Dynamical Mean Field Theory (DMFT) of correlated solids.

G.Kotliar Physics Department Center for Materials Theory

Rutgers University.

Collaborators: K. Haule (Rutgers), S. Savrasov (UC Davis)

September 2-7 (2007) Magdalen College Oxford United Kingdom

Gordon Research Conference on Solid State Chemistry II

Outline

• 1]Introduction to DMFT ideas.• 2]Application to elemental actinides, what is

valence in a correlated solid ?• 3]Application to cobaltates, why are correlation

stronger near a band insulator than near a Mott insulator? [C. Delmas talk A.Maignan talk]

• Central theme, localization-delocalization !

• 4]Application to 115’s and the tale of multiple hybridization gaps. [F. Steglich talk]

Correlated Electron Systems Pose Basic Questions in CMT

• FROM ATOMS TO SOLIDS

• How to describe electron from localized to itinerant ?

• How do the physical properties evolve ?

DMFT Spectral Function Photoemission and correlations

• Probability of removing an electron and transfering energy =Ei-Ef, and momentum k

f() A() M2

e

Angle integrated spectral Angle integrated spectral function function

( , ) ( )dkA k A 88

DMFT approximate quantum solid as atom in a medium † †

, ,

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

t c c c c U n n

† † † † †Anderson Imp 0 0 0 0 0 0 0

, , ,

( +c.c). H c A A A c c UcV c c c

1010

, ,

,

[ ] [ ]( )

[ ] [ ]spd sps spd f

f spd ff

H k H kt k

H k H k

æ ö÷ç ÷ç ÷ç ÷çè ø®

| 0 ,| , | , | | ... JLSJM g> ­> ¯> ­ ¯> >®

(GW) DFT+DMFT: determine H[k] and density and(GW) DFT+DMFT: determine H[k] and density andself consitently from a functionalself consitently from a functional

and obtain total energies. and obtain total energies. 1212

[ ]*

11

( )( ) (

,)n n

n nk

i ii t k i

V VVa a

aaaa

ew m ww m ww e

-é ùê ú+ - = +Sê ú+ - - S- ë û

å å

1( , )

( ) ( )G k i

i t k i

Spectra=- Im G(k,)

Self consistency for V and

Summary: part 1

• Gabriel Kotliar and Dieter Vollhardt, Physics Today 57, 53 (2004).

• A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, Rev. of Mod. Phys. 68, 13-125 (1996).

• G. Kotliar, S. Savrasov, K. Haule, V. Oudovenko, O. Parcollet, and C. Marianetti, Rev. of Mod. Phys. 78, 000865 (2006).

Spectral function in DMFT analogous to density in DFT

Self consistent Impurity problem, natural language to describe localization/delocalization phenomena. combines atomic physics and band theory

Systematically improvable, cluster DMFT

Recent progress in implementation

after G. Lander, Science (2003)and Lashley et. al. PRB (2006).

Mott Transition

PuPu

Mott transition across the actinides. B. Johansson Phil Mag. 30,469 (1974)]

Pu phases: A. Lawson Los Alamos Science 26, (2000)

GGA LSDA predicts Pu to be magnetic with a large moment ( ~5 Bohr) . Experimentally Pu is not magnetic. [PRB 054416(2005). Valence of Pu is controversial.

DMFT­­Phonons­in­fcc­DMFT­­Phonons­in­fcc­-Pu-Pu

  C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa)

Theory 34.56 33.03 26.81 3.88

Experiment 36.28 33.59 26.73 4.78

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

(experiments from Wong et.al, Science, 22 August 2003)

Curie-Weiss

Tc

Photoemission­of­Actinidesalpa->delta­volume­collapse­transition

Curium­has­large­magnetic­moment­and­orders­antifPu­does­is­non­magnetic.

F0=4,F2=6.1

F0=4.5,F2=7.15

F0=4.5,F2=8.11

What is the valence in the late actinides ?

Summary part 2

• Modern understanding of the Mott transition across the actinde series (B. Johanssen)

sheds light on the physics of actinides.• Important role of multiplets. Pu is non magnetic

and mixed valent element mixture of f6 and f5 • f electrons are localized in Cm f7

K. Haule and J. Shim Ref: Nature 446, 513, (2007)

DMFT study of Nax CoO2

Foo et.al. PRL 247001

CoO2NaCoO2

Assume Na patterns of Zandbergen et. al.PRB 70 024101

C. A. Marianetti and G. Kotliar Phys. Rev. Lett. 98, 176405 (2007) . A

DMFT calculations with and without disorder U=3 ev.

