Uogólnione modele t-J w związkach metali przejściowych

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JagellonianUniversity

Beyond the standard

t

J

model

Krzysztof Wohlfeld

A thesis, written under the

su-pervision of Prof. Dr. Andrzej

M.Ole±, presented inpart

full-mentof therequirementsforthe

degree of Do tor of Philosophy

inthe JagellonianUniversity

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Listof abbreviationsand remarkson notation 5

Prefa e 7

1 Motivation: The standard

t

J

model 11

1.1 TheHubbardmodel . . . 11

1.2 The anoni alperturbationexpansion . . . 12

1.3 Thestandard

t

J

Hamiltonian . . . 15

2 Explaining harge order in Sr

14−x

Ca

x

Cu

24

O

41

19 2.1 Introdu tion. . . 19

2.2 The

t

J

modelfor oupledladders . . . 22

2.3 Themodel. . . 24

2.3.1 The

t

J

V1

V2

Hamiltonian. . . 24

2.3.2 Thesuperex hangeterm . . . 26

2.3.3 Thekineti energyterm . . . 27

2.3.4 Theintraladderrepulsiveterm

V1

. . . 30

2.3.5 Theinterladderrepulsiveterm

V2

. . . 34

2.4 Methodandresults. . . 36

2.4.1 Theslave-bosonapproa h . . . 36

2.4.2 Themean-eldapproximation. . . 38

2.4.3 Thegroundstateproperties . . . 40

2.5 Dis ussion . . . 42

2.5.1 Validityoftheresults . . . 42

2.5.2 `Rigidity'oftheZhang-Ri esinglets . . . 43

2.5.3 RungstatesorZhang-Ri esinglets . . . 48

2.6 Con lusions . . . 51

2.7 Posts riptum: destabilizingeven-period-CDWstateinatoy-model 53 3 Verifyingthe idea oforbitally indu edholelo alization 55 3.1 Introdu tion. . . 55

3.2 The

t2g

orbital

t

J

modelwiththree-siteterms . . . 59

3.3.1 The

t2g

orbital

t

J

3.3.3 TheIsingsuperex hangeterm. . . 63

3.3.4 Thethree-siteterms . . . 64

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3.4.2 Theself- onsistentBornapproximation . . . 67

3.4.3 Thespe tralfun tions andquasiparti leproperties . . . . 69

3.5 Dis ussion . . . 71

3.5.2 Understandingthe1Ddispersion . . . 74

3.5.3 Renormalizationofthethree-siteterms . . . 79

3.7 Posts riptum: photoemissionspe traofvanadatesanduorides . 84 4 Understanding hole motionin LaVO

3

89 4.1 Introdu tion. . . 89

4.2 The

t2g

spin-orbital

t

J

modelwiththree-siteterms . . . 92

4.3.1 The

t2g

spin-orbital

t

J

4.3.3 Thespin-orbitalsuperex hangeterms . . . 95

4.3.4 Thethree-siteterms . . . 96

4.4 Methodand results. . . 98

4.4.1 Theslave-fermionapproa h . . . 98

4.4.2 Theself- onsistentBornapproximation . . . 102

4.4.3 Thespe tralfun tions andquasiparti leproperties . . . . 105

4.5 Dis ussion . . . 110

4.5.2 Theroleof thejointspin-orbitaldynami s. . . 112

4.5.3 Suppressionofquantumu tuations . . . 115

4.7 Posts riptum: spin,orbitalandspin-orbitalpolarons . . . 119

Summary 125

A The ontinued fra tion methodforthe 1Dorbital model 129

B The ee tive polaronmodel foruorides 133

Bibliography 144

Stresz zenie 145

The author's list of publi ations 147

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remarks on notation 1Done-dimensional 2Dtwo-dimensional 3Dthree-dimensional AFantiferromagneti AOalternatingorbital

CDW hargedensitywave

FMferromagneti

FOferro-orbital

LOW linearorbitalwave

LSWlinearspinwave

SCBAself- onsistentBornapproximation

VCAvariational luster approa h

Throughoutthethesis:

(i)weuse

H

(possiblywith someindi es) to denote anytypeof the Hubbard Hamiltonian,

(ii) we use

H

(possibly with some indi es) to denote any omponent of the (standardorextended)

t

J

model,

(iii) we use

H

ef f

(possibly with some indi es) to denote any omponent of

the ee tive model obtained from the (standard or extended)

t

J

model by introdu ingslavefermionsorslavebosons,

(iv) the main Hamiltonians of the hapters (Hubbard,

t

J

, and possibly the ee tiveone)donothaveanyindex,

(v)thelatti e onstantissetto unity,

(vi)

P

hiji

meanstakingsummationoverthebondformedbetweensite

i

and

j

. Despite theabovementioned ommonfeatures of thenotation used in the

thesisthenotationin ea h hapteris independentoftheother haptersandis

logi ally onsistentonlywithinea h hapter.

We all the spin

t

J

model of Refs. [1, 2, 3℄the standard

t

J

model [see Eq. (1.22) in this thesis℄to distinguishit from variousother

t

J

typemodels dis ussedinthisthesis.

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Inthisthesiswedis ussandpresentsolutionsof threerelatedproblems whi h

ariseinstrongly orrelatedele tronsystems:

1. Explaining hargeorderinSr

14−x

Ca

x

Cu

24

O

41

.Therstproblem on- erns the explanationof the pe uliar hargeorder observed experimentallyat

lowtemperature

T = 20

Kin the oupledladdersCu

2

O

5

in Sr

14−x

Ca

x

Cu

24

O

41

[4,5,6, 7,8℄. On theonehand,theresonantsoft x-rays attering showsthat

the hargeorderthere is formed bya harge density wave(CDW) phasewith

oddperiodandisstablefor

x = 0

and

x = 11

inSr

14−x

Ca

x

Cu

24

O

41

presumably dueto theon-site Coulombrepulsion [7,8℄. Ontheotherhand,aCDW phase

withevenperiodhasnotbeenobservedinthesesystems[8℄. Thesearestriking

resultsasthey ontradi tthetheoreti alpredi tionofastableCDWphasewith

evenperiodfor

x = 4

andnoCDWorderforothervaluesof

x

[9,10,11℄. 2. Verifyingthe ideaoforbitallyindu edholelo alization.Thenext

prob-lem is more general and `tou hes' the idea that the mere presen e of orbital

degenera yin thetransitionmetaloxides ouldleadtothehole onnementin

thestrongly orrelatedele tronsystem. Thisidea anbeba kedbythefollowing

fa ts: (i)themanganitesshowa olossalmagnetoresistiveee t [12,13,14,15℄

whi h anbeattributedtotheorbitaldegenera y[16,17,18℄,(ii)thetransition

metaloxideswithorbitaldegenera y(e.g. manganitesorvanadates)havemu h

more stable insulating phases in the regime of hole doping [15, 19℄ than the

uprateswithout orbitaldegenera y [20℄. However,in strongly orrelated

sys-temswithoutorbitaldegenera y(anddes ribedbythesimpleHubbardmodel)

the hole had beenthought to be lo alized for a very long time [21℄ and only

mu hlater[22,23℄itwasshownthattheholewasmobile. Thissuggeststhatthe

veri ationoftheideaoforbitallyindu edholelo alizationshouldbeperformed

rather arefully.

3. Understandinghole motion inLaVO

3

.Thelast problemisdevotedto theunderstandingofthebehaviourofthesingleholedopedintothe

ab

planeof LaVO

3

. ThissystemisaMottinsulatorandsuperex hangeintera tionsstabilize the spinantiferromagneti (AF) and alternating orbital (AO) ordered ground

state [19, 24, 25℄. The problem whi h arises here anbe in short formulated

asfollows: upondopingthisplanewithholes(whi hispossiblebysubstituting

lantanium forstrontiumin La

1−x

Sr

x

VO

3

)theorbital dynami sseemsto inu-en etheholemotionmu hmorethanthespindynami s(see onje turein the

Introdu tiontoChapter5ofthisthesisbasedontheexperimentalresultsfrom

Ref. [19℄). Thus, the question is: why the spin dynami s is quen hed in the

holedopedAFandAOstate.

Common feature of the three problems. Although all of the three topi s

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them. 1

However,a loserlook(takeninthe onse utive hapters)willshowthat

thethreesimplestmodels,formulatedtosolvetheseproblems,willhavealotin

ommon. A tually,allthree of themwill turnoutto bemerelyamoreorless

elaborateversionofthestandard

t

J

model[1,2,3℄althoughthestandard

t

J

modelitselfwillbe omeevidentnottobeenoughtoexplainthesephenomena.

