What can gauge-gravity duality teach us about condensed...
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What can gauge-gravity duality teach us about
condensed matter physics ?
HARVARD
Twelfth Arnold Sommerfeld Lecture Series January 31 - February 3, 2012
sachdev.physics.harvard.eduThursday, February 2, 2012
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Liza Huijse Max Metlitski Brian Swingle
Rob Myers Ajay Singh
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1. The superfluid-insulator quantum phase transition
A. Field theory
B. Holography
2. Strange metals
A. Field theory
B. Holography
Thursday, February 2, 2012
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1. The superfluid-insulator quantum phase transition
A. Field theory
B. Holography
2. Strange metals
A. Field theory
B. Holography
Thursday, February 2, 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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Excitations of the insulator:
S =�
d2rdτ�|∂τψ|2 + v2|�∇ψ|2 + (g − gc)|ψ|2 +
u
2|ψ|4
�
M.P. A. Fisher, P.B. Weichmann, G. Grinstein, and D.S. Fisher, Phys. Rev. B 40, 546 (1989).
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g
T
gc
0
InsulatorSuperfluid
Quantumcritical
TKT
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g
T
gc
0
InsulatorSuperfluid
Quantumcritical
TKT
CFT3 at T>0
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Quantum critical transport
M.P.A. Fisher, G. Grinstein, and S.M. Girvin, Phys. Rev. Lett. 64, 587 (1990) K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).
σ =Q2
h× [Universal constant O(1) ]
(Q is the “charge” of one boson)
Transport co-oefficients not determinedby collision rate, but by
universal constants of nature
Conductivity
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Describe charge transport using Boltzmann theory of in-teracting bosons:
dv
dt+
v
τc= F.
This gives a frequency (ω) dependent conductivity
σ(ω) =σ0
1− iω τc
where τc ∼ �/(kBT ) is the time between boson collisions.
Also, we have σ(ω → ∞) = σ∞, associated with the den-sity of states for particle-hole creation (the “optical con-ductivity”) in the CFT3.
Quantum critical transport
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Quantum critical transport
Describe charge transport using Boltzmann theory of in-teracting bosons:
dv
dt+
v
τc= F.
This gives a frequency (ω) dependent conductivity
σ(ω) =σ0
1− iω τc
where τc ∼ �/(kBT ) is the time between boson collisions.
Also, we have σ(ω → ∞) = σ∞, associated with the den-sity of states for particle-hole creation (the “optical con-ductivity”) in the CFT3.
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Boltzmann theory of bosons
σ0
σ∞
ω
1/τc
Re[σ(ω)]
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g
T
gc
0
InsulatorSuperfluid
Quantumcritical
TKT�ψ� �= 0 �ψ� = 0
So far, we have described the quantum critical point usingthe boson particle and hole excitations of the insulator.
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However, we could equally well describe the conductivity
using the excitations of the superfluid, which are vortices.
These are quantum particles (in 2+1 dimensions) which
described by a (mirror/e.m.) “dual” CFT3 with an emer-
gent U(1) gauge field. Their T > 0 dynamics can also be
described by a Boltzmann equation:
Conductivity = Resistivity of vortices
g
T
gc
0
InsulatorSuperfluid
Quantumcritical
TKT�ψ� �= 0 �ψ� = 0
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g
T
gc
0
InsulatorSuperfluid
Quantumcritical
TKT�ψ� �= 0 �ψ� = 0
However, we could equally well describe the conductivity
using the excitations of the superfluid, which are vortices.
These are quantum particles (in 2+1 dimensions) which
described by a (mirror/e.m.) “dual” CFT3 with an emer-
gent U(1) gauge field. Their T > 0 dynamics can also be
described by a Boltzmann equation:
Conductivity = Resistivity of vortices
Thursday, February 2, 2012
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Boltzmann theory of bosons
σ0
σ∞
ω
1/τc
Re[σ(ω)]
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Boltzmann theory of vortices
σ∞1/τcv
1/σ0v
Re[σ(ω)]
ω
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Boltzmann theory of bosons
σ0
σ∞
ω
1/τc
Re[σ(ω)]
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1. The superfluid-insulator quantum phase transition
A. Field theory
B. Holography
2. Strange metals
A. Field theory
B. Holography
Thursday, February 2, 2012
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1. The superfluid-insulator quantum phase transition
A. Field theory
B. Holography
2. Strange metals
A. Field theory
B. Holography
Thursday, February 2, 2012
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r
J. McGreevy, arXiv0909.0518
Why AdSd+2 ?
