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One-dimensional Hubbard model at quarter filling on periodic potentials C. Schuster and U. Schwingenschlögl Institut fu¨r Physik, Universität Augsburg, 86135 Augsburg, Germany (Dated: February 6, 2008) UsingtheHubbardchainatquarterfillingasamodelsystem,westudythegroundstateproperties of highly doped antiferromagnets. In particular, the Hubbard chain at quarter filling is unstable 7 against2kF-and4kF-periodicpotentials,leadingtoalargevarietyofchargeandspinorderedground 0 states. Employingthedensitymatrixrenormalizationgroupmethod,wecomparetheenergygainof 0 thegroundstateinducedbydifferentperiodicpotentials. Forinteractingsystemsthelowest energy 2 is found for a 2kF-periodic magnetic field, resulting in a band insulator with spin gap. For strong n interaction, the 4kF-periodic potential leads to a half-filled Heisenberg chain and thus to a Mott a insulating state without spin gap. This ground state is more stable than the band insulating state J caused by any non-magnetic 2kF-periodic potential. Addingmore electrons, a cluster-like ordering is preferred. 1 1 PACSnumbers: 71.10.Fd,71.10.Pm Keywords: Theoriesandmodelsofmanyelectronsystems,electronsinreduceddimensions ] l e r- I. HOLE DOPED ANTIFERROMAGNETS 8 t s . 7 t The study of low-dimensional doped antiferromagnets a m was triggered by investigations of the phase diagram of 6 high temperature superconductors. Upon doping, they - d show a transition from an antiferromagnetic to a super- ) 5 V n conducting state. On the other hand, the investigation e / o of one-dimensional compounds is also an active field. A 1 4 ( c simple form of low doping in a Cu-chain is the replace- S [ O ment of a few Cu ions by non-magnetic ions like Zn. In D 3 1 this case, the system is well described by the Heisen- v berg model with some spins removed [1]. The charge 2 5 and spin order of highly doped compounds, on the con- 3 trary, are not understood by far. A prototypical ma- 1 2 terial revealing strongly doping dependent charge order 1 0 and antiferromagnetism in a less than half-filled band is 0 7 the spin chain compound (Ca,Sr)14Cu24O41. This com- -7 -6 -5 -4 -3 -2 -1 0 1 0 posite crystal contains two different structural compo- (E - E ) (eV) t/ nents along the c-axis: Cu2O3 ladders and CuO2 chains. F a Whereas the ladders contain only Cu2+-ions, nearly all m Cu3+-ions are located on the chains [2]. Substitution of FIG.1: (Color online) Partial Cu3d densityof statesforthe CuO2-chains of Sr14Cu24O41. - calciumonthestrontiumsitesleadstoatransferofholes d from the chains into the ladders. For the Ca-rich com- n o pounds one generally assumes one hole per formula unit c on the ladders and five on the chains. Near the quarter- : filled band antiferromagnetic order is established [3]. v i DFT-LDAcalculations[4],usingtheASWmethod[5], X reveal that the ladder and chain substructures can be r treated individually. In particular, the density of states a of the chains shows a partially filled valence band, as posed charge order (cluster-like versus staggered) [8] in illustrated in Fig 1. The origin of antiferromagnetism (Ca,Sr)14Cu24O41, considering the competition between in these compounds is still unclear, and the doping de- kinetic energy, interaction and periodic distortions, on pendence of the charge order and spin gap is discussed the basis of the Hubbard model. Since no periodic lat- controversially [6]. Recent studies on the basis of the tice distortion – neither with two-site nor with four-site Heisenberg model show that a simple spin model is not periodicity – is found in the crystalstructure [9] we con- appropriateto describe the magnetization[7], indicating sider only periodic potentials. Nevertheless, our results the necessity of studying more advanced models, as e. g. arerelevantforallcorrelatedmaterialswithquarterfilled the Hubbard model. bands. Thus we aim at understanding a whole class of In this article we address the stability of the pro- materials on a common basis. 