correlated electrons

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Strongly Correlated Electron

Systems a Dynamical MeanField Perspective

G. Kotliar Physics Department and Center for

Materials Theory

Rtgers

!C"M meeting# Frontiers in Correlated Matter

Sno$mass Septem%er &''(



Strongly Correlated Electron Systems Display remar)a%le

phenomena* that cannot %e nderstood $ithin the standard model of

solids. Resistivities that rise $ithot sign of satration %eyond the

Mott limit* +e.g. ,. Ta)agi-s $or) on anadates/* temperatre

dependence of the integrated optical $eight p to high fre0ency+e.g. andermarel-s $or) on Silicides/.

Correlated electrons do 1%ig things2* large volme collapses* colossal

magnetoresitance* high temperatre spercondctivity . Properties are

very sensitive to strctre chemistry and stoichiometry* and control

parameters large non linear sscepti%ilites*etc333.

T,E 4,5



T,E ,64T,E ,64

D57"M!C"8 ME"7 F!E8D T,E6R5.

96ptimal Gassian Medim 9 : 9 8ocal ;antm Degrees of Freedom 9 : 9their interaction 9

is a good reference frame for nderstanding* and predicting physical properties

of correlated materials. Focs on local 0antities* constrct fnctionals of those 0antities* similarities $ithDFT.

,o$ to thin) a%ot their electronic

states <

,o$ to compte their properties <

Mapping onto connecting their

properties* a simpler 1reference

system2. " self consistent impritymodel

living on S!TES* 8!7KS and

P8";=ETTES......

7eed non pertr%ative tool.



What did we learn ? Schematic DMFT phase diagram

and DOS of a partially frustrated integer filled

Hubbard model and pressure driven Mott transition



!ressure driven Mott transition



,o$ do $e )no$ there is some trth in this

pictre < ;alitative Predictions erified > T$o different featres in spectra. ;asiparticles

%ands and ,%%ard %ands.> Transfer of spectral $eight $hich is non local in

fre0ency. 6ptics and Photoemission.

> T$o crossovers* associated $ith gap closreand loss of coherence. Transport.

> Mott transition endpoint* is !sing li)e* coples toall electronic properties.

> "n 1e?act nmerical approach PRG 1 recentlyfond the first order line+M. !mada/* C@DMFToffers a consistency chec).



!sing critical endpoint fondA !n &6B P.

8imelette et.al. +Science &''B/



"nomalos transfer of optical spectral

$eight* 7iSeS. Miyasa)a and Ta)agi

&'''



4hy does it $or)# Energy 8andscape of

a Correlated Material and a top to %ottom

approach to correlated materials.

Energy

Configrational Coordinate in the space of ,amiltonians

T

Single site DMFT. ,igh temperatre

niversality vs lo$ temperatre

sensitivity to detail for materials

near a temperatre@pressre driven

Mott transition



4hat did $e gain<

> Conceptal nderstanding of ho$ the electronicstrctre evolves $hen the electron goes fromlocalied to itinerant.

> =c =c&* transfer of spectral $eight* 3.> " general methodology $hich $as e?tended to

clsters +non trivialA/ and integrated into anelectronic strctre method* $hich allo$s s to

incorporate strctre and chemistry. oth areneeded a$ay from the high temperatreniversal region.



> Mott transition across the Hf-s* a veryinteresting playgrond for stdyingcorrelated electron phenomena.

> DMFT ideas have %een e?tended into

a frame$or) capa%le of ma)ing firstprinciples first principles stdies ofcorrelated materials. P Phonons.Com%ining theory and e?periments toseparate the contri%tions of differentenergy scales* and length scales to the%onding

> !n single site DMFT * spercondctivity

is an navoida%le conse0ence $hen$e try to go move from a metallic stateto a Mott inslator $here the atomshave a closed shell +no entropy/.Realiation in "m nder pressre <



DMFT Phonons in fccDMFT Phonons in fcc δδ-Pu: connect-Pu: connect

bonding to energy and length scales.bonding to energy and length scales.