Morepre isely, itwill turn outthatthe simplestmodels apableof explaining

theaboveproblems will be: (i)the

t

J

modelfor oupled laddersforthe rst problem,(ii)the

t2g

orbital

t

J

modelwiththree-sitetermsforthese ondone and(iii) the

t2g

spin-orbital

t

J

modelwiththree-sitetermsforthethird one. Thus,wewillshowthat,asthetitleofthethesissuggests,oneindeedhastogo

beyondthe standard

t

J

modeltobeableto understandthephysi sbehindall thesethree phenomena.

Aim of the thesis. The purpose of this thesis is to give answers to the

threeproblems usingtheabovementionedextensionsofthe

t

J

models. As`a side ee t'onewill see how powerfulis the on ept ofthe

t

J

model and the anoni alperturbationexpansion[1,2℄ortheZhang-Ri e s heme[26℄: merely

slightmodi ations of the model mean that it is still apable of explaining a

hugevarietyofphenomenapresentinthetransitionmetaloxides.

Stru ture of the thesis. The thesis is organized as follows. Chapter 1

ontains a preliminary material on erning the standard

t

J

model: (i) the Hubbard model, (ii) its derivation from the Hubbard model by the anoni al

perturbation expansion, and nally (iii) its form and range of appli ability.

This hapter may be easily skipped by the reader familiar with the standard

t

J

model [1, 2, 3℄, though a qui k look at this hapter would be always of great help in understanding the results presented in this thesis. Next in the

three onse utive hapters (whi h are alled the main hapters of the thesis)

wedis uss the three problems mentioned above: (i) in Chapter 2we explain

the hargeorderinSr

14−x

Ca

x

Cu

24

O

41

usingthe

t

J

modelfor oupledladders, (ii)in Chapter3weverifytheideaof orbitallyindu edholelo alization using

the

t2g

orbital

t

J

model with three-siteterms, and (iii) in Chapter 4 we try to understand holemotion in LaVO

3

using the the

t2g

spin-orbital

t

J

model with three-siteterms. Finally, in Summarywebriey dis uss thesolutions of

the problems and the ommon features of thenew

t

J

models. The thesis is supplementedbytwoappendi es(whi h ontainsomemathemati alderivation

neededinChapter3),Bibliography,`Stresz zenie'(summaryinPolish),andthe

listofpubli ationswhi hwerepublishedduringmyPhDstudies. Finally,inthe

end wementionthosepeoplewithoutwhose supportit wouldhaveneverbeen

possibleto omplete thisthesis.

The organization of material serves the main idea of the thesis. First, in

ea hof the three main hapters: (i) we dis uss theproblem in moredetailin

the introdu tion(rst se tion), (ii) we introdu e the new

t

J

model by are-fully dis ussingitsdieren es withrespe t tothe standard

t

J

model (se ond se tion),(iii)wederivethenew

t

J

modelfromtheHubbard-typemodel appro-priate for the onsidered problem using the anoni alperturbation expansion

[1,2℄or theZhang-Ri es heme[26℄(thirdse tion). Se ond,asthemethods of

solvingea h

t

J

modeldier,weintrodu etheslavebosons(Chapter2)orslave 1

Although,thereaderfamiliarwiththestrongly orrelatedele tronsystemswill

immedi-atelynotethatthe se ondand thirdproblemhas alotin ommon. SeealsoSe . 4.7fora

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presentinany

t

J

modelandonlythenwesolvetheee tivemodelwrittenin theslaveparti le languageusing themean-eldin Chapter2orself- onsistent

Born approximation(SCBA) in Chapter 3and 4(fourthse tion). Finally, we

dis usstheresultsin ludingitsvalidity(fthse tion),andwedrawsome

on- lusions(sixthse tion). Furthermore,ea hmain hapteris supplementedbya

Posts riptum (seventh se tion)inwhi h wedis uss somesideissueswhi hare

interestingbutarenot entral forthemainmessageand anbeeasily skipped

in rst reading. We would like to stress that the ability to build a ommon

stru tureofthethree main haptersree ts(pra ti ally)theabovementioned

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Motivation: The standard

t

J

model

1.1 The Hubbard model

Hamiltonian. The (ar he)typi al model whi h des ribes the strongly

orre-latedele tronsis theHubbardmodeldes ribedbytheHamiltonian[27℄

H = −t

X

hiji,σ

c

†

_iσ

cjσ

+ H.c.

+ U

X

i

ni

↑

ni

↓

,

(1.1)

where

hiji

denotes the bond formed betweensite

i

and

j

,

c

†

iσ

operator reates an ele tronat site

i

with spin

σ

, and the ele trondensity operator is dened as

niσ

= c

†

iσ

ciσ

. Here the rst term is responsible for the hopping

∝ t

of ele tronsonahyper ubi latti ewhilethese ondtermdes ribestheCoulomb

repulsion

∝ U

betweentwoele tronswithoppositespinsonthesamesite. This modelisintrodu edtodes ribea ommonsituationwhi htakespla einvarious

transition metaloxides[20℄: thelatti epotentialisverystrongand oneneeds

to al ulate theCoulombintera tion betweenele tronsin the(almost)atomi

wavefun tions. Thisleadstoamodi ationofthebareCoulombpotential: itis

short range(i.e. merelyon-site) but stronglyamplied. This naturallymeans

that thephysi al regime ofthe model iswhen

U > W

(where

W = 2zt

is the bandwidth and

z

is the oordination number for the hyper ubi latti e) and throughoutthethesiswewill assumethatoneisalwaysinthisregime.

A tuallythemoregeneraldenitionoftheHubbardmodel(1.1)would

on-tain the hemi al potential. However, it is ustomaryto omit that term and

instead to spe ify the number of ele trons per site

n

present in the system separately. This antakethevalues

0 ≤ n ≤ 2

due tothePauliprin iple.

Spa e dimensions of the latti e. Finally, let us note that the model Eq.

(1.1) anbedenedaswellintheone-dimensional(1D),two-dimensional(2D)

and three-dimensional (3D) version. However, due to its mostinteresting (in

myopinion)appli ation on ernsthe2D opperoxidelayersofhigh-

Tc

uprates [28℄. Moreover,aswewillbeinterestedeitherinlayeredstru tures(Chapter2)

orin situations where theorbital order (Chapter 3) or spinand orbital order

(Chapter 4) ontainstwospatial dimensions, we restri tthe dis ussionto the

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modelEq. (1.1)hasbeenindeedverysu essfulindes ribingvariousproperties

of the strongly orrelated ele tronsystems[28℄. However,there are twomain

drawba ksofthemodel. First,despiteitssimpli ityitishardtosolveitinthe

interestingregime

n 6= 1

astheMonteCarlosimulationsoftenbreakdowndue tothe`signproblem'whereasallothermethodsarealsonotreliableduetothe

hugedimensionsoftheHilbertspa eofthemodel(whi hin thehalf-lled ase

is

[N !/(N/2)!(N/2)!]

2

where

N

isthenumberoflatti esites)[28℄. Se ond,letus remarkthatmanysystemsaretoo ompli atedtohavetheele tron orrelations

des ribedbytheHubbardmodel inareliableway: e.g. theorbital degenera y

regime an hangethemattersdrasti ally[29℄.

On the one hand,to over ome the rst di ulty oneperforms the

anon-i al perturbation expansion 1

of the Hubbard model whi h hugely redu es the

dimensionalityoftheHilbert spa ebynegle tingthehigh-energystatesin the

regime

U > W

. This is done in the next two se tions and the model whi h is obtainedafter su h an expansionis the standard

t

J

model. On the other hand,oneshould add extratermsand/or modify thetwoexisting onesin Eq.

(1.1) to make the Hubbard model more realisti . A tually, in the nextthree

hapters ofthisthesiswewill ombine bothof theapproa hes: wewillmodify

theHubbard modelto makeitmorerealisti and redu e itto theappropriate

t

J

modelusingthe anoni alperturbationexpansion.

1.2 The anoni al perturbation expansion

Hubbard subbands.One of the main features of the model (1.1) is the split

of theHilbert spa e(spannedby theHubbardHamiltonian)into theso- alled

Hubbardsubbands[1,2,31℄. This anbeunderstoodinthefollowingway. Let

us assumethat

n ≤ 1

(the ase

n > 1

followsfrom theparti le-holesymmetry ofthemodel)andswit hothehopping

t = 0

foramoment. Thentheground state of the model will learly have no sites with two ele trons as ea h site

o upied by two ele trons osts energy

U

. This ondition denes the lowest Hubbardsubbandwithzerototalenergywhi h onsistsofall(degenerate)states

with no double o upan ies. Next, all of the states with just one single site

o upied by twoele trons (and the rest singly o upied or empty)dene the

se ond Hubbardsubbands with thetotalenergy

U

. Repeatingthis pro edure further,onesplitsuptheHilbert spa eintotheHubbardsubbandsspannedby

thestateswith

m

doublyo upiedsitesandenergy

mU

.

