Cluster Dynamical Mean Field Cluster Dynamical Mean Field Theory of the Copper Oxide Theory of the Copper Oxide Based Materials Based Materials

Gabriel KotliarGabriel Kotliar

ICAM Workshop on New Frontiers in the Phyiscs ICAM Workshop on New Frontiers in the Phyiscs of Two Diimensional Electron Systems. Buenos of Two Diimensional Electron Systems. Buenos Aires November (2011)Aires November (2011)

11

\\

Collaborators:Collaborators:

Kristjan Haule , Chuck Yee, Cedric Weber Kristjan Haule , Chuck Yee, Cedric Weber

Correlated Superconductivity Correlated Superconductivity • Order parameter transforms according to non trivial Order parameter transforms according to non trivial representation of the symmetry group of the crystal. representation of the symmetry group of the crystal.

K dependence.K dependence.

•Higher energy scales are involved. Fermi liquid Higher energy scales are involved. Fermi liquid state has not been reached, near Tc optimal. state has not been reached, near Tc optimal.

•Cluster DMFT, a natural generalization of the ME Cluster DMFT, a natural generalization of the ME theory.theory.

• Model Hamiltonian: identify common featuresModel Hamiltonian: identify common features

• System Specific Calculations: identify materials System Specific Calculations: identify materials trends.trends.

22

• Superconductivity in “simple” materials: Fermi liquid description of the normal state

+residual interaction among quasiparticles• Correlated electron materials. NEEDED: reference frame to access incoherent

(bad)metallic states • NEEDED: Bridge between atomic information and physical and spectroscopical

properties. [Structure-property relation] • MFT divides the problem into two separate parts: a) obtaining different mean field

solutions and b) evaluating their energies• . MFT provides useful concepts • MFT allows you to follow a “state” as you vary parameters.• MFT Compares different “states” of the system for the same value of parameters.

Understand “Mechanism”• MFT Separates what can be understood in models from what requires more realistic

treatment.

•

.

33DMFT strategy and Correlated Superconductivity DMFT strategy and Correlated Superconductivity

33

Cluster DMFT Cluster DMFT

Dynamical Mean Field TheoryDynamical Mean Field Theory

Maps solid to an effective quantum impurity modelMaps solid to an effective quantum impurity model

DMFT sumps up all local diagrams DMFT sumps up all local diagrams

(to all orders in perturbation theory)(to all orders in perturbation theory)

LDA+DMFTLDA+DMFT

, ,

,

[ ] [ ]( )

[ ] [ ]spd sps spd f

f spd ff

H k H kt k

H k H k

æ ö÷ç ÷ç ÷ç ÷çè ø®

0 0

0 dcff E

æ ö÷ç ÷S ç ÷ç ÷S -çè ø®

1 111

123

124

( ( , )) ( )

1( ) (cos cos )

21

( ) cos cos4

latt k

kx ky

kx ky

w m w w m

w m

w m

- -

-

-

+ - S = + - S

+ + - S +

+ + - S

44

Momentum Space Picture

T=| ↑, ↑ >T=| ↑, ↑ >

1+= 1/√2(|0, ↑> + 1+= 1/√2(|0, ↑> + | ↑, 0>)| ↑, 0>)

E=|0, 0>E=|0, 0>

S=1/√2( | ↑, ↓> S=1/√2( | ↑, ↓> -| ↑↓ >)-| ↑↓ >)

OverdopedOverdopedUnderdopedUnderdoped

Fermi Fermi LiquidLiquid

Holes in a sea of Holes in a sea of singletssinglets

Emergent low energy particle Emergent low energy particle hole symmetry !hole symmetry !

QCP K. Haule and GKQCP K. Haule and GK

PRB 76, 104509(2007).PRB 76, 104509(2007).OSMT Ferrero et. al.OSMT Ferrero et. al.

Phys. Rev. B 80, 064501 (2009)Phys. Rev. B 80, 064501 (2009)

55Link +Plaquette as reference frameLink +Plaquette as reference frame

Real Space Picture Real Space Picture

Stanescu, T. D., Stanescu, T. D., and P. and P.

Phillips,PRL Phillips,PRL 91,91,017002. (2003)017002. (2003)

Phys. Rev. Lett. 87, 047003 (2001)Phys. Rev. Lett. 87, 047003 (2001)

1111

As the apical oxygen is pulled out of the As the apical oxygen is pulled out of the plane Tcmax goes UP! plane Tcmax goes UP!

77

Theoretical issues with Andersen’s Theoretical issues with Andersen’s proposal. proposal.

Within Slave Boson MFT, t-J model. Tcmax Within Slave Boson MFT, t-J model. Tcmax decreases as t’ /t increases. decreases as t’ /t increases.

Within Hubbard model cluster DMFT studies Within Hubbard model cluster DMFT studies of superconductivity Tcmax decreases as t’/t of superconductivity Tcmax decreases as t’/t

increases. In many studies, for example in increases. In many studies, for example in CDMFTCDMFT

Earlier DMRG for ladders, White and Scalapino reached Earlier DMRG for ladders, White and Scalapino reached identical conclusion. identical conclusion. 77

Larger t’/t than cuprates but no Tc Larger t’/t than cuprates but no Tc

88

M. Greenblatt’s group (Rutgers)M. Greenblatt’s group (Rutgers)

Magnetic Magnetic Transition is Transition is

observed in trilayer observed in trilayer compoudn compoudn

Alternative view, Viktor Paardo and Warren Alternative view, Viktor Paardo and Warren Pickett. Pickett.

99

New Proposal: C. Weber C. Yee K. Haule New Proposal: C. Weber C. Yee K. Haule GK cond-mat arXiv:1108.3028GK cond-mat arXiv:1108.3028

•The main effect of the distance of the apical oxygen to the plane The main effect of the distance of the apical oxygen to the plane is to control the charge transfer gap: ed-epis to control the charge transfer gap: ed-ep

•Moving the apical oxygen away from te plane ALSO increases Moving the apical oxygen away from te plane ALSO increases tpp’ as Andersen pointed out. But the increase of tpp’ is tpp’ as Andersen pointed out. But the increase of tpp’ is detrimental to Tc. detrimental to Tc.

