typeset using JPSJ.sty <ver.1.0b> Possible Ordered States in the 2D Extended Hubbard Model Masakazu Murakami∗ 0 0 Institute for Solid State Physics, Universityof Tokyo, Roppongi, Minato-ku, Tokyo106-8666 0 (ReceivedDecember 14,1999) 2 n Possible ordered states in the 2D extended Hubbard model with on-site (U >0) and nearest- a neighbor (V) interaction are examined near half filling, with emphasis on the effect of finite V. J First, the phase diagram at absolute zero is determined in the mean field approximation. For 0 V < 0, a state where dx2−y2-wave superconductivity (dSC), commensurate spin-density-wave 3 (SDW)and π-tripletpair coexist is seen tobestabilized. Here,theimportanceof π-triplet pair onthecoexistenceofdSCandSDWisindicated. Thiscoexistentstateishamperedbythephase ] separation (PS), which is generally expected to occur in the presence of finite-range attractive el interaction, butsurvives. For V >0, a state wherecommensurate charge-density-wave(CDW), - SDW and ferromagnetism (FM) coexist is seen to be stabilized. Here, the importance of FM r on the coexistence of CDW and SDW is indicated. Next, in order to examine the effects of t s fluctuation on each mean field ordered state, the renormalization group method for the special t. case that the Fermi level lies just on the saddle points, (π,0) and (0,π), is applied. The crucial a difference from the mean field result is that superconductivity can arise even for U > 0 and m V ≥ 0, where the superconducting gap symmetry is dx2−y2-wave for U > 4V and s-wave for - U < 4V. Finally, the possibilities that the mean field coexistent states survive in the presence d of fluctuation are discussed. n o KEYWORDS: 2D extended Hubbard model, coexistent state, d-wave superconductivity, s-wave superconductiv- c ity, spin-density wave, charge-density wave, ferromagnetism, π-triplet pair, η-singlet pair, phase [ separation 2 v 3 1 nforV <0,2,3,4,5) andcommensuratecharge-andspin- 2 §1. Introduction densitywave(CDWandSDW)canappearathalffilling 4 In connection with the studies of the copper oxide n=1forV >0.2,3) However,thepropertyoftheground 0 9 high-Tc superconductors with CuO2 planes, the elec- stateforfinitecarrierdopingandtherelationshipamong 9 tronic states in two-dimensional systems has been in- various order parameters, especially between dSC and / tensively studied. Especially, the possibility of various SDW, have not been understood yet, even in the mean t a ordered states has been discussed. One characteristic field approximation. From these points of view, we will m feature is the proximity of superconductivity and an- study possible ordered states, especially possible coex- - tiferromagnetism. In our previous work (hereafter re- istence of different orders, in the 2D extended Hubbard d ferred to as I),1) we have shown in the mean field ap- model on a square lattice near half filling for U >0 and n o proximation that the coexistent state with d-wave su- V = 0, with emphasis on electrons around the saddle 6 c perconductivity(dSC),commensuratespin-density-wave points (π,0) and (0,π). : v (SDW) andπ-tripletpair canbe stabilizednearhalf fill- In 2, the extended Hubbard model is introduced and § i ingbyrepulsivebackwardscattering(’Umklapp’and’ex- itsrelationshiptoourpreviousmodelusedinIisreferred X change’) processes between electrons around the saddle to. In 3, the phase diagram at absolute zero, T =0, is r § a points(π,0)and(0,π). Asweshallshowlater,thismodel determined in the mean field approximation. In 4, the § with such a particular type of interaction have similar effectsoffluctuationonthemeanfieldorderedstatesare features to those in a square lattice model with on-site examinedbasedonthe renormalizationmethod applica- repulsion U > 0 and nearest-neighbor attraction V < 0, ble only for the special case that the saddle points (π,0) i.e., an extended Hubbard model.2) Therefore, it is inter- and (0,π) lie just on the Fermi surface. esting to examine in more detail the possibility of the §2. Extended Hubbard Hamiltonian abovecoexistentstatebyuseofthismodel. Atthesame time,theextendedHubbardmodelwithbothon-siteand The extendedHubbardHamiltonian,H =H +H + 0 U nearest-neighbor repulsion (U > 0 and V > 0), is also H , is written as follows: V of physicalinterest. In the 2D extended Hubbardmodel H = ξ c† c , (2.1a) for U > 0, it has been shown based on the mean field 0 p pσ pσ Xpσ approximation that extended-s-, p- and d-wave super- conductivitycanarisedepending onthe electrondensity U H =U n n = n n , (2.1b) U i↑ i↓ q↑ −q↓ N Xi Xq ∗E-mail: [email protected] 2 MasakazuMurakami V 1 H = n n = V n n , (2.1c) tion. We fix the electron density to n = 0.9. With this V i i+ρˆ q q −q 2 N Xiρˆ Xq choiceofparameters,theFermisurfaceofnoninteracting electronsis ofthe YBCO- orBSCCO-type and lies close where σ is the spin index taking a value of +1 ( 1) − to the saddle points (π,0) and (0,π), as shownin Fig.1. for ( ) spin, and the opposite spin to σ is denoted by σ ↑ ↓σ. N is the total number of lattice sites, ξ = p t'/t=-1/5, n=0.9 p ≡ − ǫ µ is the one-particle energy dispersion relative to p − the chemicalpotentialµ, including nearest-neighbor-(t) and next-nearest-neighbor- (t′) hopping integrals, p y ǫ = 2t(cosp +cosp ) 4t′cosp cosp , (2.1d) p x y x y − − n = n = c† c , ρˆ = xˆ, yˆ is the unit q σ qσ kσ kσ k+qσ ± ± latticePvector