Swit hingonhopping

t

obviously hangesthesituation: notonlythestates withintheHubbardsubbandarenolongerdegeneratebutmoreimportantlythe

HubbardHamiltonianarenolonger`diagonalin theHubbardsubbands'(more

pre isely thehopping

t

onne ts thestatesfromdierentHubbardsubbands). However, as longas

W < U

the Hubbard subbands do not overlap, in order to obtain the behaviour of the systemin thelowenergy limit it is enoughto

on entrateonthelowestHubbardsubbandandtreatthehoppingtothestates

fromhigherHubbardsubbandsasaperturbation.

1

Notethatthe morestandard perturbation expansionofthe Hubbardmodel[30℄,where

the entirehoppingtermistreatedasa smallperturbation,isvery tediousforthe Hubbard

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turbation expansionsets the above des ribed pro edure on the mathemati al

grounds[1, 2℄(see alsoRefs. [32, 33℄). Inthebeginningonerewritesthe

Hub-bardHamiltonian

H

in thefollowingway:

H = H

0 + H

1,

(1.2)

where

H0

des ribesthephysi swithin theHubbardsubband(

σ = −σ

¯

):

H

0 = V + T

0,

V = U

X

i

ni

↑

ni

↓

T

0 = −t

X

hiji,σ

n

(1 − n

i¯

σ)c

†

iσ

cjσ

(1 − n

j¯

σ) + ni¯

σc

†

iσ

cjσnj¯

σ

+ H.c.

o

,

(1.3)

while

H

1

is responsible forhopping pro essesbetweendierentHubbard sub-bands:

H1

= T+

+ T−

,

T+

= −t

X

hiji,σ

n

ni¯

σc

†

_iσ

cjσ(1 − n

j¯

σ) + H.c.

o

T−

= −t

X

hiji,σ

n

(1 − n

i¯

σ)c

†

_iσ

cjσnj¯

σ

+ H.c.

o

.

(1.4)

Next,thetaskisto onstru ta anoni altransformation

S

ofthe Hamilto-nian

H

˜

H = e

S

He

−S

,

(1.5)

where

S

†

_{= −S}

. If

H

˜

is al ulatedfrom the above equationexa tly then the unitarity of this transformation would mean that the observables al ulated

usingthespe trumspannedby

H

˜

willbeidenti altotheones al ulatedusing thespe trumspannedby

H

.

The expli it form of

S

is al ulated from the single requirement that the Hamiltonian

H

˜

wouldnot onne tstatesfromtwodierentHubbardsubbands. A priori this an always be done as long as the Hubbard subbands do not

overlap, i.e. when

W < U

(whi h is the ase here). Obviously, this means that the observables al ulated using the spe trum spanned by

H

˜

will not be identi al to the ones al ulated using the spe trum spanned by

H

. However, thebiggerdistan esonehasbetweentheHubbardsubbands,the moresimilar

theobservablesare. Expli itlyone al ulates

H

˜

and

S

usingthefollowingsteps ( ompareRef. [33℄):

(i)OnemakestheAnsatzthat

S

isoftheorderof

t/U

sothatone anwrite

e

S

_{= 1 + S +}

1

2 S

2 _{+ O}

t

3 U

3 .

(1.6)

Sin e

t ≪ U

thetermsoftheorder

O(

t

3 U

3 )

shouldbemu hsmallerthan

1

(e.g.

U = 12t

in thehigh-

Tc

uprates[28℄ yields

t

3 U

3

smallerthan

10 −3

) and anbe

skipped. ThenEq. (1.5) anberewrittenusingEq. (1.6)as

˜

H = H + [S, H] +

1 ₂

[S, [S, H]] + O

t

3 U

2 ,

(1.7)

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to the order

O(

t

3 U

2 )

(whi h is again enough in the regime

t ≪ U

) sin e

H

is (maximally)oftheorderof

U

.

(ii)Letusrst al ulate

S

torstorderin

t/U

[

S

(1)

℄. ThenEq. (1.7)tothe

order

O(

t

2 U

)

is

˜

H

(1)

_{= H + [S}

(1)

_{, H].}

(1.8)

Now, onedemandsthat

H

1

isnotpresentin

H

˜

(1)

: this isdue tothefa t that

with one hop one leaves the Hubbard subband under onsideration and one

prohibitsthat

H

˜

inanyorderdes ribespro esseswhi h ouplevariousHubbard subbands. Theninthisorderoneneedsto have:

T

+

+ T−

+ [S

(1)

, H] ≡ 0.

(1.9)

However,

T

+

+ T−

is

∝ t

while

S

(1)

is

∝ t/U

. Thusone anonlyhave

V

in the ommutator:

[S

(1)

_{, V] = −T+}

_{+ T−}

_.

(1.10)

One an he kthat:

S

(1)

=

1 U

(T+

− T−

),

(1.11)

fullls Eq. (1.10).

(iii) Havingdeterminded

S

to rstorderin

t/U

[

S

(1)

℄one annowpro eed

furtherand al ulate

S

to these ondorder[

S

(2)

℄. For onvenien eone denes

S

′

S

(2)

= S

(1)

+ S

′

.

(1.12)

Then

S

′

is al ulatedfrom[ ompareEq. (1.8)℄:

˜

H

(2)

_{= H0}

_{+ [S}

(1)

_{, T+}

_{+ T−}

_{] + [S}

(1)

_{, T0] +}

1

2 [S

(1)

_{, [S}

(1)

_{, V]] + [S}

′

_{, V],}

(1.13)

where we usedthe substitution

[S

′

_{, H] → [S}

′

_{, V]}

similarly aswhen goingfrom

Eq. (1.9)toEq. (1.10). NextusingEq. (1.11)weredu e Eq. (1.13)to

˜

H

(2)

= H0

+

1 U

[T+, T−

] + [S

(1)

, T0] + [S

′

, V].

(1.14)

However,the term

[S

(1)

_{, T}

_0]

is notallowedto appear in

H

˜

(2)

be auseit is

re-sponsiblefortransitionsbetweenHubbardsubbandsandoneprohibitsthat

H

˜

inanyorderdes ribespro essesbetweenvariousHubbardsubbands. Thusone

needstohave

[S

(1)

_{, T0] + [S}

′

_{, V] ≡ 0,}

(1.15)

whi hdenes

S

′

. From this equationone an al ulate

S

′

howeveritis not

needed(seebelow).

(iv)Todetermine

H

(2)

oneneedsonlytheexpli itformof

S

(1)

. Infa t,itis

straightforwardto determineitby substitutingEq. (1.15)to Eq. (1.14). One

obtains

˜

H

(2)

= H

0 +

1 U

[T

+

, T−

].

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in

t/U

andone anwrite

˜

H = T0

+ V +

1 U

[T+, T−

] + O

t

3 U

2 .

(1.17)

Hamiltonian for the lowest Hubbard subband. If one is interested in

H

˜

des ribing merelythe lowest Hubbard subband in the ase

n ≤ 1

( alled

H

), one anskip

T+T−

and

V

termsin Eq. (1.17) andonearrivesat

H = T0

−

_U

1 T−T+

+ O

t

3 U

2 .

(1.18)

One annowplugintheexpli itformsof

T

0

,

T

+

,and

T−

toobtaintheexpli it formof

H

. Thisisdoneinthenextse tion. Notethat duetotheparti le-hole symmetryasimilarHamiltonianaswrittenabovedes ribesthe ase

n > 1

.

1.3 The standard

t

J

Hamiltonian

Expli it form. After insertingEqs. (1.3-1.4)into Eq. (1.18) oneobtainsthe

expli it form of the ee tive low-energy Hamiltonian for the lowest Hubbard

subband

H = − t

X

hiji,σ

n

(1 − n

i¯

σ)c

†

_iσ

cjσ

(1 − n

j¯

σ

) + H.c.

o

−

1 ₄

J

X

hhmijii,σ,σ

′

n

(1 − n

m¯

σ

′

)c

†

_mσ

′

ciσ

′

ni¯

_σ

′

ni¯

σc

†

_iσ

cjσ(1 − n

_j¯

σ) + H.c.

o

,

(1.19)

where

hhmijii

meansthepathsbuiltof thethreenearestneighboursites. Here the rst term is responsible for hopping within the lowest Hubbard subband

while the se ond term, whi h arises from the virtual hoppings to the upper

Hubbardsubband,istheso- alledsuperex hangebterm 2

withtheenergys ale

J = 4t

2 _/U

.

Equation(1.19) anbesimplied byrepla ingtheele tronoperatorsin the

superex hangetermbythe

S = 1/2

spinoperators:

S

i

z

=

1

2 (˜

ni

↑

− ˜n

i

↓

),

S

_i

+

=˜

c

†

_i

_↑

ci

˜

_↓

,

S

_i

−

=˜

c

†

_i

_↓

ci

˜

_↑

,

(1.20) wherewedened the onstrainedele tronoperators

˜

c

†

_iσ

= c

†

iσ(1 − n

i¯

σ).

