QUANTUM COMPUTATION AND QUANTUM INFORMATION GROVER'S ALGORITHM Igor Ilijašević1 / 26.
Quantum criticality and the phase diagram of the...
Transcript of Quantum criticality and the phase diagram of the...
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Quantum criticalityand the phase diagram of
the cuprates
HARVARD
Talk online: sachdev.physics.harvard.edu
Monday, November 2, 2009
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Victor Galitski, MarylandRibhu Kaul, Harvard Kentucky
Max Metlitski, HarvardEun Gook Moon, Harvard
Yang Qi, HarvardCenke Xu, Harvard Santa Barbara
HARVARDMonday, November 2, 2009
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The cuprate superconductors
Monday, November 2, 2009
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Γ Γ
Central ingredients in cuprate phase diagram: antiferromagnetism, superconductivity, and
change in Fermi surface
Monday, November 2, 2009
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Antiferro-magnetism
Fermi surface
d-wavesupercon-ductivity
Monday, November 2, 2009
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N. E. Hussey, J. Phys: Condens. Matter 20, 123201 (2008)
Crossovers in transport properties of hole-doped cuprates
0 0.05 0.1 0.15 0.2 0.25 0.3
(K)
Hole doping
2
+ 2
or
FL?
coh?
( )S-shaped
*
-wave SC
(1 < < 2)AFM
upturnsin ( )
Monday, November 2, 2009
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0 0.05 0.1 0.15 0.2 0.25 0.3
T(K)
Hole doping x
d-wave SC
AFM
Strange metal
Crossovers in transport properties of hole-doped cuprates
Pseudo-gap
Monday, November 2, 2009
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Antiferro-magnetism
Fermi surface
d-wavesupercon-ductivity
Monday, November 2, 2009
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Antiferro-magnetism
Fermi surface
d-wavesupercon-ductivity
Monday, November 2, 2009
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Ground state has long-range Néel order
Square lattice antiferromagnet
H =∑
〈ij〉
Jij !Si · !Sj
Order parameter is a single vector field !ϕ = ηi !Siηi = ±1 on two sublattices
〈!ϕ〉 #= 0 in Néel state.
Monday, November 2, 2009
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Square lattice antiferromagnet
H =∑
〈ij〉
Jij !Si · !Sj
J
J/λ
Weaken some bonds to induce spin entanglement in a new quantum phase
Monday, November 2, 2009
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Square lattice antiferromagnet
H =∑
〈ij〉
Jij !Si · !Sj
J
J/λ
Ground state is a “quantum paramagnet”with spins locked in valence bond singlets
=1√2
(∣∣∣↑↓〉−
∣∣∣ ↓↑〉)
Monday, November 2, 2009
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Pressure in TlCuCl3
λλc
Christian Ruegg, Bruce Normand, Masashige Matsumoto, Albert Furrer, Desmond McMorrow, Karl Kramer, Hans–Ulrich Gudel, Severian Gvasaliya,
Hannu Mutka, and Martin Boehm, Phys. Rev. Lett. 100, 205701 (2008)Monday, November 2, 2009
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Classicalspin
waves
Dilutetriplongas
Quantumcritical
Neel order Pressure in TlCuCl3Christian Ruegg et al. , Phys. Rev. Lett. 100, 205701 (2008)
S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).
Canonical quantum critical phase diagram of coupled-dimer antiferromagnet
Monday, November 2, 2009
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0 0.05 0.1 0.15 0.2 0.25 0.3
T(K)
Hole doping x
d-wave SC
AFM
Strange metal
Crossovers in transport properties of hole-doped cuprates
Pseudo-gap
Monday, November 2, 2009
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0 0.05 0.1 0.15 0.2 0.25 0.3
T(K)
Hole doping x
d-wave SC
AFM
Strange metal
xm
Crossovers in transport properties of hole-doped cuprates
S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).
A. J. Millis, Phys. Rev. B 48, 7183 (1993).
C. M. Varma, Phys. Rev. Lett. 83, 3538 (1999).
Pseudo-gap
Strange metal: quantum criticality ofoptimal doping critical point at x = xm ?
