strong coupling problems in condensed matter and...
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![Page 1: Strong coupling problems in condensed matter and …qpt.physics.harvard.edu/talks/chicago_computations.pdfK. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997). In CFT3s collisions](https://reader035.vdocuments.us/reader035/viewer/2022071023/5fd7c345410a1362486ef891/html5/thumbnails/1.jpg)
Strong coupling problems in condensed matter
and the AdS/CFT correspondence
HARVARD
arXiv:0910.1139
Reviews:
Talk online: sachdev.physics.harvard.edu
arXiv:0901.4103
Thursday, November 5, 2009
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Frederik Denef, HarvardSean Hartnoll, Harvard
Christopher Herzog, PrincetonPavel Kovtun, VictoriaDam Son, Washington
Max Metlitski, Harvard
HARVARDThursday, November 5, 2009
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1. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s
2. Exact solution from AdS/CFT
3. Quantum criticality of Fermi surfaces The genus expansion
Thursday, November 5, 2009
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1. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s
2. Exact solution from AdS/CFT
3. Quantum criticality of Fermi surfaces The genus expansion
Thursday, November 5, 2009
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The Superfluid-Insulator transition
Boson Hubbard model
M.P. A. Fisher, P.B. Weichmann, G. Grinstein, and D.S. Fisher, Phys. Rev. B 40, 546 (1989).
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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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Cheng ChinJames Franck institutePhysics DepartmentChicago University
Having your cake and seeing it too - Exploring quantum criticality
and critical dynamics in ultracold atomic gases
Thursday, November 5, 2009
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Insulator (the vacuum) at large U
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Excitations:
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Excitations:
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Excitations of the insulator:
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g
T
gc
0
InsulatorSuperfluid
Quantumcritical
TKT
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g
T
gc
0
InsulatorSuperfluid
Quantumcritical
TKT
CFT3
�ψ� �= 0 �ψ� = 0
S =�
d2rdτ�|∂τψ|2 + v2|�∇ψ|2 + (g − gc)|ψ|2 +
u
2|ψ|4
�
Thursday, November 5, 2009
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g
T
gc
0
InsulatorSuperfluid
Quantumcritical
TKT
Thursday, November 5, 2009
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g
T
gc
0
InsulatorSuperfluid
Quantumcritical
TKT
Classical vortices and wave oscillations of the
condensate Dilute Boltzmann/Landau gas of particle and holes
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g
T
gc
0
InsulatorSuperfluid
Quantumcritical
TKT
CFT at T>0
Thursday, November 5, 2009
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D. B. Haviland, Y. Liu, and A. M. Goldman, Phys. Rev. Lett. 62, 2180 (1989)
Resistivity of Bi films
M. P. A. Fisher, Phys. Rev. Lett. 65, 923 (1990)
Conductivity σ
σSuperconductor(T → 0) = ∞σInsulator(T → 0) = 0
σQuantum critical point(T → 0) ≈ 4e2
h
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Quantum critical transport
S. Sachdev, Quantum Phase Transitions, Cambridge (1999).
Quantum “perfect fluid”with shortest possiblerelaxation time, τR
τR � �kBT
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Quantum critical transport Transport co-oefficients not determined
by collision rate, but byuniversal constants of nature
Electrical conductivity
σ =e2
h× [Universal constant O(1) ]
K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).Thursday, November 5, 2009
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Quantum critical transport
P. Kovtun, D. T. Son, and A. Starinets, Phys. Rev. Lett. 94, 11601 (2005)
, 8714 (1997).
Transport co-oefficients not determinedby collision rate, but by
universal constants of nature
Momentum transportη
s≡
viscosityentropy density
=�
kB× [Universal constant O(1) ]
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Quantum critical transport Euclidean field theory: Compute current correlations on R2 × S1
with circumference 1/T
1/T
R2
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Quantum critical transport Euclidean field theory: Compute current correlations on R2 × S1
with circumference 1/T
2πT
4πT
−2πT
Complex ω plane
Direct 1/N or � = 4− d expansion forcorrelators at ωn = 2πnTi, with n integer
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Quantum critical transport Euclidean field theory: Compute current correlations on R2 × S1
with circumference 1/T
2πT
4πT
−2πT
Complex ω plane
Strong coupling problem:Correlators at ω → 0, along the real axis.
