Strongly Correlated Electron Materials: A Dynamical Mean Field Theory Perspective Strongly Correlated Electron Materials: A Dynamical Mean Field Theory.

Strongly Correlated Electron Materials: A Strongly Correlated Electron Materials: A Dynamical Mean Field Theory Perspective Dynamical Mean Field Theory Perspective

Gabriel KotliarGabriel Kotliar

and Center for Materials Theory &

CPHT Ecole Polytechnique Palaiseau & SPHT CEA Saclay, France

Support : Chaire Blaise Pascal Fondation de l’Ecole Normale.

Seminaire de la federation PHYSTAT-SUD 11 Mai 2006Seminaire de la federation PHYSTAT-SUD 11 Mai 2006

Collaborators Collaborators

• A. Georges (Ecole Polytechnique)

• O. Parcollet (CEA-Saclay)

• G. Biroli (CEA-Saclay)

• M Civelli (ILL-Grenoble)

• M. Capone ( Rome )

• T. Stanescu (U. Illinois)

• K. Haule (Rutgers)

• B. Kyung, A. M. Tremblay (Sherbrook)

Correlated Electron MaterialsCorrelated Electron Materials• Not well described the “Standard Model” Solid State Physics

based on Band Theory. Reference System: QP. [Fermi Liquid Theory and Kohn Sham DFT+GW ]

• Partially filled f and d shells, some organic materials.• Spectacular “big” effects. High temperature

superconductivity, colossal magneto-resistance, huge volume collapses……………..

• Anomalous physical properties, large resistivities. Breakdown of the rigid band picture. Non trivial evolution of spectral functions.

Kappa Organics Kappa Organics

Phase diagram of (X=Cu[N(CN)2]Cl)S. Lefebvre et al. PRL 85, 5420 (2000), P. Limelette, et al. PRL 91 (2003)

F. Kagawa, K. Miyagawa, + K. Kanoda

PRB 69 (2004) +Nature 436 (2005)

Cuprate Experimental Phase diagram Cuprate Experimental Phase diagram

Damascelli, Shen, Hussain, RMP 75, 473 (2003)

Copper oxide superconducors CuOCopper oxide superconducors CuO2 2

Kappa organics Kappa organics

Y. Shimizu, et al. Phys. Rev. Lett. 91, 107001(2003)

meV 50t meV 400 U

H. Kino + H. Fukuyama, J. Phys. Soc. Jpn 65 2158 (1996), R.H. McKenzie, Comments Condens Mat Phys. 18, 309 (1998)

t’/t ~ 0.6 - 1.1

Ut

t’t’’

H ij t i,j c i c j c j

c i Uinini

Model Hamiltonians Model Hamiltonians

PerspectivePerspective

U/t

t’/t

Doping Driven Mott Doping Driven Mott TransitionTransition

Pressure Driven Pressure Driven Mott transtionMott transtion

Open Problems Controversial IssuesOpen Problems Controversial Issues

• What is the mechanism for high temperature superconductivity. Why is realized in the copper oxides?

• What are the essential low energy degrees of freedom to describe the physics of these materials at a given energy scale?

• Proper reference frame for understanding the correlated solid.

RVB physics and Cuprate SuperconductorsRVB physics and Cuprate Superconductors

• P.W. Anderson. Connection between high Tc and Mott physics. Science 235, 1196 (1987)

• Connection between the anomalous normal state of a doped Mott insulator and high Tc. t-J limit.

• Slave boson approach. <b> coherence order parameter. singlet formation order parameters.Baskaran Zhou Anderson , (1987)Ruckenstein Hirshfeld and Appell (1987) .

Other states flux phase or s+id ( G. Kotliar (1988) Affleck and Marston (1988) have point zeors.

RVB phase diagram of the Cuprate RVB phase diagram of the Cuprate Superconductors. Superexchange.Superconductors. Superexchange.

• The approach to the Mott insulator renormalizes the kinetic energy Trvb increases.

• The proximity to the Mott insulator reduce the charge stiffness , TBE goes to zero.

• Superconducting dome. Pseudogap evolves continously into the superconducting state.

