Interplay of Coulomb interactions and c-axis octahedra distortions in single-layer ruthenates

Mario Cuoco, Filomena Forte, and Canio NoceLaboratorio Regionale SuperMat, INFM-CNR, Baronissi (SA), Italy

and Dipartimento di Fisica “E. R. Caianiello,” Università di Salerno, I-84081 Baronissi (SA), Italy�Received 21 July 2006; revised manuscript received 26 September 2006; published 27 November 2006�

We use an exact-diagonalization technique to investigate the phase diagram of correlated t2g electrons withinthe single-layer Ca-based ruthenates. The interplay between the charge fluctuations, induced by the crystallinefield potential, and those due to the Coulomb correlations leads to different magnetic and orbital patterns. Weshow that the compressive octahedra distortions tend to stabilize an antiferromagnetic state with C- or G-typestructure and different orbital configurations, depending on the degree of flattening. In the case of elongatedoctahedra, as a consequence of the orbital frustration, a ground state with incomplete ferromagnetism appears,except in the limit of large Coulomb repulsion with respect to the Hund’s rule, where an antiferromagneticG-type state emerges.

DOI: 10.1103/PhysRevB.74.195124 PACS number�s�: 75.50.Ee, 71.10.�w, 75.10.�b

I. INTRODUCTION

The exotic phenomena exhibited by many transition metaloxides �TMO�1 reflect the important role played by the or-bital degree of freedom due to its strong coupling with thespin, charge, and lattice degrees of freedom via the spin-orbit, Jahn-Teller, and the Coulomb interactions.2

TMO containing metallic ions with partially filled d�t2g�orbitals, like the t2g

2 state of V3+ in the vanadates,3 the t2g1 of

the titanates,4 and the t2g4 of state of the layered ruthenates,5–7

have attracted much interest due to their rich phase diagrams,marked by unconventional magnetic and orbital phases.

Among the t2g TMO materials, here we will concentrateour attention on the case of the t2g

4 configuration for the Ca-based single-layer ruthenates. The highly extended 4d shellswould a priori suggest a weaker ratio between the intra-atomic Coulomb interaction and the electron bandwidth. Onthe other hand, the extension of the 4d shells also pointstoward a strong interaction between the d orbitals and thenearest-neighbor oxygen orbitals, implying that these TMOhave the tendency to form distorted structure with respect tothe ideal one. As a consequence, the M-O-M bond angle maybe considerably less than the ideal 180° and in turn, by nar-rowing the d bandwidth, brings the system on the verge of ametal-insulator transition or into an insulating state. Refer-ring to the deviations from the ideal structure, the c-axisdistortions turn out to be considerably relevant since theycontrol the degree of orbital fluctuations by influencing thedistribution of the charge within the three orbitals in the t2gsector. Looking at the Ca2RuO4 or the derived compoundsvia partial substitution of the Sr with Ca �Ca2−xSrxRuO4�,two basic features appear at the heart of their phenomenol-ogy: �i� the role of the c-axis compressive and the expansivedistortions of the octahedra in maximizing or quenching thelocal orbital fluctuations; �ii� the interplay of these distor-tions with the Coulomb interactions at the interface betweenthe insulator and the metal in determining the most favorablespin/orbital patterns. The relevance of these aspects for theCa2RuO4 compound can be easily understood looking at itsphase diagram.8 This compound is a Mott insulator whichexhibits G-type antiferromagnetic ordering below TN

=110 K and an insulator-metal transition at TMI=357 K onheating, the metal-nonmetal transition occurring simulta-neously with a first-order structural phase change from te-tragonal to orthorombic lattice symmetry which leads to anelongation of the RuO6 octahedra perpendicular to the RuO2layers.9–11 A puzzling aspect of this compound is representedby the charge distribution in the different orbitals of the t2gsector. The available experiments bring forward differentscenarios. Indeed, the observation of G-type antiferromag-netism has been interpreted as a preferential occupation ofthe dxy orbital �completely filled� and with the half-filled dxzand dyz bands in a state with a ferro-orbital type order.12 Onthe other hand, x-ray absorption spectroscopy reported anoccupation of 0.5 holes in the dxy orbital, bringing the atten-tion toward the interplay between the strong spin-orbit cou-pling and the distortion of RuO6 octahedra.13,14

Moreover, to explain the unexpected t2g hole distributionan orbital ordered �OO� state has been considered as a con-sequence of the combination between Coulomb interactionsand lattice effects.15 Nevertheless, such a scenario has beenconfuted and further puzzled by the observation, via resonantx-ray diffraction of an orbital ordering transition at a wavevector characteristic of the antiferromagnetic ordering.16

Structural changes may be induced by the application ofboth chemical substitution17,18 or by an externalpressure.19–21 The Sr substitution for Ca acts as an effectivenegative pressure and gives rise to a rich phase diagram too.Indeed, the Ca2−xSrxRuO4 is an insulating antiferromagnetfor x�0.2, a metallic correlated antiferromagnet for 0.2�x�0.5, a nearly ferromagnetic metal for x�0.5, and a metal-lic paramagnet at higher doping exhibiting an unconven-tional superconductivity for x=2.22 The transition from theantiferromagnetic insulator to the correlated metal, withstrong ferromagnetic fluctuations, is accompanied by a modi-fication of the RuO6 octahedron rotation/tilting and by aninversion in the c-axis distortions from compressive to ex-pansive octahedral. This is a clear manifestation of the rel-evance of the competition between the Coulomb repulsionand the octahedral deformations in this class of compounds.