•Moving the apical oxygen away from the plane reduces the Moving the apical oxygen away from the plane reduces the charge transfer gap and increases Tc.charge transfer gap and increases Tc.

Backed up by LDA+DMFT Backed up by LDA+DMFT calculations using LDA+DMFT calculations using LDA+DMFT in the LAPW and using a in the LAPW and using a downfolding method with the downfolding method with the same value of Edc for all same value of Edc for all materials materials

1010

Correlations: deriving the starting Hamiltonian to do LDA+DMFT gives parameters. Correlations: deriving the starting Hamiltonian to do LDA+DMFT gives parameters.

1111

Causation: Reducing ed-ep helps TcCausation: Reducing ed-ep helps Tc(pushing apical away from the plane)(pushing apical away from the plane)

Increasing tpp’ hurts Tc Increasing tpp’ hurts Tc Moving the apical oxygen away form the plane Moving the apical oxygen away form the plane

reduces ed-ep but also increases tpp’ . We reduces ed-ep but also increases tpp’ . We believe the reduction of ep-ed is dominant.believe the reduction of ep-ed is dominant. 1212

Damascelli et. al. RMP(2010) Damascelli et. al. RMP(2010) Armitage Fournier and Greene Armitage Fournier and Greene

RMP(2003)RMP(2003) 1313

Building phase diagram Building phase diagram magnetization at T=0 vs magnetization at T=0 vs ..

Single siteSingle site

Two siteTwo site

1414

Origin of magnetism :Comparing the AF Origin of magnetism :Comparing the AF

and the “underlying PM state and the “underlying PM state ““

<KE>sdw -<KE>pm<KE>sdw -<KE>pm

LSCO gains kinetic energy when it LSCO gains kinetic energy when it magnetizes. [Mott ] NCCO pays kinetic magnetizes. [Mott ] NCCO pays kinetic

energy [Slater ] energy [Slater ]

NCCO magnetizes to lower its NCCO magnetizes to lower its double occupancy ! Slater. double occupancy ! Slater.

pmsdwUn n Un n

Can be traced to the structure: absence of Can be traced to the structure: absence of apical oxygens reduces the charge transfer apical oxygens reduces the charge transfer

energy energy 1515

Weber Haule and GK Nature Physics Weber Haule and GK Nature Physics 10, 1038 (2010). 10, 1038 (2010).

UnderdopedUnderdoped

PCCOPCCO

BSCOBSCO

OverdopedOverdopedN. Bontemps, R.P.S.M. N. Bontemps, R.P.S.M. Lobo and A.F. Santander-Lobo and A.F. Santander-Syro,Annals of Physics, Syro,Annals of Physics,

326, 1547 (2006)326, 1547 (2006)

1616

And back to the d9 nickelatesAnd back to the d9 nickelates

Chuck Ye e et. Chuck Ye e et. al. al. 1717

Thanks for your attention!Thanks for your attention!

References.

• C.Weber, K.Haule, G.Kotliar Nature Phys. 6, 574 (2010).

• arXiv:1108.3028 Cédric Weber, Chuck-Hou Yee, Kristjan Haule, Gabriel Kotliar

• Chuck Yee et al in preparation

1818

Effective HoppingsEffective Hoppings

Hubbard model : plaquette in a medium. Hubbard model : plaquette in a medium.

Lichtenstein and Kastnelson PRB (2000)Lichtenstein and Kastnelson PRB (2000)

1616

Link DMFT. Normal state Real Space Picture. Ferrero Link DMFT. Normal state Real Space Picture. Ferrero et. al. (2010) (similar to plaquette Haule and GK) (2006) et. al. (2010) (similar to plaquette Haule and GK) (2006)

• Momentum Space Picture: High T

Singlet formation. S (singlet),T Singlet formation. S (singlet),T (triplet) N=2 singlet, triplet(triplet) N=2 singlet, triplet

E (empty) N=0E (empty) N=0

1+ states with 1 electron 1+ states with 1 electron in + orbin + orb

Underdoped region: arcs shrink as T is reduced. Overdoped Underdoped region: arcs shrink as T is reduced. Overdoped region FS sharpens as T is reducedregion FS sharpens as T is reduced. . 1717

Optimal doping: Coherence scale Optimal doping: Coherence scale seems to vanishseems to vanish

TcTc

underdopedunderdoped

overdopedoverdoped

optimallyoptimally

scattering scattering

at Tcat Tc

Particle hole symmetry is restored at critical point. Particle hole symmetry is restored at critical point. Orbital selective Mott Transition. Orbital selective Mott Transition.

Superexchange Mechanism?Superexchange Mechanism? . . K. Haule and GK K. Haule and GK Phys. Rev. Phys. Rev. B 76, 104509 (2007).B 76, 104509 (2007).Ex= Jij(< Si. Sj >s- < Si . Sj>n)/t

D.J. Scalapino and S.R. White, Phys. Rev. B 58, D.J. Scalapino and S.R. White, Phys. Rev. B 58, 8222 (1998).8222 (1998).

How is the energy distributed How is the energy distributed in q and w ?in q and w ?

Reminiscent of PW Anderson RVB Science 235, 1196 (1987) and Reminiscent of PW Anderson RVB Science 235, 1196 (1987) and slave boson picture slave boson picture G. Kotliar and J. Liu P.RB 38,5412 (1988)

Expts; Dai et.al. Expts; Dai et.al. 1818

Weber Yee Haule GK cond-matWeber Yee Haule GK cond-mat

Phys. Rev. Lett. 87, 047003 (2001)Phys. Rev. Lett. 87, 047003 (2001)

Reducing ed=ep helps TcReducing ed=ep helps Tc

Increasing tpp’ hurts Tc Increasing tpp’ hurts Tc

Moving the apical oxygen reduces ed-ep but also incrases tpp’Moving the apical oxygen reduces ed-ep but also incrases tpp’

Realistic DMFT as a conceptual tool and a computational tool Realistic DMFT as a conceptual tool and a computational tool

DMFT (simple yet accurate ? ) reference frame to think about DMFT (simple yet accurate ? ) reference frame to think about electrons in solids and compute their properties. electrons in solids and compute their properties.