Monday, November 2, 2009
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0 0.05 0.1 0.15 0.2 0.25 0.3
T(K)
Hole doping x
d-wave SC
AFM
xs
Strange metal
Only candidate quantum critical point observed at low T
Spin density wave order presentbelow a quantum critical point at x = xswith xs ≈ 0.12 in the La series of cuprates
Monday, November 2, 2009
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Antiferro-magnetism
Fermi surface
d-wavesupercon-ductivity
Monday, November 2, 2009
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Antiferro-magnetism
Fermi surface
d-wavesupercon-ductivity
Monday, November 2, 2009
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“Large” Fermi surfaces in cuprates
Γ
Hole states
occupied
Electron states
occupied
Γ
H0 = −∑
i
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Spin density wave theory
The electron spin polarization obeys〈
!S(r, τ)〉
= !ϕ(r, τ)eiK·r
where !ϕ is the spin density wave (SDW) order parameter,and K is the ordering wavevector. For simplicity, we con-sider K = (π, π).
Monday, November 2, 2009
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Spin density wave theory
Spin density wave Hamiltonian
Hsdw = !ϕ ·∑
k,α,β
c†k,α!σαβck+K,β
Diagonalize H0 + Hsdw for !ϕ = (0, 0, ϕ)
Ek± =εk + εk+K
2±
√(εk − εk+K
2
)+ ϕ2
Monday, November 2, 2009
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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, November 2, 2009
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Increasing SDW order
ΓΓΓ Γ
Hole pockets
Electron pockets
Large Fermi surface breaks up intoelectron and hole pockets
Hole-doped cuprates
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).
Monday, November 2, 2009
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Spin density wave theory in hole-doped cuprates
A. J. Millis and M. R. Norman, Physical Review B 76, 220503 (2007). N. Harrison, Physical Review Letters 102, 206405 (2009).
Incommensurate order in YBa2Cu3O6+x
ΓΓ
Monday, November 2, 2009
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Increasing SDW orderIncreasing SDW order
ΓΓΓΓ
Electron pockets
Hole pockets
Electron-doped cuprates
Large Fermi surface breaks up intoelectron and hole pockets
D. Senechal and A.-M. S. Tremblay, Physical Review Letters 92, 126401 (2004)J. Lin, and A. J. Millis, Physical Review B 72, 214506 (2005).
Monday, November 2, 2009
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Nd2−xCexCuO4
T. Helm, M. V. Kartsovnik, M. Bartkowiak, N. Bittner,
M. Lambacher, A. Erb, J. Wosnitza, and R. Gross,
Phys. Rev. Lett. 103, 157002 (2009).
Increasing SDW orderIncreasing SDW order
Quantum oscillations
Monday, November 2, 2009
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Nature 450, 533 (2007)
Quantum oscillations
Monday, November 2, 2009
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Nature 450, 533 (2007)
Quantum oscillations
Monday, November 2, 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
Monday, November 2, 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).
Monday, November 2, 2009
http://arxiv.org/find/cond-mat/1/au:+Daou_R/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Daou_R/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Chang_J/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Chang_J/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+LeBoeuf_D/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+LeBoeuf_D/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Cyr_Choiniere_O/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Cyr_Choiniere_O/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Cyr_Choiniere_O/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Cyr_Choiniere_O/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Laliberte_F/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Laliberte_F/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Doiron_Leyraud_N/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Doiron_Leyraud_N/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Doiron_Leyraud_N/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Doiron_Leyraud_N/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Ramshaw_B/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Ramshaw_B/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Liang_R/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Liang_R/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Bonn_D/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Bonn_D/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Hardy_W/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Hardy_W/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Taillefer_L/0/1/0/all/0/1http://arxiv.org/find/cond-mat/1/au:+Taillefer_L/0/1/0/all/0/1
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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)
Monday, November 2, 2009
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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)
Monday, November 2, 2009
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Evidence for connection between linear resistivity andstripe-ordering in a cuprate with a low Tc
Linear temperature dependence of resistivity and change in the Fermi surface at the pseudogap critical point of a high-Tc superconductorR. Daou, Nicolas Doiron-Leyraud, David LeBoeuf, S. Y. Li, Francis Laliberté, Olivier Cyr-Choinière, Y. J. Jo, L. Balicas, J.-Q. Yan, J.-S. Zhou, J. B. Goodenough & Louis Taillefer, Nature Physics 5, 31 - 34 (2009)
Magnetic field of upto 35 T
used to suppress superconductivity
Monday, November 2, 2009
http://www.nature.com.ezp-prod1.hul.harvard.edu/nphys/journal/v5/n1/full/nphys1109.htmlhttp://www.nature.com.ezp-prod1.hul.harvard.edu/nphys/journal/v5/n1/full/nphys1109.htmlhttp://www.nature.com.ezp-prod1.hul.harvard.edu/nphys/journal/v5/n1/full/nphys1109.htmlhttp://www.nature.com.ezp-prod1.hul.harvard.edu/nphys/journal/v5/n1/full/nphys1109.html
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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
Monday, November 2, 2009
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Antiferro-magnetism
Fermi surface
d-wavesupercon-ductivity
Monday, November 2, 2009
-
Antiferro-magnetism
Fermi surface
d-wavesupercon-ductivity
Monday, November 2, 2009
-
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
Monday, November 2, 2009
-
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 superconductivityhides the critical point x = xm
Fluctuating, paired Fermi
pockets
Monday, November 2, 2009
-
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
Monday, November 2, 2009
-
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
Monday, November 2, 2009
-
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)