G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988)

Related approach using wave functions:T. M. Rice group. Zhang et. al. Supercond Scie Tech 1, 36 (1998, Gross Joynt and Rice (1986) M. Randeria

N. Trivedi , A. Paramenkanti PRL 87, 217002 (2001)

PWA:Nature Physics PWA:Nature Physics 2, 138 (2006) , 138 (2006)

A crude version of this theory was published in 1988 by Zhang and co-authors (Supercond. Sci. Technol. 1, 36–38; 1988), based partly on my earlier ideas, and in a similar paper, Kotliar and Liu came to the same conclusions independently (Phys. Rev. B 38, 5142–5145; 1988). But the successes weren't then recognized because experiments were too primitive.

It has been my (published) opinion for years that the cause of high-temperature

superconductivity is no mystery. We now have a workable theory — not just for

calculating the broad outlines (the transition temperature Tc, energy-gap shape, effect of doping, pseudogap temperature) but details

of the anomalous phenomenology.

Problems with the approach.Problems with the approach.

• Neel order. How to continue a Neel insulating state ? Need to treat properly finite T.

• Temperature dependence of the penetration depth [Wen and Lee , Ioffe and Millis ] . Theory:[T]=x-Ta x2 , Exp: [T]= x-T a.

• Doping and polarization dependence of the Raman scattering intensity. [LeTacon et.al. 2006]

• Mean field is too uniform on the Fermi surface, in contradiction with ARPES.

• Difficulties in describing quantitavely the incoherent regime.

Develoment of cluster DMFT may solve some of these problems.!!

ApproachApproach

• Leave out inhomogeneous states and ignore disorder. • Study minimal model of a doped Mott insulator to

understand the electronic structure in that regime. • Approach the problem directly from finite temperatures,not

from zero temperature. Address issues of finite frequency –temperature crossovers.

• Compare with experiments. Reconsider.

• CDMFT has made this program feasible. Reference frame describes coherent and incoherent regimes on the same footing. • The framework and the resulting equations are very non

trivial to solve and to interpret. Very exciting time.

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).

Reviews: A. Georges W. Krauth G.Kotliar and M. Rozenberg RMP (1996)

Limit of large lattice coordinationLimit of large lattice coordination

1~ d ij nearest neighborsijt

d

† 1~i jc c

d

†

,

1 1~ ~ (1)ij i j

j

t c c d Od d

~O(1)i i

Un n

Metzner Vollhardt, 89

1( , )

( )k

G ki i

Neglect k dependence of irreducible quantities such as self energy Muller-Hartmann 89

3 1~ [ ] ij ij ijG t

d

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),……………………………………...

CDMFTCDMFT: 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); G. Kotliar S. Savrasov K. Haule O. Parcollet V. Udovenko and C. Marianetti RMP in Press.

Various cluster approaches, DCA momentum spcace. Cellular DMFT G. Kotliar et.al. Various cluster approaches, DCA momentum spcace. Cellular DMFT G. Kotliar et.al. PRL (2004). O Parcollet G. Biroli and G. Kotliar B 69, 205108 (2004)PRL (2004). O Parcollet G. Biroli and G. Kotliar B 69, 205108 (2004)

T. D. Stanescu and G. Kotliar T. D. Stanescu and G. Kotliar cond-mat/0508302cond-mat/0508302

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

Single site DMFT and kappa organics. Qualitative phase Single site DMFT and kappa organics. Qualitative phase

diagram Coherence incoherence crosoverdiagram Coherence incoherence crosover. .

Finite T Mott tranisiton in CDMFT Finite T Mott tranisiton in CDMFT O. Parcollet O. Parcollet

G. Biroli and GK PRL, 92, 226402. (2004))G. Biroli and GK PRL, 92, 226402. (2004))

CDMFT results Kyung et.al. (2006)CDMFT results Kyung et.al. (2006)

Evolution of the spectral function Evolution of the spectral function at low frequency.at low frequency.

( 0, )vs k A k

If the k dependence of the self energy is weak, we expect to see contour lines corresponding to t(k) = const and a height increasing as we approach the Fermi surface.

k

k2 2

k

Ek=t(k)+Re ( , 0)

= Im ( , 0)

( , 0)Ek

k

k

A k

Evolution of the k resolved Spectral Evolution of the k resolved Spectral Function at zero frequency. (Function at zero frequency. (Parcollet Biroli and GK Parcollet Biroli and GK

PRL, 92, 226402. (2004)) )PRL, 92, 226402. (2004)) ) ( 0, )vs k A k

Uc=2.35+-.05, Tc/D=1/44. Tmott~.01 W

U/D=2 U/D=2.25

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)

