Coexistence of Pairing Tendencies and Ferromagnetism in a Doped Two-Orbital Hubbard Model on Two-Leg Ladders J. C. Xavier,1 G. Alvarez,2 A. Moreo,3 and E. Dagotto3 1Instituto de F´ısica, Universidade Federal de Uberlaˆndia, Caixa Postal 593, 38400-902 Uberlaˆndia, MG, Brazil 2Computer Science & Mathematics Division and Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 3Department of Physics, University of Tennessee, Knoxville, TN 37996 and Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge,TN 37831 0 1 (Dated: January 18, 2010) 0 UsingtheDensityMatrixRenormalizationGroupandtwo-legladders,weinvestigateanelectronic 2 two-orbitalHubbardmodelincludingplaquettediagonalhoppingamplitudes. Ourgoal istosearch n for regimes where charges added to the undoped state form pairs, presumably a precursor of a a superconducting state. For the electronic density ρ = 2, i.e. the undoped limit, our investigations J show a robust (π,0) antiferromagnetic ground state, as in previous investigations. Doping away 8 fromρ=2andforlargevaluesoftheHundcouplingJ,aferromagneticregionisfoundtobestable. 1 Moreover, when the interorbital on-site Hubbard repulsion is smaller than the Hund coupling, i.e. for U′ < J in the standard notation of multiorbital Hubbard models, our results indicate the ] coexistence of pairing tendencies and ferromagnetism close to ρ=2. These results are compatible l e withpreviousinvestigations usingonedimensionalsystems. Althoughfurtherresearch isneededto - clarify iftherangeofcouplingsusedhereisofrelevanceforrealmaterials, suchassuperconducting r t heavy fermions or pnictides, our theoretical results address a possible mechanism for pairing that s may beactive in thepresence of short-range ferromagnetic fluctuations. . t a PACSnumbers: 71.10.Fd,71.27.+a,74.20.-z m - d I. INTRODUCTION possibleregionfortherobustcoexistenceof(spintriplet) n pairing together with magnetic order is where ferromag- o netism develops. This is in qualitative agreement with c It is widely believed that magnetism is a fundamen- [ previousinvestigationscarriedoutusingone-dimensional tal ingredient to explain the origin of high-temperature systems.11Oureffortshouldbeconsideredsimplyaspro- 1 superconductivity in several materials. In fact, there viding the first steps in relating pairing and magnetism v is experimental evidence that the superconductivity in in a complex two-orbital model via computational tech- 7 many heavy fermion (HF) compounds is mediated by 9 niquesonladdergeometries. Antiferromagneticorder,as spin fluctuations.1–4 Mechanisms for superconductivity 0 found in the pnictides, could also be favorable for pair- based on antiferromagnetism have been extensively dis- 3 ing tendencies in the spin singlet channel, as discussed 1. ctourssseadsfwoerllt.h5eRCecue-nbtalsye,dcohnisgihd-etreambpleeeraxtcuitreemseunptehrcaosnbdeuecn- recently.12,13 0 generatedbythediscoveryofhigh-temperaturesupercon- 0 Inprinciple,atheoreticalinvestigationbasedonmodel 1 ductivity in the ironpnictides.6 Except for the cuprates, Hamiltonians for strongly correlated materials starts : the iron-basedsuperconductorsnowhavethe highestsu- with an effective tight-binding model, containing the v perconducting (SC) critical temperature T of any ma- i c minimumingredientstodescribethephysicsofthemate- X terial (see for example Ref. 7). As in HF systems and rialsunder investigation. However,evenif a well-defined r cuprate superconductors, in the pnictides there is also reasonable model is used, it is still highly non-trivial to a evidence that the superconductivity is not mediated by extractthegroundstatepropertiesoftheseeffectivemod- the electron-phononinteraction.8,9 els in two or three