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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.
Why AdSd+2 ?
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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.
Why AdSd+2 ?
Thursday, February 2, 2012
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AdS4 theory of “nearly perfect fluids”
C. P. Herzog, P. K. Kovtun, S. Sachdev, and D. T. Son,
Phys. Rev. D 75, 085020 (2007).
To leading order in a gradient expansion, charge transport inan infinite set of strongly-interacting CFT3s can be described byEinstein-Maxwell gravity/electrodynamics on AdS4-Schwarzschild
SEM =
�d4x
√−g
�− 1
4e2FabF
ab
�.
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To leading order in a gradient expansion, charge transport inan infinite set of strongly-interacting CFT3s can be described byEinstein-Maxwell gravity/electrodynamics on AdS4-Schwarzschild
SEM =
�d4x
√−g
�− 1
4e2FabF
ab
�.
AdS4 theory of “nearly perfect fluids”
R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, 066017 (2011)
We include all possible 4-derivative terms: after suitable field
redefinitions, the required theory has only one dimensionless
constant γ (L is the radius of AdS4):
SEM =
�d4x
√−g
�− 1
4e2FabF
ab+
γL2
e2CabcdF
abF cd
�,
where Cabcd is the Weyl curvature tensor.
Stability and causality constraints restrict |γ| < 1/12.The parameters e2 and γ can be determined from OPE co-
efficients of CFT of interest.
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AdS4 theory of strongly interacting “perfect fluids”
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
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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 strongly interacting “perfect fluids”
• The γ > 0 result has similarities tothe quantum-Boltzmann result fortransport of particle-like excitations
R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, 066017 (2011)Thursday, February 2, 2012
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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 strongly interacting “perfect fluids”
• The γ < 0 result can be interpretedas the transport of vortex-likeexcitations
R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, 066017 (2011)Thursday, February 2, 2012
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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 strongly interacting “perfect fluids”
R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, 066017 (2011)
• The γ = 0 case is the exact result for the large N limitof SU(N) gauge theory with N = 8 supersymmetry (theABJM model). The ω-independence is a consequence ofself-duality under particle-vortex duality (S-duality).
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New insights and solvable models for diffusion and transport of strongly interacting systems near quantum critical points
The description is far removed from, and complementary to, that of the quantum Boltzmann equation which builds on the quasiparticle/vortex picture.
Prospects for experimental tests of frequency-dependent, non-linear, and non-equilibrium transport
Quantum criticality and conformal field theories
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1. The superfluid-insulator quantum phase transition
A. Field theory
B. Holography
2. Strange metals
A. Field theory
B. Holography
Thursday, February 2, 2012
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1. The superfluid-insulator quantum phase transition
A. Field theory
B. Holography
2. Strange metals
A. Field theory
B. Holography
Thursday, February 2, 2012
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• Consider an infinite, continuum,
translationally-invariant quantum system with a glob-
ally conserved U(1) chargeQ (the “electron density”)
in spatial dimension d > 1.
• Describe zero temperature phases where d�Q�/dµ �=0, where µ (the “chemical potential”) which changes
the Hamiltonian, H, to H − µQ.
• Compressible systems must be gapless.
• Conformal quantum matter is compressible in
d = 1, but not for d > 1.
Compressible quantum matter
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• Consider an infinite, continuum,
translationally-invariant quantum system with a glob-
ally conserved U(1) chargeQ (the “electron density”)
in spatial dimension d > 1.
• Describe zero temperature phases where d�Q�/dµ �=0, where µ (the “chemical potential”) which changes
the Hamiltonian, H, to H − µQ.
• Compressible systems must be gapless.
• Conformal quantum matter is compressible in
d = 1, but not for d > 1.
Compressible quantum matter
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The only compressible phase of traditional condensed matter physics which does not break the translational or U(1) symmetries
is the Landau Fermi liquid
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Graphene
Conformal quantum matter
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Compressible quantum matter
Fermi Liquid with a
Fermi surface
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• The only low energy excitations are long-lived quasiparticlesnear the Fermi surface.
