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

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

(DMFT) Perspective.(DMFT) Perspective. Gabriel KotliarGabriel Kotliar

and Center for Materials Theory

$upport : NSF -DMR DOE-Basic Energy Sciences

Cornell Ithaca NY November 27 2007Cornell Ithaca NY November 27 2007

11

Outline Outline

• Introduction to strongly correlated materials.

• Brief overview of Dynamical Mean Field Theory.

• Application to heavy fermions: a case study of CeIrIn5 [with K. Haule and J. Shim, Science Express Nov 1st (2007) ]

• Conclusions – and some thought about the 5f elemental metals.

.Interactions renormalize away (Landau) . Band Theory: electrons as waves

Electrons in a Solid:the Standard Model Electrons in a Solid:the Standard Model

•Quantitative Tools. Density Functional Theory Kohn Sham

(1964)

2 / 2 ( )[ ] KS kj kj kjV r r y e y- Ñ + =

Rigid bands , optical transitions , thermodynamics, transport………

Static Mean Field Theory.

22

Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation theory in the screened interactions (Hedin 1965)

Self Energy Self Energy

M VanShilfgaarde et. al. PRL M VanShilfgaarde et. al. PRL 9696, , 226402 (2006)226402 (2006)

33

Correlated Electron Systems Pose Basic Correlated Electron Systems Pose Basic Questions in CMTQuestions in CMT

• FROM ATOMS TO SOLIDS

• How to describe electron from localized to itinerant ?

• How do the physical properties evolve ?

Strong Correlation Problem:where Strong Correlation Problem:where the standard model failsthe standard model fails

• Fermi Liquid Theory works but parameters can’t be computed in perturbative theory.

• Fermi Liquid Theory does NOT work . Need new concepts to replace of rigid bands !

• Partially filled d and f shells. Competition between kinetic and Coulomb interactions.

• Breakdown of the wave picture. Need to incorporate a real space perspective (Mott).

• Non perturbative problem.

44

Strongly correlated materials do “big” thingsStrongly correlated materials do “big” things

• Huge volume collapses Pu …….• Masses as large as 1000 me

(heavy fermions UPt3, CeIrIn5…..

• High Temperature Superconductivity. 150 K Ca2Ba2Cu3HgO8 .• Large thermoelectric response in NaxCo2O4

• Large change in resistivity. MIT in TM oxides (V2O3, VO2, LaSrMnO3……..)

• …………………..55

Hubbard model Hubbard model

† †

, ,

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

t c c c c U n n

U/t

Doping or chemical potential

Frustration tij

T temperature

66

†i i ic c n

Mott-Hubbard PhysicsMott-Hubbard PhysicsBaBa

Real space picture

High T : local moments

Low T: spin orbital order

1

T

HH HH HH HHHH++

Excitations: Excitations: Excitations: adding (removing ) e, Upper Hubbard

band.

77

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).

†

0 0 0

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

s st m tt

t t ­ ¯

¶+ D-

¶- +òò ò

,ij i j i

i j i

J S S h S- -å å eMF offhH S=-† †

, ,

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

t c c c c U n n

*

( )V Va a

a a

ww e

D =-å

† † † † †Anderson Imp 0 0 0 0 0 0 0

, , ,

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

A(A())

1010

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

Dynamical Mean Field TheoryDynamical Mean Field Theory

• Weiss field is a function. Multiple scales in strongly correlated materials.

• Exact in the limit of large coordination (Metzner and Vollhardt 89) , kinetic and interaction energy compete on equal footing.

• Immediate extension to real materials

, ,

, 22

[ ] [ ]( )

[ ] [ ]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> ­> ¯> ­ ¯> >®

DFT+DMFTDFT+DMFT1212

DMFT Spectral Function Photoemission and DMFT Spectral Function Photoemission and correlationscorrelations

• 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

Evolution of the DOS. Theory and experimentsEvolution of the DOS. Theory and experiments

( )A 1313

, '( )RL RLA

Summary: DMFTSummary: DMFTSelf consistent Impurity problem, natural language to quantify localization/delocalization phenomena.