Monday, November 2, 2009
-
E. Demler, S. Sachdevand Y. Zhang, Phys.Rev. Lett. 87,067202 (2001).
E. G. Moon andS. Sachdev, Phy.Rev. B 80, 035117(2009)
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.
Criticality of the coupled dimer antiferromagnet at x=xs
Classicalspin
waves
Dilutetriplon
gas
Quantumcritical
Neel order
Fluctuating, paired Fermi
pockets
Monday, November 2, 2009
-
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.
Criticality of the topological change in Fermi surface at x=xm
E. Demler, S. Sachdevand Y. Zhang, Phys.Rev. Lett. 87,067202 (2001).
E. G. Moon andS. Sachdev, Phy.Rev. B 80, 035117(2009)
Fluctuating, paired Fermi
pockets
Increasing SDW orderIncreasing SDW order
Monday, November 2, 2009
-
Small Fermipockets with
pairing fluctuationsLargeFermi
surface
StrangeMetal
Magneticquantumcriticality
Spin gapThermallyfluctuating
SDW
d-waveSC
T
Fluctuating, paired Fermi
pockets
Monday, November 2, 2009
-
Small Fermipockets with
pairing fluctuationsLargeFermi
surface
StrangeMetal
Magneticquantumcriticality
Spin gapThermallyfluctuating
SDW
d-waveSC
T
Fluctuating, paired Fermi
pockets
E. Demler, S. Sachdev and Y. Zhang,Phys. Rev. Lett. 87, 067202 (2001).
Monday, November 2, 2009
-
Small Fermipockets with
pairing fluctuationsLargeFermi
surface
StrangeMetal
Magneticquantumcriticality
Spin gapThermallyfluctuating
SDW
d-waveSC
T
Fluctuating, paired Fermi
pockets
E. Demler, S. Sachdev and Y. Zhang,Phys. Rev. Lett. 87, 067202 (2001).
H
SC
M"Normal"
(Large Fermisurface)
SDW(Small Fermi
pockets)
SC+SDWHc2
Monday, November 2, 2009
-
Small Fermipockets with
pairing fluctuationsLargeFermi
surface
StrangeMetal
Magneticquantumcriticality
Spin gapThermallyfluctuating
SDW
d-waveSC
T
Fluctuating, paired Fermi
pockets
E. Demler, S. Sachdev and Y. Zhang,Phys. Rev. Lett. 87, 067202 (2001).
H
SC
M"Normal"
(Large Fermisurface)
SDW(Small Fermi
pockets)
SC+SDWHc2
Quantum oscillations
Monday, November 2, 2009
-
Small Fermipockets with
pairing fluctuationsLargeFermi
surface
StrangeMetal
Magneticquantumcriticality
Spin gapThermallyfluctuating
SDW
d-waveSC
T
Fluctuating, paired Fermi
pockets
E. Demler, S. Sachdev and Y. Zhang,Phys. Rev. Lett. 87, 067202 (2001).
H
SC
M"Normal"
(Large Fermisurface)
SDW(Small Fermi
pockets)
SC+SDW
Hsdw
Monday, November 2, 2009
-
Small Fermipockets with
pairing fluctuationsLargeFermi
surface
StrangeMetal
Magneticquantumcriticality
Spin gapThermallyfluctuating
SDW
d-waveSC
T
Fluctuating, paired Fermi
pockets
E. Demler, S. Sachdev and Y. Zhang,Phys. Rev. Lett. 87, 067202 (2001).
H
SC
M"Normal"
(Large Fermisurface)
SDW(Small Fermi
pockets)
SC+SDW
Hsdw
Monday, November 2, 2009
-
Small Fermipockets with
pairing fluctuationsLargeFermi
surface
StrangeMetal
Magneticquantumcriticality
Spin gapThermallyfluctuating
SDW
d-waveSC
T
Fluctuating, paired Fermi
pockets
E. Demler, S. Sachdev and Y. Zhang,Phys. Rev. Lett. 87, 067202 (2001).
H
SC
M"Normal"
(Large Fermisurface)
SDW(Small Fermi
pockets)
SC+SDW
Hsdw
Neutron scattering & muon resonance
Monday, November 2, 2009
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B. Lake, G. Aeppli, K. N. Clausen, D. F. McMorrow, K. Lefmann, N. E. Hussey, N. Mangkorntong, M. Nohara, H. Takagi, T. E. Mason, and A. SchröderScience 291, 1759 (2001).