2 II. PERIODIC DISTORTIONS IN doping and potential strength, using the density matrix INTERACTING CHAINS renormalization group method [30]. Thus, our calculations give detailed insights into the properties of hole- Charge order – independent of magnetic ordering – doped antiferromagnets. in less than half-filled bands is found, for example, in III. MODELS organic charge transfer salts, like the Bechgaard salts (TMTSF)2X,withX=PF6,AsF6,ClO4,ReO4,orBr,or In the following section we first discuss the properties their sulfur analogs (TMTTF)2X. These systems, which of non-interacting electrons on a chain with different pe- consistofstacksoforganicmoleculesformingweaklycou- riodic potentials and the phase diagram of the homoge- pledone-dimensionalchains,exhibitalargevarietyoflow neous Hubbard chain. In the last subsection we discuss temperature phases[10, 11]. They have beenstudied us- some basic features of the periodic Hubbard model. ingapurelyelectronicmodel,namelytheextendedHub- bard model with next-nearest-neighbor interaction. Re- centresultshavebeenobtainedbymean-fieldapproaches A. Peierls model [12] and exact diagonalization [13]. On the other hand, chargeorderisoftenconnectedto astructuraltransition and hence to electron-phonon coupling. The coupling of The Peierls model is the prototypical model to study aone-dimensionalmetaltoanelasticlatticeresultsinan the coupling of non-interacting electrons to the lattice. instability towards a periodic lattice distortion commen- The Hamiltonian is given by surate to the band-filling, known as Peierls transition. Withinalattice model,the Peierlstransitioniscaptured N by a modulated hopping term in the Hamiltonian of the HPeierls =− ti c+i,σci+1,σ +h. c. , (1) non-interactingsystem[14]. Aperiodicpotential,instead i,σ X (cid:0) (cid:1) of a modulated hopping, has similar effects on the elec- where we consider a modulated hopping term with t = tronic structure ofthe one-dimensionalmetal, leading to i t[1+ucos(Qa i)]. N denotes the number of lattice sites a band insulator. · andathelatticeconstant. Inaddition,N isthenumber The interplay of electron-lattice and electron-electron e of electrons on the chain and Q the wave vector of the interaction is conveniently studied in the framework of periodicdistortion. Incaseofacommensuratedistortion the Hubbard model. With interaction, the Peierls and we have Q = 2mk , m = 1,2,.... The dimerized chain, the so-called ionic model show different ground-state F Q=π/a, at half filling, n=N /N =1, wasinvestigated phase diagrams [16]. Many results are available for the e by Su, Schrieffer, and Heeger (SSH) [14]. Due to the half-filled Hubbard model with periodic hopping or potential, forwhich the transitionfroma bandinsulator to periodic distortion a band gap, ∆= 4ut, opens at k∆ = a Mott insulator has been studied in [20, 21, 22, 23, 24]. Q/2. Since at half filling we have k∆ = kF, the system is insulating. As is characteristic for a band insulator, On the other hand, away from half filling no insulator- the gaps for chargeand spin excitations are identical. In insulator-transition is present, but a variety of ordering the following, we concentrate on the quarter-filled band processes have to be taken into account [25]. The sta- and 4a-periodic distortions, because in case of the 2a- bility of the different ground states of the quarter-filled periodic distortion the quarter-filled system is metallic, Hubbard model with respect to a periodic potential has beeninvestigated[26,28]lessextensivelyascomparedto since k∆ =kF. 6 the half-filled model. The phase diagram of the quarter- To begin, we determine the energy bands and the filled Peierls-Hubbard model is discussed in [29]. Hence, energy gain due to a periodic modulation of the hop- we restrict our calculations to the ionic potentials. ping parameter with Q = π/2a. For quarter filling, We access the stability regions of the ionic Hubbard kF = π/4a, this system shows the Peierls transition, model by introducing several potentials connected with since Q = π/2a = 2kF. The Fourier transformation of differentorderingprocessesofchargeandspindegreesof thehoppingtermyields,inanalogywiththeSSH-model, freedom into the Hamiltonian. In particular, we inves- the expression tigate