B
A
Entanglement entropy of Fermi surfaces
B. Swingle, Physical Review Letters 105, 050502 (2010)Y. Zhang, T. Grover, and A. Vishwanath, Physical Review Letters 107, 067202 (2011)
P
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Logarithmic violation of “area law”: SE = CE 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.
The prefactor CE is independent of the shape of the entangling
region, and dependent only on IR features of the theory.
B
A P
Entanglement entropy of Fermi surfaces
B. Swingle, Physical Review Letters 105, 050502 (2010)Y. Zhang, T. Grover, and A. Vishwanath, Physical Review Letters 107, 067202 (2011)
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g
T
gc
0
InsulatorSuperfluid
Quantumcritical
TKT
�c
Quantum state with
complex, many-body,
“long-range” quantum entanglement
h i 6= 0 h i = 0
S =
Zd2rdt
⇥|@t |2 � c2|rr |2 � V ( )
⇤
V ( ) = (�� �c)| |2 + u�| |2
�2
� ⇠ U/t
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• Entanglement entropy obeys SE = aP � �, where
� is a shape-dependent universal number associated
with the CFT3.
Entanglement at the quantum critical point
B
A P
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
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depth ofentanglement
D-dimensionalspace
Tensor network representation of entanglement at quantum critical point d
G. Vidal, Phys. Rev. Lett. 99, 220405 (2007)
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depth ofentanglement
D-dimensionalspaceA
d
Tensor network representation of entanglement at quantum critical point
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depth ofentanglement
D-dimensionalspaceA
Most links describe entanglement within A
d
Tensor network representation of entanglement at quantum critical point
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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
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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:0905.1317
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depth ofentanglement
D-dimensionalspace
Tensor network representation of entanglement at quantum critical point d
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String theory near a D-brane
depth ofentanglement
D-dimensionalspace
Emergent direction of AdS4
d
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depth ofentanglement
D-dimensionalspace
Tensor network representation of entanglement at quantum critical point
Emergent direction of AdS4 Brian Swingle, arXiv:0905.1317
d
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(a) Entanglement
(b) Holography, entanglement, and CFTs
(c) Generalized holography beyond CFTs
(d) Holography of strange metals
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(a) Entanglement
(b) Holography, entanglement, and CFTs
(c) Generalized holography beyond CFTs
(d) Holography of strange metals
![Page 25: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/25.jpg)
Field theories in d + 1 spacetime dimensions arecharacterized by couplings g which obey the renor-malization group equation
udg
du= �(g)
where u is the energy scale. The RG equation islocal in energy scale, i.e. the RHS does not dependupon u.
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u
xi
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u
xi
Holography
Key idea: ) Implement u as an extra dimension,
and map to a local theory in d+2 spacetime dimensions.
We identify the extra-dimensional co-ordinate r = 1/u.
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r
Holography
xi
Key idea: ) Implement u as an extra dimension,
and map to a local theory in d+2 spacetime dimensions.
We identify the extra-dimensional co-ordinate r = 1/u.
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r
Holography
xi
For a relativistic CFT in d spatial dimensions, the
proper length, ds, in the holographic space is fixed by
demanding the scale transformation (i = 1 . . . d)
xi ! ⇣xi , t ! ⇣t , ds ! ds
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r
Holography
xi
This gives the unique metric
ds
2=
1
r
2
��dt
2+ dr
2+ dx
2i
�
This is the metric of anti-de Sitter space AdSd+2.