Compare different “states” of the system for the same value of Compare different “states” of the system for the same value of parameters. parameters. Understand Mechanism for ordering , magnetic, Understand Mechanism for ordering , magnetic, superconducting, exotic, ………. superconducting, exotic, ……….

Bridge between atomic information and physical and Bridge between atomic information and physical and spectroscopical properties. [Structure-Property relation spectroscopical properties. [Structure-Property relation Learning --> Design ? ] Learning --> Design ? ]

Qualitative and quantitative system specific results gives us Qualitative and quantitative system specific results gives us confidence in the method. Many examples (sp, 3d,4d, 5d, 4f, confidence in the method. Many examples (sp, 3d,4d, 5d, 4f,

5f…) 5f…)

NEEDED: improvments in obtaning and NEEDED: improvments in obtaning and understanding the solutions of the DMFT euqations. understanding the solutions of the DMFT euqations.

Fluctuations around DMFT> Fluctuations around DMFT>

2323

Optical Spectral Weights in LSCO and Optical Spectral Weights in LSCO and NCCO (up to 1.5 ev)NCCO (up to 1.5 ev)

Cedric Weber, Kristjan Haule, Gabriel Kotliar Nature Cedric Weber, Kristjan Haule, Gabriel Kotliar Nature Physics 10, 1038 (2010). Comanac et. al. Nature Physics 10, 1038 (2010). Comanac et. al. Nature

PhysicsPhysicsNature Phys. 4, 287290 (2008).Nature Phys. 4, 287290 (2008).

2020

Cuprates SuperconductorsCuprates Superconductors

• Plaquette DMFT reasonable reference frame to think about the qualitative physics of cuprates, starting from high temperatures.

• High Tc materials. are near the single site DMFT Mott boundary. LSCO more correlated than NCCO, role of apical oxygens.

• High temperature superconductivity occurs in the region where neither wave/itinerant nor localized/ particle picture fully applies.

2222

Cuprates : fundamental questions Cuprates : fundamental questions Mechanism of SuperconductivityMechanism of Superconductivity

1515

•SC emerges from doping RVB paired spins SC emerges from doping RVB paired spins in Mott insulator. in Mott insulator.

[ P. W. Anderson, [ P. W. Anderson, ScienceScience 235235, 1196 (1987), 1196 (1987)

• Predicted d-wave symmetry of order paramteterPredicted d-wave symmetry of order paramteter

and the pseudogap at low doping. G. Kotliar and and the pseudogap at low doping. G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988)J. Liu Phys.Rev. B 38,5412 (1988)

K >K > -K>-K>

•Quasiparticles glued by spin fluctuations.Quasiparticles glued by spin fluctuations.

• Predicted d-wave symmetry of order Predicted d-wave symmetry of order parameter . D. Scalapino , D. Pines parameter . D. Scalapino , D. Pines

Shining light on correlated electrons. Shining light on correlated electrons. Optical conductivity. Failure of the StandardOptical conductivity. Failure of the StandardModel: Anomalous Spectral Weight TransferModel: Anomalous Spectral Weight Transfer

Optical Conductivity Optical Conductivity Schlesinger Schlesinger t.al (1993) t.al (1993)

0( )d

= Neff (T, )depends on T = Neff (T, )depends on T

Very Non local transfer of spectral weight in FeSiVery Non local transfer of spectral weight in FeSiD. Van der Marel et.al (2005) [ 1 ev 800 cm-1]D. Van der Marel et.al (2005) [ 1 ev 800 cm-1]

Weight does not recover up to 5 Weight does not recover up to 5 ev. ev.

66Other probes for correlated electrons X-rays, neutrons, electrons, the kitchen sink, Other probes for correlated electrons X-rays, neutrons, electrons, the kitchen sink,

theory ……….theory ……….

Slave boson MFT. Slave boson MFT.

• SC order and Tc decrease as x decreases.

• Low doping . pseudogap with D wave symmetry .

• D wave symmetry of the SC OPD wave symmetry of the SC OP

•VVFF is weakly dependent on is weakly dependent on dopingdoping, , . .

• Coherence incoherence crossover Coherence incoherence crossover on the overdoped side. on the overdoped side.

G. Kotliar and J. Liu

PRB 38,5412 (1988)

singlet pairing O

itinerac P

P

y Ob

D

0b ¹

• Related T=0 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)

1818

• "The ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe..The constructionist hypothesis breaks down when confronted with the twin difficulties of scale and complexity. At each level of complexity entirely new properties appear. Psychology is not applied biology, nor is biology applied chemistry. We can now see that the whole becomes not merely more, but very different from the sum of its parts."(Anderson 1972)

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

““A crude version of this theory was published in 1988 by Zhang and co-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 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 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 conclusions independently (Phys. Rev. B 38, 5142–5145; 1988). But the successes weren't then recognized because experiments were too successes weren't then recognized because experiments were too

primitive.” primitive.”

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

no mystery. We now have a workable theory — no mystery. We now have a workable theory — not just for calculating the broad outlines (the not just for calculating the broad outlines (the

transition temperature transition temperature TTc, energy-gap shape, effect c, energy-gap shape, effect of doping, pseudogap temperature) but details of of doping, pseudogap temperature) but details of

the anomalous phenomenology.”the anomalous phenomenology.”

don’don’tt

theory was theory was

Cluster (link/plaquette)DMFT, promising new avenueCluster (link/plaquette)DMFT, promising new avenue

1919

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), 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),……………………………………...