B. Lake, H. M. Rønnow, N. B. Christensen, G. Aeppli, K. Lefmann, D. F. McMorrow, P. Vorderwisch, P. Smeibidl, N. Mangkorntong, T. Sasagawa, M. Nohara, H. Takagi, and T. E. Mason, Nature 415, 299 (2002)
Monday, November 2, 2009
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Monday, November 2, 2009
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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, November 2, 2009
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0.3 0.4 0.5 0.6 0.7
0
500
1000
1500
0.3 0.4 0.5 0.6 0.7
0
500
1000
1500
0.4 0.5 0.6
1500
1600
1700
0.3 0.4 0.5 0.6 0.7
-500
0
500
0.4 0.5 0.6
0.4
0.5
0.6
0.7
QH (r.l.u.)
I(14.9T,2K) ! I(0T,2K)
(d)
(c)
H = 14.9T
(b)
(a)
ba
I (14.9T, 2K) ! I (0T, 2K)
H = 0T
Inte
nsity (
counts
/ ~
10 m
in)
QH (r.l.u.)
H = 14.9T
2K
50K
QH (r.l.u.)
QK (r.l.u.)
Inte
nsity (
counts
/ ~
1 m
in)
Inte
nsity (
counts
/ ~
10 m
in)
Inte
nsity (
counts
/ ~
10 m
in)
QK
QH
0.3 0.4 0.5 0.6 0.7
20
40
60
80
100
(b)
(b)(a)
Inte
nsity (
a.u
.)
H = 0T
H = 13.5T
Inte
nsity (
counts
/ ~
12 m
in)
QH (r.l.u.)
H = 0T
H = 14.9T
0 1 2 3 4
100
1000
T = 2K
T = 2K
Energy E (meV)
D. Haug, V. Hinkov, A. Suchaneck, D. S. Inosov, N. B. Christensen, Ch. Niedermayer, P. Bourges, Y. Sidis, J. T. Park, A. Ivanov, C. T. Lin, J. Mesot, and B. Keimer, Phys. Rev. Lett. 103, 017001 (2009)
Monday, November 2, 2009
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Small Fermipockets with
pairing fluctuationsLargeFermi
surface
StrangeMetal
Magneticquantumcriticality
Spin gapThermallyfluctuating
SDW
d-waveSC
T
H
SC
M"Normal"
(Large Fermisurface)
SDW(Small Fermi
pockets)
SC+SDW
Fluctuating, paired Fermi
pockets
Monday, November 2, 2009
-
Small Fermipockets with
pairing fluctuationsLargeFermi
surface
StrangeMetal
Magneticquantumcriticality
Spin gapThermallyfluctuating
SDW
d-waveSC
T
H
SC
M"Normal"
(Large Fermisurface)
SDW(Small Fermi
pockets)
SC+SDW
Theory of theonset of d-wave
superconductivityfrom a
large Fermi surface
Fluctuating, paired Fermi
pockets
Monday, November 2, 2009
-
Increasing SDW order
ΓΓΓ Γ
Spin-fluctuation exchange theory of d-wave superconductivity in the cuprates
!ϕ
D. J. Scalapino, E. Loh, and J. E. Hirsch, Phys. Rev. B 34, 8190 (1986) K. Miyake, S. Schmitt-Rink, and C. M. Varma, Phys. Rev. B 34, 6554 (1986)
Fermions at the large Fermi surface exchangefluctuations of the SDW order parameter !ϕ.
Monday, November 2, 2009
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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
D. J. Scalapino, E. Loh, and J. E. Hirsch, Phys. Rev. B 34, 8190 (1986) K. Miyake, S. Schmitt-Rink, and C. M. Varma, Phys. Rev. B 34, 6554 (1986)
Monday, November 2, 2009
-
:
• Tc increases upon approaching the SDW transition.
• Pairing from SDW fluctuations: SDW and SC ordersdo not compete, but attract each other.
Ar. Abanov, A. V. Chubukov and J. Schmalian, Advances in Physics 52, 119 (2003).
