2. coupled dimer antiferromagnet 3. honeycomb lattice ... · 2. coupled dimer antiferromagnet cft3:...
TRANSCRIPT
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1. Introduction to the Hubbard model Superexchange and antiferromagnetism
2. Coupled dimer antiferromagnet CFT3: the Wilson-Fisher fixed point
3. Honeycomb lattice: semi-metal and antiferromagnetism CFT3: Dirac fermions and the Gross-Neveu model
4. Quantum critical dynamics AdS/CFT and the collisionless-hydrodynamic crossover
5. Hubbard model as a SU(2) gauge theorySpin liquids, valence bond solids: analogies with SQED and SYM
Outline
Monday, June 14, 2010
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6. Square lattice: Fermi surfaces and spin density waves Fermi pockets and Quantum oscillations
7. Instabilities near the SDW critical point d-wave superconductivity and other orders
8. Global phase diagram of the cuprates Competition for the Fermi surface
Outline
Monday, June 14, 2010
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6. Square lattice: Fermi surfaces and spin density waves Fermi pockets and Quantum oscillations
7. Instabilities near the SDW critical point d-wave superconductivity and other orders
8. Global phase diagram of the cuprates Competition for the Fermi surface
Outline
Monday, June 14, 2010
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H = −�
i<j
tijc†iαcjα + U
�
i
�ni↑ −
1
2
��ni↓ −
1
2
�− µ
�
i
c†iαciα
tij → “hopping”. U → local repulsion, µ → chemical potential
Spin index α =↑, ↓
niα = c†iαciα
c†iαcjβ + cjβc
†iα = δijδαβ
ciαcjβ + cjβciα = 0
The Hubbard Model
Will study on the honeycomb and square lattices
Monday, June 14, 2010
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H = −�
i<j
tijc†iαcjα + U
�
i
�ni↑ −
1
2
��ni↓ −
1
2
�− µ
�
i
c†iαciα
tij → “hopping”. U → local repulsion, µ → chemical potential
Spin index α =↑, ↓
niα = c†iαciα
c†iαcjβ + cjβc
†iα = δijδαβ
ciαcjβ + cjβciα = 0
The Hubbard Model
Will study on the honeycomb and square lattices
Monday, June 14, 2010
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Fermi surface+antiferromagnetism
Γ
Hole states
occupied
Electron states
occupied
Γ
The electron spin polarization obeys�
�S(r, τ)�
= �ϕ(r, τ)eiK·r
where K is the ordering wavevector.
+
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Γ
Fermi surfaces in electron- and hole-doped cupratesHole states
occupied
Electron states
occupied
ΓEffective Hamiltonian for quasiparticles:
H0 = −�
i<j
tijc†iαciα ≡
�
k
εkc†kαckα
with tij non-zero for first, second and third neighbor, leads to satisfactory agree-
ment with experiments. The area of the occupied electron states, Ae, from
Luttinger’s theory is
Ae =
�2π
2(1− p) for hole-doping p
2π2(1 + x) for electron-doping x
The area of the occupied hole states, Ah, which form a closed Fermi surface and
so appear in quantum oscillation experiments is Ah = 4π2 −Ae.
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Spin density wave theory
In the presence of spin density wave order, �ϕ at wavevector K =(π,π), we have an additional term which mixes electron states withmomentum separated by K
Hsdw = �ϕ ·�
k,α,β
c†k,α�σαβck+K,β
where �σ are the Pauli matrices. The electron dispersions obtainedby diagonalizing H0 +Hsdw for �ϕ ∝ (0, 0, 1) are
Ek± =εk + εk+K
2±
��εk − εk+K
2
�+ ϕ2
This leads to the Fermi surfaces shown in the following slides forelectron and hole doping.
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Increasing SDW order
ΓΓΓ
S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).
Γ
Hole pockets
Electron pockets
Hole-doped cuprates
Monday, June 14, 2010
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Increasing SDW order
ΓΓΓ
S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).
Γ
Hole pockets
Electron pockets
Hole-doped cuprates
Monday, June 14, 2010
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Increasing SDW order
ΓΓΓ
S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).
Γ
Hole pockets
Electron pockets
Hole-doped cuprates
Hot spots
Monday, June 14, 2010
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Increasing SDW order
ΓΓΓ
S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).
Γ
Hole pockets
Electron pockets
Hole-doped cuprates
Fermi surface breaks up at hot spotsinto electron and hole “pockets”
Hole pockets
Hot spots
Monday, June 14, 2010
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Increasing SDW order
ΓΓΓ
S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).