andP 0 p p x V =V(cosq +cosq ). (2.1e) q x y The energy dispersion ǫ has two independent saddle Fig. 1. TheFermisurfaceintheabsenceofinteractionfort′/t= p points,(π,0)and(0,π). Inthispaper,wefixt′/t= 1/5 −1/5andn=0.9. − with t > 0, in which case the Fermi surface in the ab- sence of interaction approaches (π,0) and (0,π) as the hole doping rate,δ 1 n,is increasedfromhalf filling, ≡ − 3.1 Nearest-Neighbor Attraction V <0 δ =0. WestartwiththecaseV <0. Aswesawin 2,theor- Here, we examine the relationship between the ’g- § deredstateswithdSC,SDW(=antiferromagnetism,AF) ´ology’modelusedinIandthepresentextendedHubbard model.6) In I, we have treated the backward scattering andπ-tripletpairisexpected. Weconsiderthefollowing order parameters, withlargemomentumtransferbetweenelectronsaround (π,0) and (0,π), i.e., ’Umklapp’ (g ) and ’exchange’ 3⊥ <c c > σs +q cos(QR ), (3.1a) iσ i+ρˆ,σ ρˆ ρˆ i (g ) processes, and considered three types of the scat- ≡ 1⊥ n tering processes, i.e., (1) Cooper-pair, (2) density-wave <niσ > +σmcos(QRi), (3.1b) ≡ 2 and (3)π-pair channels.1) (Here we denote the Hamilto- where Q = (π,π), s = s and q = q . s (q ) nian for these channels as H , H and H .) Especially −ρˆ ρˆ −ρˆ ρˆ ρˆ ρˆ 1 2 3 stands for a spin-singlet (triplet) pair of two electrons for the repulsive case, g > 0 and g > 0, we have 3⊥ 1⊥ with a total momentum 0 (Q) and total spin S = 0 shown that the coexistent state with dSC, SDW and π- (S = 1 and S = 0), i.e., Cooper-pair (π-triplet pair), tripletpair canbe stabilizednear halffilling atlowtem- z and m for the local staggered spin moment. We take perature. However,the above effective interactionis too s = s = s = real and q = q = q = real, simplifiedinthat(a)forwardscatteringprocessesarenot xˆ yˆ 0 xˆ yˆ 0 − − included and (b) the k-dependence of interaction is ig- i.e., consider dx2−y2wave pairing, which is favored near half filling.2) While s describes dSC, q describes an nored. Therefore, by transforming Hi (i = 1,2,3), into 0 0 electron-pair of p p -wave symmetry.7,8) This can be real-space representation, we will obtain a well-defined x y − easily seen by writing the operator of π-triplet pair in modelonasquarelattice. Ifwekeepon-siteandnearest- k-space, neighbordensity-densityCoulombinteraction,weobtain 1 Oˆ w c c , (3.2a) π p −p+Qσ pσ g ≡ √2 H 3⊥ n n α n n ,(2.2a) Xpσ 1 i↑ i↓ iσ jσ ∼ 2N(cid:26) − (cid:27) Xi <Xij>Xσ 1 = w c c , (3.2b) H g⊥ n n n n , (2.2b) √2Xpσ Q2+p Q2−pσ Q2+pσ 2 i↑ i↓ iσ jσ ∼ 2N(cid:26) − (cid:27) Xi <Xij>Xσ where wp cospx cospy is the dx2−y2-wave factor ∝ − and < Oˆ > q . The orbital function as a function g π 0 1⊥ ∝ H3 ∼ 2N(cid:26)Xi ni↑ni↓−α<Xij>Xσ niσnjσ(cid:27),(2.2c) opfrotphoertrieolantailvetomsionmpentusminpp o.f two electrons, wQ2+p, is x y − where g g +g , < ij > stands for a bond con- The order parameter in the mean filed Hamiltonian ⊥ 3⊥ 1⊥ necting site≡i and its nearest-neighbor site j and α = are given as follows in terms of s0, q0 and m, 16/π4 0.164. Each H is seen to describe on-site re- pulsion∼andnearest-neighiborattractionforg ,g >0. ∆dSC =−2|V|s0, ∆π =−2|V|q0, (3.3a) 3⊥ 1⊥ Therefore,alsointheextendedHubbardmodelforU >0 ∆SDW =Um, (3.3b) and V <0, the coexistent state with dSC, SDW and π- where ∆ and ∆ include only V because s and q triplet pair is expected to be stabilized. dSC π ρˆ ρˆ are defined on a bond, and ∆ does not include V SDW §3. Mean Field Analysis because <ni >= σ <niσ > is independent of m. First, we determine the phase diagram at absolute The pure π-tripPlet pairing state with ∆π 6= 0 and ∆ = ∆ = 0 is always energetically unfavorable zero,T =0,nearhalffillinginthemeanfieldapproxima- dSC SDW PossibleOrderedStates inthe2DExtended HubbardModel 3 compared with the pure dSC state with ∆ = 0 and ted line. It is seen that the coexistent state is severely dSC 6 ∆ =∆ =0. However,sincethecoexistenceofdSC suppressed by PS but survives. This PS is expected to SDW π and SDW (∆ = 0 and ∆ = 0) generally results be suppressed if we take the long-range Coulomb repul- dSC SDW 6 6 innonzerot (andnonzero∆ here),theself-consistency sion into account. Here, for simplicity, we consider the 0 π ofmeanfieldcalculationrequiresthe considerationof π- next-nearest-neighbor density-density repulsion V′ > 0, triplet pair into accountfrom the outset.1,9) The impor- tantfactthatthecoexistenceofspin-singletCooper-pair V′ 1 and SDW always leads to nonzero spin-triplet pair am- HV′ = 2 nini+ˆl = N Vq′nqn−q, (3.4a) plitude withfinite totalmomentumhadbeenrecognized Xiˆl Xq by Psaltakis et al. in a slightly different context.10) The V′ =2V′cosq cosq , (3.4b) q x y close relationship among the order parameters of dSC, SDW and π-triplet pair is discussed in Appendix A. where ˆl = (xˆ+yˆ), (xˆ yˆ), and incorporate V′ into ± ± − The mean field phase diagram in the plane of U and the RPA calculation, i.e., we replace Vq with Vq +Vq′. V is shown in Fig. 2. While the dSC state is stabi- We note that this replacement, which does not alter the | | lized for small U/t, the coexistent state with dSC, SDW mean field equations, can bring about the nontrivial ef- andπ-tripletpairispossibleforlargeU/t,andthephase fect on κ in the coexistent state, from eq. (B.10a). For boundarybetweenthesetwostatesisshownbysolidline. V′ = V /2,the PStransitionline is shifted, asshownin | | Although the pure π-triplet pairing state cannot be