dimensions using unbiased numerical Magnetism and superconductivity can appear in dif- methods. In fact, at present there are no accurate tech- ferent ways. In some HF compounds, superconductiv- niques to study Hubbard-like models in dimensions two ity and antiferromagnetic (AFM) order co-exist,1 while and three. Thus, in order to get at least some insight for the cuprates the superconductivity emerges after the on the ground states properties of theses models, it is long-range AFM order is destroyed by doping.5 In some common practice to study the model Hamiltonians in HF systems, it is the superconductivity and ferromag- quasi-one-dimensional geometries. In particular, a very netism(FM) (asopposedto AFM order)that co-exist.10 popular route that has been used for several theoretical Inthiswork,wewillbeinterestedindetectingaclearev- investigations is to study strongly correlated electronic idence of pairing of extra charges that are added to the systems using“ladder”geometries.14 The N-leg ladders undopedlimitwheremagneticorderexists. Usingatwo- consist of N chains of length L coupled by some param- orbitalmodelandtwo-legladders,itwillbeshownthata eter (as, for example, fermionic hopping terms). The 2 two-dimensional system can in principle be obtained by II. MODEL considering the limits of both N and L sent to infinity, although in practice this is difficult to do. This ladder- In these studies, we have considered the following based procedure has been used to investigate models for Hamiltonian defined on a two-leg ladder geometry the high temperature superconductors15 and for the HF systems.4,16 Some important results were obtained with H = tλ,λ′ d† d +H.c. thismethod. Forexample,researchbasedonmicroscopic − X γ,γ′(cid:16) j,γσ,λ j+1,γ′σ,λ′ (cid:17) models for the high T superconductors,5 as well as re- j,σ,γ,γ′,λ,λ′ c mseeadrciahteodnbHyFanmtiofedrerlosm,4aignndeitciactfleutchtautatsiuonpserccaonndbuecsttivabitiy- − X t˜γ,γ′(cid:16)d†j,γσ,1dj,γ′σ,2+H.c.(cid:17) j,σ,γ,γ′ lized, in agreement with several experiments. Thus, the + U ρ ρ +(U′ J/2) ρ ρ use of ladders appears to be an important ingredient to X j,γ↑,λ j,γ↓,λ − X j,x,λ j,y,λ unveil dominant ground state tendencies. Moreover, the j,γ,λ j,λ hopping amplitudes that will be used in our investiga- 2J S S tions below include next-nearest-neighbor diagonal hop- − X j,x,λ· j,y,λ j,λ pings that are only possible when plaquettes exist in the + J (d† d† d d +H.c), (1) lattice under consideration. X j,x↑,λ j,x↓,λ j,y↓,λ j,y↑,λ j,λ Note that microscopic models that may present superconductivity induced by antiferromagnetism, such as where d†j,γσ,λ creates an electron with spin projection σ the one-orbital Hubbard model and the Kondo Lattice in the orbital γ = x,y (dxz and dyz, respectively) at model, have been extensively studied by several au- the rung j and leg λ = 1,2, Sj,γ,λ is the electron spin thors. However, microscopic models for superconductiv- density operator, ρ = d† d , and ρ = j,γσ,λ j,γσ,λ j,γσ,λ j,γ,λ ity in a ferromagnetic spin background have been much d† d . less explored, with the exception of studies using one- Pσ j,γσ,λ j,γσ,λ dimensionalchains.11 This may be causedin partby the The hopping amplitudes are: tλx,,xλ =t˜y,y =−t1, tλy,,yλ = perception that superconductivity and ferromagnetism, t˜x,x = −t2, tγ1,,2γ = tγ2,,1γ = −t3, tγ1,,2γ′ = −tγ2,,1γ′ = −t4, if as opposed to antiferromagnetism, can not coexist.17 γ =γ′,andzerootherwise. Toavoidaproliferationofex- 6 However,this perceptionhas beenchallengedby the dis- traparametersinouranalysis,wehavedecidedtofixthe covery of superconductivity and FM in the HF com- values of these hoppings from considerations previously pounds UGe 10 