Thursday, November 5, 2009
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Quantum critical transport Euclidean field theory: Compute current correlations on R2 × S1
with circumference 1/T
2πT
4πT
−2πT
Complex ω plane
Strong coupling problem:Correlators at ω → 0, along the real axis.
Thursday, November 5, 2009
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Density correlations in CFTs at T >0
Two-point density correlator, χ(k, ω)
Kubo formula for conductivity σ(ω) = limk→0
−iω
k2χ(k, ω)
For all CFT2s, at all �ω/kBT
χ(k, ω) =4e2
hK
vk2
v2k2 − ω2; σ(ω) =
4e2
h
Kv
−iω
where K is a universal number characterizing the CFT2 (the levelnumber), and v is the velocity of “light”.
This follows from the conformal mapping of the plane to the cylin-der, which relates correlators at T = 0 to those at T > 0.
Thursday, November 5, 2009
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Conformal mapping of plane to cylinder with circumference 1/T
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Conformal mapping of plane to cylinder with circumference 1/T
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Density correlations in CFTs at T >0
Two-point density correlator, χ(k, ω)
Kubo formula for conductivity σ(ω) = limk→0
−iω
k2χ(k, ω)
For all CFT2s, at all �ω/kBT
χ(k, ω) =4e2
hK
vk2
v2k2 − ω2; σ(ω) =
4e2
h
Kv
−iω
where K is a universal number characterizing the CFT2 (the levelnumber), and v is the velocity of “light”.This follows from the conformal mapping of the plane to the cylin-der, which relates correlators at T = 0 to those at T > 0.
No hydrodynamics in CFT2s.Thursday, November 5, 2009
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Density correlations in CFTs at T >0
Two-point density correlator, χ(k, ω)
Kubo formula for conductivity σ(ω) = limk→0
−iω
k2χ(k, ω)
For all CFT3s, at �ω � kBT
χ(k,ω) =4e2
hK
k2
√v2k2 − ω2
; σ(ω) =4e2
hK
where K is a universal number characterizing the CFT3, and v isthe velocity of “light”.
Thursday, November 5, 2009
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Density correlations in CFTs at T >0
Two-point density correlator, χ(k, ω)
Kubo formula for conductivity σ(ω) = limk→0
−iω
k2χ(k, ω)
K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).
However, for all CFT3s, at �ω � kBT , we have the Einstein re-lation
χ(k, ω) = 4e2χcDk2
Dk2 − iω; σ(ω) = 4e2Dχc =
4e2
hΘ1Θ2
where the compressibility, χc, and the diffusion constant Dobey
χ =kBT
(hv)2Θ1 ; D =
hv2
kBTΘ2
with Θ1 and Θ2 universal numbers characteristic of the CFT3
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Density correlations in CFTs at T >0
K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).
In CFT3s collisions are “phase” randomizing, and lead torelaxation to local thermodynamic equilibrium. So thereis a crossover from collisionless behavior for �ω � kBT , tohydrodynamic behavior for �ω � kBT .
σ(ω) =
4e2
hK , �ω � kBT
4e2
hΘ1Θ2 ≡ σQ , �ω � kBT
and in general we expect K �= Θ1Θ2 (verified for Wilson-Fisher fixed point).
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1. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s
2. Exact solution from AdS/CFT
3. Quantum criticality of Fermi surfaces The genus expansion
Thursday, November 5, 2009
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1. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s
2. Exact solution from AdS/CFT
3. Quantum criticality of Fermi surfaces The genus expansion
Thursday, November 5, 2009
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Field theories in D spacetime dimensions are char-
acterized by couplings g which obey the renormal-
ization group equation
udg
du= β(g)
where u is the energy scale. The RG equation is
local in energy scale, i.e. the RHS does not depend
upon u.