which potential leads to the largest energy gain, and analyze against which potential or order the system ck ilsishmeodstinunnasttuabrele.,Iinndoiucratcianlgcuwlahtiicohnso,rwdeercwonilclebnteraetsetaobn- HPeierls = c+k c+k+Q c+k+2Q c+k+3Q thˆαβ cckk++2QQ  , theionicHubbardmodelatquarterfilling,forbothweak Xk (cid:16) (cid:17)  ck+3Q  and strong interaction. We determine the ground state  (2) and the groundstate energy as a function ofinteraction, where the matrix hˆ is given by αβ 3 2coska u(eika+ie−ika) u(eika ie−ika) − − − u(ieika e−ika) 2sinka u(e−ika+ieika) hˆαβ = − u(ie−−ika eika) − 2coska u(eika+ie−ika). (3) − u(e−ika+ieika) u(e−ika ieika) 2sinka   −    The eigenvalues of this matrix are 2 ε(k)= t 2 1+2u2 cos22ka+4u2(1+sin22k) , ± s ± 1 (cid:18) q (cid:19) Wcos(πi) (4) Wcos(πi/2) compare [31]. The four energy bands are separated by the energy gaps (k) 0 ǫ ∆1 =4ut, ∆2 =2u2t, and ∆3 =∆1, (5) where ∆1 = ε2(k = kF) ε1(k = kF) and ∆2 = ε3(k = -1 − 2kF) ε2(k =2kF). Theenergygainofthefermions,for − the quarter-filledcase,obtainedbysumming upthe con- tributionsofalloccupiedstates,isfoundtobe u2tlnu, -2 ∝ -π/a -π/2a 0 π/2a π/a as for half filling. k FIG. 2: (Color online) One-particle energy ε(k) versus mo- B. Ionic model mentum k, where W =0.1t and the horizontal lines indicate theFermi levelof thehalf-filled and quarter-filledband. The Another type of periodic distortion is given by local straight line corresponds to the Q = π/a-periodic potential, potentials, resulting in the so-called ionic model the dashed line to the Q = π/2a-periodic potential. The reduction of theBrillouin zone of the clean model is obvious. N N N Hionic =−t c+i,σci+1,σ+h. c. + Wini− hiSiz . Xi,σ (cid:0) (cid:1) Xi Xi Diagonalizing the resulting Hamilton matrix, as de- (6) scribed above, yields the dispersions Again, for the half-filled band the same calculation as before can be performed for a periodic potential, with a), d) ε(k)= t 2+w2/2 4cos2(2k)+2w2+w4/4, ∆ =∆ =2W. Incaseoftheionicmodelforaquarter- c s ± ± filled systemwe compareseveralpotentials, whichcorre- q p spond to different charge and spin patterns: b) ε(k)= t 2+w2/2 4cos2(2k)+4w2, ± ± a) First, we study a simple potential with 2kF(4a)- q p period, given by W = W cos(πi/2). Here, c) ε(k)= t√w2+4cos2k, i ± the charge order for zero hopping is given by 1/0.5/0/0.5 electrons per site, periodically contin- withw=W/t. Fig. 2showsthedispersionforpotentials ued. Spin order is not established. a)andc). In comparisonto the Peierlsmodel not only a gapopensatk =Q/2butthelowerandupperbandsare b) A modification of a) given by Wi =W cos(πi/2)+ shifteddownandup,respectively. Incaseofasmallper- Wcos(π(i+1)/2),whichleadstotwooccupiedand turbation,thedispersionissimilarforthePeierlsandthe twounoccupiedsites(“cluster”)[13],i.e.a1/1/0/0 ionicmodel,aswellasforpotentialsa)andb). Potential charge order. a) and d) are equivalent in the non-interacting case. The energy gain in the quarter filled band is found to c) A charge order with a 1/0/1/0 pattern, i. e. a be about W1.65 in cases a), b), and d). For small W, 4k (2a)-periodic order, is forced by a potential F − the energy gain here is nearly the same in case a) and with W = W cos(πi). With interaction we like- i b), whereas it becomes weaker in case b) than in case a) wise expect magnetic ordering. at W 0.5t. In case c), the energy gain is quadratic, d) In addition, a local magnetic field with hi = E(W)≈E(0)= 8W2. It mainlytraces backto the band Wcos(πi/2)(ni,↑ ni,↓) enforces a /0/ /0 pat- shift. − 3 − ↑ ↓ tern. As in c), we expect a 2kF(4a)-period for the Accordingly, the gaps are given by spinsandcorrespondinglya4k (2a)-periodforthe F charges. a), d) ∆1 =W, ∆2 =√2W, ∆3 =W; 4 b) ∆1 W for small W, ∆1 ∆∞1 for strong W, is smooth and continuous [18], whereas a third phase – ∼ → ∆2 =√2 W/2, ∆3 =∆1; a spontaneous dimerized insulating phase – is found in − the ionicHubbardmodel. Within this phase,chargeand c) ∆1 =0, ∆2 =W, ∆3 =0. spingaparenon-zero,andthedimerizationoperatorhas a non-zero expectation value. In fact, the extension of this phase is discussed controversially. C. Hubbard model The situation in the quarter