(1.21)

2

Notethatweuseherethe term`superex hange'insteadofthe moreproper`kineti

(16)

Thenoneobtainsthe2Dversionofthestandard

t

J

Hamiltonian[1, 2,3℄

H = −t

X

hiji,σ

(˜

c

†

_iσ

˜

cjσ

+ H.c.) + J

X

hiji

Si

_{· S}

j

−

1

4 ˜

ni˜

nj

,

(1.22) where

ni

˜

= ˜

c

†

i

_↑

˜

ci

↑

+ ˜

c

†

i

_↓

˜

ci

↓

andweassumed that

m

= j

in Eq. (1.19). The 1D and 3Dversionof thestandard

t

J

Hamiltonian follow in anaturalwayfrom theaboveequation.

The kineti and superex hange terms. The rst term

∝ t

des ribes the hoppingofele tronsinthe onstrainedHilbertspa ewithnodoubleo upan ies

(i.e. the lowest Hubbard subband). Thus, it an be viewed as an ee tive

hoppingofholesassu hahoppingofele tronsispossibleonlyifthereisahole

atthesitetowhi htheele tronhops. Notethattheoperators

c

˜

†

iσ

donotfulll the fermioni ommutation rules [32℄. Thus one annot treatthese obje tsas

ele tronsande.g. one annotintrodu etheFermienergyormomentuminthis

ase. Therefore,even withoutthe se ond term(as obtainedfor

U → ∞

),Eq. (1.22) onstitutesanontrivialproblem.

These ondterm

∝ J

des ribestheintera tionbetweenthespinswhi hisof theAF hara tersin e

J > 0

. Themeaningof thisterm anbeeasilyseenin thehalf-lled ase(

n = 1

)whenEq. (1.22)redu estotheHeisenberg Hamilto-nian sin ethenthere arenoholesin thesystemandthekineti termdoesnot

ontribute. Thusinsteadofhavingstrongly orrelatedele trons,see Eq. (1.1),

oneisleftwithintera tingspindegreesoffreedomasthe hargedegreesof

free-domareintegratedout. Thisstrikingresultmeansthattheintera tionsareso

stronginthis ase(dueto

U > W

intheHubbardmodel)thattheele tronsare lo alized ( hargedegreesoffreedom arefrozen) and onlythevirtualhoppings

ofele trons(des ribedby

T+

and

T−

pro esses)leadto a`residual'intera tion between ele tronspins. Thisis the physi al explanation of the anoni al

per-turbationexpansion. Notealso,thatnaturallythedimensionalityoftheHilbert

spa eis nowredu ed: e.g. inthehalf-lled asethereare onlyspindegreesof

freedomandthedimensionoftheHilbert spa eis

2 N

.

The three-site terms. Theassumption

m

= j

needs further explanation. It means that the ele tron, whi h is virtually ex ited to the upper Hubbard

subband by

T

+

pro ess, returns (by the

T−

pro ess) to the same site from whereitwasex itedinthelowest Hubbardband. Thus,oneomitsherethe

so- alledthree-siteterms. These ontributemerelyifthere areholesinthelowest

Hubbard band sin e the ele tron ex ited from site

j

in the lowest Hubbard subband anreturn to adierentsite

m

in thelowest Hubbardsubband only when there is ahole on site

m

(be auseotherwise adouble o upan y would be reatedwhi hisprohibitedinthelowestHubbardsubband). Thussimilarly

as the kineti term

∝ t

in Eq. (1.22) the three-site terms will des ribe the hopping of holesin the lowest Hubbard subband. However,unlike thekineti

termtheys aleas

∝ J

. Thus, altogetherthethree-siteterms ontributetothe totalenergyofthesystemas

∝ Jδ

where

δ

isthenumberofholesinthesystem. If

δ ≪ 1

(whi h is thetypi alregime for the

t

J

model) and sin e

J < t

(as

t ≪ U

), thenthis ontributiontothetotalenergyis verysmall. In parti ular, itismu hsmallerthanboththe ontributionofthekineti term

∝ tδ

andthe superex hangeterm

∝ J(1 − δ)

2

.

Appli ation. Theappli ationofthe

t

J

modelfollowsfrom twofa ts: (i) asshown above,in thelow energybut strongly orrelated regime,it des ribes

(17)

easierto solvethantheHubbardmodel sin ethedimensionalityof itsHilbert

spa eis onsiderablyredu edin omparisonwiththeoneoftheHubbardmodel.

Thelatterproperty meansthat: (i) allthenumeri al al ulations,su h asthe

Lan zosor exa tdiagonalizationte hniques aremoreeasily done, and(ii) the

spinsaremu heasiertotreatanalyti allyasthegroundstatesofthespinmodels

are typi ally more lassi al [33℄. Consequentlythere havebeena tremendous

numberofpapersonthe

t

J

model,itssolutions,andappli ations. Forfurther detailswereferto thereview arti lesofRef. [28℄or[20℄orto Ref. [32℄ forthe

(18)

(19)

Explaining harge order in

Sr

14 −x

Ca

x

Cu

24

O

41

This hapterisbasedonthefollowingpubli ations: (i)K.Wohlfeld,`DopedSpin

Ladder: Zhang-Ri eSingletsorRung- entredHoles?',AIPConferen e

Pro eed-ings918,337-341(2007);(ii)K.Wohlfeld,A.M.Ole±,G.A.Sawatzky,`Origin

of hargedensitywaveinthe oupledspinladdersofSr

14−x

Ca

x

Cu

24

O

41

', Phys-i al ReviewB 75,180501(R)/1-4 (2007); (iii)K. Wohlfeld, A.M. Ole±,G. A.

Sawatzky,`Thet-J-VModelforCoupledLadders',inpreparationtobesubmitted

toPhysi al ReviewB (Rapid Communi ation).

2.1 Introdu tion

Crystalstru tureofSr

14−x

Ca

x

Cu

24

O

41

.Thetelephonenumber ompound,as Sr

14−x

Ca

x

Cu

24

O

41

isoften alled dueto its hemi alformulawhi hresembles atelephone number14-24-41, is a layered material with two distin tly

dier-ent types of 2D opper oxide planes separated by Sr/Ca atoms [4℄: (i) the

planes withalmost de oupledCuO

2

hains and (ii) theCu

2

O

3

planes formed by Cu

2

O

5

oupled ladders (see Fig. 2.1). Although in prin iple there ould be some intera tion between the ladder subsystem, the hain subsystem and

theSr/Ca atoms 1

we would assumethat theladdersubsystem anbetreated

independently, i.e. the Hamiltoniansfor ea h subsystem are independent one

fromanother,ex eptforthe hemi alpotentialwhi hshould bedeterminedto

onserveaparti ularnumberofele tronsinthewhole3D rystal(seebelow).

Numberof arriersinSr

14−x

Ca

x

Cu

24

O

41

.The ompli ated hemi al for-mulaleadstotheproblemswithdeterminingthenumberofele tronspresentin

thesystem. Letusrst on entrateonthe

x = 0

ase. A tually, theioni pi -turesuggeststhatonehasintheformulaunit: 14Sr

2+

ions,24Cu

2+

ionsand 41O

2−

ionswithalloftheseionshavinglledshells,ex eptfor opper(where

1

Inparti ularthe substitution ofstrontium by al ium yieldsstru turalmodulationsin

theladdersubsystem,seeRef. [34℄. However,thismodulationgrowswith al iumdoping

x

and annotexplaintheonsetof hargeorderforsmall

x

andlarge

x

(whilethe hargeorder isunstableforintermediate

x

),seedis ussionbelow. Furthermore,theinuen eofthe hain subsystemontheladder subsystem anberedu edtothe hainsbeingthe hargereservoir

(20)

Figure 2.1: Left panel: the 3D stru ture of Sr

14

Cu

24

O

41

. Right panel: the Cu

2

O

5

oupled ladders whi h form one of the two types of planes in Sr

14

Cu

24

O

41

. Thebigyellowspheresdepi t opperatoms, thebigredspheres strontiumatoms,thesmallblue spheresoxygenatoms. Both panelsare

repro-du edafter Ref. [4℄.

the

3d

shellisnaturallyunlled). Thus,oneobtainsfromtheioni pi turethat there is onehole per Cu

2+

ion, 2

similarly asin the CuO

2

planes of La

2

CuO

4

[28℄.

However,oneseesthat su hioni pi ture onsiderationsleadtothe6extra

holes present in the formula unit and the ompound is self-doped already at

x = 0

. As the forumula unit onsists of 7 Cu

2

O

3

units in the ladder plane, 14 strontiumatoms and10 CuO

2

units in the hainplane, anaturalquestion arises: how these 6 extra holes are distributed between the ladders and the

hains. A tually, the answerto this question is nontrivial(see Refs. [35, 36,

37℄ for various s enarios) and it was only re ently that the x-ray absorption

spe tros opyresultssuggested[9℄ thatthere are2.8extraholesin theformula

unit in theladders(whi h meansthat there are0.2holesper oppersite)and

3.2extraholesintheformulaunitinthe hains(i.e. 0.32holesper oppersite).