C. A. Marianetti and G. Kotliar Phys. Rev. Lett. 98, 176405 (2007)

x=.33 QP dispersion DMFT LDA

References: part 3

•   C. Marianetti, G. Kotliar, and G. Ceder, Nature Materials 3, 627 - 631 (2004).

• C. A. Marianetti and G. Kotliar Phys. Rev. Lett. 98,176405 (2007)

• C. Marianetti, K. Haule and O Parcollet cond-mat (2007)

Alternative theory : low spin to high spin Khaliullin Phys. Rev. Lett. 96, 216404 (2006)

Summary part 3

• What is the minimal model of the cobaltes ? • t2g orbitals + binary potential a see which

results of the Li /Na vacancy .• Why are correlations stronger near a band

insulator than near a Mott insulator ?• U < Uc2 , hole moves in a restricted space

(where potential is low) and is strongly correlated.

• DMFT calculations account for the Curie Weiss phase and the Fermi liquid phase

Conclusion

• DMFT as a technique, makes contact with experiments, total energies, phonons, photoemission, ARPES,optics,…thermopower…..neutron scattering …..

• Concepts, atom in a quantum medium, Weiss field, local spectral function, A(), three peak structure,transfer of spectral weight , valence histogram, [bridges between atomic physics and band theory ]

• Under constant development, but already gives some exciting results.

• Actinides, phonons, role of multiplets, spectral signatures, Pu as mixed valent metal.

• Cobaltates, key role of inhomogeneities bringing correlations near a (correlated) insulator. DMFT treatment of an alloy.

Conclusions :chemistry brings out different aspects of localization delocalization physics.

•115’s delocalization transition as a function of T. Spectral function as a coherence order parameters. Multiple hybridization gaps.

Thanks!

• Acknowldegment. NSF-DMR. DOE-BES.

• Collaborators:K. Haule, C. Marianetti, J. Shim, and S. Savrasov

Conclusions :chemistry brings out different aspects of localization delocalization physics.

Ce

In

Ir

CeIn

In

Crystal­structure­of­115’s­­CeMIn5 M=Co, Ir, Rh­

CeIn3­layer

IrIn2­layer

IrIn2­layer

Tetragonal­crystal­structure

4­in­plane­In­neighbors

8­out­of­plane­in­neighbors

3.27au

3.3 au

CeRhIn5: TN=3.8 K; 450 mJ/molK2 CeCoIn5: Tc=2.3 K; 1000 mJ/molK2;

CeIrIn5: Tc=0.4 K; 750 mJ/molK2

CeMIn5 M=Co, Ir, Rh

out­of­plane

in-plane

ARPESFujimori,­2006

Angle­integrated­photoemission­

Experimental­resolution­~30meV,­theory­predicts­3meV­broad­band

Surface­sensitive­at­122eV

Very slow crossover!

T*

Slow­crossover­more­consistent­with­NP&F­

T*

coherent­spectral­

weight

T

NP&F:­Nakatsuji,Pines&Fisk,­2004

Buildup­of­coherence­in­single­impurity­case

TK

coherent­spectral­

weight

T

scattering­rate

coherence­peak

Buildup­of­coherence

Crossover­around­50K

•At­300K­very­broad­Drude­peak­(e-e­scattering,­spd­lifetime~0.1eV)­•At­10K:­

•very­narrow­Drude­peak•First­MI­peak­at­0.03eV~250cm-1

•Second­MI­peak­at­0.07eV~600cm-1

Optical­conductivity­in­LDA+DMFT

CeIn

In

Multiple­hybridization­gaps

300K

e V

10K

•Larger gap due to hybridization with out of plane In•Smaller gap due to hybridization with in-plane In

non-f­spectra

Summary part 4

• 115’s model systems to study the evolution of the f electron as a function of temperature

• Multiple hybridization gaps in optics.

• Very different Ce-In hybridizations with In

out of plane being larger.