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zr
AdS4R
2,1
Minkowski
CFT3
AdS/CFT correspondence
xi
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zr
AdS4R
2,1
Minkowski
CFT3
A
xi
Holography and Entanglement
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zr
AdS4R
2,1
Minkowski
CFT3
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.
xi
Holography and Entanglement
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zr
AdS4R
2,1
Minkowski
CFT3
AMinimal
surface area measures
entanglement entropy
S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 18160 (2006).
xi
Holography and Entanglement
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zr
AdS4R
2,1
Minkowski
CFT3
A
• Computation of minimal surface area yields
SE = aP � �,where � is a shape-dependent universal number.
xi
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
Holography and Entanglement
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(a) Entanglement
(b) Holography, entanglement, and CFTs
(c) Generalized holography beyond CFTs
(d) Holography of strange metals
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(a) Entanglement
(b) Holography, entanglement, and CFTs
(c) Generalized holography beyond CFTs
(d) Holography of strange metals
![Page 38: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/38.jpg)
r
xi
Generalized holography
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r
xi
Consider a metric which transforms under rescaling as
xi ! ⇣ xi, t ! ⇣
zt, ds ! ⇣
✓/dds.
Recall: conformal matter has ✓ = 0, z = 1, and the metric is
anti-de Sitter
Generalized holography
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r
xi
ds
2 =1
r
2
✓� dt
2
r
2d(z�1)/(d�✓)+ r
2✓/(d�✓)dr
2 + dx
2i
◆
The most general such metric is
Generalized holography
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ds
2 =1
r
2
✓� dt
2
r
2d(z�1)/(d�✓)+ r
2✓/(d�✓)dr
2 + dx
2i
◆Generalized holography
This is the most general metric which is invariant under the
scale transformation
xi ! ⇣ xi
t ! ⇣
zt
ds ! ⇣
✓/dds.
This identifies z as the dynamic critical exponent (z = 1 for
“relativistic” quantum critical points). We will see shortly
that ✓ is the violation of hyperscaling exponent.
We have used reparametrization invariance in r to define it so
that it scales as
r ! ⇣
(d�✓)/dr .
L. Huijse, S. Sachdev, B. Swingle, Physical Review B 85, 035121 (2012)
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ds
2 =1
r
2
✓� dt
2
r
2d(z�1)/(d�✓)+ r
2✓/(d�✓)dr
2 + dx
2i
◆
L. Huijse, S. Sachdev, B. Swingle, Physical Review B 85, 035121 (2012)
At T > 0, there is a “black-brane” at r = rh.
The Beckenstein-Hawking entropy of the black-brane is thethermal entropy of the quantum system r = 0.
The entropy density, S, is proportional to the“area” of the horizon, and so S ⇠ r�d
h
r
Generalized holography
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ds
2 =1
r
2
✓� dt
2
r
2d(z�1)/(d�✓)+ r
2✓/(d�✓)dr
2 + dx
2i
◆
L. Huijse, S. Sachdev, B. Swingle, Physical Review B 85, 035121 (2012)
r
Under rescaling r ! ⇣(d�✓)/dr, and the
temperature T ⇠ t�1, and so
S ⇠ T (d�✓)/z= T deff/z
where ✓ = d�de↵ , the “dimension deficit”, is now identified
as the violation of hyperscaling exponent.
Generalized holography
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L. Huijse, S. Sachdev, B. Swingle, Physical Review B 85, 035121 (2012)N. Ogawa, T. Takayanagi, and T. Ugajin, JHEP 1201, 125 (2012).
ds
2 =1
r
2
✓� dt
2
r
2d(z�1)/(d�✓)+ r
2✓/(d�✓)dr
2 + dx
2i
◆
The null energy condition (stability condition for gravity)
yields a new inequality
z � 1 +
✓
d
The non-Fermi liquid in d = 2 has ✓ = d�1, and this implies
z � 3/2. So the lower bound is precisely the value obtained
for the non-Fermi liquid!
Generalized holography
![Page 45: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/45.jpg)
L. Huijse, S. Sachdev, B. Swingle, Physical Review B 85, 035121 (2012)N. Ogawa, T. Takayanagi, and T. Ugajin, JHEP 1201, 125 (2012).
ds
2 =1
r
2
✓� dt
2
r
2d(z�1)/(d�✓)+ r
2✓/(d�✓)dr
2 + dx
2i
◆
The null energy condition (stability condition for gravity)
yields a new inequality
z � 1 +
✓
d
The non-Fermi liquid in d = 2 has ✓ = d�1, and this implies
z � 3/2. So the lower bound is precisely the value obtained
for the non-Fermi liquid!