In what follows, we adopt the latter results as they seem to agree best with

otherexperimentaldataforthissystem[9℄.

Let us now turn to the

x 6= 0

ase. Here, the ioni pi ture suggeststhat againthereare6extraholesin theformulaunit: thisisbe ause al iumis

iso-valentwithstrontium. However,ithasbeensuggestedthatintrodu ing al ium

leadsto the gradual in reaseof thenumber ofthese extraholesin the ladder

subsystem[9℄. Indeedthesamex-rayabsorptionspe tros opyresultsasforthe

x = 0

ase[9℄ revealed that for the interesting ase (see below)of

x = 4

the numberof holesin theladdersis 3.4(i.e. a. 0.24per oppersite) and 2.6in

the hains (i.e. a. 0.26per oppersite) whilefor

x = 11

thenumberofholes 2

Sin eitiseasiertotalkaboutoneholeper oppersitethanabout9ele tronsper opper

(21)

Figure 2.2: The intensity of the s attering at the oxygen

K

`mobile arrier peak' (528.6 eV, see Ref. [8℄ for detailed explanation) in theresonant soft

x-rays attering for variousvaluesof al ium doping

x

in Sr

14−x

Ca

x

Cu

24

O

41

at temperature

T = 20

K.CDWisobservedfor

x = 0

(withperiod

λ = 5

,depi ted as

LL

= 1/5

onthegure)and

x = 11

(withperiod

λ = 3

,depi tedas

LL

= 1/5

onthegure). Asmallintensityisalsovisible for

x = 10

andevensmallerfor

x = 12

whi h also orresponds to a (small) CDW with period

λ = 3

. For

0 < x < 6

noree tionsareobservedandinparti ularnoCDWisseenat

x = 4

where

nh

= 1.24

would suggesta CDW with period

λ = 4

(

LL

= 1/4

) to be stable. Thegureisreprodu edafterRef. [8℄.

intheladdersis4.4(i.e. a. 0.31per oppersite)and1.6inthe hains(i.e. a.

0.16per oppersite).

Pe uliar hargeorderintheladdersubsystem.Whiletheladdersubsystem

exhibitsthenon-BCSsuper ondu tingphasefor

x = 13.6

underpressurelarger than3GPa[38℄, in broadrangeof

x

andunder normalpressureaspin-gaped insulating CDW states was dis overedin the ladders [5,6℄. By means of the

resonantsoft x-ray s attering it was found [7℄ that this CDW stateis driven

bymany-bodyintera tions(presumablyjustCoulombon-siteintera tionssin e

the long-rangeintera tions are s reened in opperoxides [39℄), and it annot

be explained bya onventional Peierls me hanism. Hen e, the observed

om-petition betweentheCDW phase(also referredto asthe`hole rystal'due to

itsele troni origin) and super ondu ting statesin spin laddersresembles the

onebetweenstripesandthesuper ondu tingphaseinCuO

2

planesofahigh-

Tc

super ondu tor[40℄. Thisiswhytheproblem oftheoriginoftheCDW phase

in the ladder subsystem of Sr

14−x

Ca

x

Cu

24

O

41

is both generi and of general interest.

Furthermore,re entlyit wasfound [8℄ that the onlystable CDW statesin

thelowtemperatureregime (

T = 20

K) arewith period

λ = 5

for

x = 0

, and with period

λ = 3

for

x = 11

(and with amu h smaller intensity for

x = 10

and

12

),seeFig. 2.2. Evenmorestrikingresultsshowthat su h aCDWorder ould not bestable for

1 ≤ x ≤ 5

, see also Fig. 2.2. These striking results, whi h ontradi ttheprevioussuggestion[6℄thattheCDWordero ursin the

(22)

entirerange of

0 ≤ x < 10

, need to be explainedby onsidering hole density per oppersitein reasingwith

x

. Aswrittenabove

x = 0

asewithCDWstate withperiod

λ = 5

orrespondsto

nh

= 1.20

(totalnumberof holesper opper ion)while the

x = 11

asewith CDW order withperiod

λ = 3

orrespondsto

nh

= 1.31

. Interestingly,the

x = 4

ase(withnoCDW phase) orrespondsto

nh

= 1.24

,i.e. tothe asewherethenumberofextradopedholesisvery lose to

1/4

andwhereone ouldintuitivelyexpe taCDWstatewithperiod

λ = 4

. Main goals of the hapter. The main aim of this hapter is to explain

theoreti ally (at temperature

T = 0

K) the onset of the CDW order in the telephone number ompound foronlysele tedvaluesof

x

while usingamodel whi hmerely ontainson-siteCoulombintera tions. Inparti ularthequestions

tobeansweredinthis hapterare: (i)whattheproper

t

J

modelforthe oupled Cu

2

O

5

ladders,whi hwouldariseduetotheon-siteCoulombintera tions,looks like,and(ii)whether thismodel anexplaintheonsetoftheCDWphasewith

parti ularperiodsforparti ularvaluesof

x

.

Stru ture of the hapter. The hapter is organized as follows. In Se .

2.2 westarttheanalysis bylookingat theanti ipatedfeatures ofthenew

t

J

model whi h is derived in Se . 2.3. Next, we solve the model for the three

interesting hole dopings

nh

= 4/3

,

nh

= 5/4

and

nh

= 6/5

: (i) using the slave boson language we redu e the model to the ee tive Hamiltonian with

the onstraintsof `no double o upan ies' (always presentin any

t

J

model) releasedseeSe . 2.4.1,(ii)weintrodu ethemean-eldapproximationforthe

ee tive Hamiltoniansee Se . 2.4.2, (iii) wesolvethe mean-eld equations

onanite meshof

k

points(Se . 2.4.3). InSe . 2.5theresultsare dis ussed, withaparti ularemphasisontheapproximationsmadeinobtainingthe orre t

t

J

model. Finally, wedrawsome on lusions in Se . 2.6 and adda pe uliar exampleofatoy-modelfor oupled hainsin whi h theeven-period CDW an

be omeunstablein thePosts riptumin Se . 2.7.

2.2 The

t

J

model for oupled ladders

`Rough'predi tionsofthe new

t

J

model.Letusrstlookattheanti ipated features ofthenew

t

J

modelwithout goingdeeplyinto mathemati aldetails (su h al ulationswill beperformedinthenextse tion). A tually,thebiggest

problem withderivingsu h amodelis thatthe Cu

2

O

5

oupledladdersbelong to a lassof opperoxideswhi hare lassiedas hargetransfersystems[41℄.

On theonehand,in these systemstheHubbard repulsion

U

betweenholesin the

3d

orbitalsonthe oppersitesisstillthelargestenergys aleinthesystem anditismu hbiggerthanthelargesthopping

tpd

betweenthe opper

3dx

2 −y

2

andtheoxygen

2pσ

orbitals[39℄. Ontheotherhand,theon-siteenergies

∆

for theholesin theoxygen

2pσ

orbitalsaresmallerthantheHubbardrepulsion

U

onthe oppersites[39℄. Therefore,whenthenumberofholesisbiggerthanone

perone opperion,someholestendtoo upyoxygensites. Thus,unlikeinthe

Mott-Hubbardsystem, heretheoxygen atoms annot be easily integratedout

andtheHubbardmodel( alledthenthe hargetransfermodel[42,43℄)should

not only ontain orbitals on the opper sites but also the ones on theoxygen

sites [41℄. Nevertheless, Zhang andRi e [26℄ showedthat for theCuO

2

plane itis stillpossibletointegrateouttheoxygenatoms andthe

t

J

model, whi h results from su h an itinerant model, is apable of des ribing the low energy

(23)

Figure2.3: Theartist'sviewoftheCDWwithperiod

λ = 4

forasingleladderas obtainedfromthedensitymatrixrenormalizationgroup al ulationsforthe

t

J

modelonasingleladderwith

J = 0.25t

[10℄. Bla klled ir lesdepi ta opper siteo upiedbyahole,unlled ir lesdepi ta oppersitewiththeZhang-Ri e

singlet entredaroundit,i.e. where theextrahole(situated symmetri allyon

thefour oxygen sites surrounding the oppersite) formed asinglet statewith

theholeonthe oppersite. Inthiswaynumberofholes

nh

= 1.25

intheCu

2

O

5

singleladder orresponds to the

n = 0.75

lling(numberof spins per site) in the

t

J

modelonthetwo-legladder. Thegureisreprodu edafterRef. [10℄.

physi sof hargetransfersystems. Notehowever,thatthemeaningof

J

isthen dierentand

J 6= 4t

2 _/U

.