Larger frustration: t’=.9t U=16tLarger frustration: t’=.9t U=16tn=.69 .92 .96n=.69 .92 .96

M. Civelli M. CaponeO. Parcollet and GK M. Civelli M. CaponeO. Parcollet and GK

PRL (20050PRL (20050

Spectral shapes. Large Doping Stanescu Spectral shapes. Large Doping Stanescu and GK cond-matt 0508302and GK cond-matt 0508302

Small Doping. T. Stanescu and GK cond-Small Doping. T. Stanescu and GK cond-matt 0508302matt 0508302

Photoemission spectra near Photoemission spectra near the antinodal direction in a the antinodal direction in a

Bi2212 underdoped sample. Bi2212 underdoped sample. Campuzano et.alCampuzano et.al

EDC along different parts of the zone, from Zhou et.al.

Lower Temperature, AF and SCLower Temperature, AF and SCM. Capone and GK, Kancharla et. al.M. Capone and GK, Kancharla et. al.

AF

AF+SC

SC

AFSC

• Can we continue the superconducting state towards the Mott insulating state ?

For U > ~ 8t YES.

For U ~ < 8t NO, magnetism really gets in the way.

Connection between superconducting and Connection between superconducting and normal state properties. K. Haule andnormal state properties. K. Haule and

G. Kotliar cond-mat (2006)G. Kotliar cond-mat (2006)

• Elucidate how the spin superexchange energy and the kinetic energy of holes changes upon entering the superconducting state!

• Mechanism of superconductivity. Temperature dependence of optical spectral weigths.

RESTRICTED SUM RULESRESTRICTED SUM RULES

0( ) ,eff effd P J

iV

, ,eff eff effH J P

2

0( ) ,

ned P J

iV m

Low energy sum rule can have T and doping dependence . For nearest neighbor it gives the kinetic energy.

, ,H hamiltonian J electric current P polarization

Below energy

2

2

kk

k

nk

Treatement needs refinementTreatement needs refinement

• The kinetic energy of the Hubbard model contains both the kinetic energy of the holes, and the superexchange energy of the spins.

• Physically they are very different.

• Experimentally only measures the kinetic energy of the holes.

K. Haule (2006)K. Haule (2006)

DcaDca vs vs CDMFTCDMFT critical critical temperatures for the t-J temperatures for the t-J

model. model.

Energy Balance between the normal and Energy Balance between the normal and superconducing state.superconducing state.

. Spectral weight integrated up to 1 eV of the three BSCCO . Spectral weight integrated up to 1 eV of the three BSCCO films. a) under-films. a) under-

doped, Tc=70 K; b) optimally doped, Tc=80 K; c) ∼doped, Tc=70 K; b) optimally doped, Tc=80 K; c) ∼overdoped, Tc=63 K; the fulloverdoped, Tc=63 K; the full

symbols are above Tc (integration from 0+), the open symbols symbols are above Tc (integration from 0+), the open symbols below Tc, (integrationfrom 0, including th weight of the below Tc, (integrationfrom 0, including th weight of the

superfuid).superfuid).

H.J.A. Molegraaf et al., Science 295, 2239 (2002). A.F. Santander-Syro et al., Europhys. Lett. 62, 568 (2003). Cond-mat 0111539. G. Deutscher et. A. Santander-Syro and N. Bontemps. PRB 72, 092504(2005) . Recent review:

Mott Phenomeman and High Temperature Superconductivity Mott Phenomeman and High Temperature Superconductivity Studied minimal model of a doped Mott insulator within Studied minimal model of a doped Mott insulator within

plaquette Cellular DMFT ?plaquette Cellular DMFT ?

• Rich Structure of the normal state and the interplay of the ordered phases.

• Work needed to reach the same level of understanding of the single site DMFT solution.

• A) Either that we will account semiquantitatively for the large body of experimental data once we study more realistic models of the material.

• B) Or we do not, in which case other degrees of freedom, or inhomgeneities or long wavelength non Gaussian modes are essential as many authors have surmised.

• Too early to tell, talk presented some evidence for A.

.

Collaborators Collaborators

• O. Parcollet (CEA-Saclay)

• G. Biroli (CEA-Saclay)

• M Civelli (ILL-Grenoble)

• M. Capone ( Rome )

• T. Stanescu (U. Illinois)

• K. Haule (Rutgers)

• B. Kyung, A. M. Tremblay (Sherbrook)