and URhGe.18 Moreover, SC and FM usedinthepnictidecontext. Ouruseofmodelsoriginally 2 were also observed19 in the d-band metal ZrZn . devised for pnictides is simply based on the pragmatic 2 observation that some pairing tendencies and ferromag- Motivatedbythediscoveryofsuperconductivityinthe netic regions at large J were already observed in those HFcompoundUGe ,afewyearsagoKarchevet al. pro- models.12,13 It should be clear though, that our research 2 posed a one-band model to study the coexistence of su- is mainly motivated by heavy fermion phenomenology. perconductivity and ferromagnetism.20 However, other Following this strategy, then the hopping amplitudes researchershavearguedthatthetreatmentusedtoinves- t1, t2, t3, and t4 are obtained using the Slater-Koster tigatethatmodel,aswellasthemodelitself,werenotap- tight-binding scheme, and they are given by12,13,25 propriatetodescribethecoexistenceofthesephases.21–23 t = 2 b2 a2+g2 /∆ 1 pd − (cid:0) − (cid:1) Duetothelackofstudiesofmicroscopicmodelsforsu- t = 2 b2 a2 g2 /∆ 2 pd perconductivity mediated by ferromagnetic fluctuations − (cid:0) − − (cid:1) t = b2+a2 g2 /∆ beyond one-dimensional chains,11 in this work we have 3 −(cid:0) − (cid:1) pd decided to investigate a microscopic ladder model where t4 = ab g2 /∆pd, (2) −(cid:0) − (cid:1) thecoexistenceofsuperconductivityandferromagnetism appears possible. In fact, it will be shown below that where the Fe-As hopping amplitudes are a = the two-orbital model that has been originally proposed 0.324(pdσ) 0.374(pdπ), b = 0.324(pdσ)+0.123(pdπ), − to describe the low-energy physics of the iron-based and g = 0.263(pdσ) + 0.31(pdπ). ∆ is the energy pd superconductors12,13,24 actually leads to the coexistence difference between the p and d levels. We have set of pairing and spin ferromagnetic tendencies. Qualita- (pdσ)2/∆ = 1 to fix the energy scale, and we use pd tiveourresultsarecompatiblewiththosereportedusing pdπ/pdσ = 0.2, as previously discussed.12,13 Regard- chains.11Althoughthemodelconsideredheremaynotbe ing the coup−lings U, J, and J′, note that they are not a proper effective model for superconducting ferromag- independent, but they are assumed to satisfy the rela- nets such as UGe , we believe that the mechanism that tion U = U′+2J, which is strictly valid within a cubic 2 bounds together the charges carriers (see section III.B) environment for the full t sector.26,27 For an explicit 2g is so simple and generic that our calculations may also derivation of this relation in the case of manganites see apply in a variety of other models as well. Ref. 28. 3 We have investigatedthe model defined above using a 10 (a) two-legladder of size 2 L,by means ofthe Density Ma- 2x12 q =0 trix Renormalization G×roup (DMRG) technique,29 un- 8 2x12 qy=π der open boundary conditions (OBC), and keeping up y U =-0.5 2x16 q =0 to m=1400 states per block in the final DMRG sweep. ) 6 eff y We have carried out 6 13 sweeps, and the discarded qx 2x16 q =π owuerigDhtMwRaGs tpyproiccaeldlyur∼1e0,−t6h−e−c1e0n−te9rabtlothckesfianrael scwomeeppo.seInd S( 4 J=1.5 2x24 qyy=0 2x24 q =π of 16 states. We have used a FORTRAN DMRG code y to calculate most results, and a C++ DMRG code for 2 additional validation.30 We have also confirmed some of our results against Lanczos Exact Diagonalization tech- 0 1 1.5 2 π niques, when possible. q / x In this work, we will focus on the region of Hubbard 5 andHundparameterswhereUeff U 3J =U′ J <0, (b) ≡ − − although a few results for positive values of U will eff be presented as well. As shown below, in the region 5 where Ueff < 0 there is a robust evidence of binding of ) 4 holes/electrons close to density ρ = 2. In this regime 0 4.96 , U′ < J. This inequality may not seem realistic at first π ( U =-0.5 sight, since the on-site spin triplet formation favored by S eff 4.92 J=1.5 J (Hund’s rules) has its origin in the alleviation of the 3 2x8 Coulombic energy penalization caused by U′. However, 4.88 -2 -1 0 1 2 while in the full five-orbital model for Fe-based com- U poundsU′ 0. Neutron scattering measurements for pnic- afinitecluster,andthiswouldbeindicativethateffective eff tides also show a similar spin order.31 We have observed attractive forces are present in the system. that the stripe-like AFM structure appears only when Beforepresentingournumericalresultsforthebinding the plaquette-diagonal hopping amplitudes (t and t ) energies,letusfirstconsiderthefollowingstrongcoupling 3 4 areofvalue similarasthose ofthe nearest-neighborhop- regime defined by U = 3J U = J U′ 1. As eff − − − ≫ ping amplitudes. If we force t = t =0 then the peak it will be argued below, and as it was discussed in pre- 3 4 in the spin structure factor S(q) appears at wavevector vious investigations for one-dimensionalchains,11 in this q=(π,π). Note also that for the two-leg geometry, the regime the binding of hole/electrons is clearly present. (π,0) AFM state is not, naturally, degenerate with the For completeness, let us address this limit in detail, (0,π) AFM state. Due to this fact, a study in a ladder although it is clear that large J compared with U′ leads geometry could make a better connection with the two- to aneffective attractiveinteraction.11 Letus denote the dimensionalresultsofthepnictidematerialswhere(π,0) statesoftheone-siteproblemviathesymbol sy ,where is favored over (0,π) by a lattice distortion. (cid:16)sx(cid:17) the “arrows” s (s ) represent the electrons (with their x y For quasi-one-dimensional systems, a true long-range spins projections) at the orbital x(y). For this one-site magnetic order is replaced by a power-law decay of the problemwithtwoelectronsandthelimitconsideredhere spin-spin correlations. Thus, in order to analyze the where J is large, the ground state energy is degener- range of the magnetic order, we have also investigated ate. Its value is e = U , and the three corresponding 2 eff the spin-spin correlation function along one of the legs eigenstates are ↓ , ↑ , and [ ↑ + ↓ ]/√2. Since (say, leg 1) defined as (cid:16)↓(cid:17) (cid:16)↑(cid:17) (cid:16)↓(cid:17) (cid:16)↑(cid:17) U 1, we can approximate the Hamiltonian as eff Cspin(l)= M1 X (cid:10)Siz,1Sjz,1(cid:11), (4) Ha−te)=grHo≫kuinnedticst+ateHiennter∼gyHoifntt,haentdwoth-leeg(hmigohdleyl dbeegcoenmeers- |i−j|=l E(0)=2L e =2L U . 2 eff × × where M is the number of site pairs (i,j) satisfying l = Below, the dominant spin arrangement in the ground state at density ρ=2 is shown to guide the discussion i j . Inpractice,wehaveaveragedoverallpairsofsites | − | separated by distance l, in order to minimize boundary ↓ ↑ ↓ ↑ ↓ effects (afew sites atthe edgeswerealsodiscardedwhile (cid:16)↓(cid:17) (cid:16)↑(cid:17) (cid:16)↓(cid:17) (cid:16)↑(cid:17) (cid:16)↓(cid:17)  , implementing this averagingprocedure). ↓ ↑ ↓ ↑ ↑ In Fig. 1(c), the spin correlation function C (j) is (cid:16)↓(cid:17) (cid:16)↑(cid:17) (cid:16)↓(cid:17) (cid:16)↑(cid:17) (cid:16)↑(cid:17)  spin shown for the 2 24 cluster and using U = 0.5. The × eff − using only the up and down projections of the spin one dashedlineisafitofthenumericaldatawiththefunction states at every site for simplicity. Such a state is to be cos(πx) expected since for J 1 the alignment of the two spins C˜spin =a x1/3 . (5) at the two orbitals i≫n the same site will occur, and as U 2J 1 then having two electrons in the same or- Similar results were found for the 2 16 cluster. The