Fermi Liquid with a
Fermi surface
Compressible quantum matter
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• Luttinger relation: The total “volume (area)” A enclosedby the Fermi surface is equal to �Q�.
A
Fermi Liquid with a
Fermi surface
Compressible quantum matter
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The Fermi Liquid (FL)
L = f†σ
�∂τ − ∇2
2m− µ
�fσ + 4 Fermi terms
A =�f†σfσ
�= �Qσ�
Gf =1
ω − vF (k − kF ) + iω2
A
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=
• Model of a spin liquid (“Bose metal”): couple fermions to adynamical gauge field Aµ.
L = f†σ
�∂τ − iAτ − (∇− iA)2
2m− µ
�fσ
The Non-Fermi Liquid (NFL)
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The simplest example of an exotic compressible phase
(a “strange metal”) is realized by
fermions with a Fermi surface coupled to an Abelian
or non-Abelian gauge field.
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The theory of this strange metal is strongly coupled in two spatial dimensions, and the traditional field-
theoretic expansion methods break down.
S.-S. Lee, Phys. Rev. B 80, 165102 (2009)M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075127 (2010)
The simplest example of an exotic compressible phase
(a “strange metal”) is realized by
fermions with a Fermi surface coupled to an Abelian
or non-Abelian gauge field.
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A
L = f†σ
�∂τ − iAτ − (∇− iA)2
2m− µ
�fσ
The Non-Fermi Liquid (NFL)• The location of the Fermi surfaces is well defined,
and the Luttinger relation applies as before.
• The singularity of the Green’s function upon ap-proaching the Fermi surface is described by thescaling form
G−1f = q1−ηF (ω/qz)
where q = |k|−kF is the distance from the Fermisurface. So fermions disperse transverse to theFermi surface with dynamic exponent z.
• To three-loop order, we find η �= 0 and z = 3/2.
• The entropy density of the non-Fermi liquid S ∼T 1/z, because the density of fermion states is ef-fectively one dimensional.
A
A =�f†σfσ
�= �Qσ�
M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075127 (2010)Thursday, February 2, 2012
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A
L = f†σ
�∂τ − iAτ − (∇− iA)2
2m− µ
�fσ
The Non-Fermi Liquid (NFL)• The location of the Fermi surfaces is well defined,
and the Luttinger relation applies as before.
• The singularity of the Green’s function upon ap-proaching the Fermi surface is described by thescaling form
G−1f = q1−ηF (ω/qz)
where q = |k|−kF is the distance from the Fermisurface. So fermions disperse transverse to theFermi surface with dynamic exponent z.
• To three-loop order, we find η �= 0 and z = 3/2.
• The entropy density of the non-Fermi liquid S ∼T 1/z, because the density of fermion states is ef-fectively one dimensional.
A
A =�f†σfσ
�= �Qσ�
M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075127 (2010)Thursday, February 2, 2012
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A
L = f†σ
�∂τ − iAτ − (∇− iA)2
2m− µ
�fσ
The Non-Fermi Liquid (NFL)
A
A =�f†σfσ
�= �Qσ�
M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075127 (2010)
• The location of the Fermi surfaces is well defined,and the Luttinger relation applies as before.
• The singularity of the Green’s function upon ap-proaching the Fermi surface is described by thescaling form
G−1f = q1−ηF (ω/qz)
where q = |k|−kF is the distance from the Fermisurface. So fermions disperse transverse to theFermi surface with dynamic exponent z.
• To three-loop order, we find η �= 0 and z = 3/2.
• The entropy density of the non-Fermi liquid S ∼T 1/z, because the density of fermion states is ef-fectively one dimensional.
Thursday, February 2, 2012
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A
L = f†σ
�∂τ − iAτ − (∇− iA)2
2m− µ
�fσ
The Non-Fermi Liquid (NFL)
A
A =�f†σfσ
�= �Qσ�
M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075127 (2010)
• The location of the Fermi surfaces is well defined,and the Luttinger relation applies as before.