Combines atomic physics and band theory

Systematically improvable, cluster DMFT

Implementation

CeRhIn5CeRhIn5: : TNTN=3.8 K; =3.8 K; 450 mJ/molK2 450 mJ/molK2 CeCoIn5CeCoIn5: : TcTc=2.3 K; =2.3 K; 1000 mJ/molK2; 1000 mJ/molK2; CeIrIn5CeIrIn5: : TcTc=0.4 K; =0.4 K; 750 mJ/molK2 750 mJ/molK2

CeMIn5 M=Co, Ir, Rh

out­of­plane

in-plane

Ce

In

Ir

•Ir­atom­is­less­correlated­than­Co­or­Rh­(5d­/­3d­or­4d)

•CeIrIn5­is­more­itinerant(coherent)­than­Co­(further­away­from­QCP)

Why­CeIrIn5?

Phase­diagram­of­115’s­

Generalized Anderson Lattice ModelGeneralized Anderson Lattice Model

† † †

,

†

,

, ,

,

( )( )

. .

ij ij i j j ii j

f i j i ii i

i ALMij ji j

f t c c c c

V c c c H

f U n n

f

66

•High­temperature­Ce-4f­local­moments

•Low­temperature­–­Itinerant­heavy­bands

CC++ffff++

S. Doniach, 1978.

† † †

,

†

,

, ,

,

( )( )

. .

ij ij i j j ii j

f i j i ii i

i ALMij ji j

f t c c c c

V c c c H

f U n n

f

e- 1/ JKT De

2NT J

JJkk= V= V22//ffKondo ExchangeKondo Exchange

Kondo scaleKondo scale

RKKY scale RKKY scale

DONIACH PHASE DIAGRAMDONIACH PHASE DIAGRAM

Angle­integrated­photoemission­

Experimental­resolution­~30meVSurface­sensitivity­at­122­­ev­,­theory­predicts­3meV­broad­band

Expt Fujimori et al., PRB 73, 224517 (2006) P.R B 67, 144507 (2003).

­Theory:­LDA+DMFT,­impurity­solvers­­­SUNCA­and­CTQMC­­Shim­Haule­and­GK­­(2007)

Very slow crossover!

T*

Slow­crossover­pointed­out­by­NPF­2004­

Buildup­of­coherence­in­single­impurity­case

TK

cohere

nt­sp

ect

ral­

weig

ht

T scattering­rate

coherence­peak

Buildup­of­coherence

Crossover­around­50K

Consistency with the phenomenological Consistency with the phenomenological approach of NPFapproach of NPF

Remarkable­agreement­with­Y.­Yang­&­D.­Pines­cond-mat/0711.0789!

+C

DMFT is not a single impurity calculationDMFT is not a single impurity calculation

Auxiliary­impurity­problem:

High-temperature­­given­mostly­by­LDA

low­T:­Impurity­hybridization­affected­by­the­emerging­coherence­of­the­lattice­

(collective­phenomena)

Weiss­field temperature­dependent:

Feedback effect on makes the crossover from incoherent to coherent state very slow!

high­T

low­T

DMFT­SCC:

Momentum­resolved­total­spectratrA(,k)

Fujimori,­PRB­

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

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

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

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

•Second­MI­peak­at­0.07eV~600cm-1

Optical­conductivity­in­LDA+DMFT­

Expts:­­F.­P.­Mena,­D.­van­der­Marel,­J.­L.­Sarrao,­PRB 72,­045119­(2005).16.­K.­S.­Burch­et al.,­PRB 75,­054523­(2007).17.­E.­J.­Singley,­D.­N.­Basov,­E.­D.­Bauer,­M.­B.­Maple,­PRB 65,­161101(R)­(2002).

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

T=10K T=300Kscattering­rate~100meV

Fingerprint­of­spd’s­due­to­hybridization

Not­much­weight

q.p. bandSO

DMFT­-Momentum­resolved­Ce-4f­spectra

Af(,k)

Hybridization­gap

DMFT­qp­bands

LDA­bands LDA­bands DMFT­qp­bands

Quasiparticle­bands

three­bands,­Zj=5/2~1/200

Quantum Phase Transition: Kondo Quantum Phase Transition: Kondo Breakdown vs SDW. Breakdown vs SDW.