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Field theories in D spacetime dimensions are char-
acterized by couplings g which obey the renormal-
ization group equation
udg
du= β(g)
where u is the energy scale. The RG equation is
local in energy scale, i.e. the RHS does not depend
upon u.
Key idea: ⇒ Implement u as an extra dimen-sion, and map to a local theory in D+1 dimensions.
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At the RG fixed point, β(g) = 0, the D dimen-sional field theory is invariant under the scale trans-formation
xµ → xµ/b , u→ b u
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At the RG fixed point, β(g) = 0, the D dimen-sional field theory is invariant under the scale trans-formation
xµ → xµ/b , u→ b u
This is an invariance of the metric of the theory in
D + 1 dimensions. The unique solution is
ds2=
� u
L
�2dxµdxµ + L2 du2
u2.
Or, using the length scale z = L2/u
ds2= L2 dxµdxµ + dz2
z2.
This is the space AdSD+1, and L is the AdS radius.
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Bonus: AdSD+1 is a solution of Einstein’s equations
with a negative cosmological constant, and is a sym-
metric space; the full group of symmetries of the
metric is SO(D + 1, 1) (in Euclidean signature)
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Bonus: AdSD+1 is a solution of Einstein’s equations
with a negative cosmological constant, and is a sym-
metric space; the full group of symmetries of the
metric is SO(D + 1, 1) (in Euclidean signature)
SO(D+1, 1) is the group of conformal transforma-
tions in D dimensions, and relativistic field theo-
ries at the RG fixed point are conformally invari-
ant.
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At T > 0, the Euclidean field theory is on thecylinder RD−1×S1, where the time co-ordinate isperiodic under τ → τ + 1/T .
1/T
RD−1
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At T > 0, the Euclidean field theory is on thecylinder RD−1×S1, where the time co-ordinate isperiodic under τ → τ + 1/T .
1/T
RD−1
Solving Einstein’s equations with a negative cosmological constantwe have the solution
ds2 =L2
z2
�f(z)dτ2 + d�x2 +
dz2
f(z)
�; f(z) = 1−
�z
zH
�D
This is a AdS-Schwarzschild black hole with a horizon at z = zH .This space is periodic in τ with period 1/T for
T =d
4πzH
Thursday, November 5, 2009
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SU(N) SYM3 with N = 8 supersymmetry
• Has a single dimensionful coupling constant, e0, which flowsto a strong-coupling fixed point e0 = e∗0 in the infrared.
• The CFT3 describing this fixed point resembles “critical spinliquid” theories.
• This CFT3 is the low energy limit of string theory on anM2 brane. The AdS/CFT correspondence provides a dualdescription using 11-dimensional supergravity on AdS4×S7.
• The CFT3 has a global SO(8) R symmetry, and correlatorsof the SO(8) charge density can be computed exactly in thelarge N limit, even at T > 0.
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SU(N) SYM3 with N = 8 supersymmetry
• The SO(8) charge correlators of the CFT3 are given by the
usual AdS/CFT prescription applied to the following gauge
theory on AdS4:
S = − 1
4g24D
�d4x√−ggMAgNBF a
MNF aAB
where a = 1 . . . 28 labels the generators of SO(8). Note that
in large N theory, this looks like 28 copies of an Abelian gauge
theory.
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P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, 085020 (2007)
Imχ(k, ω)/k2 ImK√
k2 − ω2
Collisionless to hydrodynamic crossover of SYM3
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P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, 085020 (2007)
Imχ(k, ω)/k2
ImDχc
Dk2 − iω
Collisionless to hydrodynamic crossover of SYM3
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Universal constants of SYM3
σ(ω) =
4e2
hK , �ω � kBT
4e2
hΘ1Θ2 , �ω � kBT
χc =kBT
(hv)2Θ1
D =hv2
kBTΘ2
K =√
2N3/2
3
Θ1 =8π2√
2N3/2
9
Θ2 =3
8π2
P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, 085020 (2007) C. Herzog, JHEP 0212, 026 (2002)
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Electromagnetic self-duality
• Unexpected result, K = Θ1Θ2.
