Kinks in the dispersion of strongly correlatedelectrons

Krzysztof Byczuk

Institute of Physics, EKM, Augsburg University, Germany

April 20th, 2007

K. Byczuk, M. Kollar, K. Held, Y.-F. Yang, I.A. Nekrasov, Th. Pruschke, D. VollhardtNature Physics 3, 168 (2007)

Collaboration M. Kollar, D. Vollhardt, Augsburg, Germany

K. Held, Y.-F. Yang, Stuttgart, Germany

I. Nekrasov, Ekaterinburg, Russia

T. Pruschke, Gottingen, Germany

Support from SFB 484

Standard model of quantum many-body system

E(k)

k

quasiparticlequasiholeholonspinonplasmonmagnonphononpolaritonexcitonanyong-on...

emergent particles

(i) well defined dispersion relation

(ii) long (infinite) life-time

(iii) proper set of quantum numbers

(iv) statistics

Dispersions and kinks

Coupling/hybridization

between different particles/modes

E(k)

k

E(k)

k

Df. kinks are abrupt slope changes in the dispersion relations

anticrossing, lifting degeneracy, ...

Provide information on modes and couplings

Dispersions and kinks - coupling to bosons

E(k)

electron

k

boson

energy of a kink is related to energy of a bosonic fluctuation

Dispersion of correlated electrons

One-particle spectral function - excitations at and

Dispersion relation

Dispersion relation is experimentally measured

Angular Resolved Photoemission Spectroscopy

;

<

=KY

H

I

T

(OHFWURQ

HQHUJ\DQDO\]HU

&U\VWDO

k

k

k

k

1

2

3

4

Energy

Inte

nsity

energy distribution curve (EDC)

Inte

nsity

Momentum

E

E

E

E

1

2

3

4

momentum distribution curve (MDC)

energy resolution 1meV

ARPES and graphene

Dirac linear dispersion relation for graphene

cond-mat/0608069

Kinks in HTC

Kinks at meV

electron-phonon or electron-spin fluctuations coupling

cond-mat/0604284

“Waterfalls” in HTC

different HTC systems, cond-mat/0607319

Kinks seen experimentally between 300-800 meVOrigin: phonos, spin fluctuations, not known yet

Kinks orbital selective

π

γ β

π π π

α π

π

π

Γ

Χαβ

γ

Sr RuO , cond-mat/0508312

Kink at 30meV in -band only

More examples of kinks in ARPES

SrVO

, cond-mat/0504075

Kinks seen experimentally at 150 meVPure electronic origin?

Kinks in LDA+DMFT study of SrVO

plain band model with local correlations, no other bosons, ... but kinks!

≥21.510.50-0.5-1-1.5≤-2

ener

gy [e

V]

RΓXMΓ

-1

-0.5

0

0.5

1

1.5

log A(k,ω) [1/eV]

0.60.40.2

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7Energy, eV

-6

-5

-4

-3

-2

-1

0

1

2

Σ(ω

), eV

Real partImaginary part

-0.5 0 0.5

Energy, eV

-0.5

0

0.5

ReΣ

(ω),

eVSrVO3

LDA+DMFT(QMC) self-energy

V3d(t2g

) orbitals

I.A. Nekrasov et al., cond-mat/0508313, PRB (2006)

Not found in SIAM with simple hybridization function! DMFT self-consistency effect

New purely electronic mechanism in strongly correlated systems

characteristic energy scale

range of validity for Fermi liquid theory

Hubbard model for strongly correlated electrons

t

U

t

! !! !! ! " "" "" "# ## ## #

$ $$ $$ $% %% %% % & && && &' '' '' ' ( (( (( () )) )) ) * ** ** *+ ++ ++ +

, ,, ,, ,- -- -- - . .. .. ./ // // / 0 00 00 01 11 11 1 2 22 22 23 33 33 34 44 44 45 55 55 5

67676767676767676767676

87878787878787878787878

979797979797979797979

:7:7:7:7:7:7:7:7:7:7:

; ;; ;; ;; ;< << << << < = == == == => >> >> >> > ? ?? ?? ?? ?@ @@ @@ @@ @ A AA AA AA AB BB BB BB B

InIn

Out

TIME

DC DC E DC DC F

Local Hubbard physics

All what we know about Hubbard model

Solved in limit (non-interacting limit)

Dispersion relation

Spectral function - one-particle excitations

Density of states (DOS) - thermodynamics

k

ω

A(k, )ω

ω

ωN( )

All what we know about Hubbard model

Solved in limit (atomic limit)

Real self-energy

Spectral function

Green function and self-energy are local,i.e. independent

k

ω

A(k, )ω

ω

ωN( )LHB UHB

U

U

Weakly correlated system

≥210-1-2≤-3

ener

gy [e

V]

RΓXMΓ

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

log A(k,ω) [1/eV]

Fermi liquid : for

for

Kinks due to strong correlations

≥10.50-0.5-1-1.5≤-2

ener

gy [e

V]

RΓXMΓ

-0.2

-0.1

0

0.1

0.2

log A(k,ω) [1/eV]

0.60.4

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

ω qp

U

Ω

Fermi liquid : for

Different renormalization : for

Mathematical explanation of kinks within DMFT

DMFT self-consistency condition

Three-peak structure sufficient condition

Fermi-liquid for

Microscopic predictions

Starting from:

- bare dispersion relation

we predict that:

Microscopic predictions Kink position

Intermediate energy regime

Change in the slope interaction independent

Curvature of the kink

Sharpness of the kink

Sharper for stronger

Outlook: possible origin of the “waterfalls”

“Waterfalls”: kinks at ω? ≈ 300-400 meV in cuprates

• crossover to Hubbard bands? Wang et al. (2006)

• U t ⇒ dispersion goes from central peak to Hubbard bandK. Byczuk, M. Kollar (unpublished)

Σ(ω) = Σ0 +Σ1

ω+O

(1

ω2

)

⇓

EUHB,LHBk

≈1

2

[εk ±

√ε2

k+ cU2

] -2 -1.5 -1 -0.5 0 0.5 10

2

4

6

8

10U=8t=2, n=0.79

-2 -1.5 -1 -0.5 0 0.5 10

2

4

6

(0,0)-(0,π)

(0,0)-(π,π)

(0,π)

(0,0)

(π,π)

(0,0)

Crossover to Hubbard bands

Hubbard model, square lattice, DMFT(NRG), U = 8t, n = 0.79

• ImΣ decays faster than ReΣ

• for large energies: Ek approaches EUHB,LHBk

• waterfalls from central peak to LHBK. Byczuk, M. Kollar (unpublished)Y.-F. Yang, K. Held (unpublished)

Conclusions Strong correlations (three peak spectral function) a sufficient condition for

electronic kinks

Energy scale for electronic kinks determined by Fermi-liquidrenormalization and bare (LDA) density of states

sets the energy scale for Fermi-liquid regime where for

Beyond Fermi-liquid regime the dispersion is still renormalized and useful

for where the offset and determined by

and

Electronic kinks are within cluster extension of DMFT (DCA)

Electronic kinks and waterfalls are generic feature of strongly correlatedsystems