Γ
Hole pockets
Electron pockets
Hole-doped cuprates
Fermi surface breaks up at hot spotsinto electron and hole “pockets”
Hot spots
Monday, June 14, 2010
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arXiv:0912.3022
Fermi liquid behaviour in an underdoped high Tc superconductor
Suchitra E. Sebastian, N. Harrison, M. M. Altarawneh, Ruixing Liang, D. A. Bonn, W. N. Hardy, and G. G. Lonzarich
Evidence for small Fermi pockets
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Increasing SDW order
ΓΓΓ
S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).
Γ
Hole pockets
Electron pockets
Large Fermi surface breaks up intoelectron and hole pockets
Hole-doped cuprates
Monday, June 14, 2010
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Increasing SDW order
ΓΓΓ
S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).
Γ
Hole pockets
Electron pockets
Hole-doped cuprates
�ϕ
�ϕ fluctuations act on thelarge Fermi surface
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Start from the “spin-fermion” model
Z =�DcαD�ϕ exp (−S)
S =�
dτ�
k
c†kα
�∂
∂τ− εk
�ckα
− λ
�dτ
�
i
c†iα�ϕi · �σαβciβeiK·ri
+�
dτd2r
�12
(∇r �ϕ)2 +�ζ2
(∂τ �ϕ)2 +s
2�ϕ2 +
u
4�ϕ4
�
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� = 1
� = 2
� = 4
� = 3
Low energy fermionsψ�
1α, ψ�2α
� = 1, . . . , 4
Lf = ψ�†1α
�ζ∂τ − iv�
1 · ∇r
�ψ�
1α + ψ�†2α
�ζ∂τ − iv�
2 · ∇r
�ψ�
2α
v�=11 = (vx, vy), v�=1
2 = (−vx, vy)Monday, June 14, 2010
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Lf = ψ�†1α
�ζ∂τ − iv�
1 · ∇r
�ψ�
1α + ψ�†2α
�ζ∂τ − iv�
2 · ∇r
�ψ�
2α
v1 v2
ψ2 fermionsoccupied
ψ1 fermionsoccupied
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Lf = ψ�†1α
�ζ∂τ − iv�
1 · ∇r
�ψ�
1α + ψ�†2α
�ζ∂τ − iv�
2 · ∇r
�ψ�
2α
v1 v2
“Hot spot”
“Cold” Fermi surfaces
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Order parameter: Lϕ =1
2(∇r �ϕ)
2+
�ζ2
(∂τ �ϕ)2
+s
2�ϕ2
+u
4�ϕ4
Lf = ψ�†1α
�ζ∂τ − iv�
1 · ∇r
�ψ�
1α + ψ�†2α
�ζ∂τ − iv�
2 · ∇r
�ψ�
2α
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Lf = ψ�†1α
�ζ∂τ − iv�
1 · ∇r
�ψ�
1α + ψ�†2α
�ζ∂τ − iv�
2 · ∇r
�ψ�
2α
Order parameter: Lϕ =1
2(∇r �ϕ)
2+
�ζ2
(∂τ �ϕ)2
+s
2�ϕ2
+u
4�ϕ4
“Yukawa” coupling: Lc = −λ�ϕ ·�ψ�†
1α�σαβψ�2β + ψ�†
2α�σαβψ�1β
�
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Hertz theoryIntegrate out fermions and obtain non-local corrections to Lϕ
Lϕ =12
�ϕ2�q2 + γ|ω|
�/2 ; γ =
2πvxvy
Exponent z = 2 and mean-field criticality (upto logarithms)
Lf = ψ�†1α
�ζ∂τ − iv�
1 · ∇r
�ψ�
1α + ψ�†2α
�ζ∂τ − iv�
2 · ∇r
�ψ�
2α
Order parameter: Lϕ =1
2(∇r �ϕ)
2+
�ζ2
(∂τ �ϕ)2
+s
2�ϕ2
+u
4�ϕ4
“Yukawa” coupling: Lc = −λ�ϕ ·�ψ�†
1α�σαβψ�2β + ψ�†
2α�σαβψ�1β
�
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Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004).