Structure Property Relation in Correlated Systems:Structure Property Relation in Correlated Systems: c axis optics in YBCO. c axis optics in YBCO.

Compare with experimentsCompare with experiments

C axis optical conductvity C axis optical conductvity SpectraSpectra

Structure Structure

Sanibel 2008Sanibel 2008

Ratio ARatio ASS/A/ANN

Ratio more universal,Ratio more universal,

more symmetricmore symmetric

With decreasing doping gap With decreasing doping gap

increases, coherence peaks increases, coherence peaks

less sharp->Non BCSless sharp->Non BCS

McElroy,.. JC Davis, McElroy,.. JC Davis,

PRL 94, 197005 (2005)PRL 94, 197005 (2005)

Exp:Bi2212 with STMExp:Bi2212 with STM

Alternative explanation Fang, et.al. Alternative explanation Fang, et.al. PRLPRL vol 96, 017007 (2006). vol 96, 017007 (2006).

Early studies Hubbard model : plaquette in a Early studies Hubbard model : plaquette in a medium. medium.

Lichtenstein and Kastnelson PRB (2000) Stanescu, T. D., and P. Phillips, 2003, Phys. Rev. Lichtenstein and Kastnelson PRB (2000) Stanescu, T. D., and P. Phillips, 2003, Phys. Rev. Lett. Lett. 91,91,017002. DCA in 2x2 Jarrell, M., T. Maier, et. al. 2001, Europhys. Lett. 017002. DCA in 2x2 Jarrell, M., T. Maier, et. al. 2001, Europhys. Lett. 56, 563.56, 563.

2020

• QUALITATIVE INSIGHTS: a) Strongly Frequency QUALITATIVE INSIGHTS: a) Strongly Frequency Dependent Spin and Orbital Exchange SplittingDependent Spin and Orbital Exchange Splittingb) Spin splitting large at high frequency. b) Spin splitting large at high frequency. Orbital splitting large at low frequency. c) Qualitative Orbital splitting large at low frequency. c) Qualitative difference between BaFe2As2 and Oxides. difference between BaFe2As2 and Oxides.

Z. P. Yin, KH, G. Kotliar Nature Physics in press . Z. P. Yin, KH, G. Kotliar Nature Physics in press .

A. Georges and G. Kotliar PRB 45, A. Georges and G. Kotliar PRB 45, 6479 (1992).6479 (1992).

DMFT self consistency condition DMFT self consistency condition

1( , )

( )k

G k ii i

*

( )V Va a

a a

ww e

D =-å

impG ( )1

[ ]( ) ( )[ ][ ]n imp

nk ni k i

iww e w

=-

DS D-å

DMFT DMFT

Collective field describing the localization delocalization phenomenaCollective field describing the localization delocalization phenomena

Can be sublattice dependent, spin dependent, superconducting.. Can be sublattice dependent, spin dependent, superconducting.. … …

( )wD

† †

, ,

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

t c c c c U n n

Hubbard Model Hubbard Model

† † † † †Anderson Imp 0 0 0 0 0 0 0

, , ,

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

88

,ij i j i

i j i

J S S h S- -å å

eMF offhH S=-

Require that the effective Require that the effective medium produces the best medium produces the best possible local spectral function. possible local spectral function.

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), 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),……………………………………...

Why do we need a MFT of correlated materials ?Why do we need a MFT of correlated materials ?• Few exact solutions available.• Need for simplification, understanding, design.• Separate essential ingredients [e.g. phonons, orbitals, structure

etc. ] responsible for an effects. • Separate long wavelength non linearities (fluctuations, collective

modes, defects) from local physics. • Bridge between atomic information and physical and

spectroscopical properties. [Structure-Property relation Design] • Compare different “states” of the system for the same value of

parameters. Understand the Mechanism

.

DMFT Strategy and IdeasDMFT Strategy and Ideas• Breaks problems in two parts a) study of mean

field states from b ) evaluation of their energies.

• Second step, and detailed comparison experiments s within realistic implementions of electronic structure, e.g. LDA+DMFT.

• Qualitative lessons can be drawn from a) applied to simple models.

• Tools to think about correlated materials, e.g. Weiss fields, valence histograms, impurity model reference frames. etc.

• Locality assumption exact at high T.

Even within the same scheme at low T. In some region of Even within the same scheme at low T. In some region of parameters of the Hamiltonian, there are at low temperature parameters of the Hamiltonian, there are at low temperature many many solutions to the DMFT equations with different many many solutions to the DMFT equations with different

broken symmetries and ever increasing unite cells. broken symmetries and ever increasing unite cells.

Landscape of DMFT Solutions Problem Landscape of DMFT Solutions Problem

DifficultiesDifficulties

•2x2 cluster DMFT equations are considerably harder to 2x2 cluster DMFT equations are considerably harder to solve and to interpret than single site DMFT. solve and to interpret than single site DMFT.

•Uniqueness: no unique formulation of cluster DMFT Uniqueness: no unique formulation of cluster DMFT

Technical IssuesTechnical Issues

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

• Tc controlled by J. • Trvb, onset of spin pairing.• < b>, TBE , coherence

temperature, formation of QP.. • Superconducting dome.

Pseudogap evolves into SC• Problems: a) poor description

of the incoherent part b) MFT too uniform c) other states i.e. AF.

• Restricted form of the electron self energy.