Approaching the onset of antiferromagnetism in the spin-fluctuation theory
Monday, November 2, 2009
-
:
• Tc increases upon approaching the SDW transition.
• Pairing from SDW fluctuations: SDW and SC ordersdo not compete, but attract each other.
Ar. Abanov, A. V. Chubukov and J. Schmalian, Advances in Physics 52, 119 (2003).
Approaching the onset of antiferromagnetism in the spin-fluctuation theory
Monday, November 2, 2009
-
H
SC
M"Normal"
(Large Fermisurface)
SDW(Small Fermi
pockets)
SC+SDW
Small Fermipockets with
pairing fluctuationsLargeFermi
surface
StrangeMetal
Magneticquantumcriticality
Spin gapThermallyfluctuating
SDW
d-waveSC
T
Fluctuating, paired Fermi
pockets
Monday, November 2, 2009
-
H
SC
M"Normal"
(Large Fermisurface)
SDW(Small Fermi
pockets)
SC+SDW
Small Fermipockets with
pairing fluctuationsLargeFermi
surface
StrangeMetal
Magneticquantumcriticality
Spin gapThermallyfluctuating
SDW
d-waveSC
T
Theory of theonset of d-wave
superconductivityfrom small
Fermi pockets
Fluctuating, paired Fermi
pockets
Monday, November 2, 2009
-
H
SC
M"Normal"
(Large Fermisurface)
SDW(Small Fermi
pockets)
SC+SDW
Small Fermipockets with
pairing fluctuationsLargeFermi
surface
StrangeMetal
Magneticquantumcriticality
Spin gapThermallyfluctuating
SDW
d-waveSC
T
Fluctuating, paired Fermi
pockets
Physics ofcompetition:
d-wave SC andSDW “eat up’same pieces ofthe large Fermi
surface.
B. Kyung, J.-S. Landry, andA.-M. S. Tremblay,Phys. Rev. B 68, 174502 (2003).
Monday, November 2, 2009
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Begin with SDW ordered state, and rotate to a framepolarized along the local orientation of the SDW order !̂ϕ
(c↑c↓
)= R
(ψ+ψ−
); R† !̂ϕ · !σR = σz ; R†R = 1
Theory of underdoped cuprates
Increasing SDW order
ΓΓΓ Γ
H. J. Schulz, Physical Review Letters 65, 2462 (1990)B. I. Shraiman and E. D. Siggia, Physical Review Letters 61, 467 (1988).
J. R. Schrieffer, Journal of Superconductivity 17, 539 (2004)Monday, November 2, 2009
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Theory of underdoped cuprates
With R =(
z↑ −z∗↓z↓ z∗↑
)or !̂ϕ = z∗α!σαβzβ
the theory is invariant under
zα → eiθzα ; ψ+ → e−iθψ+ ; ψ− → eiθψ.−
We obtain a U(1) gauge theory of
• bosonic neutral spinons zα;
• spinless, charged fermions ψ±;
• an emergent U(1) gauge field Aµ.
X.-G. Wen, Phys. Rev.B 39, 7223 (1989).P. A. Lee, Phys. Rev. Lett.63, 680 (1989).R. Shankar, Phys. Rev.Lett. 63, 203 (1989).L. B. Ioffe and P. B. Wieg-mann, Phys. Rev. Lett.65, 653 (1990).R. K. Kaul, A. Kolezhuk,M. Levin, S. Sachdev, andT. Senthil, Phys. Rev. B75 , 235122 (2007).
Monday, November 2, 2009
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Increasing SDW order
Γ
Pairing across Tc
Focus on pairing near (π, 0), (0, π), where ψ± ≡ g±,and the electron operators are
(c1↑c1↓
)= Rz
(g+g−
);
(c2↑c2↓
)= Rz
(g+−g−
)
Rz ≡(
z↑ −z∗↓z↓ z∗↑
).
Electron c1α,spinless fermion g±
Electron c2α,spinless fermion g±
R. K. Kaul, M. Metlitksi, S. Sachdev, and Cenke Xu, Phys. Rev. B 78, 045110 (2008).
Monday, November 2, 2009
-
Increasing SDW order
Γ
Pairing across Tc
Focus on pairing near (π, 0), (0, π), where ψ± ≡ g±,and the electron operators are
(c1↑c1↓
)= Rz
(g+g−
);
(c2↑c2↓
)= Rz
(g+−g−
)
Rz ≡(
z↑ −z∗↓z↓ z∗↑
).