Integrate out fermions and obtain non-local corrections to Lϕ
Lϕ =12
�ϕ2�q2 + γ|ω|
�/2 ; γ =
2πvxvy
Exponent z = 2 and mean-field criticality (upto logarithms)
Lf = ψ�†1α
�ζ∂τ − iv�
1 · ∇r
�ψ�
1α + ψ�†2α
�ζ∂τ − iv�
2 · ∇r
�ψ�
2α
Order parameter: Lϕ =1
2(∇r �ϕ)
2+
�ζ2
(∂τ �ϕ)2
+s
2�ϕ2
+u
4�ϕ4
“Yukawa” coupling: Lc = −λ�ϕ ·�ψ�†
1α�σαβψ�2β + ψ�†
2α�σαβψ�1β
�
OK in d = 3, but higher order terms contain an
infinite number of marginal couplings in d = 2
Hertz theory
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Perform RG on both fermions and �ϕ,using a local field theory.
Lf = ψ�†1α
�ζ∂τ − iv�
1 · ∇r
�ψ�
1α + ψ�†2α
�ζ∂τ − iv�
2 · ∇r
�ψ�
2α
Order parameter: Lϕ =1
2(∇r �ϕ)
2+
�ζ2
(∂τ �ϕ)2
+s
2�ϕ2
+u
4�ϕ4
“Yukawa” coupling: Lc = −λ�ϕ ·�ψ�†
1α�σαβψ�2β + ψ�†
2α�σαβψ�1β
�
Monday, June 14, 2010
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6. Square lattice: Fermi surfaces and spin density waves Fermi pockets and Quantum oscillations
7. Instabilities near the SDW critical point d-wave superconductivity and other orders
8. Global phase diagram of the cuprates Competition for the Fermi surface
Outline
Monday, June 14, 2010
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6. Square lattice: Fermi surfaces and spin density waves Fermi pockets and Quantum oscillations
7. Instabilities near the SDW critical point d-wave superconductivity and other orders
8. Global phase diagram of the cuprates Competition for the Fermi surface
Outline
Monday, June 14, 2010
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Superconductivity by SDW fluctuation
exchange
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Physical Review B 34, 8190 (1986)
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Increasing SDW order
Spin density wave theory in hole-doped cuprates
ΓΓΓ Γ
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Increasing SDW order
ΓΓΓ Γ
Spin-fluctuation exchange theory of d-wave superconductivity in the cuprates
�ϕ
David Pines, Douglas Scalapino
Fermions at the large Fermi surface exchangefluctuations of the SDW order parameter �ϕ.
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Pairing by SDW fluctuation exchange
We now allow the SDW field �ϕ to be dynamical, coupling to elec-trons as
Hsdw = −�
k,q,α,β
�ϕq · c†k,α�σαβck+K+q,β .
Exchange of a �ϕ quantum leads to the effective interaction
Hee = −12
�
q
�
p,γ,δ
�
k,α,β
Vαβ,γδ(q)c†k,αck+q,βc†p,γcp−q,δ,
where the pairing interaction is
Vαβ,γδ(q) = �σαβ · �σγδχ0
ξ−2 + (q−K)2,
with χ0ξ2 the SDW susceptibility and ξ the SDW correlation length.
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BCS Gap equation
In BCS theory, this interaction leads to the ‘gapequation’ for the pairing gap ∆k ∝ �ck↑c−k↓�.
∆k = −�
p
�3χ0
ξ−2 + (p− k−K)2
�∆p
2�
ε2p + ∆2
p
Non-zero solutions of this equation require that∆k and ∆p have opposite signs when p− k ≈ K.
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Increasing SDW order
++_
_
Γ
d-wave pairing of the large Fermi surface
�ck↑c−k↓� ∝ ∆k = ∆0(cos(kx)− cos(ky))
�ϕ
K
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� = 1
� = 2
� = 4
� = 3
Low energy fermionsψ�
1α, ψ�2α
� = 1, . . . , 4
d-wave pairing in the theory of hotspots
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Hot spots have
strong instability to
d-wave pairing near
SDW critical point.
This instability is
stronger than the
BCS instability of a
Fermi liquid.
Pairing order parameter: εαβ�ψ31αψ
11β − ψ3
2αψ12β
�
d-wave pairing in the theory of hotspots
ψ31
ψ32
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ψ31 ψ3
2 ψ31 ψ3
2 ψ31
d-wave Cooper pairing instability in particle-particle channel
+�ϕ
+- +-
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Similar theory applies to the pnictides, and leads to s± pairing.
kx
ky
1
23
4 3̄
4̄
Q = ( !, 0)
!
2̄
1̄
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Emergent Pseudospin symmetry
Continuum theory of hotspots in invariant under:
�ψ�↑
ψ�†↓
�→ U �
�ψ�↑
ψ�†↓
�
where U � are arbitrary SU(2) matrices which can bedifferent on different hotspots �.