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

Theory 34.56 33.03 26.81 3.88

Experiment 36.28 33.59 26.3 4.8

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

(e!eriments "rom #ong et$al, Science, 22 A%g%st 2003)



ig 0estion# $ill $e %e nearly as sccessfl

in or attemps to nderstand and predict+some / physical properties of correlated

materials* $ith DMFT* as $e have %een for

$ea)ly correlated materials sing+ appro?imate DFT and pertr%ation theory in

screened Colom% interactions eg.G4 /<



6ne dimensional ,%%ard model & site +8!7K/ CDMFT compare $ith ethe "nats* .

Kancharla C. olech and GK PR IJ* 'JH' +&''B/M.CaponeM.Civelli Kancharla

C.Castellani and GK P. R 69*H'H +&''(/

U/t=4.

" rapidly convergent algorithm <



8in)s* Ti&6B # Colom% and

Paling

C.E.Rice et all, Acta Cryst B33, 1!" #1$%%&

LTS 250 K, HTS 750 K.

E l ti f th ) l d S t l



Evoltion of the ) resolved Spectral

Fnction at ero fre0ency. +Parcollet iroli and

GK PR8* &* &&I('&. +&''(// / ( 0! )"# $ A & ω =

Uc=2.35+-.05, Tc/D=1/44

U/D=2 U/D=2.25



U/t=8, t’= -0.3

Density= 0.88, 0.89, 0.9, 0.91, 0.922,

0.96, 0.986, 0.988, 0.989, 0.991,

0.993

U/t=16,t’= +0.9

=nderlying normal state

of the ,%%ard model

near the Mott transition*+force the 4eiss field to

its paramagnetic vale/*

TL' ED soltion of the

C@DMFT e0ations. M.

Civelli* M. Capone* 6.

Parcollet and GK



"pproaching the Mott transition#

pla0ette Cdmft.> ;alitative effect* momentm space

differentiation. Formation of hot cold regions isan navoida%le conse0ence of the approach tothe Mott inslating stateA

> D $ave gapping of the single particle spectra asthe Mott transition is approached.

> Stdy the 1normal state2 of the ,%%ard model.

General phenomena* %t the location of the coldregions depends on parameters. Civelli CaponeParcollet and Kotliar



4here do $e go no$ <

> 6ne can stdy a large nm%er of e?perimentallyrelevant pro%lems $ithin the single siteframe$or).

> Contine the methodological development* $eneed toolsA

> Solve the CDMFT Mott transition pro%lem onthe pla0ette pro%lem* hard* %t it is a significantimprovement* the early mean field theories $hile)eeping its physical appeal.

> Stdy material trends* ma)e contact $ithphenomenological approaches* dopedsemicondctors +hatt and Sachdev/* heavyfermions * H-s+7a)atsNi* Pines and Fis) /33



Mott transition into an open +right/ and closed +left/ shell

systems. !n single site DMFT* spercondctivity mst intervene

%efore reaching the Mott inslating state.Capone et. al. AmAt room pressure a loalise! "#6 system$%="/2. & = -' = 3( )

= 0 apply pressure *

S S

= =

.γ T8og&O:

=c

γ Q+=c@=/

SL'

<<<



"mericim nder pressre O.C.

Grivea? O. Re%iant G. 8ander



Evoltion of the Spectral Fnction $ith

Temperatre

%noma&o# tran#er o #petra& *ei+ht onnete, to the

proximity to the -#in+ ott en,point (/ot&iar an+e n,

oener+Phys. Rev. Lett. 84, 5180 (2000)

" ti l ti i ti



"ns$er# catiosly optimistic yes*

%t it needs a lot of $or).

> Focs on short distance intermediateenergy scale properties. Method isdesigned for that

> 7eed analytic :nmerical $or).Connection $ith other approachesQDMRG

> 7eed adaptive ) space.

