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Electronic structure, non-local correlation effects and superconductivity in cuprates
Alexander Lichtenstein University of Hamburg
In collaboration with
H. Hafermann, A. Rubtsov, and M. Katsnelson
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Outline
• What is new in HTSC? • From electronic structure to Hubbard model • Antiferromagnetism and Superconductivity • d-wave: BSE
• Conclusions
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Tem
pera
ture
X 0.15
Can we control superconductivity with light ?
Yes - Andrea Cavalleri, MPI Hamburg
AFM d-wave
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Can we selectively quench stripes distortion?
Andrea Cavalleri MPI Hamburg
D. Fausti, Science, 331, 189 (2011)
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YBCO: Driving Apical Oxygen Motion
Andrea Cavalleri MPI Hamburg
S. Kaiser, et al., arXiv:1205.4661
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HTSC: from LDA to 1-band model
O.K. Andersen, at al J. Phys. Chem. Solids 56, 1573 (1995)
From LDA “Chemistry” to
Low-energy TB-model
t’/t= -0.3 for YBCO
EF
EF
NMTO-orbitals O.K. Andersen, et al Phys. Rev. B 62, R16219 (2000)
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General Cluster Idea One-band Habbard model on Lattice
Exact solurion:
Approximate self-energy:
N=1 single-site DMFT N=4 plaquette CDMFT
MIT Mott Transition Paramagnetic Insulator
d-wave HTSC Antiferromagnetism CDW
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Dynamical Mean Field Theory
Σ Σ Σ
Σ
Σ
Σ
ΣΣ
U
U
G( ’)τ−τ
ττ’
W. Metzner and D. Vollhardt, PRL(1989) A. Georges et al., RMP 68, 13 (1996)
( ) ( ) ( ) ( )[ ]1
0ˆˆˆ1ˆ
−
∑ Σ−−+Ω
=BZ
knnn ikHiIiG ωωµω
( ) ( ) ( )nnn iiGi ωωω Σ+= −− ˆˆˆ 110G
( ) ( ) ( )nnnnew iGii ωωω 110
ˆˆˆ −− −=Σ G
)()()()()( 10 τττττττττ σσ
↓↑−+ ∫∫∫ +′′−′−= nUndccddSeff G
ˆ G τ − ′ τ ( )= −1Z
D[c,c +]∫ c(τ )c +( ′ τ )e−Seff
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Cluster DMFT scheme
( ) ( ) ( ) ( )[ ]1
0ˆˆˆ1ˆ
−
∑ Σ−−+Ω
=BZ
knnn ikHiIiG ωωµω
( ) ( ) ( )nnn iiGi ωωω Σ+= −− ˆˆˆ 110G
( ) ( ) ( )nnnnew iGii ωωω 110
ˆˆˆ −− −=Σ G
Seff = − dτd ′ τ ∫∫ cIσ+ (τ )GIJ
−1(τ − ′ τ )cJσ ( ′ τ ) + dτ∫ UnI↑(τ )nJ↓(τ )
ˆ G IJ τ − ′ τ ( )= −1Z
D[c,c +]∫ cI (τ)cJ+( ′ τ )e−Seff
A.L., M. Katsnelson, PRB 62, R928368, (2000) G. Kotliar, et al RMP 78, 865 (2006)
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Cluster Impurity Problem Super-impurity partition function:
I=(1,2,3,4)
Local plaquette Green-function:
Bath Green-fanction matrix:
CTQMC: Exact solution of S-imp:
New self-energy matrix: CDMFT: Self-consistent condition:
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Imputity solver: miracle of CT-QMC
Interaction expansion CT-INT: A. Rubtsov et al, JETP Lett (2004)
Hybridization expansion CT-HYB: P. Werner et al, PRL (2006)
E. Gull, et al, RMP 83, 349 (2011)
Efficient Krylov scheme: A. Läuchli and P. Werner, PRB (2009)
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CT-QMC: random walks in the k space
Step k+1 Step k-1
1
1
k
k
w Dk D
+
+
1k
kk Dw D
−
Acceptance ratio
0 20 40 60
0
decrease increase
Maximum at 2UNβ
k-1 k+1
Z=… Zk-1 + Zk + Zk+1+ ….
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Comparison of different CT-QMC
Σ Σ Σ
Σ
Σ
Σ
ΣΣ
U
U
G( ’)τ−τ
ττ’
Ch. Jung, unpublished
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Scaling of CT-QMC
Temperature Interactions
CT-QMC review :E. Gull et al. RMP (2011)
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Phase diagram of Hubbard model
H. Park et al PRL (2008) C-DMFT with CT-QMC
Uc=9.35t Uc=6.05t
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ARPES of HTSC Z.X. Shen (Stanford)
M. Civelli et al PRL (2005) CDMFT
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Cluster-DMFT Plaquette hopping matrix
Supercell Green Function
where
Where Self-energy matrix for plaquette has the form:
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Coexistence of AFM and d-wave
0 1
2 3
A.L. and M.Katsnelson, PRB (2000)
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AFM and d-wave in HTSC
A.L. and M.Katsnelson, PRB 62, R9283 (2000)
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S. Kancharla et al, PRB (2008) M. Jarrell et al, EPL (2001)
CDMFT and DCA: phase diagram
CDMFT DCA
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Superconducting order parameter
Emanuel Gull et al. to be published, CT-QMC and Dynamical Cluster Approximation
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Beyond DMFT: Dual Fermion scheme
A. Rubtsov, et al, PRB 77, 033101 (2008)
General Lattice Action
Reference system: Local Action with hybridization ∆ω
Lattice-Impurity connection:
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Dual Fermions Gaussian path-integral
With new Action:
Diagrammatic:
gω and χν,ν‘,ω from DMFT impurity solver
-1
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Convergence of Dual Fermions: 2d
=1
H. Hafermann, et al. PRL102, 206401 (2009)
DMFT DF
LDFA
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Cluster Dual Fermions: 1d-test, n=1
H. Hafermann, et al. JETP Lett (2007)
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Pseudogap in HTSC: Ladder-DF
H. Hafermann, et al. PRL102, 206401 (2009)
n=1
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Bethe-Salpeter Equations
Non-local susceptibility with vertex corrections
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Susceptibility: 2d – Hubbard model
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DF: Ladder Approximation
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Bethe-Salpeter equation: pp-channel
AFM dx2-y2
p
U=W t’/t=-0.3 x=15%
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d-wave symmetry of the eigenfunction
0
max
min
U/W=1 t’/t=-0.3
H. Hafermann, et al, J. Supercond. Nov. Magn. 22, 45 (2008)
DB?
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Conclusions • Antiferromagnetism and d-wave
superconductivity obtained in cluster-DMFT • Pseudogap and Fermi-arcs describe well in
ladder DF-scheme • Realistic multiorbital LDA+DF for correlated
higher-Tc materials is a next challenge