Electron c1α,spinless fermion g±
Electron c2α,spinless fermion g±
R. K. Kaul, M. Metlitksi, S. Sachdev, and Cenke Xu, Phys. Rev. B 78, 045110 (2008).
Monday, November 2, 2009
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Fluctuating pocket theory forelectrons near (0, π) and (π, 0)
Attractive gauge forces lead to simple s-wave pairing of the g±
〈g+g−〉 = ∆
For the physical electron operators, this pairing implies
〈c1↑c1↓〉 = ∆〈|zα|2
〉
〈c2↑c2↓〉 = −∆〈|zα|2
〉
i.e. d-wave pairing !
R. K. Kaul, M. Metlitksi, S. Sachdev, and Cenke Xu, Phys. Rev. B 78, 045110 (2008).
Monday, November 2, 2009
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Increasing SDW order
++_
_
Γ
Strong pairing of the g± electron pockets
g±
〈g+g−〉 = ∆
Monday, November 2, 2009
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〈f+1(k)f−1(−k)〉 ∼ (kx − ky)J〈g+g−〉;〈f+2(k)f−2(−k)〉 ∼ (kx + ky)J〈g+g−〉;〈f+1(k)f−2(−k)〉 = 0,
Increasing SDW order
f±v ++_
_
Γ
Weak pairing of the f± hole pockets
Monday, November 2, 2009
-
V. B. Geshkenbein, L. B. Ioffe, and A. I. Larkin, Phys. Rev. B 55, 3173 (1997).
Γ
-2e bosons at antinodes,+e fermion “arcs” at nodes,and proximity “Josephson”
coupling
Similar features in our theoryV. Galitski and S. Sachdev, Physical Review B 79, 134512 (2009).
Monday, November 2, 2009
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• d-wave superconductivity.
• Nodal-anti-nodal dichotomy: strong pairing near (π, 0),(0, π), and weak pairing near zone diagonals.
• Tc decreases as spin correlation increases (competingorder effect).
• Shift in quantum critical point of SDW ordering:gauge fluctuations are stronger in the superconduc-tor.
• After onset of superconductivity, monopoles condenseand lead to confinement and nematic and/orvalence bond solid (VBS) order.
Features of superconductivity
V. Galitski and S. Sachdev, Physical Review B 79, 134512 (2009).
Monday, November 2, 2009
-
• d-wave superconductivity.
• Nodal-anti-nodal dichotomy: strong pairing near (π, 0),(0, π), and weak pairing near zone diagonals.
• Tc decreases as spin correlation increases (competingorder effect).
• Shift in quantum critical point of SDW ordering:gauge fluctuations are stronger in the superconduc-tor.
• After onset of superconductivity, monopoles condenseand lead to confinement and nematic and/orvalence bond solid (VBS) order.
Features of superconductivity
Eun Gook Moon and S. Sachdev, Physical Review B 80, 035117 (2009).
Monday, November 2, 2009
-
• d-wave superconductivity.
• Nodal-anti-nodal dichotomy: strong pairing near (π, 0),(0, π), and weak pairing near zone diagonals.
• Tc decreases as spin correlation increases (competingorder effect).
• Shift in quantum critical point of SDW ordering:gauge fluctuations are stronger in the superconduc-tor.
• After onset of superconductivity, monopoles condenseand lead to confinement and nematic and/orvalence bond solid (VBS) order.
Features of superconductivity
R. K. Kaul, M. Metlitksi, S. Sachdev, and Cenke Xu, Phys. Rev. B 78, 045110 (2008).Monday, November 2, 2009
-
H
SC
M"Normal"
(Large Fermisurface)
SDW(Small Fermi
pockets)
SC+SDW
Small Fermipockets with
pairing fluctuationsLargeFermi
surface
StrangeMetal
Magneticquantumcriticality
Spin gapThermallyfluctuating
SDW
d-waveSC
T
Fluctuating, paired Fermi
pockets
Monday, November 2, 2009
-
H
SC
M"Normal"
(Large Fermisurface)
SDW(Small Fermi
pockets)
SC+SDW
Small Fermipockets with
pairing fluctuationsLargeFermi
surface
StrangeMetal
d-waveSC
T
Tsdw
R. K. Kaul, M. Metlitksi, S. Sachdev, and Cenke Xu, Physical Review B 78, 045110 (2008).
Fluctuating, paired Fermi
pockets
VBSand/ornematic
Monday, November 2, 2009
-
H
SC
M"Normal"
(Large Fermisurface)
SDW(Small Fermi
pockets)