Generalized holography
![Page 46: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/46.jpg)
L. Huijse, S. Sachdev, B. Swingle, Physical Review B 85, 035121 (2012)N. Ogawa, T. Takayanagi, and T. Ugajin, JHEP 1201, 125 (2012).
ds
2 =1
r
2
✓� dt
2
r
2d(z�1)/(d�✓)+ r
2✓/(d�✓)dr
2 + dx
2i
◆
Application of the Ryu-Takayanagi minimal area formula to a dual
Einstein-Maxwell-dilaton theory yields
SE ⇠
8<
:
P , for ✓ < d� 1
P lnP , for ✓ = d� 1
P ✓/(d�1) , for ✓ > d� 1
.
The non-Fermi liquid has log-violation of “area law”, and this ap-
pears precisely at the correct value ✓ = d� 1!
Moreover, the co-e�cient of P lnP computed holographically is in-
dependent of the shape of the entangling region just as expected for
a circular Fermi surface!!
Generalized holography
![Page 47: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/47.jpg)
L. Huijse, S. Sachdev, B. Swingle, Physical Review B 85, 035121 (2012)N. Ogawa, T. Takayanagi, and T. Ugajin, JHEP 1201, 125 (2012).
ds
2 =1
r
2
✓� dt
2
r
2d(z�1)/(d�✓)+ r
2✓/(d�✓)dr
2 + dx
2i
◆
Application of the Ryu-Takayanagi minimal area formula to a dual
Einstein-Maxwell-dilaton theory yields
SE ⇠
8<
:
P , for ✓ < d� 1
P lnP , for ✓ = d� 1
P ✓/(d�1) , for ✓ > d� 1
.
The non-Fermi liquid has log-violation of “area law”, and this ap-
pears precisely at the correct value ✓ = d� 1!
Moreover, the co-e�cient of P lnP computed holographically is in-
dependent of the shape of the entangling region just as expected for
a circular Fermi surface!!
Generalized holography
![Page 48: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/48.jpg)
L. Huijse, S. Sachdev, B. Swingle, Physical Review B 85, 035121 (2012)N. Ogawa, T. Takayanagi, and T. Ugajin, JHEP 1201, 125 (2012).
ds
2 =1
r
2
✓� dt
2
r
2d(z�1)/(d�✓)+ r
2✓/(d�✓)dr
2 + dx
2i
◆
Application of the Ryu-Takayanagi minimal area formula to a dual
Einstein-Maxwell-dilaton theory yields
SE ⇠
8<
:
P , for ✓ < d� 1
P lnP , for ✓ = d� 1
P ✓/(d�1) , for ✓ > d� 1
.
The non-Fermi liquid has log-violation of “area law”, and this ap-
pears precisely at the correct value ✓ = d� 1!
Moreover, the co-e�cient of P lnP computed holographically is in-
dependent of the shape of the entangling region just as expected for
a circular Fermi surface!!