sta- Fig.2,fromdottedlinetobrokenline. Itisseenthatthe bilized, π-triplet pair can condensate as a result of the long-rangeCoulombinteractiondoesleadtothesuppres- coexistence of dSC and SDW. Since there is attractive sion of the PS, which is less prominent in the coexistent interactionfor spin-triplet channelin the presentmodel, state than in the dSC state. thecoexistentregionofdSCandSDWiswidenedbythe inclusion of π-triplet pair. We note that the Fermi sur- = + face remains in the SDW state near half filling. As we saw in I, when SDW appears first as the temperature is lowered, the coexistent state can be stabilized at lower temperature, especially at T =0, near half filling. Fig. 3. TheRPAdiagramforchargesusceptibility. n=0.9, t'/t=-1/5 4 Thecalculationofκinthe RPAhadbeencarriedover 3.5 by Micnas et al.,2) only in the normal state. Dagotto et 3 dSC+AF+p- triplet al.3) haveshownbasedonquantumMonteCarlo(QMC) 2.5 AF simulationthat(1)PSdrasticallyreducesthe sizeofthe 2 meanfielddSCregionand(2)theenhancementofdx2−y2 1.5 pairing correlation itself is not found. This QMC result of(2)isdifferentfromthepresentmeanfieldcalculation. 1 dSC N WenotethatthisQMCcalculationislimitedtothecase 0.5 ofhalf filling n=1 and relativelyhightemperatureT = 0 0 0.5 1 1.5 2 2.5 t/6. |V|/t The coexistence of dSC and SDW near half filling has beenfoundalsointhet-J modelintheslave-bosonmean Fig. 2. ThemeanfieldphasediagramatT =0forU >0,V <0, field approximation11) and by use of variational Monte n = 0.9 and t′/t = −1/5. Solid line stands for the boundary Carlo calculation (VMC),12,13,14) and in the repulsive between the coexistent state and dSC state in the mean field Hubbard model (V = 0) by use of VMC.13,15) In these approximation. Dotted and broken lines stand for the bound- studies, however, π-triplet pair has not been taken into ary in the right side of which the phase separation occurs in therandomphaseapproximationforV′=0andV′=|V|/2,re- account. Theeffectofπ-tripletpaironthecoexistenceof spectively,whereV′isthenext-nearest-neighbordensity-density dSC and SDW has been recently examined by Arrachea interaction. N stands forthenormalstate. et al.9) based on a generalized Hubbard model. In the repulsive Hubbard model, the nearest-neighbor hopping term is modified as the correlatedone, Generally, in the presence of finite-range attractive dbyentshitey-pdheansseityseipnatrearaticotnion(P,tSh)e. Isnysotermdercatnobeexahmaminpeerthede Hch =−<Xij>σnc†iσcjσ +c†jσciσo PS transition, we calculate the charge compressibility, t (1 n )(1 n )+t n n AA iσ jσ BB iσ jσ κ, given by static and uniform charge susceptibility, in ×{ − − +t [n (1 n )+(1 n )n ] , (3.5) the ground state, i.e., dSC or coexistent state. This AB iσ jσ iσ jσ − − } phase boundary is determined from κ−1 = 0. Here we where three hopping integrals,t , t and t , incor- AA BB AB use the random phase approximation (RPA), and take porate many-body effects into one-particle hopping pro- the RPA diagram, shown in Fig. 3, into account. The cesses phenomenologically. t and t do not change AA BB explicit form of κ in the RPA is shown in Appendix the number of doubly occupied sites, and t does, as AB B. The PS transition line is shown in Fig. 2 by dot- 4 MasakazuMurakami showninFig.4. ItistobenotedthatH canberewrit- and spin on a bond (i,i+ρˆ), ch ten as <jc >= <jc > gcos(QR ), (3.8a) H = c† c +c† c i,i±xˆ − i,i±yˆ ∝ i ch iσ jσ iσ jσ <js >= <js > lcos(QR ), (3.8b) <Xij>σn o i,i±xˆ − i,i±yˆ ∝ i t+t (n +n )+t n n , (3.6a) where 2 iσ jσ 3 iσ jσ ×{− } where jiν,i+ρˆ≡i vν(c†iσci+ρˆ,σ−c†i+ρˆ,σciσ), (3.9) Xσ t t , t t t , t 2t t t . (3.6b) ≡ AA 2 ≡ AA− AB 3 ≡ AB− AA− BB and vc = 1 and vs = σ. In the OAF (SN) state, the Thet termcanbealsodeducedfromthebareCoulomb localstaggeredcurrentofcharge(spin)circulatesaround 2 interaction16,17,18) or by including the effects of phonon the plaquettes, as schematically shown in Fig. 5, and in the antiadiabatic approximation M 0 (where M the bond-ordered wave (BOW) does not exist, i.e., < is the phonon mass),19) and the t3 term→describes the c†iσci+ρˆ,σ+c†i+ρˆ,σciσ >≡0. three-body interaction. Arrachea et al. have shown in the meanfield approximationfort >t =t (i.e., AB AA BB t <0andt = 2t >0)andt′ =0thatthecoexistence 2 3 2 − of dSC and SDW is possible but prevented by π-triplet pair (and ruled out for large U), due to repulsive spin- triplet pairing interaction. Hence, the effect of π-triplet paironthecoexistenceofdSCandSDWisdifferentfrom that in the present extended Hubbard model. Fig. 5. TheOAF(SN)state inwhichthelocalstaggeredcharge i j i j (spin)currentcirculates aroundtheplaquettes. The order parameters in the mean field Hamiltonian t t are given as follows in terms of f, p, m, g and l, AA BB i j i j ∆ =Uf, (3.10a) FM ∆ =(8V U)p, ∆ =Um, (3.10b) CDW SDW − ∆ =2Vg, ∆ =2Vl, (3.10c) OAF SN t t AB AB where ∆ and ∆ include only V because g and OAF SN ρˆ l are defined on a bond, and ∆ does not include ρˆ FM Fig. 4. Thecorrelatedhoppingprocesses. V because < n >= < n > is independent of f. i σ iσ The order parametersPof CDW, SDW, OAF and SN are closelyrelatedtoeachother,whichisshowninAppendix A. 3.2 Nearest-Neighbor Repulsion V >0 ThemeanfieldphasediagramintheplaneofU andV Next we treat the case V > 0. In this case, not only isshowninFig.6. Wehavefoundthatacoexistentsolu- charge- and spin-density-wave states (CDW and SDW), tion