Although the above mentioned redu tion of the harge transfer model to

the standard

t

J

model is done for the CuO

2

plane [26℄, a similar derivation shouldin prin iplebepossibleforasingleCu

2

O

5

ladder. Thedieren ewould bethatinthis aseonewillbeleftwitha

t

J

modeldenedonatwo-legladder but otherwise the

t

J

model would be exa tlythe same asthestandard one, known from Chapter 1. Indeed, it is widely believed [11, 44℄ that a two-leg

ladderdes ribedbythe

t

J

model apturestheessentialphysi alpropertiesof theplanewithCu

2

O

5

laddersinSr

14−x

Ca

x

Cu

24

O

41

. Furthermoresu hamodel hasbeenextensivelystudied (seee.g. Refs. [45, 46, 47, 48, 49℄): in parti ular

Whiteet al. [10℄foundusingthedensitymatrixrenormalizationgroup,thata

CDW of period

λ = 4

isthe (possibly spingaped) groundstateat

nh

= 1.25

(

n = 0.75

llinginthe

t

J

model,see aption ofFig. 2.3anddis ussionin the end of Se . 2.3.3 for further details). Besides, only re ently it wasshown in

Ref. [11℄ that aCDW ispossiblefor su h amodelmerelyfor numberof holes

nh

= 1.25

(

n = 0.75

)and

nh

= 1.5

(

n = 0.5

).

Reason for wrong predi tions. However, one immediately sees that the

aboveresultsare totallyin ompatiblewiththeexperimental onesdes ribedin

Se . 2.1: theretheCDWwasstableintheCu

2

O

5

laddersinSr

14−x

Ca

x

Cu

24

O

41

for

nh

= 1.31

(

x = 11

) and

nh

= 1.2

(

x = 0

) whereas it wasnot stable for

nh

= 1.24

(

x = 4

), i.e. around the only point (apart from

nh

= 1.5

) where thedensitymatrixrenormalizationgrouppredi tedtheCDWtobestable. One

may thus wonder what may be wrong with the above

t

J

model? A tually, it is easy to see that the validity of the

t

J

model for the plane with Cu

2

O

5

ladders is far from obvious due to the spe i geometry. In parti ular: (i)

unlike the CuO

2

plane of a high-

Tc

super ondu tor, a single Cu

2

O

5

ladder la ks the

D4h

symmetry whi h makes the Zhang-Ri e derivation [26℄ of the

t

J

model questionable and (ii) Cu

2

O

5

spin ladders are oupled through the on-site Coulomb intera tionsbetweenholesin dierentO(

2p

)orbitals, so new intera tions ouldarise.

(24)

`granted', i.e. assuming that the derivation of the

t

J

model from the harge transfermodelvalidfortheCuO

2

wouldworkalsoforthe oupledCu

2

O

5

ladders andwouldgivea

t

J

modelonthetwo-legladder,oneshouldfollowtheZhang and Ri e s heme [26℄ step-by-stepin the aseof thisspe i laddergeometry.

Morepre isely,oneshouldtakethe hargetransfermodelfor opperoxideplanes

[42, 43℄,adopt it tothe oupled Cu

2

O

5

ladders,and thenfollowingtheZhang andRi es heme[26℄derivetheproper

t

J

model. Wepresentsu haderivation in thenextse tion.

2.3 The model

2.3.1 The

t

J

V

1 V

2

Hamiltonian

The Hubbard-type model. Asthestartingpointwe hoosetheHubbard-type

model relevant for the harge transfer systems (and thus alled also harge

transfermodel[41℄). Itfollowsfromthemultiband hargetransferHamiltonian

[41℄ and is adapted to the Cu

2

O

5

oupled ladder geometry, similarly as the oneintrodu edearlierforCuO

2

planes[42℄orCuO

3

hains[50℄, thestru tural units of high-

Tc

super ondu tors. As parameters the harge transfer model in ludes: the energy for oxygen

2p

orbital

∆

(measured with respe t to the energyfor the

3d

orbital), the

d

-

p

hopping

tpd

betweenthenearest neighbour opperandoxygensites,andtheon-siteCoulombrepulsion

U

(

Up

)onthe opper (oxygen)sites,respe tively. Notethatthe hargetransferregimenaturallyleads

to

∆ < U

sin e otherwise the oxygen atoms ould be easily integrated out. Indeed,thetypi alparametersare

U ∼ 8t

pd

,

∆ ∼ 3t

pd

, and

Up

∼ 3t

pd

,seee.g. Ref. [51℄. Thenthemodelin holenotationreads,

H = −tpd

n X

iσ

− d

†

iLσ

x

iLσ

+ d

†

iRσ

x

iRσ

+ d

†

iLσ

b

iσ

− d

†

iRσ

b

iσ

+

H. .

+

X

iασ

d

†

_iασ

y

iασ

− d

†

i+1,ασ

y

iασ

+

H. .

o

+∆

nX

iα

niαx

+ niαy

+ ε

X

i

nib

o

+Up

nX

iα

n

_iαx↑

n

_iαx↓

+ n

_iαy↑

n

_iαy↓

+

X

i

n

_ib↑

n

_ib↓

o

+Up

n

(1 − 2η)

X

iασ

niαxσ

ni ¯

¯

αy ¯

σ

+ niαyσ

ni ¯

¯

αx¯

σ

+ (1 − 3η)

X

iασ

niαxσ

ni ¯

¯

αyσ

+ niαyσ

ni ¯

¯

αxσ

o

+U

X

iα

n

_iαd↑

n

_iαd↓

,

(2.1)

wherethephasesoftheorbitalswereexpli itlytakenintoa ountinthehopping

elements,theindex

α ∈ {R, L}

denotestherightorleftlegoftheladder(

R = L

¯

and

L = R

¯

),and

σ = −σ

¯

for

σ ∈ {↑, ↓}

. Theparameter

η = JH

/Up

∼ 0.2

stands for arealisti value ofHund's ex hange onoxygen ions (

Up

is theintraorbital repulsion)[39℄. Besides,

ε ∼ 0.9

yieldsthe orre torbitalenergy(

ε∆

)atbridge positionsontherungof theladder[52℄but, unlessexpli itlystateddierently,

(25)

y

x

d

y

d

x

b

Figure2.4: Three oupledCu

2

O

5

ladders. Only orbitalswhi h arein ludedin themodel Eq. (2.1) areshown,see text. The dottedline depi ts theunit ell

ofthesingleladderunder onsideration;it onsistsofsevenorbitals.

we will assume that

ε = 1

for simpli ity (see also Se . 2.5.3 for a detailed dis ussion on this issue). The model of Eq. (2.1) in ludes seven orbitals per

Cu

2

O

5

ladderunit ell(seeFig. 2.4): twoCu(

3dx

2 −y

2 ≡ d

)orbitalsonthe

R/L

leg,twoO(

2py

≡ y

) orbitals onthe

R/L

leg,twoO(

2px

≡ x

)side orbitals on the

R/L

leg,andoneO(

2px

≡ b

)bridgeorbitalontherungoftheladder.

SpatialdimensionoftheHubbard-typemodel.Itshouldbeemphasizedthat

thetermsin thefth andsixth line ofEq. (2.1)a ountforinterladder

inter-a tion theholeswithin twodierent orbitalson agiven oxygen ion in aleg

belong totwoneighbouringladders(shownaswhite/greyorbitalsin Fig. 2.4),

and are des ribed by harge operators

niαx(y)σ

with/without bar sign in Eq. (2.1). Thus,inprin ipleoneshoulddenetwootherHamiltonians

H

whi h de-s ribethetwoneighbouringladdersandfromwhi hone andetermine

˜

niαx(y)σ

. Then, these twoHamiltonians will be again oupled to twoHamiltoniansand

soon. In what follows, we will impli itly assumethat su h Hamiltoniansare

indeed dened and when needed we will use this feature to solvethe oupled

ladderproblem. Obviously,su hanotationisnotveryelegant. An alternative

s enariowouldbetodeneasingleHamiltonianforalltheladdersintheplane

however, thiswould ompli atethe notationeven moreand, in myopinion,

would notmakethephysi smoretransparent.

Central Hamiltonian of the hapter. ApplyingtheZhang-Ri e pro edure

[26℄adoptedtothegeometryof oupledladdersandnitevalueofthe

intera -tion

Up

weobtainthefollowing

t

J

modelwithintraladderintera tion

V1

and interladderintera tion

V2

(therefore alledalso

t

J

V1

V2

Hamiltonian):

H = Ht

+ HJ

+ HV

1 + HV

2 .

(2.2)

Here

Ht

stands for the kineti term[see Eq. (2.13) in Se . 2.3.3℄,

HJ

is the superex hange term [see Eq. (2.5) in Se . 2.3.2℄, while

HV

1

and

HV

2

are the

(26)

Note that,during thepro eduresuggestedbyZhangandRi enotonlythe

Hamiltonian hanges but also theform and number of arriers hanges asthe

form of the Hilbert spa e is hangeddrasti ally [26℄. Whereas in the harge

transfer model we denote by

nh

the number of holes per opper site, in the

t

J

V1

V2

model the lling(number ofspins persite) is

n = 2 − n

h

, see also dis ussionintheendofSe . 2.3.3.

