Nodal/Anti-nodal Dichotomy and the Energy-Gaps of a doped Mott …corpes07/BEITRAEGE/civelli.pdf ·...
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Nodal/Anti-nodal Dichotomy and the Energy-Gaps of a doped Mott Insulator arXiv:0704.1486v1 [cond-mat.str-el] Marcello Civelli April 2007 Collaborators: B.G. Kotliar, T.D Stanescu, M. Capone, A. Georges, K. Haule, O. Parcollet
Transcript of Nodal/Anti-nodal Dichotomy and the Energy-Gaps of a doped Mott …corpes07/BEITRAEGE/civelli.pdf ·...
Nodal/Anti-nodal Dichotomy and the Energy-Gaps of a doped Mott Insulator
arXiv:0704.1486v1 [cond-mat.str-el]
Marcello CivelliApril 2007
Collaborators:
B.G. Kotliar, T.D Stanescu, M. Capone, A. Georges, K. Haule, O. Parcollet
OUTLINEH-Tc Superconductors:
Strongly correlated many-body physics DMFTfundamental anisotropic properties DMFT not enough !
DMFT Cellular –DMFT
2D Hubbard Model: Normal state- Mott transitiond-wave SC statenodal/antinodal dichotomytwo nodal/antinodal energy-scales
H-Tc Superconductors Materials1986 Bednorz and Muller
L a, S r
O
C u
No FL
FL
SC
PG
AF
Tc
T*
x0.150.0 under doped over doped
T
doping0
T
SC
TC
AF
pgFL
FL?
15%
Reservoir:La, Sr
OCu
2D physics! Simple theoretical Hubbard Model
MO
TT in
sula
tor
Strongly Correlated Mott Physics
Kinetic term Local repulsionCompeting!
t U
Uk k
εk εk
εFεF
Why Dynamics so important?e.g. Density of states- Halfilling-vary interaction U
metal
insulator
HOW?
CompetitionHigh-low energy physics!
Density ofStates A(ω)
ω
Dynamical Mean Field Theory
Selfconsistency
AndersonImpurityModel
For a review: A. Georges et al. Rev. Mod. Phys. 68, 13 1996
non-interacting interacting
Remarks DMFT:
What if Σ is k dependent?(finite D..)
DMFT is exact for Dimension =∞:i) Mapping to Impurity model exactii) Local Self-energy Σ≠Σ(k)
exact self-consistency conditionBUT
Experimental Evidence: ARPES
D.S Marshall et al Phys. Rev. Lett. 76, 4841 (1996)
(π,0)
(π,π)
(0,0)
FS
Quasiparticle !
?!
Simple FLbreaks!
Density ofStates A(ω)
ω
ky
kx
QuasiparticlePeak
ARPES in the first quadrant of BZ
hole doped
e- doped
*ARPES:Armitage, Damacelli, Z. Shen, K. Shen, Hussain,Campuzano, Norman, Randeira
A(k,ω→0)= -1/π G(k,ω→0) *
!Quasiparticle
Peak
K-dependent Properties
Local correlations
needed
DMFTNot
Enough !
Describing K-dependence is a fundamental ingredient
Can we improve a LOCAL DMFT?
Cellular Dynamical Mean Field Theory
Selfconsistency
AIM
G. Kotliar et al. Phys. Rev Let. 87, 186401 (2001)
Lattice→ Superlattice
ClusterImpurity
Remarks CDMFT:Good:
Σ=Σ(k) Real SystemsD finite
Price: i) Mapping to Impurity Model not exactii) Self-energy Σ(k) approx.
OUTLINEH-Tc Superconductors:
Strongly correlated many-body physics DMFTfundamental anisotropic properties DMFT not enough !
DMFT Cellular –DMFT
2D Hubbard Model: Normal state- Mott transitiond-wave SC statenodal/antinodal dichotomytwo nodal/antinodal energy-scales
2D Hubbard Model with next-nearest neighbor hopping t’Impurity considered:
CMFT output few parameters: cluster self-energy Σ11, Σ12, Σ13
Σano→ < c1↑ c2↓ >
Preserve square lattice symmetries
Allows to describe d-wave superconductorbroken simmetry
EigenValues of the cluster ImΣµν
kY
(0,0) π
π ΣA ΣB
ΣC ΣA
kx
eigenvaluesh-doped e-doped
All information in the Green’s function
Extracting k-dependence ---- for example Σk
A(k,ω→0)= -1/π G(k,ω)Experiments Theory
Mott transition in the Normal State:hot/cold Spots
CDMFT cluster-local
theory
USE most localquantities to
extract k-quantities!
How do we extract k-quantities?
Truncated Fourier Expansion
e.g.
T.Stanescu et al. Ann. of Phys., Vol. 321, 2006, p.1682 T. Stanescu and G. Kotliar, Phys. Rev. B 74, p.125110
cumulant
Σ1) candidate
2) candidate Mcluster lattice
QMC TEST ON 1D CHAINcourtesy of B. Kyung 0 1 2 3
MGreen’s function
Σ
24 sites!
TEST: insulating state 2D
ω
DOS
M
Σ
ΣStatesIn the
Mott gap!
ΣM Gk(ω)periodize
localK-spacecluster
INSULATOR
Normal state at small doping
Den
sity M
Σ
M works better!
5% doping
MΣ
G
CONSEQUENCE IS A
PSEUDO-GAP STATE!
A
A
C
C
B
A(k,ω)Exp.