• This is traced to a four -dimensional electromagnetic
self-duality of the theory on AdS4. In the large Nlimit, the SO(8) currents decouple into 28 U(1) cur-
rents with a Maxwell action for the U(1) gauge fields
on AdS4.
• This special property is not expected for generic CFT3s.
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C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981)Thursday, November 5, 2009
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C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981)Thursday, November 5, 2009
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Electromagnetic self-duality
• Unexpected result, K = Θ1Θ2.
• This is traced to a four -dimensional electromagnetic
self-duality of the theory on AdS4. In the large Nlimit, the SO(8) currents decouple into 28 U(1) cur-
rents with a Maxwell action for the U(1) gauge fields
on AdS4.
• This special property is not expected for generic CFT3s.
• Although there is no boson-vortex self-duality at the
Wilson-Fisher fixed point, the applicability of AdS/CFT
suggests that the conductivity may be close to its
self-dual value, σ ≈ 4e2/h.
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Electromagnetic self-duality
• Unexpected result, K = Θ1Θ2.
• This is traced to a four -dimensional electromagnetic
self-duality of the theory on AdS4. In the large Nlimit, the SO(8) currents decouple into 28 U(1) cur-
rents with a Maxwell action for the U(1) gauge fields
on AdS4.
• This special property is not expected for generic CFT3s.
• Although there is no boson-vortex self-duality at the
Wilson-Fisher fixed point, the applicability of AdS/CFT
suggests that the conductivity may be close to its
self-dual value, σ ≈ 4e2/h.
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1. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s
2. Exact solution from AdS/CFT
3. Quantum criticality of Fermi surfaces The genus expansion
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1. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s
2. Exact solution from AdS/CFT
3. Quantum criticality of Fermi surfaces The genus expansion
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Small Fermipockets with
pairing fluctuationsLargeFermi
surface
StrangeMetal
Magneticquantumcriticality
Spin density wave (SDW)
Spin gap
Thermallyfluctuating
SDW
d-wavesuperconductor
Theory of quantum criticality in the cuprates
Competition between SDW order and superconductivitymoves the actual quantum critical point to x = xs < xm.
Fluctuating, paired Fermi
pockets
E. Demler, S. Sachdevand Y. Zhang, Phys.Rev. Lett. 87,067202 (2001).
E. G. Moon andS. Sachdev, Phy.Rev. B 80, 035117(2009)
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FluctuatingFermi
pocketsLargeFermi
surface
StrangeMetal
Spin density wave (SDW)
Theory of quantum criticality in the cuprates
Underlying SDW ordering quantum critical pointin metal at x = xm
Increasing SDW orderIncreasing SDW order
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Nematic order in YBCOV. Hinkov, D. Haug, B. Fauqué, P. Bourges, Y. Sidis, A. Ivanov, C. Bernhard, C. T. Lin, and B. Keimer , Science 319, 597 (2008)
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Nematic order in YBCO
V. Hinkov, D. Haug, B. Fauqué, P. Bourges, Y. Sidis, A. Ivanov, C. Bernhard, C. T. Lin, and B. Keimer , Science 319, 597 (2008)
Thursday, November 5, 2009
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Broken rotational symmetry in the pseudogap phase of a high-Tc superconductorR. Daou, J. Chang, David LeBoeuf, Olivier Cyr-Choiniere, Francis Laliberte, Nicolas Doiron-Leyraud, B. J. Ramshaw, Ruixing Liang, D. A. Bonn, W. N. Hardy, and Louis TailleferarXiv: 0909.4430
S. A. Kivelson, E. Fradkin, and V. J. Emery, Nature 393, 550 (1998).
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“Large” Fermi surfaces in cuprates
Γ
Hole states
occupied
Electron states
occupied
Γ
H0 = −�
i<j
tijc†iαciα ≡
�
k
εkc†kαckα
The area of the occupied electron/hole states:
Ae =�
2π2(1− x) for hole-doping x
2π2(1 + p) for electron-doping p
Ah = 4π2 −Ae
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Fermi surface with full square lattice symmetry
Quantum criticality of Pomeranchuk instability
x
y
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G(k,ω) =Z
ω − vF (k − kF )− iω2F�
k−kFω
� + . . .