• The singularity of the Green’s function upon ap-proaching the Fermi surface is described by thescaling form
G−1f = q1−ηF (ω/qz)
where q = |k|−kF is the distance from the Fermisurface. So fermions disperse transverse to theFermi surface with dynamic exponent z.
• To three-loop order, we find η �= 0 and z = 3/2.
• The entropy density of the non-Fermi liquid S ∼T 1/z, because the density of fermion states is ef-fectively one dimensional.
Thursday, February 2, 2012
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Study the large N limit of a SU(N) gauge field coupled to
adjoint (matrix) fermions at a non-zero chemical potential
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1. The superfluid-insulator quantum phase transition
A. Field theory
B. Holography
2. Strange metals
A. Field theory
B. Holography
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1. The superfluid-insulator quantum phase transition
A. Field theory
B. Holography
2. Strange metals
A. Field theory
B. Holography
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r
J. McGreevy, arXiv0909.0518
Holography of non-Fermi liquids
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Consider the following (most) general metric for theholographic theory
ds2 =1
r2
�− dt2
r2d(z−1)/(d−θ)+ r2θ/(d−θ)dr2 + dx2
i
�
This metric transforms under rescaling as
xi → ζ xi
t → ζz t
ds → ζθ/d ds.
This identifies z as the dynamic critical exponent (z = 1for “relativistic” quantum critical points).
What is θ ? (θ = 0 for “relativistic” quantum criti-cal points).
Holography of non-Fermi liquids
Thursday, February 2, 2012
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Consider the following (most) general metric for theholographic theory
ds2 =1
r2
�− dt2
r2d(z−1)/(d−θ)+ r2θ/(d−θ)dr2 + dx2
i
�
This metric transforms under rescaling as
xi → ζ xi
t → ζz t
ds → ζθ/d ds.
This identifies z as the dynamic critical exponent (z = 1for “relativistic” quantum critical points).
What is θ ? (θ = 0 for “relativistic” quantum criti-cal points).
Holography of non-Fermi liquids
L. Huijse, S. Sachdev, B. Swingle, arXiv:1112.0573Thursday, February 2, 2012
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Consider the following (most) general metric for theholographic theory
ds2 =1
r2
�− dt2
r2d(z−1)/(d−θ)+ r2θ/(d−θ)dr2 + dx2
i
�
This metric transforms under rescaling as
xi → ζ xi
t → ζz t
ds → ζθ/d ds.
This identifies z as the dynamic critical exponent (z = 1for “relativistic” quantum critical points).
What is θ ? (θ = 0 for “relativistic” quantum criti-cal points).
Holography of non-Fermi liquids
Thursday, February 2, 2012
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At T > 0, there is a “black-brane” at r = rh.
The Beckenstein-Hawking entropy of the black-brane is the
thermal entropy of the quantum system r = 0.
The entropy density, S, is proportional to the
“area” of the horizon, and so S ∼ r−dh
r
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r
At T > 0, there is a “black-brane” at r = rh.
The Beckenstein-Hawking entropy of the black-brane is the
thermal entropy of the quantum system r = 0.
The entropy density, S, is proportional to the
“area” of the horizon, and so S ∼ r−dh
Under rescaling r → ζ(d−θ)/dr, and thetemperature T ∼ t−1, and so
S ∼ T (d−θ)/z
So θ is the “violation of hyperscaling” exponent.Thursday, February 2, 2012
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Holography of non-Fermi liquids
ds2 =1
r2
�− dt2
r2d(z−1)/(d−θ)+ r2θ/(d−θ)dr2 + dx2
i
�
L. Huijse, S. Sachdev, B. Swingle, arXiv:1112.0573
A non-Fermi liquid has gapless fermionicexcitations on the Fermi surface, whichdisperse in the single transverse directionwith dynamic critical exponent z, with en-tropy density ∼ T 1/z. So we expect com-pressible quantum states to have an effec-tive dimension d− θ with
θ = d− 1
S.-S. Lee, Phys. Rev. B 80, 165102 (2009)M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075127 (2010)
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Logarithmic violation of “area law”: SEE =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.
B
A
Entanglement entropy of Fermi surfaces
D. Gioev and I. Klich, Physical Review Letters 96, 100503 (2006)B. Swingle, Physical Review Letters 105, 050502 (2010)
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Logarithmic violation of “area law”: SEE =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.
