Challenges in Strongly Correlated Electron Challenges in Strongly Correlated Electron Systems: A Dynamical Mean Field Theory Systems: A Dynamical Mean Field Theory

Perspective Perspective Gabriel KotliarGabriel Kotliar

and Center for Materials Theory &

CPHT Ecole Polytechnique Palaiseau & SPHT CEA Saclay, France

$upport : NSF -DMR DOE-Basic Energy Sciences . Chaire Blaise Pascal Fondation de l’Ecole Normale.

EPS 21st General Conference of the Condensed matter EPS 21st General Conference of the Condensed matter Division and DPG Spring MeetingDivision and DPG Spring Meeting March 26 - 31, 2006 In Dresden, Germany March 26 - 31, 2006 In Dresden, Germany

Fermi Liquid Theory (Landau 1957)

Density Functional Theory (Kohn Sham 1964)2 / 2 ( )[ ] KS kj kj kjV r r y e y- Ñ + =

+ [ - ]KSV10KSG 1G

Static Mean Field Theory.

Starting point for perturbation theory in the screened interactions (Hedin 1965)

Strong Correlation Challenges Strong Correlation Challenges Approach fails for strongly correlated systems

•Fermi Liquid Parameters Non Perturbative.

•Regimes where FLT Does NOT Apply. Need new concepts to replace rigid KS bands !

Mott transition: Mott transition: how does the electron go from the localized to itinerant ?

Matsuura et. al.Matsuura et. al.(2000)(2000)

-(BEDT-TTF)2Cu[N(CN)2]Cl LLefevre et.al.

(2000)Limelette et al.,(2003)Kagawa et al. (2003)

Dynamical Mean Field Theory. Cavity Construction.Dynamical Mean Field Theory. Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992).A. Georges and G. Kotliar PRB 45, 6479 (1992).

Inspiration: Weiss (1907), Onsager (1936), DMFT for spin Inspiration: Weiss (1907), Onsager (1936), DMFT for spin glasses, Fermions in d=glasses, Fermions in d=∞∞ Metzner and Vollhardt (1989) Metzner and Vollhardt (1989)

Reviews: A. Georges W. Krauth G.Kotliar and M. Rozenberg RMP (1996)G. Kotliar and D. Vollhardt Physics Today (2004).

Classical case Quantum case

A. Georges, G. Kotliar (1992)

Mean-Field : Classical vs QuantumMean-Field : Classical vs Quantum

†

0 0 0

( )[ ( ' ] ( '))o o o oc c U n nb b b

s st m tt

t t ­ ¯

¶+ D-

¶- +òò ò

( )wD

†( )( ( ) )) (

MFo n oo n n Sc i c iG i s ss ww w D=- á ñ

( )

(()

)

11

([ ]

)[ ]n

n

kn

G i

G it ki m

w

wwD

D

=- - +

å

,ij i j i

i j i

J S S h S- -å å

eMF offhH S=-

effh

00 ( )MF effH hm S=á ñ

ijff jj

e mh J h= +å

† †

, ,

( )( )ij ij i j j i i ii j i

t c c c c U n n

Easy!!!

0 [ ]S th heffbá ñ=

Hard!!!QMC: J. Hirsch R. Fye (1986)NCA : T. Pruschke and N. Grewe (1989)PT : Yoshida and Yamada (1970)NRG: Wilson (1980)

• Pruschke et. al Adv. Phys. (1995) • Georges et. al RMP (1996)

IPT: Georges Kotliar (1992). .QMC: M. Jarrell, (1992), NCA T.Pruschke D. Cox and M. Jarrell

(1993),ED:Caffarel Krauth and Rozenberg (1994)Projective method: G Moeller (1995). NRG: R. Bulla et. al. PRL 83, 136 (1999),……………………………………...

DMFT Qualitative Phase diagram of a DMFT Qualitative Phase diagram of a frustrated Hubbard model at integer fillingfrustrated Hubbard model at integer filling

T/W Synthesis:Synthesis:

Brinkman RiceBrinkman Rice

HubbardHubbard

Castellani et.al.Castellani et.al.