Generalized holography
![Page 49: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/49.jpg)
(a) Entanglement
(b) Holography, entanglement, and CFTs
(c) Generalized holography beyond CFTs
(d) Holography of strange metals
![Page 50: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/50.jpg)
(a) Entanglement
(b) Holography, entanglement, and CFTs
(c) Generalized holography beyond CFTs
(d) Holography of strange metals
![Page 51: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/51.jpg)
Begin with a CFT
![Page 52: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/52.jpg)
Holographic representation: AdS4
S =
Zd
4x
p�g
1
22
✓R+
6
L
2
◆�
A 2+1 dimensional
CFTat T=0
![Page 53: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/53.jpg)
Holographic representation: AdS4
A 2+1 dimensional
CFTat T=0
ds
2 =
✓L
r
◆2 dr
2
f(r)� f(r)dt2 + dx
2 + dy
2
�
with f(r) = 1
![Page 54: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/54.jpg)
Apply a chemical potential
![Page 55: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/55.jpg)
This is to be solved subject to the constraint
Aµ(r ! 0, x, y, t) = Aµ(x, y, t)
where Aµ is a source coupling to a conserved U(1) current Jµ
of the CFT3
S = SCFT + i
ZdxdydtAµJµ
At non-zero chemical potential we simply require A⌧ = µ.
AdS4 theory of “nearly perfect fluids”To leading order in a gradient expansion, charge transport in
an infinite set of strongly-interacting CFT3s can be described by
Einstein-Maxwell gravity/electrodynamics on AdS4-Schwarzschild
SEM =
Zd
4x
p�g
� 1
4g
24
FabFab
�.
![Page 56: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/56.jpg)
AdS4 theory of “nearly perfect fluids”To leading order in a gradient expansion, charge transport in
an infinite set of strongly-interacting CFT3s can be described by
Einstein-Maxwell gravity/electrodynamics on AdS4-Schwarzschild
SEM =
Zd
4x
p�g
� 1
4g
24
FabFab
�.
This is to be solved subject to the constraint
Aµ(r ! 0, x, y, t) = Aµ(x, y, t)
where Aµ is a source coupling to a conserved U(1) current Jµ
of the CFT3
S = SCFT + i
ZdxdydtAµJµ
At non-zero chemical potential we simply require A⌧ = µ.
![Page 57: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/57.jpg)
+
++
++
+Electric flux
hQi6= 0
The Maxwell-Einstein theory of the applied chemical potential yields a AdS4-Reissner-Nordtröm black-brane
S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, Physical Review B 76, 144502 (2007)
Er = hQiEr = hQi
r
S =
Zd
4x
p�g
1
22
✓R+
6
L
2
◆� 1
4g24FabF
ab
�
![Page 58: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/58.jpg)
+
++
++
+Electric flux
hQi6= 0
The Maxwell-Einstein theory of the applied chemical potential yields a AdS4-Reissner-Nordtröm black-brane
Er = hQiEr = hQi
r
ds
2 =
✓L
r
◆2 dr
2
f(r)� f(r)dt2 + dx
2 + dy
2
�
with f(r) =⇣1� r
R
⌘2✓1 +
2r
R
+3r2
R
2
◆and R =
p6Lg4µ
, and A⌧ = µ
⇣1� r
R
⌘
![Page 59: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/59.jpg)
+
++
++
+Electric flux
hQi6= 0
The Maxwell-Einstein theory of the applied chemical potential yields a AdS4-Reissner-Nordtröm black-brane
At T = 0, we obtain an extremal black-brane, with
a near-horizon (IR) metric of AdS2 ⇥R
2
ds
2=
L
2
6
✓�dt
2+ dr
2
r
2
◆+ dx
2+ dy
2
r
T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv:0907.2694
![Page 60: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/60.jpg)
Einstein-Maxwell-dilaton theory
Electric flux
C. Charmousis, B. Gouteraux, B. S. Kim, E. Kiritsis and R. Meyer, JHEP 1011, 151 (2010).S. S. Gubser and F. D. Rocha, Phys. Rev. D 81, 046001 (2010).N. Iizuka, N. Kundu, P. Narayan and S. P. Trivedi, arXiv:1105.1162 [hep-th].
Er = hQi
Er = hQi
r
S =
Zd
d+2x
p�g
1
22
✓R� 2(r�)2 � V (�)
L
2
◆� Z(�)
4e2FabF
ab
�
with Z(�) = Z0e↵�, V (�) = �V0e
���, as � ! 1.