Electron Green’s function in Fermi liquid (T=0)
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Electron Green’s function in Fermi liquid (T=0)
G(k,ω) =Z
ω − vF (k − kF )− iω2F�
k−kFω
� + . . .
Green’s function has a pole in the LHP at
ω = vF (k − kF )− iα(k − kF )2
+ . . .
Pole is at ω = 0 precisely at k = kF i.e. on a sphere of
radius kF in momentum space. This is the Fermi surface.
Re(ω)
Im(ω)
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Fermi surface with full square lattice symmetry
Quantum criticality of Pomeranchuk instability
x
y
Thursday, November 5, 2009
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Spontaneous elongation along x direction:Ising order parameter φ > 0.
Quantum criticality of Pomeranchuk instability
x
y
Thursday, November 5, 2009
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Quantum criticality of Pomeranchuk instability
x
y
Spontaneous elongation along y direction:Ising order parameter φ < 0.
Thursday, November 5, 2009
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λλc
Pomeranchuk instability as a function of coupling λ
�φ� = 0 �φ� �= 0
Quantum criticality of Pomeranchuk instability
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λλc
T
�φ� = 0 �φ� �= 0
Quantum criticality of Pomeranchuk instability
Phase diagram as a function of T and λ
Quantumcritical Tc
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λλc
T
�φ� = 0 �φ� �= 0
Quantum criticality of Pomeranchuk instability
Phase diagram as a function of T and λ
Quantumcritical Tc
Classicald=2 Isingcriticality
Thursday, November 5, 2009
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λλc
T
�φ� = 0 �φ� �= 0
Quantum criticality of Pomeranchuk instability
Phase diagram as a function of T and λ
Quantumcritical Tc
D=2+1 Ising
criticality ?
Thursday, November 5, 2009
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Quantum criticality of Pomeranchuk instability
Effective action for Ising order parameter
Sφ =�
d2rdτ�(∂τφ)2 + c2(∇φ)2 + (λ− λc)φ2 + uφ4
�
Thursday, November 5, 2009
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Quantum criticality of Pomeranchuk instability
Effective action for Ising order parameter
Sφ =�
d2rdτ�(∂τφ)2 + c2(∇φ)2 + (λ− λc)φ2 + uφ4
�
Effective action for electrons:
Sc =�
dτ
Nf�
α=1
�
i
c†iα∂τ ciα −�
i<j
tijc†iαciα
≡Nf�
α=1
�
k
�dτc†kα (∂τ + εk) ckα
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Quantum criticality of Pomeranchuk instability
�φ� > 0 �φ� < 0
Coupling between Ising order and electrons
Sφc = − γ
�dτ φ
Nf�
α=1
�
k
(cos kx − cos ky)c†kαckα
for spatially independent φ
Thursday, November 5, 2009
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Quantum criticality of Pomeranchuk instability
�φ� > 0 �φ� < 0
Coupling between Ising order and electrons
Sφc = − γ
�dτ
Nf�
α=1
�
k,q
φq (cos kx− cos ky)c†k+q/2,αck−q/2,α
for spatially dependent φ
Thursday, November 5, 2009
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Quantum criticality of Pomeranchuk instability
Sφ =�
d2rdτ�(∂τφ)2 + c2(∇φ)2 + (λ− λc)φ2 + uφ4
�
Quantum critical field theory
Z =�DφDciα exp (−Sφ − Sc − Sφc)
Sc =Nf�
α=1
�
k
�dτc†kα (∂τ + εk) ckα
Sφc = − γ
�dτ
Nf�
α=1
�
k,q
φq (cos kx− cos ky)c†k+q/2,αck−q/2,α
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Quantum criticality of Pomeranchuk instability
Hertz theory
Integrate out cα fermions and obtain non-local correctionsto φ action
δSφ ∼ Nfγ2
�d2q
4π2
�dω
2π|φ(q, ω)|2
� |ω|q
+ q2�
+ . . .