Kotliar RuckensteinKotliar Ruckenstein

FujimoriFujimori

Interaction with Experiments. Interaction with Experiments. V2O3:Anomalous transfer of spectral weightV2O3:Anomalous transfer of spectral weight

M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkine J Honig and P

Metcalf Phys. Rev. Lett. 75, 105 (1995)

T=170T=170

T=300T=300

. Photoemission measurements and TheoryPhotoemission measurements and TheoryV2O3 V2O3 Mo, Denlinger, Kim, Park, Allen, Sekiyama, Yamasaki, Mo, Denlinger, Kim, Park, Allen, Sekiyama, Yamasaki, Kadono, Suga, Saitoh, Muro, Metcalf, Keller, Held, Eyert, Anisimov, Kadono, Suga, Saitoh, Muro, Metcalf, Keller, Held, Eyert, Anisimov,

Vollhardt PRL . (2003Vollhardt PRL . (2003))NiSxSeNiSxSe1-x1-xMatsuura Watanabe Kim Doniach Shen Thio Bennett (1998)Matsuura Watanabe Kim Doniach Shen Thio Bennett (1998)

Poteryaev et.al. (to be published)Poteryaev et.al. (to be published)

Spinodals and Ising critical endpoint. Spinodals and Ising critical endpoint. Theory: Theory: Castellani et.al PRL 43, 1957 (1979); Castellani et.al PRL 43, 1957 (1979);

Kotliar et.al. PRL84, 5180 (2003) Kotliar et.al. PRL84, 5180 (2003) Observation in VObservation in V22OO3 3 :: P. Limelette et.al. Science 302, 89 (2003)P. Limelette et.al. Science 302, 89 (2003)

• Further extensions, clusters, GW+DMFT …

Electronic Structure Meets DMFTElectronic Structure Meets DMFT

abcdU U®• LDA+DMFT

V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). Lichtentsein and Katsnelson. PRB 57,6884 (1998).

Almbladh et.al.(1999), Chitra and Kotliar (2000) (2001).

Savrasov Kotliar and Abrahams (2001)

,[ ] [ , ]dft sdft locG Ur rG ¾¾®G

( ) ( )lda dct k H k E-®

Large Number of Groups and Many Compounds have been studied.

• Functional formulations, life without U

,[ ]sdft loc locG W¾¾®G

Localization Delocalization in the Localization Delocalization in the ActinidesActinides

after G. Lander, Science (2003).

The­f­electrons­in­Plutonium­­are­close­to­a­localization-delocalization­transition­(Johansson, 1974) .

Mott Transition

Modern understanding of this phenomena using functional Modern understanding of this phenomena using functional approach toDMFT. Savrasov et.al. (2001-2005)approach toDMFT. Savrasov et.al. (2001-2005)

DMFT Phonons in fcc DMFT Phonons in fcc -Pu-Pu

( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)

Experiments at the European Synchrotron Radiation Facility, (Wong, Krisch, Farber, Occelli, Schwartz,Chiang,Wall, Boro and Xu et.al, Science, 22 August 2003)

Cluster DMFTCluster DMFT: removes limitations of single site DMFTlimitations of single site DMFT

11 23

24

( , ) (cos cos )

cos coslatt k kx ky

kx ky

wS =S +S +

+S

•No k dependence of the self energy.

•No d-wave superconductivity.

•No Peierls dimerization.

•No (R)valence bonds.

Reviews: Reviews: Georges et.al. RMP(1996). Th. Maier, M. Jarrell, Th.Pruschke, M.H. Hettler RMP (2005); Kotliar Savrasov et. .al. RMP in Press.

U/t=4.

Two Site Cellular DMFTTwo Site Cellular DMFT (G.. Kotliar et.al. PRL (2001)) in the 1D in the 1D Hubbard modelHubbard model M.Capone M.Civelli V. Kancharla C.Castellani and GK PRB

69,195105 (2004)

Doping Driven Mott transiton at low temperature, in 2d Doping Driven Mott transiton at low temperature, in 2d ((U=16 t=1, t’=-.3U=16 t=1, t’=-.3 ) Hubbard model ) Hubbard model

Spectral Function A(k,Spectral Function A(k,ω→ω→0)= -1/0)= -1/ππ G(k, G(k, ωω →→0) vs k0) vs kK.M. Shen et.al. 2004

2X2 CDMFT

Nodal Region

Antinodal Region

Civelli et.al. PRL 95 (2005)Civelli et.al. PRL 95 (2005)

Conclusion Conclusion

• Controlled, first principles, many body studies of correlated materials.

• Finite T Mott transition in 3d . Single site DMFT worked well!

• Lower T, 2d ? Will CDMFT on a plaquette help us generate the right concepts?

• New RG methods built around DMFT ?