2.3.2 The superex hange term

Single ladderin the undoped ase. Intheso- alled half-lled ase(i.e. when

there is just onehole per opper site) and in the harge transferregime (see

above),the hargetransfermodel(2.1) anbeeasilyredu edtotheHeisenberg

modelusingtheperturbationtheorytofourthorderin

tpd

[53℄. 3

Thisisbe ause,

when

tpd

= 0

theholesarelo alized onthe oppersites,while forsmall

tpd

in omparison with the other energy s ales in the in the hargetransfer system

theholesperformmerelyvirtualex itationswhi hinvolvethedoublyo upied

opperoroxygensite. Thus,the hargedegreesarefrozenandoneisleftmerely

withspindegreesoffreedom,somewhatsimilarlyasinthehalf-lled aseofthe

HubbardmodelofChapter1. Oneobtains[53℄:

HJ(nh

= 1) = J

X

iα

S

iα

· Si+1,α

−

1

4 + J

X

i

S

iR

· SiL

−

1

4 ,

(2.3)

wherethesuperex hange onstantfornite

Up

ase[53℄ is

J =

4t

4 pd

∆

2

1 U

+

2 2∆ + Up

.

(2.4)

Spe i geometryof oupledladders.Thereadermaywonder,whetherthe

geometryof oupledladders ouldinuen etheaboveresult. Indeed,there

ex-istsa

90

0

superex hangepro essbetweentheholesontwoneighbouringladders

whi h involves

niαx(y)σ

˜

operators. However, a ording to the Goodenough-Kanamori-Anderson rules [54, 55, 56℄ su h a superex hange pro ess [whi h is

ferromagneti (FM) in ontrast to the above AF intera tion℄ is mu h weaker

than the superex hange along the

180

0

path in the single ladder and an be

negle ted. Thus,Eq. (2.3)shouldalsobevalidfor oupledladders.

Coupled ladderin the doped ase.Although whenthesystemisnot

half-lledthereareotherpro esseswhi h ontributetothelowenergy

t

J

Hamilto-nian(seebelow),theaboveresult anbeextendedtothedoped ase. A tually,if

thereisnoholeononeofthesitesformingabondbetweenthe oppersites,then

thesuperex hangepro essdoesnoto ur. One aneasily he kthatEq. (2.3)

for this parti ular bond doesnot ontribute to the

t

J

Hamiltonian provided one hangesitinto:

HJ

= J

X

iα

S

iα

· S

i+1,α

−

1

4 niαd

˜

ni+1,αd

˜

+ J

X

i

S

iR

· S

iL

−

1

4 niRd

˜

niLd

˜

.

(2.5) 3

Notethat inthe half-lled asethere isno needtoperformthe anoni alperturbation

(27)

symmetri (

|Piασi

) antisymmetri singleoxygen singlet

−8(t1

+ t2) + 2t3

−4t1

+ 2t3

−2(t1

+ t2) + 2t3

triplet

0 −4t

1

0

Table2.1: Bindingenergyofthesingletandtripletstateformedbythethehole

onthe oppersiteandtheextradopedholeinoneofthethreevariousoxygen

states: (i)symmetri plaquette

|Piασi

state,(ii)antisymmetri plaquettestate with a similar ombination of oxygen orbitals asin

|P

iασ

i

but with the same signs before ea h oxygen orbital, and (iii) single oxygen orbital state. Here

t1

= t

2 pd

/∆ ∼ t

pd/3

,

t2

= t

2 pd/(U − ∆) ∼ t

pd/5

,

t3

= t

2 pd

Up/(∆

2 + Up

∆) ∼ t

pd

/6

wheretheestimationsfollowfromthetypi al hargetransferparameters[51℄.

Here tilde above the numberoperator denotes the fa t that the double

o u-pan iesonthe oppersitesareprohibitedin thelowenergylimitofthe harge

transfersystem.

2.3.3 The kineti energy term

Finite ontribution onlyfor doped ase.Asdes ribedabove,in thehalf-lled

asethe holeslo alize on the opper sites with the hargedegrees offreedom

entirely gone and one is left with the Heisenberg Hamiltonian for the spins.

Thus, there is no kineti term at all in the half-lled ase and it ould only

ontributeinthedoped aseduetotherestri tedhopping.

Zhang-Ri es hemeneeded.Inthedoped aseasigni antproblemarises:

wheretheextraholedopedintothehalf-lledsystemgoes. A tually,if

tpd

= 0

, thentheholewillforsurelo alizeatoneoftheoxygensitesastheon-siteenergy

∆

is smallerthan therepulsion betweentwoholesat the same oppersite

U

. Therefore,in thisregime one annot integrateouttheoxygensites. Itmaybe

expe tedthatsu hstateswilldominatealsofornite

tpd

.

A tually,fornite

tpd

intheCuO

2

plane,ito ursthattheholealso tends tolo alizeonoxygensitesbutformsape uliarboundstatewiththenearbyhole

on the oppersite the so- alledZhang-Ri e singlet [26℄. Wenow onstru t

su h a state step-by-step for the oupled ladder ase(again starting with the

singleladderandonlylaterondis ussingthe oupledladderproblem),seeFig.

2.5foranartist'sviewoftheresultobtainedinthisse tion.

NonorthogonalZhang-Ri esinglets.First,itisevidentthatpla ingahole

ontheoxygensiteandaligningitsspinintheAF-waywithrespe ttothespin

of the hole on the opper site, one an gain some energy due to the virtual

hoppingpro essesbysmallbutnite

tpd

(intheferromagneti asesu h harge ex itations are notallowed due to Pauli prin iple). Se ond, however, one an

gain even more binding energy if one uses the possibility of forming a phase

oherent state out of the four oxygen orbitals surrounding the opper. More

pre iselyito ursthatthesingletstateformedbyaholeonthe oppersiteand

ahole in oneof the followingsymmetri plaquette state (dierentfor the left

andrightlegoftheladder):

(28)

ladder (the state depi tedby adotted ring). Large (small)arrowsdepi t the

hole spins for +1.0 (+0.25) harge. The red arrowsstand for spins of doped

holeswhiletheblue arrowsshowthespinsintheundoped ase.

or

|P

iRσ

i =

1

2 (−x

†

iRσ

+ b

†

iσ

+ y

i−1,Rσ

†

− y

iRσ

†

)|0i,

(2.7)

hasabindingenergyof

−8(t1

+ t2) + 2t3

. A tually, thisbinding energyisnot only negative and hugein omparison with the ee tive hopping (whi h is of

the order of

t1

or

t2

[26℄) but it is also mu h bigger than the binding energy of some other possiblebound statesformed by a hole onthe opper site and

oxygen site,see Table2.1. Itmaybeveriedthat nite

Up

,not onsideredin the Zhangand Ri e paper[26℄, whi h results in nite

t3

term (see Table 2.1) does not hange qualitativelythe largebinding energy of a symmetri singlet

state.

At this stage one an already imaginethat all of the doped holes(if their

numberissmallerthanthenumberof oppersites)shouldbeabletoformsu h

symmetri singletstatesinthe hargetransfersystemsanditwouldbepossible

tointegrateoutoxygensitesentirely. Althoughthis onje turewillturnoutto

betrue,it annotbedoneso easily. Aqui klookatEqs. (2.6-2.7)revealsthat

theabovesymmetri singlet statesarenonorthogonal(they ouldbe alledthe

nonorthogonal Zhang-Ri e singlets) as theneighbouring states share ommon

oxygenorbitals.

OrthogonalizedZhang-Ri esinglets.Thetaskisnowtomakethestates

de-nedinEqs. (2.6-2.7)orthogonal. Thisisdonebythefollowingtransformation

in thesingleladder ase:

|φ

lLσ

i =

1 N

X

jk

e

ikl

e

−ikj

(αk

|P

jLσ

i + β

k

|P

jRσ

i),

(2.8)

and

|φ

lRσ

i =

1 N

X

jk

(29)

αk

=

q

1 1 −

1

2 cos k −

1

4 +

q

1 1 −

1

2 cos k +

1

4 ,

(2.10) and

βk

=

q

1 1 −

1

2 cos k −

1

4 −

q

1 1 −

1

2 cos k +

1

4 .

(2.11)

One an he kthat the`extended'symmetri states

|φiασi

areindeed orthogo-nal.