ob- bit∼al is n≫ot allowed. Of course, any other configuration × served power-law decay suggests that a two-dimensional (suchasa fully ferromagneticstate)is alsoequallylikely system with the same model and parameters would de- as the one shown in the figure for U 1. However, eff − ≫ velop long-rangemagnetic order at zero temperature. the hopping terms will lift the large degeneracy,and our numericalresultsfor the spin-spincorrelationspresented before indicate that hopping terms favorthe (π,0) AFM B. Doping with two holes or electrons configuration, for a wide range of couplings in the undoped limit. For this reason, here we have chosen to Let us now consider the effect of doping with charges present the dominant spin arrangement of the ground this ladder system. If tendencies toward the pairing of state as a stripe-like AFM state, but it could be fully theextrachargesareunveiled,theywouldbeanindicator FM as well. Regardless of this detail, the arguments we that this model could become superconducting in a two will use below to obtain the energies E(n) will hold true dimensionalgeometry,for the couplings here considered. both for a stripe-like AFM state, as well as for a FM Letus startwith the calculationofthe binding energy state. of two doped holes/electrons. This binding energy is de- Let us now add two extra electrons to the undoped fined as ∆ = E(2)+E(0) 2E(1), where E(n) is the system. In the limit being considered here, where the b − groundstate energywith (4L+n) holes/electrons(4L is hopping amplitudes are negligible, the best way for the the number of electrons corresponding to the“undoped” systemtominimizeitsenergyistohavethetwoelectrons limit where there is anelectronper orbitalandper site). located on the same site, since in this way less on-site On a finite system the binding energy ∆ > 0 is posi- ferromagnetic links are broken. The on-site Hubbard U b tive if the electrons/holes do not form a bound state,5 energy penalization is the same whether the doubly oc- while in the thermodynamic limit ∆ should vanish in cupiedorbitalsareatthesamesiteornot,andsinceU′is b the absence of pairing. On the other hand, if the extra negligible with respect to J in the limit considered,then holes/electronsformaboundstate,then∆ <0evenon the(effectivelyattractive)Hundcouplingdeterminesthe b 5 location of the extra charge, leading to the double occu- 10 (a) pation of both orbitals at the same site.11 In this case, 2x3 two doped electrons the dominant spin arrangementof the ground state is 8 FM 6 ↓ ↑ ↓↑ ↑ ↓ (cid:16)↓(cid:17) (cid:16)↑(cid:17) (cid:16)↓↑(cid:17) (cid:16)↑(cid:17) (cid:16)↓(cid:17). J ↓ ↑ ↓ ↑ ↑ 4 (cid:16)↓(cid:17) (cid:16)↑(cid:17) (cid:16)↓(cid:17) (cid:16)↑(cid:17) (cid:16)↑(cid:17) Once again, a stripe-like AMF backgroundis used, since 2 ∆b<-0.1 for J 1 and close to the density ρ = 2 our numerical total spin changes continuously ∼ data show (see Fig. 4(a)) that the (π,0) AFM order is 0 the dominant one, but it could have been FM as well. 0 0.2 0.4 0.6 3J-U In the limit of couplings considered here, the ground state energy for the doped two-electronsystem is (b) E(2)=(2L+5)Ueff +8J. (6) 8 2x3 two doped holes Note that if the two extra electrons that are in the state 6 ↓↑ were to move in opposite directions (after consid- FM (cid:16)↓↑(cid:17) ering the presence of small but nonzero hopping terms) J4 theywouldbreaktwoferromagneticon-sitelinks. Sucha ∆ <-0.1 statewouldhavealargeenergyanditistherefore“forbid- b 2 den”for U 1. However, if the two extra electrons eff move“tog−ether≫”(i.e. ↓) in the same direction,forming a ↓ total spin changes continuously spin triplet, then no other on-site FM links are broken. 