• SDW picture. Focus on order parameters. Neglect changes in the electronic structure.[Hertz, Morya]

• Kondo breakdown scenario. Drastic changes in the electronic structure. [Doniach] [ Coleman, Pepin, Paul, Senthil, Sachdev, Vojta, Si ]

3

2. mod 2(2 )

FSc f

Vn n

p= +

V> VcV> Vc

3

2. mod 2(2 )

FSc

Vn

p=

Neglect Magnetic orderNeglect Magnetic order

V < Vc V < Vc c

3mod 1= n

(2 )FS

c fV

n np

= +Magnetic OrderMagnetic Order

.

DMFT: consider the underlying DMFT: consider the underlying paramagnetic solution. Study finite T. paramagnetic solution. Study finite T.

Kondo Breakdown as an Orbitally selective Mott Kondo Breakdown as an Orbitally selective Mott Transition. [L. DeMedici, A. Georges GK and S. Transition. [L. DeMedici, A. Georges GK and S. Biermann PRL (2005), C. Pepin (2006) , L. DeLeo Biermann PRL (2005), C. Pepin (2006) , L. DeLeo M. Civelli and GK ]M. Civelli and GK ]

• Analogous situation to the Mott transition. Mott / Slater.

• f localization - Jump in the Fermi volume-Jump in DeHaas VanAlven frequencies.

• f Localized and f Itinerant phases have different compressibilities.

• Low but finite temperature aspects of the transition governed by a two impurity model.

Fermi surface changes under Fermi surface changes under pressure in CeRhInpressure in CeRhIn55

– Fermi surface reconstruction at 2.34GPa . Sudden jump of dHva frequencies

– Delocalization. Increase of electron FS frequencies . Localization decreases them.

Shishido, (2005)

localized itinerant

We can not yet address FS change with pressure

We can study FS change with Temperature -

At­high­T,­Ce-4f­electrons­are­excluded­from­the­FSAt­low­T,­they­are­included­in­the­FS

R A

R

RR

A

AA

c

2 2

LDA+DMFT (10 K)LDA LDA+DMFT (400 K)

Electron fermi surfaces at (z=Electron fermi surfaces at (z=))No c in DMFT!No c in Experiment!

Slight decrease of the electron

FS with T

LDA+DMFT (10 K)LDA LDA+DMFT (400 K)

X M

X

XX

M

MM

g h

Hole fermi surfaces at z=0Hole fermi surfaces at z=0

g h

Big change-> from small hole like to large electron like

1

Conclusion- future directionsConclusion- future directions

• Long wavelength vs short distance [ mean field ] physics in correlated materials.

• Further improvements and developments of DMFT [ CDMFT, electronic structure]

• Other systems. […..] System specific studies. Variety and universality in the localization delocalization phenomena.

• Towards a (Dynamical ) Mean Field Theory based theoretical spectroscopy.

Conclusions [115’s]Conclusions [115’s]

• DMFT in action: collective behavior of the hybridization field. Very slow crossover. Spectral evolution. Valence histograms.

• Theory/Experiment Spectroscopy. Multiple hybridization gaps in optics.

• Very different Ce-In hybridizations with In out of plane being larger.• Kondo breakdown as an orbitally selective

Mott transition. dhv orbits. • Lessons for the 5f’s. Elemental actinides.

Thanks!Thanks!

• $upport NSF-DMR.

• Collaborators: K. Haule, L. DeLeo, J. Shim, M. Civelli.

K. Haule and J. Shim and GK, Science Express Nov 1st (2007). To appear in science.

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)]

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

1414

­Gradual­decrease­of­electron­FS

Most­of­FS­parts­show­similar­trend

Big­change­might­be­expected­in­the­­plane­–­small­hole­like­FS­pockets­(g,h)­merge­into­electron­FS­1­(present­in­LDA-f-core­but­not­in­LDA)

Fermi­surface­a­and­c­do­not­appear­in­DMFT­results­

Increasing­temperature­from­10K­to­300K:­

Fermi surfacesFermi surfaces

Summary part 2Summary part 2

• Modern understanding (DMFT) of the (orbitally selective) Mott transition across the actinde series (B. Johanssen 1970 ) sheds light on 5f physics.