> 6ne can already do a lot $ith single site

DMFT in many many many materials.> Pla0ette e0ations are one order of

magnitde harder to solve.



Total Energy as a function of 'olu(e for Total Energy as a function of 'olu(e for

PuPu 4 +ev/ vs +a.. &J.& ev/

(Savrasov, Kotliar, Abrahams, 'at%re ( 200)

'on magnetic correlated state o" "cc %$

iw

ein Savrasov and Kotliar +&''(/



DMFT Phonons in fccDMFT Phonons in fcc δδ-Pu-Pu

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

Theory 34.56 33.03 26.81 3.88

Experiment 36.28 33.59 26.3 4.8

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

(e!eriments "rom #ong et$al, Science, 22 A%g%st 2003)



Epsilon Pltonim.



Phonon entropy drives the

epsilon delta phase transition> Epsilon is slightly more delocalied than delta* has

SM"88ER volme and lies at ,!G,ER energy than

delta at TL'. t it has a mch larger phonon entropy

than delta.

> "t the phase transition the volme shrin)s %t the

phonon entropy increases.

> Estimates of the phase transition follo$ing Drmont

and G. "c)land et. al. PRB."# , 184104 (2002); +and

neglecting electronic entropy/. TC I'' K.

ransverse onon a ong



ransverse onon a ong+'**/ in epsilon P in self

consistent orn appro?imation.



Mott transition into an open +right/ and closed +left/ shell

systems. !n single site DMFT* spercondctivity mst intervene

%efore reaching the Mott inslating state.Capone et. al. AmAt room pressure a loalise! "#6 system$%="/2. & = -' = 3( )

= 0 apply pressure *

S S

= =

.γ T8og&O:

=c

γ Q+=c@=/

SL'

<<<



"mericim nder pressre O.C.

Grivea? O. Re%iant G. 8ander

erie o# ro p



erie o# ro p,

o# Am

> 7ote strongly increasingresistivity as f+p/ at all T.Sho$s that moreelectrons are enteringthe condction %and

> Spercondcting at allpressre

> !ariation of rho vs. T forincreasing p.



DMFT stdy in the fcc strctre. S.

Mrthy and G. Kotliar

fcc



8D":DMFT spectra. 7otice the

rapid occpation of the fJQ& %and.



6ne electron spectra. E?periments +7egele/ and 8D":DFT

theory +S. Mrthy and GK /



Conclsion "m

> Crde 8D":DMFT calclations descri%e the crdeenergetics of the material* e0. volme* even p vs .

> Spercondctivity near the Mott transition.

Tc increases first and the decreases as $e approach the

Mott %ondary.Dramatic effect in the f %l) modle.

4hat is going on at the "m !@ "m !! %ondary <<< S%tleeffect +%l) modli do not change mch /* %t crcial

modifications at lo$ energy. Mott transition of the fJQ& %and < ;antm critical

point <#



,.;. 5an et. al. CeC&+Si&@? Ge?/.

"m nder pressre Grivea et. al.



Electronic states in $ea)ly and

strongly correlated materials

> Simple metals* semicondctors. Fermi 8i0idDescription# ;asiparticles and 0asiholes* +and their%ond states /. Comptational tool# Density fnctional

theory : pertr%ation theory in 4* G4 method.> Correlated electrons. "tomic states. ,%%ard %ands.

7arro$ %ands. Many anomalies.

> 7eed tool that treats ,%%ard %ands* and 0asiparticle

%ands* real and momentm space on the same footing.DMFTA



4ea)ly correlated electrons. F8T and DFT* and $hat goes $rong in

correlated materials.

> Fermi 8i0id . . Correspondence %et$een asystem of non interacting particles and the fll,amiltonian.

> " %and strctre is generated +Kohn Shamsystem/.and in many systems this is a goodstarting point for pertr%ative comptations of thespectra +G4/.