This leads to a critical point with dynamic critical expo-nent z = 3 and quantum criticality controlled by the Gaus-sian fixed point.
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Quantum criticality of Pomeranchuk instability
Hertz theory
Self energy of cα fermions to order 1/Nf
Σc(k, ω) ∼ i
Nfω2/3
This leads to the Green’s function
G(k, ω) ≈ 1ω − vF (k − kF )− i
Nfω2/3
Note that the order 1/Nf term is more singular in the infrared thanthe bare term; this leads to problems in the bare 1/Nf expansionin terms that are dominated by low frequency fermions.
1Nf (q2 + |ω|/q)
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Quantum criticality of Pomeranchuk instability
The infrared singularities of fermion particle-hole pairsare most severe on planar graphs: these all contribute at
leading order in 1/Nf .
1Nf (q2 + |ω|/q)
1ω − vF (k − kF )− i
Nfω2/3
Sung-Sik Lee, Physical Review B 80, 165102 (2009)
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Quantum criticality of Pomeranchuk instability
1Nf (q2 + |ω|/q)
1ω − vF (k − kF )− i
Nfω2/3
A string theory for the Fermi surface ?
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Conformal field theoryin 2+1 dimensions at T = 0
Einstein gravityon AdS4
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Conformal field theoryin 2+1 dimensions at T > 0
Einstein gravity on AdS4
with a Schwarzschildblack hole
Thursday, November 5, 2009
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Conformal field theoryin 2+1 dimensions at T > 0,
with a non-zero chemical potential, µand applied magnetic field, B
Einstein gravity on AdS4
with a Reissner-Nordstromblack hole carrying electric
and magnetic chargesThursday, November 5, 2009
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AdS4-Reissner-Nordstrom black hole
ds2 =L2
r2
�f(r)dτ2 +
dr2
f(r)+ dx2 + dy2
�,
f(r) = 1−�
1 +(r2
+µ2 + r4+B2)
γ2
� �r
r+
�3
+(r2
+µ2 + r4+B2)
γ2
�r
r+
�4
,
A = iµ
�1− r
r+
�dτ + Bx dy .
T =1
4πr+
�3−
r2+µ2
γ2−
r4+B2
γ2
�.
Thursday, November 5, 2009
![Page 85: Strong coupling problems in condensed matter and …qpt.physics.harvard.edu/talks/chicago_computations.pdfK. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997). In CFT3s collisions](https://reader035.vdocuments.us/reader035/viewer/2022071023/5fd7c345410a1362486ef891/html5/thumbnails/85.jpg)
T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv:0907.2694F. Denef, S. Hartnoll, and S. Sachdev, arXiv:0908.1788
Examine free energy and Green’s function of a probe particle
Thursday, November 5, 2009
![Page 86: Strong coupling problems in condensed matter and …qpt.physics.harvard.edu/talks/chicago_computations.pdfK. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997). In CFT3s collisions](https://reader035.vdocuments.us/reader035/viewer/2022071023/5fd7c345410a1362486ef891/html5/thumbnails/86.jpg)
Short time behavior depends uponconformal AdS4 geometry near boundary
T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv:0907.2694F. Denef, S. Hartnoll, and S. Sachdev, arXiv:0908.1788
Thursday, November 5, 2009
![Page 87: Strong coupling problems in condensed matter and …qpt.physics.harvard.edu/talks/chicago_computations.pdfK. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997). In CFT3s collisions](https://reader035.vdocuments.us/reader035/viewer/2022071023/5fd7c345410a1362486ef891/html5/thumbnails/87.jpg)
Long time behavior depends uponnear-horizon geometry of black hole
T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv:0907.2694F. Denef, S. Hartnoll, and S. Sachdev, arXiv:0908.1788
Thursday, November 5, 2009
![Page 88: Strong coupling problems in condensed matter and …qpt.physics.harvard.edu/talks/chicago_computations.pdfK. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997). In CFT3s collisions](https://reader035.vdocuments.us/reader035/viewer/2022071023/5fd7c345410a1362486ef891/html5/thumbnails/88.jpg)
Radial direction of gravity theory ismeasure of energy scale in CFT
T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv:0907.2694F. Denef, S. Hartnoll, and S. Sachdev, arXiv:0908.1788
Thursday, November 5, 2009
![Page 89: Strong coupling problems in condensed matter and …qpt.physics.harvard.edu/talks/chicago_computations.pdfK. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997). In CFT3s collisions](https://reader035.vdocuments.us/reader035/viewer/2022071023/5fd7c345410a1362486ef891/html5/thumbnails/89.jpg)
AdS4-Reissner-Nordstrom black hole
ds2 =L2
r2
�f(r)dτ2 +
dr2
f(r)+ dx2 + dy2
�,
f(r) = 1−�
1 +(r2
+µ2 + r4+B2)
γ2
� �r
r+
�3
+(r2
+µ2 + r4+B2)
γ2
�r
r+
�4
,
A = iµ
�1− r
r+
�dτ + Bx dy .