SC+SDW
Small Fermipockets with
pairing fluctuationsLargeFermi
surface
StrangeMetal
d-waveSC
T
Tsdw
R. K. Kaul, M. Metlitksi, S. Sachdev, and Cenke Xu, Physical Review B 78, 045110 (2008).
Fluctuating, paired Fermi
pockets
VBSand/ornematic
Monday, November 2, 2009
-
H
SC
M"Normal"
(Large Fermisurface)
SDW(Small Fermi
pockets)
SC+SDW
Small Fermipockets with
pairing fluctuationsLargeFermi
surface
StrangeMetal
d-waveSC
T
Tsdw
R. K. Kaul, M. Metlitksi, S. Sachdev, and Cenke Xu, Physical Review B 78, 045110 (2008).
Fluctuating, paired Fermi
pockets
VBSand/ornematic
Monday, November 2, 2009
-
Quantum phase transitions in metal
〈z〉 #= 0 ; 〈N〉 #= 0SDW order
small Fermi pockets 〈z〉 #= 0 ; 〈N〉 = 0Fermi liquidlarge Fermi surface
〈z〉 = 0 ; 〈N〉 #= 0small critical Fermi pockets
gapless U(1) photon
〈z〉 = 0 ; 〈N〉 = 0large critical Fermi surface
gapless SU(2) photons
S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, Physical Review B 80, 155129 (2009)
Increasing SDW orderIncreasing SDW order
Monday, November 2, 2009
-
Quantum phase transitions in metal
〈z〉 #= 0 ; 〈N〉 #= 0SDW order
small Fermi pockets 〈z〉 #= 0 ; 〈N〉 = 0Fermi liquidlarge Fermi surface
〈z〉 = 0 ; 〈N〉 #= 0small critical Fermi pockets
gapless U(1) photon
〈z〉 = 0 ; 〈N〉 = 0large critical Fermi surface
gapless SU(2) photons
S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, Physical Review B 80, 155129 (2009)
Fermi liquid phases
Increasing SDW orderIncreasing SDW order
Monday, November 2, 2009
-
Quantum phase transitions in metal
〈z〉 #= 0 ; 〈N〉 #= 0SDW order
small Fermi pockets 〈z〉 #= 0 ; 〈N〉 = 0Fermi liquidlarge Fermi surface
〈z〉 = 0 ; 〈N〉 #= 0small critical Fermi pockets
gapless U(1) photon
〈z〉 = 0 ; 〈N〉 = 0large critical Fermi surface
gapless SU(2) photons
S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, Physical Review B 80, 155129 (2009)
non Fermi liquid phases
Increasing SDW orderIncreasing SDW order
Monday, November 2, 2009
-
Quantum phase transitions in metal
〈z〉 #= 0 ; 〈N〉 #= 0SDW order
small Fermi pockets 〈z〉 #= 0 ; 〈N〉 = 0Fermi liquidlarge Fermi surface
〈z〉 = 0 ; 〈N〉 #= 0small critical Fermi pockets
gapless U(1) photon
〈z〉 = 0 ; 〈N〉 = 0large critical Fermi surface
gapless SU(2) photons
S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, Physical Review B 80, 155129 (2009)
Hertz-Millis-Moriya theory
Increasing SDW orderIncreasing SDW order
Monday, November 2, 2009
-
Max Metlitski
M. Metlitski and S. Sachdev, to appear
Ar. Abanov, A.V. Chubukov, and J. Schmalian, Advances in Physics 52, 119 (2003)
Sung-Sik Lee, arXiv:0905.4532.
Monday, November 2, 2009
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Hertz-Millis-Moriya (HMM) theory: mean field theory
+ Gaussian fluctuations of overdamped paramagnons.
Monday, November 2, 2009
-
Theory for the onset of spin density wave order in metals is strongly coupled in two dimensions
Monday, November 2, 2009
-
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
[(∂r !ϕ)
2 +1c2
(∂τ !ϕ)2]
Monday, November 2, 2009
-
! = 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!=12 = (−vx, vy)Monday, November 2, 2009
-
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
Monday, November 2, 2009
-
Order parameter: Lϕ =12
(∇r !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
Monday, November 2, 2009
-
Order parameter: Lϕ =12
(∇r !ϕ)2 +s
2!ϕ2 +
u
4!ϕ4
“Yukawa” coupling: Lc = −!ϕ ·(ψ!†1α!σαβψ
!2β + ψ
!†2α!σαβψ
!1β
)
Lf = ψ!†1α(ζ∂τ − iv!1 · ∇r
)ψ!1α + ψ
!†2α
(ζ∂τ − iv!2 · ∇r
)ψ!2α
Monday, November 2, 2009
-
Order parameter: Lϕ =12
(∇r !ϕ)2 +s
2!ϕ2 +
u
4!ϕ4
“Yukawa” coupling: Lc = −!ϕ ·(ψ!†1α!σαβψ
!2β + ψ
!†2α!σαβψ
!1β
)
HMM 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
Monday, November 2, 2009
-
Order parameter: Lϕ =12
(∇r !ϕ)2 +s
2!ϕ2 +
u
4!ϕ4
“Yukawa” coupling: Lc = −!ϕ ·(ψ!†1α!σαβψ
!2β + ψ
!†2α!σαβψ
!1β
)
But, higher order terms contain an infinitenumber of marginal couplings . . . . . .