Letusnotethatitisatthispointthattheequationsaretruelydistin there

thantheones onsideredbyZhangandRi einRef. [26℄: thereitwasonlyshown

howto`orthogonalize'symmetri statesforthe2D ase. Whilethatpro edure

ouldhavebeeneasilygeneralized(oroneshouldrathersay`redu ed')tothe1D

ase,theladder aserequiredamore areful onsideration. A tuallytheeasiest

waytoobtainequationsfor

αk

and

βk

istoderivethemrstformerelyasingle rungoftheladder. Inthat aseone aneasily he k

αk

= 1/

√

3 + 1/

√

5

while

βk

= 1/

√

3 − 1/

√

5

. Thenone angeneralizethisresulttothewholeladder. Finallyone anexpli itlydenetheZhang-Ri esinglets as

|ψiαi =

√

1

2 |φiα↑

d

iα↓

− φiα↓

d

iα↑i,

(2.12)

see also Fig. 2.5 for an artist's view of this state. In prin iple, one should

also he k how the binding energy hanges when the Zhang-Ri e singlets are

orthogonalized. It was shown in Ref. [26℄ that the energy splitting hanges

only slightly when the singlets are orthogonalized. Obviously, the results in

Ref. [26℄ are valid only for the 2D ase. Fortunately, asimilar result anbe

easily obtained for the 1D ase. A tually, the energy splitting between the

orthogonalized Zhang-Ri e singlets and triplets anbe dened as

16χ

2 _t1

(for

thesimplied ase

t1

= t2

and

t3

= 0

)[26℄. Thenthe ru ial onstant

χ

isvery loseto onebothin the 1D(

χ = 0.98

)andin the 2D ase(

χ = 0.96

)[26℄. One ansafely arguethat

χ

forthe ladderstakessome valuein between

0.96

and

0.98

asthereisnophysi alreasonthattheorthogonalizationpro edurefor theladderswouldleadto totallydierentbehaviourthanfor the1D hainsor

2D ase(despitetheformwhi hisslightlymore ompli atedintheladder ase).

Thus, theorthogonalizedZhang-Ri e singlets haveahugebinding energyalso

forthe ladder. In what follows, wewill referto Zhang-Ri esinglets havingin

mindmerelytheirorthogonalversion.

Kineti termforsingleladder.HavingshownthattheZhang-Ri esinglets

in thesingleladderdonotdiermu hfrom those inthe2D ase,we annow

safelyassumethat one anapplyto theladder ase alltheargumentsused in

Ref. [26℄toderivetheee tivehoppingofZhang-Ri esingletsduetonite

tpd

. Thus, weobtain,

Ht

= − t

X

iασ

˜

d

†

_iασ

di+1,ασ

˜

+ H.c.

− t

X

iσ

˜

d

†

_iRσ

diLσ

˜

+ H.c.

,

(2.13)

where again

diασ

˜

= diασ(1 − niα¯

σ)

is the restri ted fermion operator and as before

diασ

reatesaholein the oppersite

iα

. Thisfollowsfrom theee tive hopping of Zhang-Ri e singlets

t

bya hole-parti letransformation. Whilewe

(30)

singlets

t

,notethatitis onsiderablysmallerthan

tpd

( a. 30%). Notealsothat havingtwoZhang-Ri esingletsonthesamesite ostsenergy

4t2

+ 2t1

(seeRef. [26℄) and therefore we used the tilde operators above to prevent from having

twoZhang-Ri esinglets onthesamesite.

Extensionto oupledladders.Sin etheinteroxygenhopping

tpp

′

< tpd

[39℄ in opperoxidesystems,thereisnopossibilityofhoppingbetweentheladders.

Thus,theaboveresultwillalsobevalidfor oupledladdersprovidedthe

Zhang-Ri esinglets anbe onstru tedinthat ase. Thisisindeedthe ase,howeverit

issomewhatsubtleandwereferthereadertothenextse tionformoredetails.

Number of arriers inthe

t

J

V1

V2

model.Due to theZhang-Ri e pro- edure notonlythe natureof arriersbut alsotheirnumberis hangedin the

ee tive

t

J

V1

V2

model. Sin ethe numberofextra holeswhi h o upy the oxygensitesandformtheZhang-Ri esingletsisequalto

nh

− 1

per oppersite (where

nh

isthe numberof holes per oppersite), there are

ne

= nh

− 1

per siteemptystatesintheee tive

t

J

V1

V2

model. Thismeans,thatthelling

n

inthe

t

J

V1

V2

model(i.e. thenumberofspins)is

n = 1 − n

e

= 2 − n

h

per site.

2.3.4 The intraladder repulsive term

V

1

Finite

Up

and the intera tion between Zhang-Ri e singlets in 2D ase. In theoriginal ZhangandRi e paper[26℄ theintera tionon oxygensites

Up

was entirely negle ted. Here, we have already stated its rather minor role in the

stability of the Zhang-Ri e singlets (see e.g. Table 2.1 where

t3

is nite for nite

Up

aswellas dis ussioninSe . 2.3.3). However,thisisnotthefullstory [57, 58, 59℄. A tually, due to the nite

Up

the twononorthogonal Zhang-Ri e singletsrepeliftheyaresituatedonthenearestneighboursite. Thisisbe ause

these twononorthogonalZhang-Ri esinglets share a ommonoxygen siteand

the twoholessituated onthis oxygen site and belonging to twoneighbouring

nonorthogonalZhang-Ri esingletsrepel.

Obviously, this intera tion is quite redu ed as there is just

25%

proba-bility to nd a hole forming a nonorthogonal Zhang-Ri e state on the

par-ti ular oxygen site (whi h is shared with the neighbouring Zhang-Ri e

sin-glet). Indeeddetailed al ulationsfortheorthogonal Zhang-Ri e singlets,

per-formed in Refs. [58, 59℄, showedthat this repulsionis of theorder of

0.029Up

(while the not- onsidered-herenite intersite Coulomb repulsion

Vpd

between holes on oxygen sites and opper sites even further redu es this value [57℄).

Thus, the orthogonalization pro edure redu es its value from the estimated

1/2(1/4 × 1/4 + 1/4 × 1/4) = 1/32 ∼ 0.031

(the fa tor

1/2

before the equa-tionoriginatesfromthePauliprin iple)fornonorthogonalZhang-Ri esinglets.

Therefore,oneusually negle tstheee tiverepulsionbetweenholesinthe

t

J

model asitwill beat maximumof theorder of

0.2t

(for parametersfrom[39℄ where

Up

= 4.18

eV is ratherlarge) whiletypi ally

J ∼ 0.4t

in opper oxides [23℄.

Intraladder and interladder repulsion. In the oupled ladder geometry,

however,thesituation hangesdrasti ally. Although,within ea hsingleladder

the repulsion is somewhat similar as in the 2D ase (this will be alled the

intraladder repulsion, seeFig. 2.6), adistin tsituation o urs forthe oupled

(31)

neighbour Zhang-Ri e singlets. See Fig. 2.5 for further explanation of the

symbolsusedhere.

Zhang-Ri esingletsonneighbouringladders. Thisisbe ause,su hZhang-Ri e

singlets share not onebut two oxygen sites,see Fig. 2.7 in the nextse tion.

Thus, the interladderrepulsionbetweenZhang-Ri esinglets shouldnaivelybe

fourtimes 4

asbigastheintraladderrepulsionandthereforeit anhappenthat

it ouldbeoftheorderof

J

.

Cal ulation of the intraladder repulsion. Whereas the signi an e of the

interladderrepulsionisdis ussedinthenextse tion,letusnow on entrateon

therepulsion between theZhang-Ri e singlets within asingle ladder(see Fig.

2.6fortheartist'sviewoftheproblem). Thus,thetaskisto al ulaterepulsion

betweenorthogonalizedZhang-Ri esingletswithintheladderduetotheon-site

intera tion

Up

:

H

′

= Up

nX

iα

n

_iαx↑

n

_iαx↓

+ n

_iαy↑

n

_iαy↓

+

X

i

n

_ib↑

n

_ib↓

o

.

(2.14)

Thus, oneneedsto al ulate thefollowingmatrixelements:

hψsα, ψrα|H

′

_{|ψhα, ψjαi,}

_{hψsα, ψr ¯}

_α|H

′

_{|ψh ¯}

_{α, ψjαi.}

(2.15)

Let us note that the mixed termssu h asfor example

(RL, LL)

give zero in the Zhang-Ri e singlet basis they ould a priori lead to the destru tion of

the Zhang-Ri e singlets but fortunately are mu h smaller than the respe tive

bindingenergy.

Intraladderrepulsionalongtheleg.First,we al ulatethematrixelements

4

Thisisbe ause hereboththe holeswiththe sameand opposite spins anrepel:

1/4 ×

1/4 + 1/4/4 = 1/8

.However,thisfa torwillmultiplysmalleron-siterepulsion,withrespe t

totheintraladder ase,duetoHund'sex hangeandaltogetheritwillturnoutthatfor

η = 0.2

[39 ℄theinterladderrepulsionisroughlytwi estrongerthantheintraladderrepulsion.