T =1
4πr+
�3−
r2+µ2
γ2−
r4+B2
γ2
�.
Thursday, November 5, 2009
![Page 90: Strong coupling problems in condensed matter and …qpt.physics.harvard.edu/talks/chicago_computations.pdfK. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997). In CFT3s collisions](https://reader035.vdocuments.us/reader035/viewer/2022071023/5fd7c345410a1362486ef891/html5/thumbnails/90.jpg)
AdS2 x R2 near-horizongeometry
r − r+ ∼ 1ζ
ds2 =R2
ζ2
�−dτ2 + dζ2
�+
r2+
R2
�dx2 + dy2
�
Thursday, November 5, 2009
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T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv:0907.2694
Infrared physics of Fermi surface is linked tothe near horizon AdS2 geometry of
Reissner-Nordstrom black hole
Thursday, November 5, 2009
![Page 92: Strong coupling problems in condensed matter and …qpt.physics.harvard.edu/talks/chicago_computations.pdfK. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997). In CFT3s collisions](https://reader035.vdocuments.us/reader035/viewer/2022071023/5fd7c345410a1362486ef891/html5/thumbnails/92.jpg)
T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv:0907.2694
AdS4
Geometric interpretation of RG flow
Thursday, November 5, 2009
![Page 93: Strong coupling problems in condensed matter and …qpt.physics.harvard.edu/talks/chicago_computations.pdfK. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997). In CFT3s collisions](https://reader035.vdocuments.us/reader035/viewer/2022071023/5fd7c345410a1362486ef891/html5/thumbnails/93.jpg)
T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv:0907.2694
AdS2 x R2
Geometric interpretation of RG flow
Thursday, November 5, 2009
![Page 94: Strong coupling problems in condensed matter and …qpt.physics.harvard.edu/talks/chicago_computations.pdfK. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997). In CFT3s collisions](https://reader035.vdocuments.us/reader035/viewer/2022071023/5fd7c345410a1362486ef891/html5/thumbnails/94.jpg)
Green’s function of a fermion
T. Faulkner, H. Liu, J. McGreevy, and
D. Vegh, arXiv:0907.2694
G(k,ω) ≈ 1ω − vF (k − kF )− iωθ(k)
See also M. Cubrovic, J Zaanen, and K. Schalm, arXiv:0904.1993
Thursday, November 5, 2009
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Green’s function of a fermion
T. Faulkner, H. Liu, J. McGreevy, and
D. Vegh, arXiv:0907.2694
G(k,ω) ≈ 1ω − vF (k − kF )− iωθ(k)
Similar to non-Fermi liquid theories of Fermi surfaces coupled to gauge fields, and at quantum critical points
Thursday, November 5, 2009
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General theory of finite temperature dynamics and transport near quantum critical points, with
applications to antiferromagnets, graphene, and superconductors
Conclusions
Thursday, November 5, 2009
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The AdS/CFT offers promise in providing a new understanding of
strongly interacting quantum matter at non-zero density
Conclusions
Thursday, November 5, 2009