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
HMM theory
Monday, November 2, 2009
-
Order parameter: Lϕ =12
(∇r !ϕ)2 +s
2!ϕ2 +
u
4!ϕ4
“Yukawa” coupling: Lc = −!ϕ ·(ψ!†1α!σαβψ
!2β + ψ
!†2α!σαβψ
!1β
)
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
Monday, November 2, 2009
-
!ϕ propagator
fermion propagator
1N
1(q2 + γ|ω|)
1
v · q + iζω + i 1N√
γv
√ωF
(v2q2
ω
)
New infra-red singularities as ζ → 0 at higher loops(Breakdown of Migdal-Eliashberg)
Monday, November 2, 2009
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!ϕ propagator
fermion propagator
Dangerous
1N
1(q2 + γ|ω|)
1
v · q + iζω + i 1N√
γv
√ωF
(v2q2
ω
)
New infra-red singularities as ζ → 0 at higher loops(Breakdown of Migdal-Eliashberg)
Monday, November 2, 2009
-
Ignoring fermion self energy: ∼ 1N2
× 1ζ2× 1
ω
New infra-red singularities as ζ → 0 at higher loops(Breakdown of Migdal-Eliashberg)
Monday, November 2, 2009
-
Actual order ∼ 1N0
Ignoring fermion self energy: ∼ 1N2
× 1ζ2× 1
ω
New infra-red singularities as ζ → 0 at higher loops(Breakdown of Migdal-Eliashberg)
Monday, November 2, 2009
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Double line representation• A way to compute the order of a diagram.
• Extra powers of N come from the Fermi-surface
• What are the conditions for all propagators to be on the Fermi surface?
• Concentrate on diagrams involving a single pair of hot-spots
• Any bosonic momentum may be (uniquely) written as
R. Shankar, Rev. Mod. Phys. 66, 129 (1994).S. W. Tsai, A. H. Castro Neto, R. Shankar, and D. K. Campbell, Phys. Rev. B 72, 054531 (2005).
Monday, November 2, 2009
-
=
New infra-red singularities as ζ → 0 at higher loops(Breakdown of Migdal-Eliashberg)
Actual order ∼ 1N0
Singularities as ζ → 0 appear when fermions in closed blueand red line loops are exactly on the Fermi surface
Monday, November 2, 2009
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Graph is planar after turning fermion propagators also into double lines by drawing additional dotted single line loops for each fermion loop
Actual order ∼ 1N0
=
New infra-red singularities as ζ → 0 at higher loops(Breakdown of Migdal-Eliashberg)
Sung-Sik Lee, arXiv:0905.4532Monday, November 2, 2009
-
A consistent analysis requires resummation of all planar graphs
Actual order ∼ 1N0
=
New infra-red singularities as ζ → 0 at higher loops(Breakdown of Migdal-Eliashberg)
Monday, November 2, 2009
-
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
VBSand/ornematic
Monday, November 2, 2009
-
98H
SC
M"Normal"
(Large Fermisurface)
SDW(Small Fermi
pockets)
SC+SDW
LargeFermi
surface
StrangeMetal
d-waveSC
T
Tsdw
Fluctuating, paired Fermi
pockets
Fluctuating, paired Fermi
pockets
VBS and/or Ising nematic order
SDWInsulator
T*
Tc
Hc2
Monday, November 2, 2009
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Identified quantum criticality in cuprate superconductors with a critical point at optimal
doping associated with onset of spin density wave order in a metal
Conclusions
Elusive optimal doping quantum critical point has been “hiding in plain sight”.
It is shifted to lower doping by the onset of superconductivity
Monday, November 2, 2009
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Conclusions
Theory for the onset of spin density wave order in metals is strongly coupled in two dimensions
Monday, November 2, 2009