• Important role of multiplets. Pu is non magnetic and mixed valent element mixture of f6 and f5

• f electrons are localized in Cm f7 • Physics of 5f’s and 4f’s is similar but different. Main

difference, the coherence scale in 5f’s much larger, resulting in a much larger coupling to the lattice.

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

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

Photoemission­of­ActinidesPhotoemission­of­Actinidesalpa->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 ?

Electron fermi surfaces at (z=0)Electron fermi surfaces at (z=0)

LDA+DMFT (10 K)LDA LDA+DMFT (400 K)

X M

X

XX

M

MM

2 2

Slight decrease of the electron

FS with T

R A

R

RR

A

AA

3

a

3

LDA+DMFT (10 K)LDA LDA+DMFT (400 K)

Electron fermi surfaces at (z=Electron fermi surfaces at (z=))No a in DMFT!No a in Experiment!

Slight decrease of the electron

FS with T

LDA+DMFT (10 K)LDA LDA+DMFT (400 K)

X M

X

XX

M

MM

c

2 2

11

Electron fermi surfaces at (z=0)Electron fermi surfaces at (z=0)Slight decrease of the electron

FS with T

R A

R

RR

A

AA

c

2 2

LDA+DMFT (10 K)LDA LDA+DMFT (400 K)

Electron fermi surfaces at (z=Electron fermi surfaces at (z=))No c in DMFT!No c in Experiment!

Slight decrease of the electron

FS with T

LDA+DMFT (10 K)LDA LDA+DMFT (400 K)

X M

X

XX

M

MM

g h

Hole fermi surfaces at z=0Hole fermi surfaces at z=0

g h

Big change-> from small hole like to large electron like

1

Hole fermi surface at z=Hole fermi surface at z=

R A

R

RR

A

AANo Fermi surfaces

LDA+DMFT (400 K)LDA+DMFT (10 K)LDA

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 et. al. RMP (2005); Kotliar et. .al. RMP (2006).

2323

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)T. D Stanescu and GK PRB (2006)

2424

High Temperature superconductorsHigh Temperature superconductors

LeTacon et.al. Nature Physics (2006)

Raman Hg-1201

2626

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 k

K.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

• DMFT conceptual framework to think about electrons in solids.

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

• Ab-initio many body electronic structure of solids. Building theoretical spectroscopies.

• Frontier, cuprates, lower T, two dimensionality is a plaquette in a medium enough?

• Inhomogenous structure in correlated materials• New renormalizaton group methods built around

DMFT ?2828

Conclusion Conclusion

• A Few References ……

• A.Georges, G. K., W. Krauth and M. J. Rozenberg, Reviews of . Modern Physics 68, 13 (1996).

• G. K, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, C.A. Marianetti, RMP 78, 865-951, (2006).

• G. K and D. Vollhardt Physics Today, Vol 57, 53 (2004).

2929

OutlineOutline

• The standard model of solids.

• Correlated electrons and Dynamical Mean Field Theory (DMFT).

• The temperature driven Mott transition.

• Mott transition across the actinide series.

• Future Directions, cuprate superconductors and Cluster DMFT……

+ KS crystalV V 10KSG 1G

Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation theory in the screened interactions (Hedin 1965)

Self Energy Self Energy

VanShilfgaarde (2005)VanShilfgaarde (2005)

Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation theory in the screened interactions (Hedin 1965)

Self Energy Self Energy

VanShilfgaarde (2005)VanShilfgaarde (2005)

A. Georges, G. Kotliar (1992)

*

( )V Va a

a a

ww e

D =-å

latt ( ,1

G [ ]( ) [( ) ]

)[ ]n imp

nn

ik ii

ktw m

ww+ + - S

DD

=

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

G i i kw wD D=å

[ ]ijij

jm mJth hb= +ålattG ( ,

1 [ ]

( ) ( )[)