Holography of a non-Fermi liquid
![Page 61: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/61.jpg)
ds
2 =1
r
2
✓� dt
2
r
2d(z�1)/(d�✓)+ r
2✓/(d�✓)dr
2 + dx
2i
◆Holography of a non-Fermi liquid
The r ! 1 limit of the metric of the Einstein-Maxwell-
dilaton (EMD) theory has the most general form with
✓ =
d2�
↵+ (d� 1)�
z = 1 +
✓
d+
8(d(d� ✓) + ✓)2
d2(d� ✓)↵2.
![Page 62: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/62.jpg)
L. Huijse, S. Sachdev, B. Swingle, Physical Review B 85, 035121 (2012)
ds
2 =1
r
2
✓� dt
2
r
2d(z�1)/(d�✓)+ r
2✓/(d�✓)dr
2 + dx
2i
◆Holography of a non-Fermi liquid
Computation of the entanglement entropy in the EMD
theory via the Ryu-Takayanagi formula for ✓ = d� 1
yields
SE = CEQ(d�1)/dP lnP
where CE is independent of UV details.
This is precisely as expected for a Fermi surface, when
combined with the Luttinger theorem!
![Page 63: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/63.jpg)
L. Huijse, S. Sachdev, B. Swingle, Physical Review B 85, 035121 (2012)
ds
2 =1
r
2
✓� dt
2
r
2d(z�1)/(d�✓)+ r
2✓/(d�✓)dr
2 + dx
2i
◆Holography of a non-Fermi liquid
Computation of the entanglement entropy in the EMD
theory via the Ryu-Takayanagi formula for ✓ = d� 1
yields
SE = CEQ(d�1)/dP lnP
where CE is independent of UV details.
This is precisely as expected for a Fermi surface, when
combined with the Luttinger theorem!
![Page 64: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/64.jpg)
Holography of a non-Fermi liquid
Answer: need non-perturbative e↵ects of monopole operators
T. Faulkner and N. Iqbal, arXiv:1207.4208;
S. Sachdev, Phys. Rev. D 86, 126003 (2012)
Open questions:
• Is there any selection principle for the values of ✓ andz ?
• What is the physical interpretation of metallic stateswith ✓ 6= d� 1 ?
• Does the metallic state have a Fermi surface, and whatis kF ?
• Are there N2 Fermi surfaces of ‘quarks’ with kF ⇠ 1,or 1 Fermi surface of a ‘baryon’ with kF ⇠ N2 ?
• Why is kF not observable as Friedel oscillations in cor-relators of the density, and other gauge-neutral oper-ators ?
![Page 65: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/65.jpg)
Holography of a non-Fermi liquid
Answer: need non-perturbative e↵ects of monopole operators
T. Faulkner and N. Iqbal, arXiv:1207.4208;
S. Sachdev, Phys. Rev. D 86, 126003 (2012)
Open questions:
• Is there any selection principle for the values of ✓ andz ?
• What is the physical interpretation of metallic stateswith ✓ 6= d� 1 ?
• Does the metallic state have a Fermi surface, and whatis kF ?
• Are there N2 Fermi surfaces of ‘quarks’ with kF ⇠ 1,or 1 Fermi surface of a ‘baryon’ with kF ⇠ N2 ?
• Why is kF not observable as Friedel oscillations in cor-relators of the density, and other gauge-neutral oper-ators ?
![Page 66: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/66.jpg)
Holography of a non-Fermi liquid
Answer: need non-perturbative e↵ects of monopole operators
T. Faulkner and N. Iqbal, arXiv:1207.4208;
S. Sachdev, Phys. Rev. D 86, 126003 (2012)
Open questions:
• Is there any selection principle for the values of ✓ andz ?
• What is the physical interpretation of metallic stateswith ✓ 6= d� 1 ?
• Does the metallic state have a Fermi surface, and whatis kF ?
