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