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

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

Paramenkanti PRL 87, 217002 (2001) Paramenkanti PRL 87, 217002 (2001)

arXiv:1009.0271 arXiv:1009.0271

Theory : arXiv:1007.2867 Magnetism and Charge Dynamics in Iron Pnictides Z. P. Yin, K. Haule, G. Kotliar

04/22/2304/22/23 Zhiping Yin, Rutgers Zhiping Yin, Rutgers UniversityUniversity

5959

X’ Z X’ Z Y’ Y’

X’ Z X’ Z Y’ Y’

arXiv:1009.0271arXiv:1009.0271

Evolution of the correlations in Hunds Evolution of the correlations in Hunds metals metals

LDA LDA LDA+DMFT LDA+DMFT

Static Magnetic Moment is Determined by strength of Static Magnetic Moment is Determined by strength of correlatoins AND by the shape of the Fermi Surface correlatoins AND by the shape of the Fermi Surface

On First Principles Approaches to On First Principles Approaches to Materials Science Materials Science

Auguste Compte (1830). “ Every attempt to emply mathematical methods in the study of chemical questions must be considered profoundly irrational and contrary to the spirit of chemistry “

Paul Dirac (1929) “The underlying laws necessary for the Paul Dirac (1929) “The underlying laws necessary for the mathematical theory of the whole chemistry are thus completely mathematical theory of the whole chemistry are thus completely known and the difficulty is only that the exact application of these known and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble “ laws leads to equations much too complicated to be soluble “

““Approximate practical methods of applying quantum Approximate practical methods of applying quantum mechanics should be developed which can lead to an mechanics should be developed which can lead to an explanation of the main features of complex atomic systems explanation of the main features of complex atomic systems without too much computation” without too much computation”

“ …John Slater, already in 1953 , was obsessed with what I have rudely called the Great Solid State Dream machine. He envisioned that the new electronic computer could be applied to the task of automatically providing the electronic structure of any desired solid; and he literally believed that he would have the answers to any conceivable question . The latter idea was wrongheaded; but the former has gradually become a reality with the rise of what is now known as LDA. It is not enough appreciated that Slater himself provided the key element in that method. “

In the Rise of Complexity 1953-2002, PW Anderson In the Rise of Complexity 1953-2002, PW Anderson writes:writes:

But there are many cases where it fails spectacularly: essentially all of the interesting class of substances with magnetic inner shell atoms, most of which exhibit what has been called the Mott Phenomenon- a dominance of the local repulsion among inner shell electrons.

“…A new version of the Dream Machine has recently been invented which is quite successful in most of these cases-DMFT, dynamical mean field theory cooked up by Georges and Kotliar where the assumption of locality of the self energy in time is abandoned… “

Phil Anderson on the rise Phil Anderson on the rise of complexity 1953-2002of complexity 1953-2002

Correlated electrons are not well described by either Correlated electrons are not well described by either the itinerant (wave) picture, or the localized (particle) the itinerant (wave) picture, or the localized (particle) picture. Difficult non perturbative problem. picture. Difficult non perturbative problem. Theoretical ApproachesTheoretical Approaches• Phenomenology• Study of exactly soluble (by numerical and analytical

means) 1-d Hamiltonians• Approximate methods for models in 2d-3d • Guessing effective low energy theories. • Dynamical Mean Field Theory . [ Exact in the

Metzner Vollhard limit of infinite dimensions – now can bridge between structure and property of materials ]

, ,

,

[ ] [ ]( )

[ ] [ ]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> ­> ¯> ­ ¯> >®

1212

1( , )

( ) ( )G k i

i t k i

Spectra=- Im G(k,Spectra=- Im G(k,))

LDA+DMFT. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. LDA+DMFT. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). Lichtenstein and Katsnelson (1998) LDA++Phys. Cond. Mat. 35, 7359 (1997). Lichtenstein and Katsnelson (1998) LDA++

0 0

0 ff Edc

æ ö÷ç ÷S ç ÷ç ÷ç S -è ø®

abcdU U®

Conceptual Underpinning Diagrams: PT in W and G. Conceptual Underpinning Diagrams: PT in W and G.

1 1 1 10

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

2 2 C hartreeG W TrLnG Tr G G G TrLnW Tr V W W E G W

Introduce projector Gloc Introduce projector Gloc Wloc Wloc

: Chitra and Kotliar Phys. Rev. B 62, 12715 : Chitra and Kotliar Phys. Rev. B 62, 12715 (2000) and Phys. Rev.B (2001)(2000) and Phys. Rev.B (2001). .   

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0 1 2

Sigma s GW

Sigma p GW

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0 1 2

D Sigma s

D Sigma p

GW self energy for SiGW self energy for Si Beyond GWBeyond GW

Coordination SphereCoordination Sphere Coordination SphereCoordination Sphere

Proof of Principle ImplementationProof of Principle ImplementationFull implementation in the context of a a one orbital lattice model.Full implementation in the context of a a one orbital lattice model.

P Sun and G. KotliarP Sun and G. Kotliar Phys. Rev. B 66, 85120 (2002). Propose GW+DMFT . Phys. Rev. B 66, 85120 (2002). Propose GW+DMFT .

P.Sun and GK PRL (2004). Test various levels of self consistencyin Gnonloc PinonlocP.Sun and GK PRL (2004). Test various levels of self consistencyin Gnonloc Pinonloc

Test notion of locality in LMTO basis set in various materials. Test notion of locality in LMTO basis set in various materials.

N. Zeyn S. Savrasov and G. Kotliar PRL 96, 226403, 2006 N. Zeyn S. Savrasov and G. Kotliar PRL 96, 226403, 2006 N Zeyn S. Savrasov and G. K PRL 96, 226403 (2006)N Zeyn S. Savrasov and G. K PRL 96, 226403 (2006)

Still, summing all Still, summing all diagramas with diagramas with

dynamical U and dynamical U and obtaining the GW obtaining the GW starting point is starting point is

extremely expensive. So extremely expensive. So this is stillthis is still

a point of principle rather a point of principle rather than a practical tool. than a practical tool.