B
A
Entanglement entropy of Fermi surfaces
Y. Zhang, T. Grover, and A. Vishwanath, Physical Review Letters 107, 067202 (2011)Thursday, February 2, 2012
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Holography of non-Fermi liquids
ds2 =1
r2
�− dt2
r2d(z−1)/(d−θ)+ r2θ/(d−θ)dr2 + dx2
i
�
• The entanglement entropy exhibits logarithmic vio-lation of the area law only for this value of θ !
• The co-efficient of the logarithmic term is consistentwith the Luttinger relation.
• Many other features of the holographic theory areconsistent with a boundary theory which has “hid-den” Fermi surfaces of gauge-charged fermions.
θ = d− 1
Thursday, February 2, 2012
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• The entanglement entropy exhibits logarithmic vio-lation of the area law only for this value of θ !
• The metric can be realized as the solution of a Einstein-Maxwell-Dilaton theory with no explicit fermions.The density of the “hidden Fermi surfaces” of theboundary gauge-charged fermions can be deducedfrom the electric flux leaking to r → ∞.
Holography of non-Fermi liquids
ds2 =1
r2
�− dt2
r2d(z−1)/(d−θ)+ r2θ/(d−θ)dr2 + dx2
i
�
θ = d− 1
N. Ogawa, T. Takayanagi, and T. Ugajin, arXiv:1111.1023; L. Huijse, S. Sachdev, B. Swingle, arXiv:1112.0573Thursday, February 2, 2012
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• The entanglement entropy exhibits logarithmic vio-lation of the area law only for this value of θ !
• The metric can be realized as the solution of a Einstein-Maxwell-Dilaton theory with no explicit fermions.The density of the “hidden Fermi surfaces” of theboundary gauge-charged fermions can be deducedfrom the electric flux leaking to r → ∞.
Holography of non-Fermi liquids
ds2 =1
r2
�− dt2
r2d(z−1)/(d−θ)+ r2θ/(d−θ)dr2 + dx2
i
�
θ = d− 1
K. Goldstein, S. Kachru, S. Prakash, and S. P. Trivedi JHEP 1008, 078 (2010)
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• The co-efficient of the logarithmic term in the entan-glement entropy is insensitive to all short-distancedetails, and depends only upon the fermion density.
• These two methods of deducing with fermion densityare consistent with the Luttinger relation !
Holography of non-Fermi liquids
ds2 =1
r2
�− dt2
r2d(z−1)/(d−θ)+ r2θ/(d−θ)dr2 + dx2
i
�
θ = d− 1
L. Huijse, S. Sachdev, B. Swingle, arXiv:1112.0573Thursday, February 2, 2012
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• The co-efficient of the logarithmic term in the entan-glement entropy is insensitive to all short-distancedetails, and depends only upon the fermion density.
• These two methods of deducing with fermion densityare consistent with the Luttinger relation !
Holography of non-Fermi liquids
ds2 =1
r2
�− dt2
r2d(z−1)/(d−θ)+ r2θ/(d−θ)dr2 + dx2
i
�
θ = d− 1
L. Huijse, S. Sachdev, B. Swingle, arXiv:1112.0573Thursday, February 2, 2012
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N. Ogawa, T. Takayanagi, and T. Ugajin, arXiv:1111.1023; L. Huijse, S. Sachdev, B. Swingle, arXiv:1112.0573
Inequalities
ds2 =1
r2
�− dt2
r2d(z−1)/(d−θ)+ r2θ/(d−θ)dr2 + dx2
i
�
The area law of entanglement entropy is obeyed for
θ ≤ d− 1.
The “null energy condition” of the gravity theory yields
z ≥ 1 +θ
d.
Remarkably, for d = 2, θ = d − 1 and z = 1 + θ/d, we obtainz = 3/2, the same value associated with the field theory.
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Conclusions
Non-Fermi liquid 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 holography offer a remarkable new approach to
describing “strange metal” states with long-range quantum
entanglement.
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Conclusions
Presented evidence for holographic dual of a Fermi surface coupled to Abelian or non-Abelian guage fields
Thursday, February 2, 2012