J. Shim K Haule and G.K. Submitted to Science. (2007).

PRL 80, (1998) GPalsson and GK Thermoelectricity near a Mott

transition La1-xSrxTiO3

ARPESFujimori,­2006

Angle­integrated­photoemission­vs­DMFT

Experimental­resolution­~30meV,­theory­predicts­3meV­broad­band

Surface­sensitive­at­122eV

Angle­integrated­photoemission­vs­DMFT

ARPESFujimori,­2006

Nice­agreement­for­the•­Hubbard­band­position•SO­split­qp­peak

Hard­to­see­narrow­resonance­

in­ARPES­since­very­little­weight­

of­q.p.­is­below­Ef

Lower­Hubbard­band

• . arXiv:0704.1065 [pdf] – Title: Precise Control of Band Filling in NaxCoO2 – Authors: Daisuke Yoshizumi, Yuji Muraoka,

Yoshihiko Okamoto, Yoko Kiuchi, Jun-Ichi Yamaura, Masahito Mochizuki, Masao Ogata, Zenji Hiroi

– Comments: 4 pages, 5 figures, submitted to J. Phys. Soc. Jpn

– Journal-ref: J. Phys. Soc. Jpn. 76 (2007) 063705 – Subjects: Strongly Correlated Electrons (cond-mat.str-

el)

1

10

1( ) ( )

( )n nn k nk

G i ii t i

w ww m w

-

-é ùê ú= +Sê ú- + - Sê úë ûå

DMFT Impurity cavity construction

1

10

1( ) ( )

V ( )n nk nk

D i ii

w ww

-

-é ùê ú= +Pê ú- Pê úë ûå

0

1 † 10 0 ( )( )[ ] ( ) [ ( ) ( ) ]n n n n S Gi G G i c i c ia bw w w w- -S = + á ñ

†

0 0

( ) ( , ') ( ') ( , ') o o o oc Go c n n Ub b

s st t t t d t t­ ¯+òò

† †

, ,

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

t c c c c U n n

()

1 100 0 0( )[ ] ( ) [ ( ) ( ) ]n n n n Si G D i n i n iw w w w- -P = + á ñ

,ij i j

i j

V n n

0 0( , ')Do n nt t+

Functional formulation. Chitra and Kotliar Phys. Rev. B 62, 12715 (2000)

and Phys. Rev.B (2001) . 

1 †1( ) ( , ') ( ') ( ) ( ) ( )

2Cx V x x x i x x xff f y y-+ +òò ò

†( ') ( )G x xy y=- < > ( ') ( ) ( ') ( )x x x x Wff ff< >- < >< >=

Ex. Ir>=|R, > Gloc=G(R, R ’) R,R’’

1 10

1 1[ , , , ] [ ] [ ] [ ] [ ] [ , ]

2 2C hartreeG W M P TrLn G M Tr G TrLn V P Tr P W E G W

Introduce Notion of Local Greens functions, Wloc, Gloc G=Gloc+Gnonloc .

Sum of 2PI graphs[ , ] [ , , 0, 0]EDMFT loc loc nonloc nonlocG W G W G W

One can also view as an approximation to an exact Spetral Density Functional of Gloc and Wloc.

1 10

1 10

loc

loc

G

W

G M

V P1

1

( )

1

( )

lock

locq C

GH k

Wv q

M

P

0 0G V,,intM PLocal Impurity Model

Input: ,M P

Output: Self-Consistent Solution

Spectral Density Functional Theory withinLocal Dynamical Mean Field Approximation

,loc locG W 1 10

1 10

loc

loc

G

W

G M

V P1

1

( )

1

( )

lock

locq C

GH k

Wv q

M

P

0 0G V,,intM PLocal Impurity Model

Input: ,M P

Output: Self-Consistent Solution

Spectral Density Functional Theory withinLocal Dynamical Mean Field Approximation

•Full implementation in the context of a a one orbital model. P Sun and G. Kotliar Phys. Rev. B 66, 85120 (2002).

•After finishing the loop treat the graphs involving Gnonloc Wnonloc in perturbation theory. P.Sun and GK PRL (2004). Related work, Biermann Aersetiwan and Georges PRL 90,086402 (2003) .