filled band is completely different. With interaction, no insulating phase is estab- The Hubbard modelis knownto capture the interplay lished, except for very strong interaction, where higher between kinetic energy (delocalization) and interaction order terms of the Umklapp scattering can cause a Mott (localization) in electronic systems. The Hamiltonian is insulator on its own [26]. In the quarter-filled case with given by 2k -periodic potential we expect from bosonization a F transitionfromtheLuttingerliquidtothebandinsulator N N HHubb =− ti c+i,σci+1,σ+h. c. +U ni,↑ni,↓. itnemth.eTnhoeni-nitnetrearcatcitoinngmaosdiwfieelsltahseinextphoeneinnttesroafcttihneggsaypss- i,σ i X (cid:0) (cid:1) X and the energy gain. Since they become smaller with (7) increasing interaction,interaction stabilizes the band in- The Hubbard model in one dimension is exactly solv- sulator. Here, the interplay between periodic distortion able by means of the Bethe ansatz [15]. Note that in and interaction is weak, since the quarter filled model one dimension another useful formulation of the Hub- with interaction is still a metal. A similar situation is bard model is available on the basis of the bosoniza- foundforhalffilledspinlessfermionsforweakinteraction, tion technique. The low lying excitations of the non- wheretheUmklappscatteringinducestheinsulatoratfi- interactingaswellastheinteractingfermionssystemare nite interaction strength [27]. In case of a 4k -periodic sound waves, i. e. the Fermi system can be described as F potential Umklapp scattering becomes relevant [28] due a non-interacting Bose system, called a Luttinger liquid, to the doubling of the unit cell or, equivalently, due to showing spin-charge separation. In the clean case, the the reduction of the Brillouin zone, see the straight line Hubbard model has three phases. For U < 0, the spin in Fig. 1. In this case, the system undergoes a transi- excitation spectrum has a gap and the low-lying charge tionfromtheLuttingerliquidtoaMottinsulatingphase excitations can be described by those of a Luttinger liq- when the potential and the interaction are present. Po- uid. For U > 0 and away from half filling, spin and tential and interaction hence cooperate rather than op- charge excitations are those of a Luttinger liquid. The pose each other as in the half filled ionic model. last phase occurs for U > 0 and half filling, where the An analysis of the ionic Hubbard model with another charge excitations have a gap and the spin excitations 2k -periodic potential is given in [19]. The authors find are of Luttinger type. A relevant 4k -Umklapp scatter- F F thebandinsulatorforweakinteractionbutstrongpoten- ingterm,onlypresentforhalffilling,isresponsibleforthe tial. Our4k -periodicpattern,hencetheMottinsulator, Mottgapinthelatterphase. Thebosonizationtechnique F is obtained in their model in the limit of strong inter- is adequate for metallic systems or in the weak coupling action. Both phases are separated by a bond ordered regime. It is useful to determine the phase boundary phase. Thus, the model potential used in [19] reveals between metals and insulators but it is not suitable for the features of the half filled ionic Hubbard model. The distinguishing different insulating phases for intermedi- analysis,however,concentratesonlarge U W . In our ate or strong perturbations. | − | case, the 4k -periodic pattern is not a limit of the 2k - F F periodic patterns. D. Periodic Hubbard model In the following, we concentrate on the ionic Hubbard model at weak to intermediate potential strength and intermediatetostronginteraction. Inaddition,westudy A commensurateperiodic distortion– i.e. commensu- the region of stability for each potential. rate to the band filling – introduces an additional non- linear term in the bosonizedHamiltonian, which couples spin and charge degrees of freedom and destroys the in- tegrabilityofthe cleanHubbardmodel. Inthe half-filled IV. NUMERICAL RESULTS FOR THE IONIC HUBBARD MODEL AT QUARTER FILLING case we therefore find a transition between two possible insulating phases, i. e. the band (Peierls) insulator (W,ut U) with