with nonzero ∆ , ∆ and ∆ can be sta- CDW SDW FM but also orbital antiferromagnetic (OAF) and spin ne- bilizedforn=1. Thisstateisferrimagnetic,asshownin 6 matic (SN) states, in which the local staggered currents Fig. 7. We note that the coexistence of CDW and SDW of charge and spin circulate, respectively,20,21,22,23) are (∆ =0and∆ =0)generallyresultsinnonzero CDW SDW expected.24) Here we include ferromagnetism (FM) for f (andno6 nzero∆ her6e),whichhadbeenindicatedby FM the reason as we shall describe later. The order param- Dzyaloshinski˘ı25) based on a qualitative symmetry anal- eters are ysis. This is the reason why we take FM into account n from the outset. With the present choice of parameters, <n > +σf +(p+σm)cos(QR ), (3.7a) iσ i ≡ 2 pure FM state with only ∆FM =0 cannotbe stabilized, 6 but FM can arise as a result of the coexistence of CDW <c† c > (g +σl )cos(QR ), (3.7b) and SDW. In the present case, the coexistent region of iσ i+ρˆ,σ ≡ ρˆ ρˆ i CDW and SDW is widened by the inclusion of FM. We wheref,p andm arereal,g = g∗ andl = l∗. f, −ρˆ − ρˆ −ρˆ − ρˆ notethattheFermisurfaceremainsintheCDWorSDW pandmdescribesFM,CDWandSDW,respectively. We state near half filling. take gρˆ and lρˆ to be of dx2−y2wave symmetry, which is It is to be noted that neither OAF nor SN can been favorednearhalffilling.24) Inthiscase,g andl become ρˆ ρˆ stabilized solely in the mean field approximation, inde- pure imaginary, g = g = ig and l = l = il, ±xˆ − ±yˆ ±xˆ − ±yˆ pendentoft′,U,V,T andn. Thisconclusioniscontrary where g and l are real. The states with g =0 and l =0 6 6 to that ofChattopadhyayet al.24) that pure OAFor SN are called as OAF and SN ones,20,21,22,23) respectively, state has lower ground-state energy than pure CDW or inwhichthereexistthestaggeredlocalcurrentsofcharge SDW state for the half-filled case by introducing finite PossibleOrderedStates inthe2DExtended HubbardModel 5 t′. Moreover, a state where local-current (OAF or SN) U/t=4.0,V/t=1.5,n=0.9,t'/t=-1/5 anddensity-wave(CDWorSDW)coexistcannotbealso stabilized. n=0.9, t'/t=-1/5 8 6 7 6 5 SDW+CDW+FM 4 SDW 0 3 2 (0,0) (p, 0) (p /2,p /2) (0,0) 1 N CDW 0 0 0.5 1 1.5 2 Fig. 9. The energy dispersion relative to the Fermi level in the V/t coexistentstatewithCDW,SDWandFMforU/t=4.0,V/t= 1.5, n=0.9andt′/t=−1/5. Full(dotted) linesstand forthat of electrons with up (down) spin, respectively. The lower band Fig. 6. ThemeanfieldphasediagramatT =0forU >0,V >0, ofelectrons withupspinisfullyoccupied. n=0.9andt′/t=−1/5. ∆θ =0, and the magnitude of local spin moment is dif- ferent at each sublattice, as shown in Fig. 7. This ferri- magneticcoexistentstateisthe2Dversionofthatfound in the 3D Hubbard model (V = 0) which had been de- noted as the special ferrimagnetic (S.F.) state.27) In the present 2D case, this coexistent state for V = 0 can be stabilized for 12<U/t<14 (not shown in Fig. 6). ∼ ∼ Fig. 7. The ferrimagnetic coexistent state. Each lattice site is shaded according to electron density. The length of each arrow is proportional to the magnitude of local spinmoment. Lattice Fig. 10. The coexistent state with 2kF CDW and 2kF SDW siteswithlargerlocalspinmomenthavehigherelectrondensity. found in a quarter-filled 1D modified Hubbard model.26) Each latticesiteisshadedaccordingtoelectrondensity. Thelengthof eacharrow,proportionaltothemagnitudeoflocalspinmoment, For U/t = 4.0, the V dependences of ∆ , ∆ , isequalateachsite. CDW SDW ∆ and the difference between the energy of the pure FM state (CDW for U < 4V or SDW for U > 4V), E , and p that of the coexistent state (CDW+SDW+FM), E , are c §4. Renormalization Group Analysis showninFig.8. InthecoexistentstatewithCDW,SDW and FM, ∆ is larger than ∆ for U > 4V Inthelastsection,weexaminedpossibleorderedstates SDW CDW | | | | and vice versa for U < 4V, and the first-order phase for U > 0 and V = 0 in the mean field approxima- 6 transitionoccursatU =4V. ForfixedU, ∆ rapidly tion. In this section, we examine the effects of fluctu- FM | | saturates as a function of V. It is seen that the energy ation on these ordered states which are not taken into gain in the coexistent state is very small. account in the mean field calculation. As a theoret- The energy dispersion in the ferrimagnetic coexistent ical treatment beyond the mean field level, we adopt state is shown in Fig. 9. There are four energy bands, the renormalization group (RG) method for the saddle and the Fermi surface remains as in the CDW or SDW points which has been applied to the Hubbard model state. However,the lower band of electrons with major- (V =0),28,29,25,30,31,32,33)anddeterminethemostdom- ity spin (up spinfor ∆ >0)is fully occupiedandthe inantcorrelationinthenormalstate. Wealsodiscussthe FM Fermilevelcrossesonly the lowerbandof electronswith possibilityofthecoexistentstatesbeyondthemeanfield minority spin (down spin for ∆ >0). Therefore,this approximation. FM coexistent state is half metallic. The coexistence of CDW and SDW with same wave 4.1 Saddle Point Singularity vectors has also been found in a 1D modified Hubbard We consider the special case where the Fermi level in modelforaquarter-filledbandinthe meanfieldapprox- the absence of interaction lies just on the saddle points imation.26) In this coexistent state, the wave vector of QA (π,0) and QB (0,π), i.e., µ = 4t′ (n 0.83), ≡ ≡ ∼ charge and spin density, q, and