 

EDMFT loop G. Kotliar and S. Savrasov in New Theoretical Approaches to Strongly Correlated G Systems, A. M. Tsvelik Ed. 2001 Kluwer Academic Publishers. 259-301 . cond-mat/0208241 S. Y. Savrasov, G. Kotliar, Phys. Rev. B 69, 245101 (2004)

Anomalous Resistivity

2 ( )F Fe k k l

h

Maximum metallic resistivity 2

Fe k

h

Total­Energy­as­a­function­of­volume­for­Total­Energy­as­a­function­of­volume­for­Pu­Pu­ (ev) vs (a.u. 27.2 ev)

(Savrasov, Kotliar, Abrahams, Nature ( 2001)Non magnetic correlated state of fcc Pu.

iw

Zein (2005) Following Aryasetiwan Imada Georges Kotliar Bierman and

Lichtenstein. PRB 70 195104. (2004)

Pu

alpa->delta­­Photoemission­

Gouder Havela Lande

PRB(2001)r

Photoemission­Spectra[­Shim.­Haule,GK­Nature­(2007)]

alpa->delta­volume­collapse­transition

F0=4,F2=6.1

F0=4.5,F2=7.15

2020

Resistivity of Am under pressure. J. C. Griveau Rebizant Lander and Kotliar PRL 94, 097002 (2005).

Photoemission spectra using Hubbard I solver [Lichtenstein and Katsnelson, PRB 57, 6884,(1998 ),

Svane cond-mat 0508311] and Sunca . [Savrasov Haule and Kotliar cond-mat 0507552] Hubbard bands width is

determined by multiplet splittings.

CeIrIn5

1( , ) (1/ ) Im[ ]

ˆˆ ˆ( ) ( )A k

H k E iab abw p

w w d=-

- - - S +

While DMFT aims at local quantities, one can derive non local information

Photomission Spectra of Am under pressure. Sunca. Onset of mixed valence. Savrasov Haule

Kotliar PRL (2005)

Am equation of state. LDA+DMFT.New acceleration technique for solving DMFT equations S. Savrasov

K. Haule G. Kotliar PRL (2006)

The “DMFT-valence” in the late actinides

Pu phases: A. Lawson Los Alamos Science 26, (2000)

GGA LSDA predicts Pu to be magnetic with a large moment ( ~5 Bohr) . Experimentally Pu is not magnetic. [Lashley et. al. cond-matt 0410634] PRB 054416(2005).

Approach the Mott transition from the left. (delocalized side).

DMFT Qualitative Phase diagram of a frustrated Hubbard model at integer

filling

T/W

1414

Model Hamiltonians

• Tight binding form.

• Eliminate the “irrelevant” high energy degrees of freedom

• Add effective Coulomb interaction terms

Summary: part 1

LDA+DMFT Self-Consistency loop

G0 G

Im p u rityS o lver

S .C .C .

0( ) ( , , ) i

i

r T G r r i e w

w

r w+

= å

2( ) | ( ) |k xc kH k Va ac r c= - Ñ +

DMFT

U

-Edc

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å

DMFT Cavity Construction. Happy marriage of atomic and band physics.

Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, 1996 Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004). G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti Rev. Mod. Phys. 78, 865 (2006) . G. Kotliar and D . Vollhardt Physics 53 Today (2004)

1( , )

( )k

G k ii i

Extremize a functional of the local spectra. Local self energy.

A. Georges, G. Kotliar (1992)

( )wDlatt ( ,

1 G [ ]

( ) [( ) ])

[ ]n impn

n

ik ii

ktw m

ww+ + - S

DD

=

latt( ) G ([ [)] ] ,imp n nk

G i i kw wD D=å

[ ]ijij

jm mJth hb= +å

11

( ( )( )

( [))

][ ]

imp n

imp n

kn

G i

Gti

ik

w

ww -D

D

=+-

å

A(A())

1111

Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003

<l.s> in the late actinides [DMFT results: K. Haule and J. Shim ]

Total­Energy­as­a­function­of­volume­for­Total­Energy­as­a­function­of­volume­for­Pu­Pu­W (ev) vs (a.u. 27.2 ev)

(Savrasov, Kotliar, Abrahams, Nature ( 2001)Non magnetic correlated state of fcc Pu.

iw

Nick Zein Following Aryasetiwan et. al. PRB 70 195104. (2004)

Moment is first reduced by orbital spin moment compensation. The

remaining moment is screened by the spd and f electrons

Double well structure and Pu Qualitative explanation of negative thermal expansion[Lawson, A. C., Roberts J. A., Martinez, B., and Richardson, J. W., Jr. Phil. Mag. B, 82, 1837,(2002). G. Kotliar J.Low Temp. Physvol.126, 1009 27. (2002)]

Natural consequence of the conclusions on the model Hamiltonian level. We had two solutions at the same U, one metallic and one insulating. Relaxing the

volume expands the insulator and contract the metal.

