A New Two-Pole Approximation in the Hubbard Model. Metal-Ins

A new two-pole approximation, which allows to describe the transition from an insulating state to a metallic one at increase of bandwidth, and also the observable in some compounds transition from a metalic state to an insulating one with increasing temper

002 ebF 22 ]le-rts.tam-dnoc[ 1v4332000/tam-dnco:viXraphysicastatussolidi–1–RapidResearchNote

Anewtwo-poleapproximationintheHubbardmodel.Metal-insulatortransition

L.Didukh

TernopilStateTechnicalUniversity,DepartmentofPhysics56Rus’kaStr.,TernopilUA–46001,UkraineE-mail:didukh@tu.edu.te.ua(SubmittedDecember23,1999)

Two-poleapproachesintheHubbardmodel[1]andtheHubbardbandsconception(beingtheconsequenceofatwo-poleapproximation)havebeenusefulforunderstandingofthepeculiaritiesofelectricandmagneticpropertiesofnarrow-bandmaterials[1]-[13].Howeverwithintheframeworkoftwo-poleapproachesthereareseriesofissues,inparticulartheproblemofmetal-insulatortransitiondescription.

Inthispaperanewtwo-poleapproximation,whichallowstodescribethetransitionfromaninsulatingstatetoametalliconeatincreaseofbandwidth,andalsotheobservableinsomecompoundstransitionfromametalicstatetoaninsulatingonewithincreasingtemperature,ispresented.

Letustaketheoperatorsofcreationanddestructionofanelectronwithspinσ(σ=↓,↑)oni-siteintheform:

a++

iσ=diσ+hiσ,

aiσ=diσ+h+iσ,

(1)

whered++¯,diσ=aiσniσ¯,h+,hiσ=a+

projectioniσ=aiσniσ

theofelectronspinoppositeiσ=aiσ(1 niσ¯)¯denotestoσ.Betweenthed-h-operatorsiσ(1 niσ¯),andσandHubbardX-operators[14]directrelationsexist[15].TheHubbardHamiltonian[1]intermsofthed-h-operatorsiswrittenas

H=H0+H1+H1′

,

(2)

H0=

µ

2

d+iσdiσ,

(3)

H1=t

djσijσ,i d+ h+

iσiσhjσ=j

iσ

,

H1′

=t

d++

iσhjσ+h.c.,

(4)

ijσ,i =j

whereµisthechemicalpotential;Uistheintra-atomicCoulombrepulsion;tisthenearest-neighborhoppingintegral.H0describessystemintheatomiclimit,H1describeselectron

hoppingsbetweendoublyoccupiedsites(withtwoelectronsofoppositespins-doublons)andsingleoccupiedsites(the rstsuminH1)andelectronhoppingsbetweensingleoccupied

sitesandemptysites(holes)(thesecondsuminH1).H1′

describes“hybridization”betweenthe“h-band”and“d-band”(theprocessesofpaircreationandannihilationofholesanddoublons).ThestructureoftheHamiltonianoriginatestheapproximationgivenbelow.

TheGreenfunctions dp↑|d+and h++

s↑ p↑|ds↑ satisfytheequations:

(E+µ U) dp↑|d+s↑ =

n↑

A new two-pole approximation, which allows to describe the transition from an insulating state to a metallic one at increase of bandwidth, and also the observable in some compounds transition from a metalic state to an insulating one with increasing temper

physicastatussolidi

–2–

RapidResearchNote

+t

i

(h++ (1 n+

p↓hp↑di↓ hp↑dp↓hi↓p↓)di↑)|ds↑ ,

(6)

withnpσ=a+pσapσ,nσ= npσ anddp[↑A,|dB+] =AB BA.Toobtaintheclosedsystemof

equationsfortheGreenfunctions ++

s↑ and hp↑|ds↑ wesuppose

[d+p↑,H1] |ds↑ =

ε(pj) dj↑j

|d+s↑ ,

(7) [h+ |d+

p↑,H1]s↑ =

ε1(pj) hj↑j

|d+s↑ ,

(8)

whereε(pj),ε1(pj)arethenon-operatorexpressionswhicharecalculatedbythemethod

decribedinRef.[13].Atelectronconcentrationn=1(thisistheimportantsituationtostudymetal-insulatortransition)inaparamagneticstate(n↑=n↓)wehaveε(pj)=ε1(pj)=(1 2c)t,withcbeingtheconcentrationofpolarstates(holesordoublons).ForthemodelHamiltonianH0+H1thisapproximationleadstothecriterionofmetal-insulatortransition(U/2w)cr=1reproducingtheexactresultofRef.[16].

AmongtheGreenfunctionoriginatedfromtheexpressions [dp↑,H1′

] |procedure[h+1′

] of|d+d+s↑ andp↑,Htheses↑ weGreentakefunctionsintoaccountismadethe“diagonal”bymeansofGreenthemean- eldfunctiononly.approximation:

Thedecoupling n++1/2 h++

p↓hi↑|ds↑ i↑|ds↑ ,

(1 n+ 1/2 d++

p↓)di↑)|ds↑ i↑|ds↑ ;

(9)

bymakingtheseapproximationsweneglecttheprocessesdescribingthe“inter-band”hop-pingsofelectronswhichareconnectedwithspinturningoverand“inter-band”hoppingswith

creationorannihilationoftwoelectronsonthesamesite.Finally,ink-representationone-electronGreenfunctionis

aBp↑|a+k

s↑ k=

1

E E1(k)

+

2

U2+t2(k)insteadoftheexpression

A new two-pole approximation, which allows to describe the transition from an insulating state to a metallic one at increase of bandwidth, and also the observable in some compounds transition from a metalic state to an insulating one with increasing temper

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TheconcentrationofpolarstatesiscalculatedwiththehelpoftheGreenfunction

dp↑|d+s↑ k=

1

E E1(k)

+

Dk

+U

(15)43 4c

ifU/2w≤(U/2w)crwith(U/2w)cr=0.836(w=z|t|,zisthenumberofnearestneighborstoasite);forU/2w>(U/2w)crtheconcentrationofpolarstatesis

√ w+1

ln√c=

8ww+

U2+w2.(17)

vanisheswhenthecondition(U/2w)cr=0.836issatis ed.Thisvalueisclosetothere-sultof“Hubbard-III”approximation[17].Itisimportanttonotethatinthepointofgap

disappearancewehavec=0.

Energygap(17)istemperaturedependent.Thisallowstoexplainobservedinsomenarrow-bandmaterialstransitionfromametallictoaninsulatingstatewithincreasingtemperature(see,forexample[6]).

Abovewehaveconsideredthecaseofaparamagneticnarrow-bandmaterialathalf- lling(n=1).Itisinterestingtostudynon-hal lledcase(n=1);heretheenergyspectrumcanbees …… 此处隐藏：4231字，全部文档内容请下载后查看。喜欢就下载吧 ……