the phase difference be- and focus on electrons at these two saddle points on the tween CDW and SDW, ∆θ, are equal to 2k π/2 and Fermi surface, just as two Fermi points in 1D electron F ≡ π/2, respectively, and the magnitude of local spin mo- systems. The Fermi surface is shown in Fig. 11. ment is equal at each site, as shown in Fig. 10. On the other hand, in our coexistent state, q = Q (π,π) and First, we examine the behavior of the following ≡ 6 MasakazuMurakami U/t=4.0, n=0.9, t'/t=-1/5 U/t=4.0, n=0.9, t'/t=-1/5 1.5 0.015 1 /tc 0.01 /t CDW p SDW 0.5 FM 0.005 0 0 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 V/t V/t (a) (b) Fig. 8. The V dependences of (a) ∆CDW, ∆SDW ∆FM and (b)the energy difference Ep−Ec (scaled by t) between the pure state (CDWorSDW)andthecoexistent state(CDW+SDW+FM)forU/t=4.0,n=0.9andt′/t=−1/5. p t'/t=-1/5, m= 4t' logarithmically divergent, c E c E K log2 c, P log c, (4.5a) 1 ∼ 8π2t ω 1 ∼−4π2t ω py c′′ logEc for ω rE , K2 ∼(cid:26) 4π2t P1 ω for ω ≪rEcc, (4.5b) − ≫ c′ logEc for ω rE , 0 px p P2 ∼(cid:26) −4π2tK1 ω for ω ≪rEcc, (4.5c) − ≫ where E > 0 and ω > 0 (ω E ) are the ultraviolet Fig. 11. TheFermisurfaceintheabsenceofinteractionfort′/t= c ≪ c −1/5andµ=4t′,inwhichcasen∼0.83. and infrared energy cutoff, respectively, 1 c , (4.6a) ≡ √1 4r2 particle-particle (K) and particle-hole (P) correlation − 1+√1 4r2 functions, c′ log − , (4.6b) ≡ 2r Kαα′ =ql→im0Zk,ǫGα(k,ǫ)Gα′(−k+q,−ǫ), (4.1a) c′′ ≡ 21r arctan(√12r4r2), (4.6c) − Pαα′ =ql→im0Zk,ǫGα(k,ǫ)Gα′(k+q,ǫ), (4.1b) inanFdigr.≡12|.t′E|/stp.ecc,iacl′lyanfodrcs′′maasllarf,unction of r are shown for α,α′ =A,B, where c,c′′ 1, c′ logr. (4.7) ∼ ∼− 1 Gα(k,ǫ)≡ iǫ−ξQα+k, (4.2) tFhoarnrP>2 forcr ω∼ 0.207.6W, ce n>otce′ca′′n<dmPa1xisc,mc′or,ei.de.i,veπr-gpeanirt → { } is the one-particle Green function in the absence of in- susceptibility K is always less divergent than particle- 2 teraction for electrons near the saddle point Qα and holesusceptibility. Fort′/t= 0.2,c′′ <c<c′ andthey − d2k dǫ are comparable in magnitude. , (4.3) Z ≡Z (2π)2 Z 2π k,ǫ |k|<kc where the cutoff around the saddle points, k , is intro- 4.2 Renormalization Group Method for the Saddle c duced. We note that Points In the last subsection, we saw that the saddle points K K =K , K K =K , (4.4a) 1 ≡ AA BB 2 ≡ AB BA on the Fermi surface lead to logarithmic divergence of stand for Cooper- and π-pair correlation, respectively, particle-particle and particle-hole correlation functions. and This implies that the fluctuation effect becomes strong. IntheRGapproach,weassumethatthesinglerenormal- P1 ≡PAA =PBB, P2 ≡PAB =PBA, (4.4b) izationgroupvariablex≡logEωc determinethebehavior foruniformandstaggereddensity-densitycorrelation,re- of the system. The increase of x represents renormal- spectively. For µ = 4t′, these correlation functions are ization towards lower energy scale. For simplicity, we neglect (1) the deformation of the Fermi surface by in- PossibleOrderedStates inthe2DExtended HubbardModel 7 3 g˙ = (g2 +g2 )K˙ 4⊥ − 3⊥ 4⊥ 1 2.5 [g2 +g2 +2g (g g )]P˙ , (4.8g) − 1⊥ 4⊥ 2⊥ 1k− 2k 1 2 C' C g˙ = (g2 +g2 )K˙ 4k − 3k 4k 1 1.5 [g2 +(2g2 g2 )+2g (g g ) C'' − 1k 4k− 4⊥ 2k 1k− 2k 1 (g2 g2 )]P˙ , (4.8h) − 2⊥− 2k 1 0.5 whicharetobesolvedwiththeinitialconditionsg (x= is 00 0.050.1 0.150.2 0.r250.3 0.350.4 0.450.5 xi) = gi0s ( ˙ ≡d/dx), where gi0s are the bare coupling constants. For example, g has the following form to 3⊥ one loop order, Fig. 12. c,c′ andc′′ asafunctionofr. g = 2g0 g0 K 2g0 (g0 +g0 g0 )P . (4.9) 3⊥ − 3⊥ 4⊥ 1− 3⊥ 2⊥ 2k− 1k 2 By differentiating this equation by x and replace g0 by is teraction and (2) k-dependence of interaction for small g ,i.e.,the barecouplingconstantsbythe renormalized is k < kc, i.e., we consider only eight coupling constants, ones, we obtain the scaling equation eq. (4.8e). | | gis (i = 1,2,3,4 and s = , ). This interaction of the Ifwetakeg0 =g0 astheinitialconditions,therelation ⊥ k i⊥ ik g-´ology type is shown in Fig. 13. g1 and g3 (g2 and g4) gi⊥ = gik holds all through the flow. Therefore, the stand for the backward (forward) scattering processes above scaling equations are simplified as follows,33) with large (small) momentum transfer, respectively. Es- pecially, g1 and g3 describe ’exchange’ and ’Umklapp’ g˙1 =−2g1g2K˙2−2g1g4P˙1−2g1(g2−g1)P˙2(,4.10a) processes. InI,onlyg1⊥ andg3⊥ weretreatedandtaken g˙ = (g2+g2)K˙ +2g (g g )P˙ tobemomentum-independentalloverthemagneticBril- 2 − 1 2 2 4 2− 1 1 louin zone.1) −(g22+g32)P˙2, (4.10b) g˙ = 2g g K˙ 2g (2g g )P˙ , (4.10c) 3 3 4 1 3 2 1 2 − − − g˙ = (g2+g2)K˙ 4 − 3 4 1 [g2+g2+ 2g (g g )]P˙ . (4.10d) − 1 4 − 2 2− 1 1 where g g =g . i i⊥ ik ≡ Thedivergenceofg (x)atafinitexindicatestheexis- is tence of the strong coupling fixed point, i.e., signals the development of an ordered state, at finite energy scale or finite temperature. (Strictly speaking, this finite on- set temperature is an artifact of the present approxima- tion in the 2D systems, and should be interpreted as a Fig. 13. The scattering processes. Solid and dashed lines stand crossovertemperature,oracriticaltemperaturewhenfi- forelectrons nearQA=(π,0)andQB=(0,π),respectively. nite three-dimensionalityis assumed.) The properties of this strong coupling fixed point can be obtained quali- tatively from various response functions. The response We start with the renormalization of the couplings functions in