F(T,V)=Fphonons+Finvar

Optical­conductivity

Typical heavy fermion at low T:

Narrow­Drude­peak­(narrow­q.p.­band)

Hybridization­gap

k

Interband­transitions­across­hybridization­gap­->­mid­IR­peak

CeCoIn5

no­visible­Drude­peak

no­sharp­hybridization­gap

F.P.­Mena­&­D.Van­der­Marel,­2005

E.J.­Singley­&­D.N­Basov,­2002

second­mid­IR­peakat­600­cm-1

first­mid-IR­peakat­250­cm-1

T=10K T=300Kscattering­rate~100meV

Fingerprint­of­spd’s­due­to­hybridization

Not­much­weight

q.p. bandSO

Momentum­resolved­Ce-4f­spectraAf(,k)

Hybridization­gap

DMFT­qp­bands

LDA­bands LDA­bands DMFT­qp­bands

Quasiparticle­bands

three­bands,­Zj=5/2~1/200

Momentum­resolved­total­spectra A(,k)

Fujimori,­2003

LDA+DMFT­­at­10K ARPES,­HE­I,­15K

LDA­f-bands­[-0.5eV,­0.8eV]­almostdisappear,­only­In-p­bands­remain

Most­of­weight­transferred­intothe­UHB

Very­heavy­qp­at­Ef,hard­to­see­in­total­spectra

Below­-0.5eV:­almost­rigid­downshift

Unlike­in­LDA+U,­no­new­band­at­-2.5eV

Large­lifetime­of­HBs­->­similar­to­LDA(f-core)rather­than­LDA­or­LDA+U

DMFT : Evolution of the DOS.

( )A 1313

Approach Mott point from the right Am under pressureApproach Mott point from the right Am under pressure

Experimental­Equation­of­State­ (after Heathman et.al, PRL 2000)

Mott Transition?“Soft”

“Hard”

J. C. Griveau et.al. PRL 94, 097002 (2005).

The “DMFT-valence” in the late actinides

From Atoms to Solids

Band (e.g. LDA)Atomic Physics (eg. CI) )

The “DMFT-valence” in the late actinides

Late actinide issues

• All the spin density functional studies of fcc Pu and Am , using either LDA or GGA, predict magnetic long range order with a large moment. Experimentally Pu and Am are not magnetic.

• If one treats the f electrons as part of the core LDA overestimates the volume by 30%

• Valence of Pu controversy. LDA+U Schick, Havela Lichtenstein et.al. Anisimov et.al.(5f)^6. Erickson and Wills (5f)^4.

CoO2

(e

Temperature­dependence­of­the­local­Ce-4f­spectra­

•At­low­T,­very­narrow­q.p.­peak­(width­~3meV)

•SO­coupling­splits­q.p.:­+-0.28eV­­

•Redistribution­of­weight­between­the­q.p.­and­the­upper­Hubbard­band

SO

•At­300K,­only­Hubbard­bands

• arXiv:0708.2528 [pdf] – Title: Synthesis and properties of CoO2, the x

= 0 end member of the LixCoO2 and NaxCoO2 systems

– Authors: T. Motohashi, Y. Katsumata, T. Ono, R. Kanno, M. Karppinen, H. Yamauchi

– Comments: 18 pages, 5 figures – Journal-ref: Chemistry of Materials,

cm0702464 (2007)

­­Photoemission­Gouder , Havela PRB

2002, 2003

CeCoIn5

CeRhIn

5

CeIrIn

5

Tc[K] 2.3K 2.1K@p>1.5GPa

0.4K

Cv/T[mJ/molK^2]

300 50 750

Why­CeIrIn5?

Phase­diagram­of­115’s­

5f elements: actinide series5f elements: actinide series

s/cs/c AFAF FMFM

Localisation Delocalization Localisation Delocalization

1.4K1.4K 0.40.4KK

0.9K0.9K 0.8K0.8K 52K52K 25K25K 52K52K

The “DMFT-valence” in the late actinidesHaule Shim Kotliar (2006)