chargeand spin gapof the same order In the following we discuss the numerical data for the ≫ and the Mott insulator (W,ut U) with a charge gap ground state energy, the charge and spin gap, and the ≪ proportional to the interaction, but vanishing spin gap, spin-spin correlationfunction. The numericalresults are where periodic distortion and interaction oppose each obtained by the density matrix renormalization group other. However,the transitionisfundamentallydifferent method (DMRG), as implemented by Brune [32]. The for the Peierls and the ionic Hubbard model [16, 17]. In DMRG is a quasi-exact numerical method to determine the Peierls model the transition between the two phases the ground state properties, i. e. the ground state and 5 the ground state energy, of long one-dimensional (non- 0.2 integrable) systems with reasonable accuracy [30]. Re- a) gardingdifferentboundaryconditions,itisusefultotake b) into account the equivalence of fermion and spin models c) 0.15 d) andto implement the spinless-fermionmodel interms of N an equivalentspin chain, andthe Hubbard model as two / )] coupled spin chains without perpendicular XY-coupling. W ( Using the DMRG, it is possible to extend the tractable E 0.1 system lengths for the Hubbard chain to N = 60...100 − ) sites. In our simulations we perform five lattice sweeps 0 ( and keep 300 to 500 states per block. Correlation func- E [ tionscanbeobtainedwithinanerrorof10−6intheHub- 0.05 U =2t bard model when using open boundary conditions. A memory of about 700 MB is required. 0 0 0.2 0.4 0.6 0.8 1 A. Energy W2 First,we determine the groundstateenergyas afunc- 0.2 tion of W and U. The energy gain depends almost a) quadratically on the potential strength for all potential b) types,andisdominatedbythebandshift. Thenumerical c) data are depicted in Fig. 3. In case of a coupling to the 0.15 d) N chargedensity,W n ,the 2k -periodicpotentialisstabi- i i F / lizedforsmallU (casea)),andthe4k -periodicpotential )] F W (case c)) for large U, as found for the Hubbard-Holstein ( E 0.1 model[33]. Thisenergygainismainlyduetotheprefac- − tors. A more detailed analysis of the algebraic behavior ) 0 for W 0 shows that Ea W1.68 and Ec W2 if E( U = 2t,→but Ea W1.76 and∝Ec W1.76 if U =∝10t. In [ 0.05 U =6t ∝ ∝ case a) the exponent increases with interaction, in case c) it decreases. For all interactions, however, term d), h Sz, is dominant. Of course, in a real system, a poten- i i tialcoupledtothedensityismuchbiggerthanamagnetic 0 0 0.2 0.4 0.6 0.8 1 field. It leads – like potential c) – to 4k -oscillations in F the charge density and 2k -oscillations in the magneti- W2 F zation. Inthe non-interactingsystemthe datapoints for FIG. 3: (Color online) Energy gain per site versus square of potentials a) and d) lie nearly on top of each other (by the potential strength (both in units of t). We compare po- useofopenboundaryconditionstheexactequivalenceof tentialsa),b),c),andd). Thenumberofsitesvariesbetween a) and d) is waived), but in the interacting system the N = 28 and N = 60, where [E(0)−E(W)]/N is almost in- energy gain due to potential d) grows faster with inter- dependent of N. The interaction is U =2t in theupper plot action than in all other cases. To be more specific, we and U =6t in thelower plot. find E W1.62 for U =2t and E W1.44 for U =6t. d d ∝ ∝ B. Charge and spin gap for small gap ∆0 the accuracy of the DMRG away from half filling is not accurate enough and for large ∆0 we have f(N) 0 for the considered system sizes. The de- We note that the excitation gaps ≈ pendenceoftheexcitationgapsonthepotentialstrength 1 is shown in Fig. 4. In cases a), b) and d) we see that ∆c = 2[E(Ne+1)+E(Ne−1)−2E(Ne)] ∆s ∼ ∆c, according to the band insulating state. An almost linear dependence on W, as for non-interacting and particles, is recovered in the interacting system for the charge gap in cases a), b), and c). The saturation of the 1 ∆s = [E(Sz =1)+E(Sz = 1) 2E(Sz =0)] gaps in case b), as discussed in Sec. 3B, is obtained for 2 − − the spin gap in the considered parameter regime. The arecalculatedforfinitesystemsonly. Anextrapolationto influence of the interaction