the one-loop approximation are obtained in the one-loop approximation. One-loop diagrams are from one-loop diagrams shown in Fig. 15. The response shown in Fig. 14. The scaling equations are function, g1˙⊥ = 2g1⊥g2⊥K˙2 2g1⊥g4⊥P˙1 β 1 − − R = dτei0τ <T Oˆ†(τ)Oˆ >, (4.11) +2g (g g )P˙ , (4.8a) ν Z · N τ ν ν 1⊥ 1k 2k 2 0 − g˙ = 2g g K˙ 2g g P˙ where ν stands for the kind of correlation (ν =dSC, 1k 1k 2k 2 1k 4k 1 − − SDW, ), has the following form to one-loop order, +[2g (g g )+(g2 g2 ) ··· 1k 1k− 2k 1⊥− 1k 1 R =R0 + g0(R0)2, (4.12) +(g32⊥−g32k)]P˙2, (4.8b) ν ν 4 ν ν g˙ = (g2 +g2 )K˙ 2g (g g )P˙ where gν0 is the coupling constant (linear combinationof 2⊥ − 1⊥ 2⊥ 2− 4⊥ 1k− 2k 1 g0) and R0 is the simple bubble. If we differentiate this is ν −(g22⊥+g32⊥)P˙2, (4.8c) equation by x and replace gν0, Rν0 by the renormalized g˙ = (g2 +g2 )K˙ 2(g g g g )P˙ ones, gν and Rν, we obtain 2k − 1k 2k 2− 4k 1k− 4⊥ 2⊥ 1 1 −(g22k+g32k)P˙2, (4.8d) R˙ν =R˙ν0(cid:26)1+ 2gνRν(cid:27). (4.13) g3˙⊥ = 2g3⊥g4⊥K˙1 2g3⊥(g2⊥+g2k g1k)P˙2(,4.8e) This equation is to be solved with the initial condition − − − g˙ = 2g g K˙ 2(2g g g g )P˙ , (4.8f) that Rν(x = xi) 0. Since R˙ν0 is positive, Rν can 3k − 3k 4k 1− 3k 2k− 3⊥ 1⊥ 2 ∼ 8 MasakazuMurakami be divergent for g > 0 and are suppressed to zero for ν g <0. Inthispaper,weconsidertheresponsefunctions ν shown in Fig 16, where Oˆ , Oˆ and Oˆ stand for s- sSC η PS waveCooper-pair,η-singletpairwithatotalmomentum Q and total spin S = 0,34,35) and uniform charge den- sity,respectively. ThemostdivergentR isinterpreted PS to describe the phase separation (PS). The relationship among these order parameters is discussed in Appendix A. We note that each correlation is treated indepen- dently in the above procedure. Therefore, we can de- termine the most dominant susceptibility in the normal state, and cannot assess the coexistence of different or- ders. Fig. 16. Operator Oˆν, bareresponse function R0ν(>0)and cou- plingconstantgν foreachcorrelationν. Eachresponsefunction canbedivergent(orthereexistsmean-fieldsolution<Oˆν >6=0) for gν > 0. g is defined as g ≡ g1k−g2k. In the RG method where only electrons near the saddle points are taken into ac- count, the sum over p is restricted to p∼(π,0), (0,π) and the (a) (b) p-dependence ofthe dx2−y2-wavefactor wp∝cospx−cospy is ignored. Inourcalculation,wetake wp=sgn(cospx−cospy). can be rewritten as follows, U U H = n n n . (4.15) U i i i 2 − 2 Xi Xi (c) (d) If we regardthe secondterm in the r.h.s of eq. (4.15) as the chemical potential shift, we can take g0 = g0 = U i⊥ ik Fig. 14. Diagrams contributing to the one-loop order correction as the initial conditions and therefore use eq. (4.10) tocouplingconstants. as the scaling equations of the coupling constants. For the perfect nesting case r t′ /t = 0, Schulz28) and ≡ | | Dzyaloshinski˘ı25) showedthat SDW occurs,and pointed out that small deviations from half filling lead to dSC. Ledereretal.29) andFurukawaetal.33) solvedthescaling equationsforr 1andthesameresults. Especially,Fu- ≪ rukawaetal.33) indicatedthatthecorrelationofπ-triplet pairissuppressedtozero. Inthesecalculations,however, a b P andK areneglectedineq. (4.10)forthereasonthat 1 2 they are less singular than P , i.e., c,c′′ c′ for r 1, 2 ≪ ≪ Fig. 15. Diagrams contributing to the one-loop order correction in eq. (4.5) and (4.6). On the other hand, Alvarez et toresponsefunctions. al.31) solvedthe flowequationswiththe initialcondition g0 = U and g0 = 0, by neglecting the generation of i⊥ ik g andomitting particle-particlediagramK andK in ik 1 2 eq. (4.8), and showed that dSC, SDW and FM can be stabilized depending on U/t and r. 4.3 Phase Diagram Now,weshowthephasediagramintheplaneofU >0 We solve the scaling equations, eq. (4.8) and (4.13), and V for t′ = 1/5 in Fig. 17, in which the ordered with the initial conditions, state with highe−st onset temperature is shown. First, g0 =g0 U +2V =U 4V, (4.14a) we discuss the case that each ordered state is treated 1⊥ 3⊥ ≡ Q − independently, because we cannot examine the coexis- g20⊥ =g40⊥ ≡U +2V0 =U +4V, (4.14b) tence of different orders in the RG method. The crucial g0 =g0 2V = 4V, (4.14c) difference from our mean field result is that supercon- 1k 3k ≡ Q − ductivity appears even for U > 0 and V 0, i.e., dSC g0 =g0 2V =4V, (4.14d) for U > 4V and sSC for U < 4V. This re≥sult that dSC 2k 4k ≡ 0 is possible for small V > 0 as well as V = 0 near half at x = x . Here, we take x 0 and R (x ) 0 for i i ≡ ν i ≡ filling is consistent with a recent calculation based on simplicity,althoughthe solutionofthe scalingequations the fluctuation-exchange (FLEX) approximation.36) Ex- depends on the value of x and R (x ). i ν i ceptforsuperconductivity forU >0andV 0,the RG Beforeweshowourresults,werefertopreviousresults ≥ phasediagramisqualitativelysameasthemeanfieldone for U > 0 and V = 0 obtained by many authors. H U when we do not take the coexistence of different orders into account, i.e., SDW and CDW appear for U > 4V and U < 4V, respectively, and attractive V < 0 favors PossibleOrderedStates inthe2DExtended HubbardModel 9 dSCforsmall V andPSforlarge V ,respectively. With in the presentextended Hubbard model. Therefore,this | | | | regardto the correlationofπ-triplet pair,our RGcalcu- coexistent state might be expected to be stabilized also lation has shown that it can be divergent for attractive