( ) ! ( ) r r ρ ρ ↓ ↑

Γ

DMFT Cavity Constrction# " Georges and G Kotliar



DMFT Cavity Constrction# ". Georges and G. Kotliar

PR (H* I(J +&/. Figre from # G. Kotliar and D.

ollhardt Physics Today HJ*+&''(/

http#QQ$$$.physics.rtgers.edQ)otliarQR!gen.html

The self consistent imprity model is a ne$

reference system* to descri%e strongly

correlated materials.

* * * *+ +

http://www.physics.rutgers.edu/~kotliar/RI_gen.html





cl%ster cl%ster eterior eterior * * * *

−= + +

* ) cl%ster * + * *

6imp&er 7me,i'm7 8ami&tonian

cl%ster eterior eterior * * − +

Dynamical Mean FieldTheory DMFT! "a#ity"on$tr%ction& '. (eor)e$

and (. *otliar 45, 412!.



Site Cell. Celllar DMFT. C@DMFT. G. Kotliar*S.. Savrasov*

G. Palsson and G. iroli* Phys. Rev. 8ett. J* I(' +&''/

tˆ(K) hopping expressed in the superlattice notations.

:ther &#ter exten#ion# (;C% <arre&& /ri#hnamrthy!

/at#ne&#on an, ihten#tein perio,ie, #heme! =e#te,

C&#ter heme# hi&&er -n+er#ent )! a#a&ity i##e#! :.

Paro&&et! G. >iro&i an, G/ on,?matt 03058 (2003)

T$o paths for a% initio



T$o paths for a%@initio

calclation of electronic

strctre of stronglycorrelated materials

Corre&ation @ntion# Tota&

Ener+ie# et.

o,e& ami&tonian

Cry#ta& #trtre A%tomi po#ition#

DMFT idea$ can e %$ed in oth ca$e$.



8D":DMFT . "nisimov* ". Poteryaev* M. Korotin* ". "no)hin

and G. Kotliar* O. Phys. Cond. Mat. BH* JBH +J/. + ichtenstein and M.

atsnelson PR /%, 0! #1$&.

> The light* SP +or SPD/ electrons are e?tended* $elldescri%ed %y 8D" .The heavy* D +or F/ electrons arelocalied treat %y DMFT.

> 8D" Kohn Sham ,amiltonian already contains an

average interaction of the heavy electrons* s%tract thisot %y shifting the heavy level +do%le conting term/

Kinetic energy is provided %y the Kohn Sham,amiltonian +sometimes after do$nfolding /. The =

matri? can %e estimated from first principles of vie$edas parameters. Solve reslting model sing DMFT.



Functional formulation $hitra and %otliar &'(()*+ Savrasov and %otliarcond,

matt(-(.(#- &'((-*

1 B1( ) ( ! ') ( ') ( ) ( ) ( )

2+ , i f y y -

2 233 3

B( ') ( ')- . . y r y r 4- 5 6 ( ') ( ) ( ') ( ) . . . . # r r r r 5 6- 5 65 64

r!"#$, ρ!

! ! ! 0! 0 /DM1 loc loc nonloc nonloc- # - # - # Φ Φ = =:

1 1 1 1

0

1 1

4 ! 5 4 5 4 5 4 ! 52 2 + hartree- # 1rLn- 1r - - - 1rLn# 1r , # # / - #

− − − −

Γ = − − − + − + + Φ

Do%le loo in Gloc and Wloc

! it d l t %ilit f



!mprity model representa%ility of

spectral density fnctional.

R h di f th C t



R phase diagram of the Cprate

Spercondctors

> P.4. "nderson.as)aran o and

"nderson.Connection %et$eenhigh Tc and Mottphysics.

> U%V coherence orderparameter.

> K* D singlet formationorder paramters.

G. Kotliar and O. 8i Phys.Rev.

B*H(& +/



> ,igh temperatre spercondctivity is annavoida%le conse0ence of the need toconnect $ith Mott inslator that does not %rea)any symmetries to a metallic state.

> Tc decreases as the 0asiparticle reside goesto ero at half filling and as the Fermi li0idtheory is approached.