in the band insulating state thethermodynamiclimit,∆=∆0+f(N)withf(N) 0 can be summarized as follows. In case a) the charge → for N , is not performed, for the following reasons: gap becomes smaller with interaction, the spin gap even →∞ 6 1.2 C. Correlation Function a), N =28 b), N =40 1 Inorderto calculate the spin-spincorrelationfunction c), N =60 d), N =40 hSizSjzi within DMRG we again use open boundary conditions and the Parzen filter function [34] to reduce the 0.8 Friedel oscillations. In the Mott insulating regime, the ) Friedel oscillations in the spin sector are long ranged, W ( 0.6 but decay fast in spin gap systems [35]. The main ques- c ∆ tion concerns the correspondence of the spin-spin corre- 0.4 lationfunctionsofthe2a-periodicpotential(casec))and the Heisenberg model. For comparison, we calculate the spin-spin correlation function of the half-filled Hubbard 0.2 model, which can be mapped onto a Heisenberg chain in case of strong interaction. In the Heisenberg model we 0 have 0 0.2 0.4 0.6 0.8 1 W hSizS0zi= 4π(a1 i)2 + a2kF coas(2ikFa·i). · · 0.5 In gapped systems, the spin-spin correlationfunctions a), N =28 reflect the behavior of the energy gaps. Thus, the spin- b), N =40 spincorrelationfunctiondecreasesexponentiallyincases c), N =60 0.4 a) and b), which show a spin gap without spin order. d), N =40 In case d) the spins are fixed by the potential h . The i exponentially decaying part is hard to extract from the ) 0.3 mean magnetization in the numerical data due to the W incomplete suppression of the boundary oscillations. ( ∆s In the Mott insulating regime, however, the linear de- 0.2 creaseof the spin-spincorrelationfunction withdistance is obtained already for small potential W. For bet- ter comparison with the data of the half-filled Hubbard 0.1 chain, we show data for W = t in Fig. 5. Obviously, the quarter-filled Hubbard chain with 4k -periodic dis- F tortion can be mapped onto the Heisenberg model, too. 0 0.2 0.4 0.6 0.8 1 We note that in case d) already a weak potential – connected with a small spin gap – leads to significant struc- W tures in SzSz , whereas in case c) the potential has to h i ji be much stronger to yield effects of similar size. The FIG. 4: (Color online) Charge and spin gap versus potential strength (both in units of t). In the upper plot we show ∆c increase of S(q)∝q appears both in the half-filled Hub- and in the lower ∆s, comparing potentials a), b), c), and d). bard chain and in the quarter-filled Hubbard chain with The numberof sites varies between N =28 and N =60 and 4kF-periodic potential. S(q) has a sharp maximum at theinteraction strength is U =6t. q =π/2 in case d). more. In case b) the charge gap increases with inter- V. SUMMARY action for small U, but saturates for strong interaction. The spin gap decreases rapidly with interaction. In case In summary, we have studied the effects of periodic d) the charge gap increases but does not saturate. The potentials on a chain. In particular, we have considered spin gap shows a maximum for intermediate interaction the one-dimensional Hubbard model at quarter filling. strength. For U = 10t it is even smaller than in the We have compared periodic potentials yielding different non-interacting system. On the contrary,we find for po- types of behavior: a) a 2k -periodic potential leading F tentialc)alinearincreaseof∆ withU andW aswellas to a band insulator, b) a 2k -periodic potential leading c F ∆ (W,U,N) 1/U,1/W,1/N 0, indicating the Mott to a band insulatorwith cluster-likearrangementsof the s ∼ → insulatingstate. Duetothefinitesystemsize,wearenot charges, c) a 4k -periodic potential leading to a Mott F abletoobtainexponentsforsmallW. Asaconsequence, insulatorwithantiferromagneticalignmentofHeisenberg a comparison with bosonization or the energy data, see type spins on next-nearest neighbor sites, and d) a 2k - F Section A, is not possible. To conclude, for strong inter- periodic magnetic field leading to a band insulator with actionandsmallpotentialwefindalargechargegapbut antiferromagneticalignmentofIsingtypespins. Casesc) a small spin gap. and d) reveal the same charge and spin