Monday, June 14, 2010
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ψ31 ψ3
2 ψ31 ψ3
2 ψ31
d-wave Cooper pairing instability in particle-particle channel
+�ϕ
+- +-
Monday, June 14, 2010
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ψ31 ψ3
2 ψ31 ψ3
2 ψ31
Bond density wave (with local Ising-nematic order) instability in particle-hole channel
+�ϕ
+- +-
Monday, June 14, 2010
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d-wave pairing has apartner instabilityin the particle-hole
channel
Density-wave order parameter:
�ψ3†1αψ
11α − ψ3†
2αψ12α
�
ψ31
ψ32
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ψ31
ψ32
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Single ordering wavevector Q:�c†k−Q/2,αck+Q/2,α
�=
Φ(cos kx − cos ky)
Q
Q
ψ31
ψ32
ψ31
ψ32
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!1
"1
No modulations on sites. Modulated bond-densitywave with local Ising-nematic ordering:�
c†k−Q/2,αck+Q/2,α
�= Φ(cos kx − cos ky)
“Bond density” measures amplitude for electrons to be
in spin-singlet valence bond:
VBS order
Monday, June 14, 2010
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!1
"1
No modulations on sites. Modulated bond-densitywave with local Ising-nematic ordering:�
c†k−Q/2,αck+Q/2,α
�= Φ(cos kx − cos ky)
“Bond density” measures amplitude for electrons to be
in spin-singlet valence bond:
VBS order
Monday, June 14, 2010
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M. J. Lawler, K. Fujita,
Jhinhwan Lee,
A. R. Schmidt,
Y. Kohsaka, Chung Koo
Kim, H. Eisaki,
S. Uchida, J. C. Davis,
J. P. Sethna, and
Eun-Ah Kim, preprint
STM measurements of Z(r), the energy asymmetry
in density of states in Bi2Sr2CaCu2O8+δ.
Monday, June 14, 2010
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B
M. J. Lawler, K. Fujita,
Jhinhwan Lee,
A. R. Schmidt,
Y. Kohsaka, Chung Koo
Kim, H. Eisaki,
S. Uchida, J. C. Davis,
J. P. Sethna, and
Eun-Ah Kim, preprint
AC
D
STM measurements of Z(r), the energy asymmetry
in density of states in Bi2Sr2CaCu2O8+δ.
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B
M. J. Lawler, K. Fujita,
Jhinhwan Lee,
A. R. Schmidt,
Y. Kohsaka, Chung Koo
Kim, H. Eisaki,
S. Uchida, J. C. Davis,
J. P. Sethna, and
Eun-Ah Kim, preprint
AC
D
Strong anisotropy of electronic states between
x and y directions:Electronic
“Ising-nematic” order
STM measurements of Z(r), the energy asymmetry
in density of states in Bi2Sr2CaCu2O8+δ.
Monday, June 14, 2010
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6. Square lattice: Fermi surfaces and spin density waves Fermi pockets and Quantum oscillations
7. Instabilities near the SDW critical point d-wave superconductivity and other orders
8. Global phase diagram of the cuprates Competition for the Fermi surface
Outline
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6. Square lattice: Fermi surfaces and spin density waves Fermi pockets and Quantum oscillations
7. Instabilities near the SDW critical point d-wave superconductivity and other orders
8. Global phase diagram of the cuprates Competition for the Fermi surface
Outline
Monday, June 14, 2010
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Antiferro-magnetism
Fermi surface
d-wavesupercon-ductivity
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Antiferro-magnetism
Fermi surface
d-wavesupercon-ductivity
Monday, June 14, 2010
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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
T*
Monday, June 14, 2010
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Antiferro-magnetism
Fermi surface
d-wavesupercon-ductivity
Monday, June 14, 2010
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Fermi surface
d-wavesupercon-ductivity
Spin density wave
Monday, June 14, 2010
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Fermi surface
d-wavesupercon-ductivity
Spin density wave
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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
T*
Monday, June 14, 2010
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LargeFermi
surface
StrangeMetal
Spin density wave (SDW)
d-wavesuperconductor
Small Fermipockets with
pairing fluctuations
Theory of quantum criticality in the cuprates
Onset of d-wave superconductivity
hides the critical point x = xm
Fluctuating, paired Fermi
pockets
T*
Monday, June 14, 2010
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LargeFermi
surface
StrangeMetal
Spin density wave (SDW)
d-wavesuperconductor
Small Fermipockets with
pairing fluctuationsE. Demler, S. Sachdevand Y. Zhang, Phys.Rev. Lett. 87,067202 (2001).