0 Thus,to minimize the energythe two-electronsaddedto 0 0.2 0.4 3J-U the systemmustformaboundstate,atleastinthe limit where the hoppings amplitudes are very small compared with the Hubbard and Hund couplings (the same spin (c) 0.4 binding of electrons U =-1.0 J =1.5 dominantpictureworkswheninsteadofaddingelectrons eff binding of electrons U =-1.0 J=3.0 we remove two electrons, i.e. for the two-hole problem). eff 0.2 binding of holes U =0.0 J=5.0 Thisargument,knownfrompreviousinvestigationsusing eff chains,11 explains the binding of electrons in the limit binding of holes Ueff=-0.5 J=3 0 Ueff 1. More explicitly, the binding energy of the b − ≫ ∆ dopedtwo-electrons/holessysteminthelimitconsidered here is given by -0.2 ∆ = U . (7) b −| eff| -0.4 Based on the discussion above we conclude that for 0 0.1 0.2 0.3 U 1,there isanindicationofpairing,andperhaps 1/L eff − ≫ superconductivity, in the two-orbitalmodel. In fact, the on-site interorbital pairing state found here in this ex- Figure 2: (Color online). (a)-(b) Phase diagram (J vs. treme regime results to be a spin triplet and transforms −Ueff =3J−U)forthe2×3clustershowingtheregionwhere the binding energies of electrons (a) and holes (b) is smaller according to the irreducible representation A of the group D .32 Interestingly, Exact Diagonalizati2ogncalcu- than ∆b < −0.1, indicative of pairing. Above the red line, 4h the total spin saturates to its maximum value. Below this lations in two-dimensional clusters, still in the FM state line, the total spin changes continuously from the maximum of the model but U′ > J, found indications of a pairing value to zero at J=0. The blue lines show the region of J state with the same characteristics and symmetry but where ∆b < −0.1 for Ueff = −0.5. These lines are the same withelectronsatdistanceofonelatticespacingfromeach aspresentedinFig.3. (c)∆b vs. 1/Lforsomecouplings(see other.12,13 The argument presented above for this pair- legend). ing is actually valid in any dimension, and also it works for the two doped holes case, as already mentioned. To confirmtheseargumentations,∆ wascalculatednumer- relevance for real materials. The answer to this ques- b ically for large values of U . An excellent agreement tion appears to be positive. In fact, the binding of elec- eff − between the numerical data and the analytic expression trons/holes has been observed numerically for U . 0 eff (Eq. (7)) was found. andseveralvaluesofthecouplingJ,asshowninFig. 2.33 Nowthe crucialquestioniswhetherthereisbindingof More specifically, Figs. 2(a)-(b) show the region in the holes/electrons for values of U that may be of more J (3J U)planewherethebindingenergiesofelectrons eff − − − 6 15 C. Phase Diagram U =-0.5 2x2 eff 2x3 InFig.3thephasediagram(J vs. density)ofthetwo- 2x4 leg ladder for U = 0.5 and several system sizes (see 10 2x6 eff − binding 2x8 legend) is presented. For ρ=2, it was observedthat the FM 2x12 totalspinis zerofor L even,and thatit canbe 0,1, or 2 J for L odd, depending of the values of J. Note that for L FM odd and ρ = 2, the (π,0) AFM configuration (+-+-+-) 5 does not have the same number of + and - spins. For 2 < ρ < 2.5 (1 < ρ < 2), a ferromagnetic phase wasfound (the regionabovethe symbols)with magnetic binding moment per site given by m = 2 ρ/2 (m = ρ/2). The 01 1.5 ρ2 2.5 3 symbolsindicatethevalueofJc w−herethetotalspinsat- urates. The criticalvalue J wasdeterminedby the level c crossingoftheenergiesinthesectorwithSz =Smax and Figure3: (Coloronline). (a)Phasediagramofthetwo-leglad- total Sz = Smax 1. Using this procedure, we were able to dermodelHamiltoniandefinedinEq.1. Theregionabovethe total − symbols is a fully-saturated ferromagnetic phase (see text). obtain Jc for large systems. For densities in the ranges Thebluelinescorrespondtoregionswhere∆b <−0.1forthe ρ . 