distribution but 7 tial b) gains more energy than the homogeneous distri- 0.2 butionofcasea),E >E >E . ForstrongW t,also n=1, W =0 b a c → case c) a 4kF pattern is favored against the homogeneous case, cased) E > E > E . Thus, for small or intermediate interac- b c a N =64,U =5t tionandpotential,thegainofenergyduetothe hopping 0.1 is stronger than the effects of the repulsive interaction i or the underlying potential. With increasing potential, 1 z3 S however,the effects of the potential dominates. zSi Onthe other hand, regardinga weak potential but in- h 0 creasing interaction, we find that a pattern related to potential a)is established only at small interaction. The cluster-like arrangementis formed at intermediate interaction (U 3 5t), but at strong interaction the 4k - F ≈ − -0.1 periodic pattern dominates. Concerning the spin chain compound (Ca,Sr)14Cu24O41, we have applied the po- 0 10 20 30 40 50 60 tential only in order to clarify the leading instability of i the system. For comparison with a real material which shows no lattice effects, the limit W 0 is relevant. z z → FIG.5: (Coloronline)Spin-spincorrelation functionhSiS31i In addition, for copper oxides usually a correlation pa- versus site i. We compare the half-filled Hubbard chain rameter of U 8eV [36] is assumed. Together with the (straight line; which is equivalent to a Heisenberg chain), band width of≈about 1 eV, see Fig. 1, we have U >15t. with the quarter-filled Hubbard chain on a 4kF-periodic po- In this parameter region, we find – neglecting the mag- tential (long-dashed line, W = t; equivalent to a half-filled netic field – the 4k -periodic charge pattern with anti- Heisenberg chain), and the quarter-filled Hubbard chain in F ferromagnetism to be the most probable ground state. a 2kF-periodic magnetic field (short-dashed line, W = 0.1t; Theenergeticorderfoundforweakpotentials extendsto equivalent to a half-filled Ising chain). In all data sets the strongpotentials. Forstronginteraction,wealwayshave interaction is U =5t. E >E >E . Thetendencyoftherepulsiveinteraction c b a to separate the charges dominates the phase diagram in differentspinexcitations. IntheHeisenbergcasethespin thisregion. Theinfluenceofthepotentialisweaker,lead- excitationsaregapless,whereasintheIsingcasetheyare ing to a preference of the cluster over the homogeneous gapped. distribution. In non-interacting systems, we find Ea = Ed > Eb > Finally, we remarkthat adding electronsstabilizes the Ec, thus the band insulator. The charge distribution is cluster formation. This observation likewise agrees with quite homogeneous, and the cluster like arrangement is experimentalresults,obtainedforthe (Ca,La)14Cu24O41 notminimalinenergy. Turningontheinteraction,poten- series,where the cluster size grows on additional charge. tiald)resultsinthelargestenergygain,whiletheorderof Therefore we conclude, that the antiferromagnetism is the remaining potentials depends on both the potential due to the band filling, thus a commensurability effect. strength and the interaction. The interaction strongly supports the spin order, where double occupancy is sup- pressed. Acknowledgement Inthefollowingdiscussionwerelyonthepotentials a) to c), where the spin-order, if present, is a result of the charge order. For weak interaction and weak potential, We thank U. Eckern for helpful discussions, and P. we recover the behavior of the non-interacting system, Brune for providing the DMRG code. Financial support with E > E > E . Turning to intermediate values by the Deutsche Forschungsgemeinschaft within SPP a b c (U 3t, W t ), the cluster-like arrangementof poten- 1073 and SFB 484 is acknowledged. ≈ ≈ [1] S. Eggert, I. Affleck, and M. D. P. Horton, Phys. Rev. S.Katano,M.Hiroi,M.Sera,andN.Kobayashi,J.Phys. Lett.89, 47202 (2002). Soc. Jpn.68, 2206 (1999). [2] N. Nu¨cker, M. Merz, C. A. Kuntscher, S. Gerhold, S. [4] U. Schwingenschlögl and C. Schuster(unpublished). Schuppler,R.Neudert,M.S.Golden,J.Fink,D.Schild, [5] V. Eyert, Int.J. QuantumChem. 77, 1007 (2000). S. Stadler, V. Chakarian, J. Freeland, Y. U. Idzerda, K. 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