, ,

,

[ ] [ ]( )

[ ] [ ]abcd

0 0

0 Uloc spd sps loc spd f

locloc f spd loc ff

W WW i

W Ww

æ ö÷ç ÷=ç ÷ç ÷çè ø

é ùê ú®ê úë û

Total energy is derived from a functional of the density and GlocTotal energy is derived from a functional of the density and Gloc

CHARGE SELF CONSISTENT LDA+DMFT. S. Savrasov GK (2002)CHARGE SELF CONSISTENT LDA+DMFT. S. Savrasov GK (2002)

1212

LDA+DMFT as an approximation to the general scheme LDA+DMFT as an approximation to the general scheme

, ,

dmft ,

0 0 [ ] [ ]

0 [ ] [ ]spd sps spd f

ff f spd ff

Vxc k Vxc k

Edc Vxc k Vxc k

æ ö æ ö÷ ÷ç ç÷ ÷S +ç ç÷ ÷ç ç÷ ÷ç çS -è ø è ø®

Recent calculations using B3LYP hybrid + DMFT for Ce2O3. D. Jacob K. Haule Recent calculations using B3LYP hybrid + DMFT for Ce2O3. D. Jacob K. Haule and GK EPL 84, 57009 (2008)and GK EPL 84, 57009 (2008)

U is parametrized in U is parametrized in terms of Slater terms of Slater

integrals F0 F2 F4 integrals F0 F2 F4 ….….

Savrasov, Kotliar, Abrahams, Nature ( 2001)Savrasov, Kotliar, Abrahams, Nature ( 2001)

LDA+DMFT Self-Consistency loop [Savrasov Kotliar 2002]LDA+DMFT Self-Consistency loop [Savrasov Kotliar 2002]

Derived from the functional. Derived from the functional.

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 k LMTOV H ka ac r c- Ñ + =

DMFTDMFT

UU

EEdcdc

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å

REVIEW : G. Kotliar S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, C.A. Marianetti, RMP REVIEW : G. Kotliar S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, C.A. Marianetti, RMP 78, 865 (2006). 78, 865 (2006).

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

Savrasov, Kotliar, Abrahams, Nature ( 2001)Savrasov, Kotliar, Abrahams, Nature ( 2001)

Non magnetic correlated state of fcc Pu. Non magnetic correlated state of fcc Pu.

iw

N, Zein Following Aryasetiwan Imada N, Zein Following Aryasetiwan Imada Georges Kotliar Bierman and Georges Kotliar Bierman and

Lichtenstein. PRB 70 195104. (2004)Lichtenstein. PRB 70 195104. (2004)

PuPu

Superexchange Mechanism . Superexchange Mechanism . K. Haule and GK K. Haule and GK Phys. Rev. B 76, 104509 (2007). Compare “normal “ and SC state at Phys. Rev. B 76, 104509 (2007). Compare “normal “ and SC state at

the same tempearture!the same tempearture!

Reminiscent of Reminiscent of PW Anderson RVB Science 235, 1196 (1987) and PW Anderson RVB Science 235, 1196 (1987) and slave boson picture slave boson picture G. Kotliar and J. Liu P.RB 38,5412 (1988) 3131

Neutron spectroscopy with Neutron spectroscopy with LDA+DMFT LDA+DMFT

TheoryTheory: H. Park : H. Park , K. Haule and , K. Haule and

GK GK Experiments: L Harriger H. Luo M. Liu T. Experiments: L Harriger H. Luo M. Liu T.

Perring C Frost H. Ju M. Norman and Perring C Frost H. Ju M. Norman and Pengcheng Dai : arXiv:1011.3771Pengcheng Dai : arXiv:1011.3771

CUPRATESCUPRATES

Bath Bath

11 22

Return to Return to models, models,

Hubbard, t-JHubbard, t-J

Link DMFT Ferrero et. al. Link DMFT Ferrero et. al. Europhys. Lett. 85, 57009 Europhys. Lett. 85, 57009 (2009) Stanescu and (2009) Stanescu and Phillips P RB,69, 245104 Phillips P RB,69, 245104 (2004). (2004).

Plaquette DMFT: Plaquette DMFT:

Lichtenstein and Lichtenstein and Kastnelson PRB (2000)Kastnelson PRB (2000)

T. Maier, et. al. 2001, T. Maier, et. al. 2001, Europhys. Lett. Europhys. Lett. 56, 563.56, 563.

Sordi et.al. . Sordi et.al. . arXiv:1002.2960arXiv:1002.2960

Civelli et. al. Phys. Rev. Lett. Civelli et. al. Phys. Rev. Lett. 100, 046402 (2008) 100, 046402 (2008)

Haule and Kotliar Phys. Rev. Haule and Kotliar Phys. Rev. B 76, 104509 (2007)B 76, 104509 (2007)

Kinetic Energy Kinetic Energy ExchanExchange ge

EnergyEnergy

Real Real SpaceSpace

Momentum Momentum SpaceSpace

2626

Early studies of plaquette and link DMFT of Early studies of plaquette and link DMFT of Hubbard Hubbard

Lichtenstein and Kastnelson PRB (2000) Stanescu, T. D., and P. Phillips, 2003, Phys. Rev. Lichtenstein and Kastnelson PRB (2000) Stanescu, T. D., and P. Phillips, 2003, Phys. Rev. Lett. Lett. 91,91,017002. DCA in 2x2 Jarrell, M., T. Maier, et. al. 2001, Europhys. Lett. 017002. DCA in 2x2 Jarrell, M., T. Maier, et. al. 2001, Europhys. Lett. 56, 563.56, 563.

Electron and Hole Doped Electron and Hole Doped Cuprates : Similar but Yet Cuprates : Similar but Yet

Different, why?Different, why?C. Weber et.al. Nature C. Weber et.al. Nature

Physics 2010Physics 2010

NCCO : Robust AF PhaseNCCO : Robust AF Phase

Comensurate MagnetismComensurate Magnetism

Lower TcLower Tc

T^2 resistivity.T^2 resistivity.

Non monotonic angle dependence of Non monotonic angle dependence of SC order parameter ……… SC order parameter ………

Review: Armitage Fournier Review: Armitage Fournier Green (2009)Green (2009)

Apical Apical oxygenoxygen

1188

DMFT studies of copper oxides DMFT studies of copper oxides

• Strength of correlations (as quantified by single site DMFT) the most fundamental difference between NCCO and LSCO compounds. NCCO ( < c2 )and LSCO ( > c2)straddle the Zaanen Sawatsky Allen localization delocalization boundary. Traced to the absence of apical oxygen in NCCO (structure property relation).