>Early on* acconted for the most salient featresof the phase diagram. d@$ave spercondctivity*anomalos metallic state* psedo@gap state



Pro%lems $ith the approach.

> 7meros other competing states. Dimer phase*%o? phase * staggered fl? phase * 7eel order*

> Sta%ility of the psedogap state at finitetemperatre.

> Missing finite temperatre . flctations of slave%osons *

> Temperatre dependence of the penetrationdepth 4en and 8ee * !offe and Millis Theory#

∀ ρTL?@Ta ?& * E?p# ρTL ?@T a.> Theory has niform on the Fermi srface* in

contradiction $ith "RPES.

E l ti f th t l



Evoltion of the spectral

fnction at lo$ fre0ency.( 0! )"# $ A & ω =

the deendence o the $el ener)y i$

6ea, 6e e7ect to $ee conto%rline$ corre$ondin) to 8 = con$tand a hei)ht increa$in) a$ 6earoach the Fermi $%race.

9t%dy a model o aa or)anic$.

$

$

2 2

$

E$Ct($)Ae ( ! 0)

C -m ( ! 0)

( ! 0)E$

&

&

A &

ω µ

γ ω

γ ω

γ

Σ = −

Σ =

= =+

Keeps all the goodies of the slave %oson mean



Keeps all the goodies of the slave %oson mean

field and ma)e many of the reslts more solid

%t also removes the main difficlties. > Can treat coherent and incoherent spectra.

> 7ot only spercondctivity* %t also thephenomena of momentm space differentiation

+formation of hot and cold regions on the Fermisrface/ are navoida%le conse0ence of theapproach to the Mott inslator.

> Can treat dynamical flctations %et$een

different singlet order parameters.> Srprising role of the off diagonal self energy$hich renormalies t-.

Spectral Evoltion at TL' half filling fll



Spectral Evoltion at T ' half filling fllfrstration figre from .an4 5. o7ener4 .

:otliar ;' <0,16661993

> Spectra of the strongly

correlated metallic

regime contains %oth0asiparticle@li)e and

,%%ard %and@li)e

featres.

> Mott transition is driven%y transfer of spectral

$eight.

E l ti f th S t l F ti ith



Evoltion of the Spectral Fnction $ith

Temperatre

%noma&o# tran#er o #petra& *ei+ht onnete, to the

proximity to the -#in+ ott en,point (/ot&iar an+e n,

oener+ Phys. Rev. Lett. 84, 5180 (2000)



Conse0ences for the optical condctivity Evidence

for ;P pea) in &6B from optics.

. oener+ G. /ot&iar . /aDeter G Thoma# ;. ap$ine < oni+ an, P

eta& Phy#. e". ett. 5! 105 (1995)

"nomalos transfer of optical



"nomalos transfer of optical

spectral $eight &6B

oener+ G. /ot&iar an, . /aDter Phy#. e". > 54! 8452 (1996).

. oener+ G. /ot&iar . /aDeter G Tahoma# ;. ap$i$ne < oni+an, P eta& Phy#. e". ett. 5! 105 (1995)



* .l!ri!4e, )., :ornelsen, :.,>an4, ?.,>illiams, ).,



l!ri!4e, )., :ornelsen, :.,>an4, ?.,>illiams, ).,

@rou, A., an! >atins, D., Sol State $omm, J, "83

1991.

" l R i ti it d M tt



"nomalos Resistivity and Mott

transition 7i Se&@? S?

"ro$$o#er rom Fermi li:%id to ad metal to$emicond%ctor to arama)netic in$%lator.



(ET) F tt t iti



-n#&atin+

anion &ayer

κ ?(ET)2F are aro## ott tran#itionET

X-1

[(ET)2]+1on,tin+

ET &ayer

t’

t

mo,e&e, to trian+&ar &attie

X- Ground

State

U/t t’/t

Cu2(CN)3Mottinsuator

!"2 1"#$

Cu[N(CN)2]C Mott

insuator

%"& #"%&

Cu[N(CN)2]'r SC %"2 #"$!

