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Correlated Electrons
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7/18/2019 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 &''(
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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
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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.
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What did we learn ? Schematic DMFT phase diagram
and DOS of a partially frustrated integer filled
Hubbard model and pressure driven Mott transition
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!ressure driven Mott transition
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,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).
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!sing critical endpoint fondA !n &6B P.
8imelette et.al. +Science &''B/
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"nomalos transfer of optical spectral
$eight* 7iSeS. Miyasa)a and Ta)agi
&'''
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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
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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$ay from the high temperatreniversal region.
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> Mott transition across the Hf-s* a veryinteresting playgrond for stdyingcorrelated electron phenomena.
> DMFT ideas have %een e?tended into
a frame$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 $here the atomshave a closed shell +no entropy/.Realiation in "m nder pressre <
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DMFT Phonons in fccDMFT Phonons in fcc δδ-Pu: connect-Pu: connect
bonding to energy and length scales.bonding to energy and length scales.
C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa)
Theory 34.56 33.03 26.81 3.88
Experiment 36.28 33.59 26.3 4.8
( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)
(e!eriments "rom #ong et$al, Science, 22 A%g%st 2003)
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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 /<
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6ne dimensional ,%%ard model & site +8!7K/ CDMFT compare $ith ethe "nats* .
Kancharla C. olech and GK PR IJ* 'JH' +&''B/M.CaponeM.Civelli Kancharla
C.Castellani and GK P. R 69*H'H +&''(/
U/t=4.
" rapidly convergent algorithm <
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8in)s* Ti&6B # Colom% and
Paling
C.E.Rice et all, Acta Cryst B33, 1!" #1$%%&
LTS 250 K, HTS 750 K.
E l ti f th ) l d S t l
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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
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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
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"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
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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 $ithphenomenological approaches* dopedsemicondctors +hatt and Sachdev/* heavyfermions * H-s+7a)atsNi* Pines and Fis) /33
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Mott transition into an open +right/ and closed +left/ shell
systems. !n single site DMFT* spercondctivity mst intervene
%efore reaching the Mott inslating state.Capone et. al. AmAt room pressure a loalise! "#6 system$%="/2. & = -' = 3( )
= 0 apply pressure *
S S
= =
.γ T8og&O:
=c
γ Q+=c@=/
SL'
<<<
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"mericim nder pressre O.C.
Grivea? O. Re%iant G. 8ander
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Evoltion of the Spectral Fnction $ith
Temperatre
%noma&o# tran#er o #petra& *ei+ht onnete, to the
proximity to the -#in+ ott en,point (/ot&iar an+e n,
oener+Phys. Rev. Lett. 84, 5180 (2000)
" ti l ti i ti
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"ns$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 $ith single site
DMFT in many many many materials.> Pla0ette e0ations are one order of
magnitde harder to solve.
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Total Energy as a function of 'olu(e for Total Energy as a function of 'olu(e for
PuPu 4 +ev/ vs +a.. &J.& ev/
(Savrasov, Kotliar, Abrahams, 'at%re ( 200)
'on magnetic correlated state o" "cc %$
iw
ein Savrasov and Kotliar +&''(/
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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$al, Science, 22 A%g%st 2003)
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Epsilon Pltonim.
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Phonon entropy drives the
epsilon delta phase transition> Epsilon is slightly more delocalied than delta* has
SM"88ER volme and lies at ,!G,ER energy than
delta at TL'. t it has a mch larger phonon entropy
than delta.
> "t the phase transition the volme shrin)s %t the
phonon entropy increases.
> Estimates of the phase transition follo$ing Drmont
and G. "c)land et. al. PRB."# , 184104 (2002); +and
neglecting electronic entropy/. TC I'' K.
ransverse onon a ong
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ransverse onon a ong+'**/ in epsilon P in self
consistent orn appro?imation.
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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$%="/2. & = -' = 3( )
= 0 apply pressure *
S S
= =
.γ T8og&O:
=c
γ Q+=c@=/
SL'
<<<
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"mericim nder pressre O.C.
Grivea? O. Re%iant G. 8ander
erie o# ro p
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erie o# ro p,
o# Am
> 7ote strongly increasingresistivity as f+p/ at all T.Sho$s that moreelectrons are enteringthe condction %and
> Spercondcting at allpressre
> !ariation of rho vs. T forincreasing p.
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DMFT stdy in the fcc strctre. S.
Mrthy and G. Kotliar
fcc
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8D":DMFT spectra. 7otice the
rapid occpation of the fJQ& %and.
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6ne electron spectra. E?periments +7egele/ and 8D":DFT
theory +S. Mrthy and GK /
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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 <#
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,.;. 5an et. al. CeC&+Si&@? Ge?/.
"m nder pressre Grivea et. al.
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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
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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
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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.
* * * *+ +
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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!.
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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
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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$.
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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.
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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
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!mprity model representa%ility of
spectral density fnctional.
R h di f th C t
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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(& +/
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> ,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
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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
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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
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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
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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
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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)
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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
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"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)
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* .l!ri!4e, )., :ornelsen, :.,>an4, ?.,>illiams, ).,
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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
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"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.
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(ET) F tt t iti
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-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"
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. *ond-4at/###&&5 Phys. Rev. Lett. 85, 5420 (2000)
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> Theoretical isse# is there a Mott transition
in the integer filled ,%%ard model* and is it
$ell descri%ed %y the single site DMFT <
Evoltion of the spectral
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Evoltion of the spectral
fnction at lo$ 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.
$
$
2 2
$
E$Ct($)Ae ( ! 0)
C -m ( ! 0)
( ! 0)E$
&
&
A &
ω µ
γ ω
γ ω
γ
Σ = −
Σ =
= =+
"pproaching the Mott transition#
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"pproaching the Mott transition#
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 $eapproched the inslator
Mechanism for hot spot formation# nn
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Mechanism for hot spot formation# nn
self energy A General phenomena.
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Conclsion.
> Mott transition srvives in the clster setting.
Role of magnetic frstration.
> Srprising reslt# formation of hot and cold
regions as a reslt of an approach to theMott transition. General reslt <
> =ne?pected role of the ne?t nearest
neigh%or self energy. CDMFT a ne$ $indo$to e?tend DMFT to lo$er temperatres.
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Conclsion
> DMFT mapping onto 1self consistent impritymodels2 offer a ne$ 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
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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%
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= 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 $ith intersite Colom%
U/t=16,t’= +0.9
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U/t=8, t’= -0.3
Density= 0.88, 0.89, 0.9, 0.91, 0.922,0.96, 0.986, 0.988, 0.989, 0.991,
0.993
,
=nderlying normal state
of the ,%%ard model
near the Mott transition*+force the 4eiss field to
its paramagnetic vale/*
TL' ED soltion of the
C@DMFT e0ations. M.
Civelli* M. Capone* 6.
Parcollet and GK
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=QtLI [email protected] nL.H and t-L. nL.H
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!nsights into the differences %et$een
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!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
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and DOS of a partially frustrated integer filled
Hubbard model and pressure driven Mott transition