][ ]n impn

n

ii k i

ktw m

ww+ + - S

DD

=

11

( ( )( )

( [))

][ ]

imp n

imp n

kn

G i

Gti

ik

w

ww -D

D

=+-

å

A(A())

Raman Hg-1201 LeTacon et.al. Nature Physics (2006)

Doping decreases

Anti-Nodal Nodal

0

1

2

0 200 400 600 8000

5

10

0

5

10

0

5

T = 90 K T = 20 K

Opt.95 K

Raman Shift (cm-1)

T = 90 K T = 20 K

Und.78 K

Und.89 K

T = 90 K T = 20 K

Und.63 K

T = 90 K T = 20 K

T = 90 K T = 20 K

Ov.92 K

0

5

10

0 200 400 600 8000

5

0

5

10

0

5

10

0

2

505­cm-1

T = 90 K T = 20 K

Opt.95 K

Raman Shift (cm-1)

T = 90 K T = 20 K

Und.89 K

Und.92 K

612­cm-1

"(

) (u

.a.)

T = 90 K T = 20 K

Und.78 K

661­cm-1

T = 90 K T = 20 K

417­cm-1

T = 90 K T = 20 K

Ov.92 K

B1g B2g

High Temperature superconductorsHigh Temperature superconductors

Mott transition: Mott transition: evolution of the electron from itinerant to localized ? How

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) 99

Interaction with Experiments. Photoemission Three Interaction with Experiments. Photoemission Three peak strucure. peak strucure. V2O3:Anomalous transfer of spectral V2O3:Anomalous transfer of spectral

weightweight

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

1515

. 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)1616

Spinodals and Ising critical endpoint. Spinodals and Ising critical endpoint. Observation in VObservation in V22OO3 3 :: P. Limelette et.al. Science 302, 89 (2003)P. Limelette et.al. Science 302, 89 (2003)

Critical endpoint Critical endpoint

Spinodal Uc2Spinodal Uc2

1717

Spectral Function and PhotoemissionSpectral Function and Photoemission

• 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

Georges Kotliar (1992)Georges Kotliar (1992)

DDMFT approximate quantum solid as atom in a mediumMFT 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

Chitra and Kotliar Chitra and Kotliar PRB 62, 12715 (2000) PRB (2001)P.Sun and GK (2005) Zein PRB 62, 12715 (2000) PRB (2001)P.Sun and GK (2005) Zein

et.al.et.al. PRL PRL 96, 96, 226403 (2006)). See also Bierman Aryasetiwan and Georges. 226403 (2006)). See also Bierman Aryasetiwan and Georges.

Ir,>=|R, > Gloc=G(R, R’ ’ ) R,R’

1

2

1

1 ( ) Hartreecryst

Coulomb

VG i V

W

r

V P

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

[ , ] [ , , 0, 0]DMFT loc loc nonloc nonlocG W G W G W

Electronic structure problem: compute <r|G|r’> and <r|W|r’> given structure

[ , ] sum all 2PIgraphs= +

+

G W

G

Crossover­scale­~50K

in-plane

out­of­plane•Low­temperature­–­Itinerant­heavy­bands

•High­temperature­Ce-4f­local­moments

ALM­in­DMFTSchweitzer&Czycholl,1991

Coherence­crossover­in­experiment­

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

de Haas-van Alphen experimentsde Haas-van Alphen experiments

LDA (with f’s in valence) is reasonable for CeIrIn5

Haga et al. (2001)

Experiment LDA

Ce

In

Ir

CeIn

In

Crystal­structure­of­115’s­­CeMIn5 M=Co, Ir, Rh­

CeIn3­layer

IrIn2­layer

IrIn2­layer

Tetragonal­crystal­structure

4­in­plane­In­neighbors

8­out­of­plane­in­neighbors

3.27au

3.3 au

Fermi surfaces of CeFermi surfaces of CeM M In5 In5 within LDA (P. Oppeneer)within LDA (P. Oppeneer)

Localized 4f:LaRhIn5, CeRhIn5

Shishido et al. (2002)

Itinerant 4f :CeCoIn5, CeIrIn5

Haga et al. (2001)