in the extended Hubbard model for U <0 and V >0 in V < 0 and large V but is always subdominant. Sim- the mean field approximation. Moreover, based on the | | ilarly, FM cannot be the most dominant solely. These above discussion, it might be expected to survive in the results are also consistent with our mean field ones. presenceoffluctuationfornotonlyU <0butalsoU 0, ≥ Next, we discuss the possibility of the coexistence of atlowertemperatureintheCDWregioninFig.17. This different orders at low temperature, especially at T =0. will be reported elsewhere. It is very important that our RG calculation shows the Finally, we refer to the ambiguities of the above RG existence of a region where the onset temperature of method. Since there exist not only log- but also log2- SDW or CDW becomes highest, as shown in Fig. 17. divergence in the particle-particle and particle-hole cor- Since our mean field calculation in 3.1 or 3.2 shows relation functions, eq. (4.5), we cannot safely take the § § that the Fermi surface remains in the SDW and CDW limit ω 0 in the scaling equations of coupling con- → states near half filling, we might expect to find a sec- stants and response functions, eq. (4.8) and (4.13), i.e., ond phase transition at lower temperature in such SDW it is not clear at all whether the above RG treatment is and CDW states. Therefore, at lower temperature in valid or not. In fact, the solution of eq. (4.8) and (4.13) the SDW regionin Fig. 17, (1) the coexistent state with depends on the initial value x . If we consider only the i dSC, SDW and π-triplet pair found for V < 0 in the most singular log2 term in K (and P for r = 0) and 1 2 mean field approximation might be expected to survive take y x2 = log2 Ec as a new scaling variable,28) we ≡ ω fornotonlyV <0butalsoV 0,and(2)the ferrimag- can safely take ω 0 limit in the scaling equations of ≥ → netic coexistent state with CDW, SDW and FM found coupling constants. In this case, the above RG method for U > 4V > 0 in the mean field approximation might mightcorrespondtoaparquetsummationofleadinglog2 be expected to survive. In fact, as we have pointed out divergences, rather than renormalizationprocedure.37) in 3.1,VMCcalculationsforU >0andV =0showthe § §5. Conclusion and Discussion coexistence of dSC and SDW at low temperature,13,15) although π-triplet pair has been neglected. Similarly, at We have studied in detail possible ordered states, es- lowertemperatureintheCDWregioninFig.17,(3)the peciallypossiblecoexistenceofdifferentorders,nearhalf ferrimagneticcoexistentstatewithCDW,SDWandFM fillinginthe2DextendedHubbardmodelwithon-sitere- found for 4V > U > 0 in the mean field approximation pulsion U >0 and nearest-neighbor interaction V, with might be expected to survive. Especially, π-triplet pair emphasisonelectronsaroundthesaddlepoints(π,0)and (FM), which cannot be stabilized solely in the parame- (0,π). ter region considered in the present mean field and RG First,wehavedeterminedthephasediagramatT =0 approximation,might be expected to arise as a result of in the mean field approximation. For V < 0, we have the coexistence of dSC and SDW (CDW and SDW). In shown that the coexistent state with dSC, SDW and π- order to assess the effect of π-triplet pair (FM) on the triplet pair can be stabilized near half filling. Here, we coexistenceofdSCandSDW(CDWandSDW),weneed have indicated the following important fact which has another theoretical treatment. often been neglected in previous studies: when we dis- cuss the coexistence of dSC and SDW, it is necessary t'/t=-1/5 to take π-triplet pair into account from the outset, be- 4 cause in general the coexistence of dSC and SDW results 3.5 in π-triplet pair and is affected by π-triplet pair.1) Espe- 3 SDW cially, π-triplet pair, which cannot condensate solely in 2.5 the present model, can arise through the coexistence of U/t 2 dSC CDW dSC and SDW. Since the phase separation (PS) is gen- 1.5 erally expected to occur in the presence of finite-range PS 1 attractive interaction such as V < 0, we have examined theeffectofPSonthemeanfieldgroundstateintheran- 0.5 sSC dom phase approximation (RPA). The coexistent state 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 with dSC, SDW and π-triplet pair is severely hampered V/t by PS but survives, and that the long-range Coulomb repulsion such as next-nearest-neighbor density-density Fig. 17. TheRGphasediagramfort′/t=−1/5andµ=4t′. repulsion leads to the suppression of PS. On the other hand, for V > 0, we showed that a ferrimagnetic co- existent state with commensurate charge-density-wave Here, we refer to the possibility of the coexistence of (CDW), SDW and ferromagnetism (FM) can be stabi- sSC and CDW. In the g-´ology model in I, it can be eas- lized near half filling. Here, we have indicated the fol- ily shown that the coexistent state with sSC, CDW and lowing important fact: when we discuss the coexistence η-singlet pair can be stabilized near half filling at low of CDW and SDW, it is necessary to take FM into ac- temperatureforg <0andg <0.6) Asseenfromeq. count from the outset, because in general the coexistence 3⊥ 1⊥ (2.2), this case corresponds to that of U <0 and V >0 of CDW and SDW results in FM and is affected by FM. Especially, FM, which cannot be stabilized solely with 10 MasakazuMurakami the present choice of parameters, can