•Good agreement with many experiments follow from a simple [ plaquette/link Good agreement with many experiments follow from a simple [ plaquette/link /site ] reference frame./site ] reference frame.

•In general, better modeling with DMFT (more sites, more orbitals etc ) better In general, better modeling with DMFT (more sites, more orbitals etc ) better

resultsresults..

•Power of mean field theory : comparing “normal “ and magnetic Power of mean field theory : comparing “normal “ and magnetic states, comparing “normal” and superconducting statesstates, comparing “normal” and superconducting states

2525

Iron PnictidesIron Pnictides

Iron Pnictides- Chalcogenides Iron Pnictides- Chalcogenides

FeSeFeSe1-0.081-0.08, (Tc=27K @ 1.48GPa), , (Tc=27K @ 1.48GPa),

Mizuguchi et.al., arXiv: 0807.4315Mizuguchi et.al., arXiv: 0807.4315

(Fe++)(Se__)(Fe++)(Se__)

Ba++(Fe++)2(As---)2 Ba++(Fe++)2(As---)2

PGPG

Iron pnictides Iron pnictides

(electron, hole, isovalent doping)(electron, hole, isovalent doping)

AFM: Antiferromagnetic metalAFM: Antiferromagnetic metal

SC: SuperconductorSC: Superconductor

T: TetragonalT: Tetragonal

O: OrthorombicO: Orthorombicholehole

electronelectronisovalentisovalent

Similarity/differences with Similarity/differences with cupratescuprates

Basic QuestionsBasic Questions

Relevant degreens of freedom, effective hamiltonians

• Strength of the correlations . Localized vs itinerant Fe d electrons

• Mechanism of the superconductivity and magnetism…….

• New arena to test the LDA+DMFT methodology [ with and without experimental informantion!]

Critical Critical endpoint endpoint

Spinodal Uc2Spinodal Uc22020

P. Limelette P. Limelette et.al. et.al. Science Science 302,302,

89 (2003)89 (2003)

T=170T=170

T=300T=300

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

Phys. Rev. Lett. 75, 105 (1995) Mo, Denlinger, Kim, Park, Allen, Mo, Denlinger, Kim, Park, Allen, Sekiyama, Yamasaki, Kadono, Suga, Sekiyama, Yamasaki, Kadono, Suga, Saitoh, Muro, Metcalf, Keller, Held, Saitoh, Muro, Metcalf, Keller, Held,

Eyert, Anisimov, Vollhardt PRL . Eyert, Anisimov, Vollhardt PRL . (2003(2003))

High temperature universality High temperature universality and V2O3and V2O3

Coherence Incoherence CrossoverCoherence Incoherence CrossoverHubbard U is not the “relevant” parameter.Hubbard U is not the “relevant” parameter.

The The Hund’s couplingHund’s coupling brings correlations! brings correlations!

Specific heat within LDA+DMFTSpecific heat within LDA+DMFT

for LaOfor LaO1-0.11-0.1FF0.10.1FeAs FeAs at U=4eV at U=4eV

LDA valueLDA value

For J=0 there is negligible mass enhancement at U~W!For J=0 there is negligible mass enhancement at U~W!

K. Haule and G. Kotliar K. Haule and G. Kotliar cond-mat arXiv:0805.0722 cond-mat arXiv:0805.0722

,­,­

LaOLaO1-0.11-0.1FF0.10.1FeAsFeAs

nn

wc=3000cmwc=3000cm-1 ~.-1 ~. .3 ev .3 ev

Nature Physics 5, 647 (2009) Nature Physics 5, 647 (2009) M. M. Qazilbash,1,, J. J. Hamlin,1 R. E. Baumbach,1 Lijun M. M. Qazilbash,1,, J. J. Hamlin,1 R. E. Baumbach,1 Lijun Zhang,2 D. J. Singh,2 M. B. Maple,1 and D. N. Basov1Zhang,2 D. J. Singh,2 M. B. Maple,1 and D. N. Basov1

Photoemission reveals now Z ~ .3Photoemission reveals now Z ~ .3

Freq. dep. U matrix well parametrized by F0 F2 F4Freq. dep. U matrix well parametrized by F0 F2 F4

F0 = 4:9 eV, F2 = 6:4 eV and F4 = 4:3 eV., nc=6.2F0 = 4:9 eV, F2 = 6:4 eV and F4 = 4:3 eV., nc=6.2

Z =1/2 for x2- y2 and z2 , Z =1/3 f xz; yz zx orbitals.Z =1/2 for x2- y2 and z2 , Z =1/3 f xz; yz zx orbitals.

F0 = 4:9 eV, F2 = 6:4 eV F4 = 4:3 eV., nc=6.2F0 = 4:9 eV, F2 = 6:4 eV F4 = 4:3 eV., nc=6.2

Exp: W.Z. Hu et.al., PRL 101, 257005 (2008)Exp: W.Z. Hu et.al., PRL 101, 257005 (2008)

LDA+DMFT calculations Kutepov Haule Savrasov and Kotliar PRB LDA+DMFT calculations Kutepov Haule Savrasov and Kotliar PRB (2010). Mass renormalization = 3 without satellites (2010). Mass renormalization = 3 without satellites

LDA+DMFT Magnetic moment .95 muB LDA+DMFT Magnetic moment .95 muB Expt 1 muB Expt 1 muB

LL

EXPT: Hu, W. Z. et al. Phys. Rev. Lett. 101, 257005. (2008).EXPT: Hu, W. Z. et al. Phys. Rev. Lett. 101, 257005. (2008).