Cu(NCS)2 SC $"! #"!

Cu(CN)[N(CN)2]

SC $"! #"$!

*(CN)2 2, SC $"$ #"$#

3 SC $"& #"&!

Prof. Kanoda =. To)yo

Mott transition in a.ered or*ani ondutors S 0eere et a"



. *ond-4at/###&&5 Phys. Rev. Lett. 85, 5420 (2000)



> Theoretical isse# is there a Mott transition

in the integer filled ,%%ard model* and is it

$ell descri%ed %y the single site DMFT <





fnction at lo$ fre0ency.( 0! )"# $ A & ω =

the deendence o the $el ener)y i$

6ea, 6e e7ect to $ee conto%rline$ corre$ondin) to 8 = con$tand a hei)ht increa$in) a$ 6earoach the Fermi $%race.

$

$

2 2

$

E$Ct($)Ae ( ! 0)

C -m ( ! 0)

( ! 0)E$

&

&

A &

ω µ

γ ω

γ ω

γ

Σ = −

Σ =

= =+





pla0ette Cdmft.> ;alitative effect* momentm space

differentiation. Formation of hot coldregions is an navoida%le conse0ence of

the approach to the Mott inslating stateA> D $ave gapping of the single particle

spectra as the Mott transition isapproached..

> S0are symmetry is restored as $eapproched the inslator

Mechanism for hot spot formation# nn



Mechanism for hot spot formation# nn

self energy A General phenomena.



Conclsion.

> Mott transition srvives in the clster setting.

Role of magnetic frstration.

> Srprising reslt# formation of hot and cold

regions as a reslt of an approach to theMott transition. General reslt <

> =ne?pected role of the ne?t nearest

neigh%or self energy. CDMFT a ne$ $indo$to e?tend DMFT to lo$er temperatres.



Conclsion

> DMFT mapping onto 1self consistent impritymodels2 offer a ne$ 1reference frame2* to thin)a%ot correlated materials and compte theirphysical properties.Formal parallel $ith DFT.

> .Pla0ettes@Kappa organics@,ot and coldregions.

> Titanim ses0io?ides. Dynamical Paling

Goodenogh mechanism.> Sites. Phonons in Pltonim. Mott transitionacross the actinide series.

Paling and Colom% Ti&6BS



Paling and Colom% Ti&6BS.

Poteryaev S. 8ichtenstein and GK PR8 +&''(/

Dynamical (oodeno%)h-;oni) a%lin)ict%re

&site@Clster DMFT $ith intersite Colom%&site@Clster DMFT $ith intersite Colom%



= L &* O L '.H* 4 L '.H

W L &' e@* 8T strctre

= L &* O L '.H* 4 L '.H

W L ' e@* ,T

strctre

&site Clster DMFT $ith intersite Colom%&site Clster DMFT $ith intersite Colom%

U/t=16,t’= +0.9



U/t=8, t’= -0.3

Density= 0.88, 0.89, 0.9, 0.91, 0.922,0.96, 0.986, 0.988, 0.989, 0.991,

0.993

,

=nderlying normal state

of the ,%%ard model

near the Mott transition*+force the 4eiss field to

its paramagnetic vale/*

TL' ED soltion of the

C@DMFT e0ations. M.

Civelli* M. Capone* 6.

Parcollet and GK



=QtLI [email protected] nL.H and t-L. nL.H



!nsights into the differences %et$een



!nsights into the differences %et$een

electron and hole doped cprates <

> t- U' has an nderlying normal state $ith

;P arond +piQ&* piQ&/. This is a state

$hich can natrally evolve into the d@$avespercondctor.

> t-Vo has the 0asiparticles arond +pi*'/*

does not connect smoothly $ith the SC.

What did we learn ? Schematic DMFT phase diagram



and DOS of a partially frustrated integer filled

Hubbard model and pressure driven Mott transition

correlated electrons

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