arise through the Appendix A: Symmetry among Various Order coexistenceofCDWandSDW.Itistobenotedthatthe Parameters abovemeanfieldcoexistentstatesnearhalffillingcanbe We examine the relationship among the order param- stabilizedat low temperature, especially atT =0,when eters as shown in Fig. 16. First, the operators of dSC, CDW or SDW, in which the Fermi surface remains near SDW and π-triplet pair are equivalent in that they are half filling, arises first at high temperature. ’rotated’ into each other, In order to examine the effects of fluctuation on the mean field ordered states, we have adopted the RG [OˆSDW,OˆdSC]=2Oˆπ, [Oˆπ†,OˆSDW]=2Oˆd†SC, (A.1a) method for the special case that the Fermi level lies just on the saddle points. We have shown that the crucial [Oˆ ,Oˆ†]=2Oˆ . (A.1b) dSC π SDW difference from our mean field result is that supercon- This relationship underlies the SO(5) theory, in which ductivity can arise even for U > 0 and V 0; dSC ≥ dSC and SDW are unified into a five-dimensional vec- and sSC for U > 4V and U < 4V, respectively. Except tor superspin and Oˆ describes excitation towards the forthisdifference,theRGphasediagramisqualitatively π SDW(dSC)directioninthedSC (SDW)groundstate.8) same as the mean field one when we do not take the co- We note that eq. (A.1b) holds due to w2 = 1, i.e., existenceofdifferentordersintoaccount,e.g., SDWand p w sgn(cosp cosp ). If w cosp cosp , it CDW can arise for U > 4V and U < 4V, respectively. p x y p x y ≡ − ∝ − is satisfied only in the long wavelength limit where only Especially,thecorrelationofπ-tripletpairorFMcannot electronsnearthesaddlepointsareimportant. Itisvery be the most dominant solely. Here, it is very important important to note that if two of the above three order that a region where the onset temperature of SDW or parameters coexist, another one also results generally. CDW becomes highest is found in the RG phase dia- As we have already shown in the mean field approxima- gram. Since the Fermi surface remains near half filling tion, the coexistent state with dSC, SDW and π-triplet intheseSDWandCDWstates,wemightexpecttofinda paircanbestabilizednearhalffillinginthe2Dextended secondphasetransitionatlowertemperature. IntheRG Hubbard model for U >0 and V <0. method,however,wecannotassesssuchpossibilities. On Next, the operators of sSC, CDW and η-singlet pair the other hand, the mean field approximation, which is are equivalent in that they are ’rotated’ into each other, often questionable for the 2D case, is of greatadvantage in that we can study the stability of coexistent states with different order parameters quantitatively. There- [Oˆ ,Oˆ† ]=2Oˆ†, [Oˆ ,Oˆ ]=2Oˆ , (A.2a) CDW sSC η η CDW sSC fore, we can conclude that our mean field calculations indicate the possibilities that (1) SDW, which has been [Oˆ† ,Oˆ ]=2Oˆ , (A.2b) showninourRGcalculationtoarisefirstathightemper- sSC η CDW ature for U > 4V, coexists with dSC and π-triplet pair, This relationship underlies the SO(3) theory, in which or with CDW and FM, at lower temperature, and that sSC and CDW are unified into a three-dimensional vec- (2) CDW, which has been shown in our RG calculation tor pseudospin and Oˆ describes excitation towards η to arise first at high temperature for U < 4V, coexists the CDW (sSC) direction in the sSC (CDW) ground with SDW and FM, at lower temperature. state.34,35) In fact, such properties are useful in the at- Throughout this letter, we have assumed YBCO-type tractive Hubbard model. It is very important to note Fermi surface by introducing t′ and consider only com- that if two of the above three order parameters coexist, mensurate(C)SDWorCDW.Nearhalffilling,however, anotheronealsoresultsgenerally. Itcanbeeasilyshown incommensurate(IC)orderingorstripeformationcanbe in the meanfield approximationthata state where sSC, expected, especially for t′ = 0 in the repulsive Hubbard CDW and η-singlet pair coexist can be stabilized near model.38,39,40) Recently, the effect of V on such stripe half filling in the g-´ology model used in I for g < 0 3⊥ states has been examined.41) The effect of IC ordering and g <0.6) 1⊥ on the stability of the coexistent states (with dSC, C- We note that similar relationships hold among dSC, SDW andπ-triplet pair,and with C-CDW, C-SDW and OAF and η-singlet pair, FM) is beyond the scope of the presentstudy. In the re- pulsive Hubbard model with t′ = 0, Giamarchi et al.42) [Oˆη†,OˆdSC]=2iOˆOAF, [OˆOAF,Oˆη]=2iOˆdSC, (A.3a) have shown that the coexistent state with dSC and C- SDW state have higher energy than that with dSC and [Oˆ† ,Oˆ ]=2iOˆ†, (A.3b) dSC OAF η IC-SDW. and among sSC, SN and π-triplet pair, With regard to dSC, we have assumed that the su- perconducitng gap symmetry is purely of dx2−y2-wave. [Oˆs†SC,Oˆπ]=2iOˆSN, [OˆsN,OˆsSC]=2iOˆπ, (A.4a) However, there are a few indications that dSC mixed with components of other symmetry can be stabi- [Oˆ†,Oˆ ]=2iOˆ† . (A.4b) lized,5,43,44,45) dependent on interaction, electron den- π SN sSC sity, etc. Suchmixed pairingstates leavemuch roomfor If w cosp cosp , eq. (A.3b) and (A.4b) hold p x y ∝ − future studies. only approximately. In each case, if two of the above TheauthorthankstoH.Fukuyama,H.KohnoandM. three order parameters coexist, another one also results Ogata for valuable discussions. generally. It can be easily shown in the mean field ap-