• Are there N2 Fermi surfaces of ‘quarks’ with kF ⇠ 1,or 1 Fermi surface of a ‘baryon’ with kF ⇠ N2 ?
• Why is kF not observable as Friedel oscillations in cor-relators of the density, and other gauge-neutral oper-ators ?
![Page 67: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/67.jpg)
Holography of a non-Fermi liquid
Answer: need non-perturbative e↵ects of monopole operators
T. Faulkner and N. Iqbal, arXiv:1207.4208;
S. Sachdev, Phys. Rev. D 86, 126003 (2012)
Open questions:
• Is there any selection principle for the values of ✓ andz ?
• What is the physical interpretation of metallic stateswith ✓ 6= d� 1 ?
• Does the metallic state have a Fermi surface, and whatis kF ?
• Are there N2 Fermi surfaces of ‘quarks’ with kF ⇠ 1,or 1 Fermi surface of a ‘baryon’ with kF ⇠ N2 ?
• Why is kF not observable as Friedel oscillations in cor-relators of the density, and other gauge-neutral oper-ators ?
![Page 68: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/68.jpg)
Holography of a non-Fermi liquid
Answer: need non-perturbative e↵ects of monopole operators
T. Faulkner and N. Iqbal, arXiv:1207.4208;
S. Sachdev, Phys. Rev. D 86, 126003 (2012)
Open questions:
• Is there any selection principle for the values of ✓ andz ?
• What is the physical interpretation of metallic stateswith ✓ 6= d� 1 ?
• Does the metallic state have a Fermi surface, and whatis kF ?
• Are there N2 Fermi surfaces of ‘quarks’ with kF ⇠ 1,or 1 Fermi surface of a ‘baryon’ with kF ⇠ N2 ?
• Why is kF not observable as Friedel oscillations in cor-relators of the density, and other gauge-neutral oper-ators ?
![Page 69: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/69.jpg)
Strongly-coupled quantum criticality leads to a novel regime of quantum dynamics without quasiparticles. !
The simplest examples are conformal field theories in 2+1 dimensions, realized by ultracold atoms in optical lattices. !
Holographic theories provide an excellent quantitative description of quantum Monte studies of quantum-critical boson models !
Exciting recent progress on the description of transport in metallic states without quasiparticles, via field theory and holography
![Page 70: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/70.jpg)
Strongly-coupled quantum criticality leads to a novel regime of quantum dynamics without quasiparticles. !
The simplest examples are conformal field theories in 2+1 dimensions, realized by ultracold atoms in optical lattices. !
Holographic theories provide an excellent quantitative description of quantum Monte studies of quantum-critical boson models !
Exciting recent progress on the description of transport in metallic states without quasiparticles, via field theory and holography
![Page 71: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/71.jpg)
Strongly-coupled quantum criticality leads to a novel regime of quantum dynamics without quasiparticles. !
The simplest examples are conformal field theories in 2+1 dimensions, realized by ultracold atoms in optical lattices. !
Holographic theories provide an excellent quantitative description of quantum Monte Carlo studies of quantum-critical boson models !
Exciting recent progress on the description of transport in metallic states without quasiparticles, via field theory and holography
![Page 72: Theory of Quantum Matter: from Quantum Fields to …qpt.physics.harvard.edu/talks/salam14_4.pdfSalam Distinguished Lectures! The Abdus Salam International Center for Theoretical Physics!](https://reader034.vdocuments.mx/reader034/viewer/2022050207/5f5a44665f64351cd516dee6/html5/thumbnails/72.jpg)
Strongly-coupled quantum criticality leads to a novel regime of quantum dynamics without quasiparticles. !
The simplest examples are conformal field theories in 2+1 dimensions, realized by ultracold atoms in optical lattices. !
Holographic theories provide an excellent quantitative description of quantum Monte Carlo studies of quantum-critical boson models !
Exciting recent progress on the description of transport in metallic states without quasiparticles, via field theory and holography