Pseudogapδ=0.05t~ 0.015 eV
δ
D.S Marshall et al Phys. Rev. Lett. 76, 4841 (1996)
T.Stanescu et al. Ann. of Phys., Vol. 321, 2006, p.1682
T. Stanescu and G. KotliarPhys. Rev. B 74, p.125110
G=0
poles
Simple case: 2D Hubbard Model t’=0
High doping
Occup.
empty
(0,0)
(π,π)(0,π)
Insulator
Occup.
empty
(0,0)
(π,π)(0,π)
FS Polesin G(k,0+)
G(k,0+)=0!
A(ω)
A(ω)
ω
ω
Line of G(k,0+)=0 at halffillingT.Stanescu et al. cond-mat/0602280
ω
A(ω)
-∆ ∆
ReG(k,0)=0 A(k,ω) is even
Half-filling nholes=1-ne-=1/2 + particle/hole symmetry
A(k,ω) = A(π-k,-ω)
(0,0)
(π,π)(0,π)
ky= π- ky
Natural continuity into a doped Mott insulator
Occup.
empty
(0,0)
(π,π)(0,π)
(0,0)
(π,π)
(0,0)
(π,π)
�����2
Π
�����2
Π
Topol.FS transit.
G(k,0+)=0!
FS Polesin G(k,0+)
A(k,0)
A(k,0)Coexistence of
poles and zeroesBROKEN FS IN A(k,0)
doping
OUTLINEH-Tc Superconductors:
Strongly correlated many-body physics DMFTfundamental anisotropic properties DMFT not enough !
DMFT Cellular –DMFT
2D Hubbard Model: Normal component- Mott transitiond-wave SC statenodal/antinodal dichotomytwo nodal/antinodal energy-scales
d-wave SC with CDMFTKancharla et al. cond-mat/0508205
AIM with d-wave SC bath
SC
Self-consistency
+
+--
d-wave SC state supported !
dSc non-zero Pd=<c↑c↓> in the 2D Hubbard Model
Pd
U=12t, t’=-0.3t
Anomalous Σano≠0, low energynon-momotonic in doping δ
0.25 0.20 0.15 0.10 0.05 0 δ
0
S.S Kancharla et al, cond-mat/0508205M. Civelli et al. cond-mat/0704.1486v1 Very important!
t’=0.0 Kancharla
Σ cluster-Eigenvalues
Im Σ→0Fermi Liquid
superconducting normal
Superconducting Green’s functionNambu’s notation
quasi-particle particle band
kx
ky
0
(π,π)
Periodization procedureNormal component set Σano=0
Better M
5% doping
Periodization procedure2 Σano≠0superconducting component
Only the nodal point is gapless
SpectrumLinearized!
Sum selectsNodal area only
Better Σ!
Periodizing recipe:
Nodal point→ periodize Σ
Anti-nodal point → periodize Mas in the normal state case
In particular d-wave gap
EXPERIMENTS: ARPES SPECTRA Tanaka et al. Science 314, 1910 (2006) doping
Optimal doping
Question: one or two energy-gaps?Tanaka et al. Science 314, 1910 (2006)
See e.g. discussion Science Dec 2006A. J. Millis, Science 314, 1888 (2006).
T. Valla et al. Science 314, 1914 (2006)
e.g. e.g.
-0.2 0 0.2ω/t
0
1
2
3
-0.2 0 0.2ω/t
1
2
3 δ=0.16δ=0.13δ=0.08δ=0.06δ=0.05δ=0.04δ=0.02
-0.2 0 0.2ω/t
0
0.5
1
1.5
2
0 0.08 0.16δ
0
0.05
0.1
0.15
0.2
Znod
Zanod
antinodal
antinodal
nodalΣano
=0AB
C
D
nodal
kx
ky
antinodal(π,π)(0,π)
(0,0)
0 0.08 0.16δ
kanod
knod
kx
2π
π4
2π
Quasi-particle Spectra CDMFT
pseudo-gap!
asimmetry
Σano=0
Σano≠0
At the node I have a quasiparticle
We can attempt a standard Fermi-Liquid Expansion at low energy
vnod
v∆
Nodal velocities- standard Fermi Liquid Analysis
lattice spacing X.J. Zhou et al.,
Nature 423, 398 (2003)
doping
EXP
Tanaka et al. Science 314, 1910 (2006)
K. McElroy et al., Phys. Rev. Lett. 94, 197005 (2005).
Local density of states
0.05 0.1 0.15δ
0
0.05
0.1
∆tot
/t∆
nor/t
∆sc
/t= ( ∆2tot
-∆2nor
)0.5
/t
∆sc
/t= Zanod
Σano/t
AntiNodalGAP
AntiNodalGAP
NodalGAP
NodalGAP
pseudoGAPTanaka et al. Science 314, 1910 (2006)
2 ENERGY GAPS !
PHOTOEMISSION EXP
From quasi-particle spectra we measure gaps!
CONCLUSIONS◊ Used Cellular DMFT to study the strongly
correlated many body systems:H-Tc Superconductor materials
◊ 2D Hubbard Model, Nomal Statea “mottness” region at small doping: PG, arcs FS
◊ Anomalous d-wave SC state - notal/antinodal dichotomy- 2 energy-gaps!: PG+ superconducting gap
Collaborstors:G. Kotliar*Tudor Stanescu~Massimo Capone^A. Georges+
O. Parcollet$
K. Haule*
* Rutgers University, NJ USA~ University of Maryland, College Park, Maryland, USA$ SPhT, CEA Saclay, Paris, France+ Ecole Polytechnique, Palaiseau, France^ University of Rome “La Sapienza” and INFM, Italy
SPECIAL THANKS TO:
P. NOZIÈRESE. KATS