1.25 and ρ & 2.5, we have not found any trace of cluster 2×3 with two doped electrons/holes, as presented in ferromagnetism. Below the FM region, it is very hard Figs. 2(a)-(b). numerically to determine the total spin with good accu- racy for large systems. However, our results for the 2 2 × and2 3clusterswithtwodopedelectrons/holessuggest × thatthe totalspinchangescontinuously,frommaximum value at J to zero at J=0. The total spin can be ex- (Fig.2(a))andholes(Fig.2(b))arelessthan-0.1,forthe c tractedfromthespinstructurefactoratq=(0,0). Fora case of a 2 3 cluster. The region where ∆ < 0.1 was × b − few setsofcouplingswe alsoobserved,throughthevalue chosentoberepresented,asopposedto∆ =0,sincepre- b of S(q=(0,0)), that in fact the total spinof the ground vious experiencein the contextofthe cuprates5 suggests state changes continuously for the 2 8 cluster as well. that this procedure effectively takes into account size ef- × These results suggest that the total spin varies continu- fects better. In practice, other values for this“cutoff”do ouslybelowthe FMregionpresentinFig.3 foranyclus- notalterourqualitativeconclusions. Similarresultswere ter sizes. Overall, our results are qualitatively compati- found also for the 2 2 cluster. × ble with those found in one-dimensionalsystems.11 Note also that tendencies to FM states at robust J were also Thus far, only small clusters have been considered be- reported via Exact Diagonalization methods on small cause the numerical analysis of large clusters would be clusters.12,13 too time-consuming, particularly with regards to calcu- We believethe FMphaseis stabilizedby amechanism latingthehundredsofpointsthatarerequiredtoextract thathasthesamecharacteristicsastheDoubleExchange comprehensive phase diagrams. However, larger system (DE) mechanism.34 In the original DE scenario, there sizes were consideredfor a few selected sets of couplings, are mobile and localized degrees of freedom. In the DE as shown in Fig. 2(c). In this figure, ∆ vs. 1/L for b mechanism these degrees of freedom are separated and some couplings is presented. Here, it is clearly observed well defined. Although we do not have localized degrees that in the bulk limit the binding energiesof addedelec- of freedom in our model, from the perspective of one trons/holesconvergetononzerovaluesforsomecoupling electronatagivenorbitalanelectronatthesamesitebut sets. Close to ρ =2, these results strongly indicate that the other orbital behaves in some respects as a localized there are pairing tendencies for U . 0 . Thus, to the eff spin. For doped systems, when an electron moves from extendthatfutureinvestigationsshowthatJ comparable to U′ is a realistic regime for effective two-orbital mod- one site to the other, in order to minimize the kinetic energy and the energy related with the Hund coupling, els,thisprovidesapossiblemechanismforpairinginreal all spins have to be aligned. materials. The blue lines in Fig. 3 are the same that were pre- In Figs. 2(a)-(b), the magnetic phase diagram for the sented in Figs. 2(a)-(b). We expect that the region of case of two doped electrons/holes is also presented. The binding extends beyond these lines up to the density regionabove the red (bold) line is a ferromagneticphase ρ = 2, forming regions in parameter space where super- withthemaximumtotalspinS =2L 1. Belowthis conductivity exists inside the phase diagram. total − line, we have observed that the total spin changes con- We have also measured the spin structure factor S(q) tinuously from zero, at J = 0, up to its maximum value away from the undoped density ρ = 2. In Fig. 4, S(q) at J . As observedin these figures and for a largeregion is presented for some particular densities for the two-leg c ofcouplings,pairing(andpresumablysuperconductivity) laddermodelwithsizeL=16,J =1.5,andU = 0.5. eff − co-exists with ferromagnetic tendencies. As can be observed in Fig. 4(a), there is still a peak at 7 7 (a) tendencies. Note that for the electron doped case, there q =0 ρ=15/8 is a small peak at q=(0,π) for densities ρ&2.2. 