E. G. Moon andS. Sachdev, Phy.Rev. B 80, 035117(2009)
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
T*
Monday, June 14, 2010
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LargeFermi
surface
StrangeMetal
Spin density wave (SDW)
d-wavesuperconductor
Small Fermipockets with
pairing fluctuationsE. Demler, S. Sachdevand Y. Zhang, Phys.Rev. Lett. 87,067202 (2001).
E. G. Moon andS. Sachdev, Phy.Rev. B 80, 035117(2009)
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
T*
Monday, June 14, 2010
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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)
T*
Monday, June 14, 2010
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Small Fermipockets with
pairing fluctuationsLargeFermi
surface
StrangeMetal
Magneticquantumcriticality
Spin density wave (SDW)
Spin gap
Thermallyfluctuating
SDW
d-wavesuperconductor
V. Galitski andS. Sachdev, Phys.Rev. B 79, 134512(2009).
E. G. Moon andS. Sachdev, Phys.Rev. B 80, 035117(2009)
Theory of quantum criticality in the cuprates
Fluctuating, paired Fermi
pockets
Physics of competition: d-wave SC and SDW“eat up” same pieces of the large Fermi surface.
T*
Monday, June 14, 2010
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Small Fermipockets with
pairing fluctuationsLargeFermi
surface
StrangeMetal
Magneticquantumcriticality
Spin gapThermallyfluctuating
SDW
d-waveSC
T
Fluctuating, paired Fermi
pockets
T*
Monday, June 14, 2010
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H
SC
M"Normal"
(Large Fermisurface)
SDW(Small Fermi
pockets)
SC+SDW
Small Fermipockets with
pairing fluctuationsLargeFermi
surface
StrangeMetal
d-waveSC
T
Tsdw
Fluctuating, paired Fermi
pockets
T*
Monday, June 14, 2010
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H
SC
M"Normal"
(Large Fermisurface)
SDW(Small Fermi
pockets)
SC+SDW
Small Fermipockets with
pairing fluctuationsLargeFermi
surface
StrangeMetal
d-waveSC
T
Tsdw
Fluctuating, paired Fermi
pockets
T*
Quantum oscillations
Monday, June 14, 2010
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H
SC
M"Normal"
(Large Fermisurface)
SDW(Small Fermi
pockets)
SC+SDW
Small Fermipockets with
pairing fluctuationsLargeFermi
surface
StrangeMetal
d-waveSC
T
Tsdw
Fluctuating, paired Fermi
pockets
T*
Monday, June 14, 2010
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J. Chang, N. B. Christensen, Ch. Niedermayer, K. Lefmann,
H. M. Roennow, D. F. McMorrow, A. Schneidewind, P. Link, A. Hiess,
M. Boehm, R. Mottl, S. Pailhes, N. Momono, M. Oda, M. Ido, and
J. Mesot, Phys. Rev. Lett. 102, 177006
(2009).
J. Chang, Ch. Niedermayer, R. Gilardi, N.B. Christensen, H.M. Ronnow,
D.F. McMorrow, M. Ay, J. Stahn, O. Sobolev, A. Hiess, S. Pailhes, C. Baines, N. Momono,
M. Oda, M. Ido, and J. Mesot, Physical Review B 78, 104525 (2008).
Monday, June 14, 2010
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H
SC
M"Normal"
(Large Fermisurface)
SDW(Small Fermi
pockets)
SC+SDW
Small Fermipockets with
pairing fluctuationsLargeFermi
surface
StrangeMetal
d-waveSC
T
Tsdw
Fluctuating, paired Fermi
pockets
T*
Monday, June 14, 2010
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G. Knebel, D. Aoki, and J. Flouquet, arXiv:0911.5223
Similar phase diagram for CeRhIn5
Monday, June 14, 2010
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Similar phase diagram for the pnictides
Ishida, Nakai, and HosonoarXiv:0906.2045v1
0 0.02 0.04 0.06 0.08 0.10 0.12
150
100
50
0
SC
Ort
AFM Ort/
Tet
S. Nandi, M. G. Kim, A. Kreyssig, R. M. Fernandes, D. K. Pratt, A. Thaler, N. Ni, S. L. Bud'ko, P. C. Canfield, J. Schmalian, R. J. McQueeney, A. I. Goldman, arXiv:0911.3136.
Monday, June 14, 2010