EXPT: Nakajima, M. et al. Phys. Rev. B 81, 104528 (2010)EXPT: Nakajima, M. et al. Phys. Rev. B 81, 104528 (2010)

Theory Yin et. al. (2010)Theory Yin et. al. (2010)

Origin of the anisotropy is electronicOrigin of the anisotropy is electronic

Optical features sharpen in the polarized spectra. Experimental predictions. Optical features sharpen in the polarized spectra. Experimental predictions. Measurements underway ( not easy !)Measurements underway ( not easy !)

Orbital polarization of the frequency dependentOrbital polarization of the frequency dependenthybridization Weiss field. Lives only at very low hybridization Weiss field. Lives only at very low

energies. energies.

Magnetic Stripe Phase of the FeAs Magnetic Stripe Phase of the FeAs materials: new insights from LDA+DMFT materials: new insights from LDA+DMFT Z. Yin K. Haule and GK [ in preparation] Z. Yin K. Haule and GK [ in preparation]

a) At low energies conductivity goes up. Rapid coherence crossover from an incoherent normal state compensates for a loss of carriers. Gain kinetic energy at very low energies!

b)For intermediate loss in carriers (kinetic energy )

Focus on changes of Neff(Focus on changes of Neff(, T) at various energy , T) at various energy scales scales in going to the magnetic state. in going to the magnetic state.

Mass­enhancement,­plasma­Mass­enhancement,­plasma­frequencyfrequency

Optical­conductivityOptical­conductivityPRB PRB 8282, 045105 (2010), 045105 (2010)

Exp: W.Z. Hu et.al., PRL 101, 257005 (2008) .Exp: W.Z. Hu et.al., PRL 101, 257005 (2008) .

Plasma frequency:Plasma frequency:

LDA ~ 2.6eVLDA ~ 2.6eV

DMFT ~ 1.6eVDMFT ~ 1.6eV

Exp ~ 1.6eVExp ~ 1.6eV

Mass enhancement of Fe-5d bands Mass enhancement of Fe-5d bands

m*/mm*/mLDALDA~3~3

Interband peak ~ 0.6eVInterband peak ~ 0.6eV

Drude weightDrude weight

U=5eV, J=0.7eVU=5eV, J=0.7eV

Theory : Fourth generation of LDA+DMFT methods and codes. Kutepov Haule Theory : Fourth generation of LDA+DMFT methods and codes. Kutepov Haule Savrasov and Kotliar (2010). Mass renormalization without satellites !Savrasov and Kotliar (2010). Mass renormalization without satellites !

Eliminate the hybridization to the semicore states Eliminate the hybridization to the semicore states included in GW but not in LDA +DMFT by rescalingincluded in GW but not in LDA +DMFT by rescaling

. Define a projector. Use the same projector in calculating the

U’s that you will use in your DMFT calculation

Go back to basics: U’s for DMFT. ( Kutepov et. al. Go back to basics: U’s for DMFT. ( Kutepov et. al. building on the PT in G and W by R. Chitra and G. K)building on the PT in G and W by R. Chitra and G. K)

Rigorous Definition of the Hubbard U and the Weiss Rigorous Definition of the Hubbard U and the Weiss Field Delta in a Solid. Kutepov et.al (2010)Field Delta in a Solid. Kutepov et.al (2010)

Approximate Wloc and Piloc using Self Approximate Wloc and Piloc using Self Consistent GW. Kutepov et. al. 2010Consistent GW. Kutepov et. al. 2010

DMFT Concepts and Tools in electronic structureDMFT Concepts and Tools in electronic structure

Valence Histograms. Describes the history of the “atom” in the solid, multiplets!

*

( )a b

ab V Va a

a a

ww e

D =-å

,[ ] [ , ]dft lda dmft locG Ur r+G ¾¾®G

Weiss Weiss field, collective Weiss Weiss field, collective hybridizationfunction, quantifies hybridizationfunction, quantifies the degree of localization the degree of localization

Functionals of density and Functionals of density and spectra give total energiesspectra give total energies

1( , )

( )k

G k ii H i

Local Self Energies and Correlated BandsLocal Self Energies and Correlated Bands

Local Spectral FunctionLocal Spectral Function

A. Georges and G. Kotliar PRB 45, A. Georges and G. Kotliar PRB 45, 6479 (1992).6479 (1992).

DMFT self consistency condition DMFT self consistency condition

1( , )

( )k

G k ii i

*

( )V Va a

a a

ww e

D =-å

impG ( )1

[ ]( ) ( )[ ][ ]n imp

nk ni k i

iww e w

=-

DS D-å

DMFT DMFT

Collective field describing the localization delocalization phenomenaCollective field describing the localization delocalization phenomena

Can be sublattice dependent, spin dependent, etc.. … Can be sublattice dependent, spin dependent, etc.. …

( )wD

† †

, ,

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

t c c c c U n n

Hubbard Model Hubbard Model

† †Anderson Imp 0

, ,

† † †0 0 0 0 0 0

,

( +c.c).

H c A A A

c c Uc c c

V

c

88

RVB phase diagram of the Cuprate Superconductors. RVB phase diagram of the Cuprate Superconductors. Correct prediction Correct prediction d wave symmetry of SC order parameter (and generic cuprate phase d wave symmetry of SC order parameter (and generic cuprate phase diagram) many years before the experiment. diagram) many years before the experiment.

• Tc controlled by J. • Trvb, onset of spin pairing.• < b>, TBE , coherence

temperature, formation of QP.. • Superconducting dome.

Pseudogap evolves into SC• Problems: a) poor description

of the incoherent part b) MFT too uniform c) other states i.e. AF. d) Restricted form of the electron self energy.

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

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

Paramenkanti PRL 87, 217002 (2001) Paramenkanti PRL 87, 217002 (2001)

Reducing ed=ep helps TcReducing ed=ep helps Tc

Increasing tpp’ hurts Tc Increasing tpp’ hurts Tc

Moving the apical oxygen reduces ed-ep but also incrases tpp’Moving the apical oxygen reduces ed-ep but also incrases tpp’