6 y q = π ρ=15/8 5 y U =-0.5 q =0 ρ=33/16 )x4 eff qy= π ρ=33/16 IV. CONCLUSION q y S( 3 J=1.5 q =0 ρ=17/8 y Using ladders, we have studied analytically and nu- 2 qy= π ρ=17/8 merically a two-orbital Hubbard model. Via the DMRG technique we were able to investigate the model defined 1 onatwo-legladdergeometryforsystemswithlinearsizes 0 up to L = 24. Our spin structure factor data show that 1 1.5 2 π q / forthe“undoped”densityρ=2,astripe-likeAFMorder x ispresent,asobservedinpreviousExactDiagonalization 3 (b) studies.12,13 We have also presented evidence for triplet q =0 ρ=7/4 pairing tendencies of added electrons/holes close to the y Ueff=-0.5 q = π ρ=7/4 density ρ = 2, in some range of couplings, in qualitative 2 qy=0 ρ=9/4 agreement with previous investigations using chains,11 ) J=1.5 y andwithExactDiagonalizationcalculationsina less ex- qx qy= π ρ=9/4 treme FM regime of models for pnictides.12,13 More pre- ( S cisely, we have found that pairing (and presumably su- 1 perconductivity)and ferromagnetismco-existfor a large region of parameters in the regime U′ < J. Even for U′ comparable to J our results still indicate a (mild) ten- dency to pairing. Whether this rangeofcouplings for U′ 0 1 1.5 2 andJ isrealizedinrealmaterials,suchasheavyfermions π q / or pnictides, is a matter to be decided via experiments, x or with the help of ab-initio computer simulations. Figure 4: (Color online). Spin-structure factor S(qx) vs. qx for the two-leg ladder system with size L=16, J =1.5, and Ueff =−0.5. (a) S(qx) for the densities ρ=15/8, ρ=33/16, Acknowledgments andρ=17/8(seelegend). (b)S(qx)forthedensitiesρ=7/4 and ρ=9/4. This research was supported by the Brazilian agen- cies FAPEMIG and CNPq, the National Science Foun- q = (π,0) for densities close to ρ = 2. Note that these dation grant DMR-0706020, the Division of Materials peaks have smaller intensity than those found for ρ = 2 Science and Engineering of the U.S. Department of En- in Fig. 4(a), for the system with size L = 16. We have ergy, and the Center for Nanophase Materials Sciences, also observed that the height of the peak at q = (π,0) sponsored by the Scientific User Facilities Division, Ba- increases with the system sizes for the densities close to sic Energy Sciences, U.S. Department of Energy, under ρ = 2. These results indicate that a stripe-like AFM contract with UT-Battelle. The authors are grateful to magnetic order also exists for densities close to ρ = 2. Maria Daghofer for providing us with Exact Diagonal- As shownin Fig. 4(b), this orderdoes notexistanymore ization data to compare our DMRG results against and for ρ & 2.2 and ρ . 1.7, at least within the precision to Fernando Reboredo and Satoshi Okamoto for useful of our calculations, and it is replaced by ferromagnetic comments. 1 N. D. Mathur, F. M. Grosche, S. R. Julian, I. R. Walker, J. Am Chem. Soc. 130, 3296 (2008). D.M.Freye,R.K.W.Haselwimmer,andG.G.Lonzarich, 7 C. Wang and et al., Europhys.Lett.83, 67006 (2008). 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Coexistence of Pairing Tendencies and Ferromagnetism in a Doped Two-Orbital Hubbard Model on Two-Leg Ladders

Similar Coexistence of Pairing Tendencies and Ferromagnetism in a Doped Two-Orbital Hubbard Model on Two-Leg Ladders