With the aim to shed light on the physics of the abovementioned single layered Ca-based ruthenate, we focus hereon the spin, orbital, and charge �SOC� patterns that develop

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into a three-orbital extended Hubbard model, both by induc-ing a deformation of the octahedron and by tuning the Cou-lomb amplitudes ranging from weak to strong coupling. Thisanalysis has been performed considering a 2�2 plaquettewithin an unbiased scheme of computation, as the exact di-agonalization method, in such a way to get more insight onthe nature of the ground state configurations, by treating thecompeting microscopic mechanisms on equal footing. Withinthis framework, we will show that an important role isplayed by the dynamics of the double occupied configura-tions, as a characterizing element in selecting quantum stateswith different spin, orbital, and charge structure. Indeed, go-ing from the flat to the elongated octahedra case, there occursa gradual charge transfer from the xy to the z sector of the t2gmanifold. Such modification of the local charge configura-tion is strongly interrelated to the strength of the Coulombparameters, and it is generally accompanied by a SOC rear-rangement of the short range quantum correlations. Theanalysis will be concentrated on the following aspects: �i� thedetermination of the relevant SOC patterns as a function ofthe Coulomb and crystalline field potentials, �ii� the study ofthe competing states at the boundary of a configuration withfull/half-filled xy ��z� sector. We will show that the compres-sive c-axis octahedral deformation generally leads to antifer-romagnetic states of C-�G�type, depending on the orbital dis-tribution of the double occupied states, with main antiferro-�ferro-�orbital correlations between those configurations,respectively.

The interplay between the Hund’s rule and the Coulombrepulsion is the key control parameter of the boundary be-tween the C- and the G-like antiferromagnetic state. The in-version of the c-axis distortions, in the regime of expansiveoctahedra, modifies the character of the magnetic exchangegiving rise to ferromagnetic states with partial spin align-ment, reflecting the subtle competition between the ferro andantiferromagnetic exchange. The nature of the ferromag-netism is strongly related to a mechanism of orbital frustra-tion due to the peculiar connectivity of the t2g orbitals and tothe two-dimensionality of the problem under examination.

The points �i� and �ii� together with the investigation ofthe origin of ferromagnetism will be then used to discuss thefeatures of the ground state for the CaRu2O4 system and itsevolution under the application of an external pressure or achemical substitution.

Finally, we will introduce and analyze an effective spin/orbital model that, in the limit of large intra-atomic interac-tion, is suitable for extracting the main qualitative featuresintrinsic of the competition between magnetic/orbital andcharge correlations.

The scheme of the paper is the following. In Sec. II weintroduce the microscopic model, while in Sec. III wepresent the results for the phase diagram in the presence ofCoulomb correlations and c-axis octahedral distortions. Sec-tion IV is devoted to the analysis of the strong couplingeffective spin-orbital model in the different regimes of crys-talline field potential and the last section contains a discus-sion on the results presented and the conclusions.

II. THE MODEL

The description of the electron dynamics in the ruthenateoxides, having four electrons in the 4d Ru bands, starts from

the consideration that the Hund’s rule coupling, maximizingthe total spin at each Ru site, is not strong enough to over-come the eg-t2g crystal field splitting, leaving empty the egsector. The constraint to accommodate four electrons in thet2g manifold imposes a double occupation in one of the threeorbitals. Furthermore, the t2g orbitals are expected to havedifferent energies because the RuO6 octahedra are deformed.Hence, the way to allocate the double occupied configurationdepends both on the character and the strength of the octa-hedral deformation, and on the amplitude of the Coulombinteractions compared to the bare kinetic energy.

A basic model Hamiltonian capturing the above featuresis built up by different contributions that reproduce the cor-related local dynamics of electrons and electrons coupled tothe structural distortions in the t2g manifold:

H = Hkin + Hel−el + Hcf . �1�

The first term in Eq. �1� is the kinetic operator definingthe in-plane connectivity between the Ru t2g orbitals via theoxygen ions,

Hkin = − t�ij,�

�di��† dj�� + H.c.� �2�

di��† being the creation operator for an electron with spin � at

the i site in the � orbital. The hopping amplitude is assumedto be t for all the orbitals in the t2g manifold, due to thesymmetry relations of the connections via oxygen � ligands.One important aspect of the single particle connectivity isthat for each direction there are only two active hoppingswithin the t2g sector. Hence, for the two-dimensional system,the xy orbital has a link both along the x and y direction,while the xz �yz� are connected only on the x�y� axis, respec-tively.

The second term in H stands for the local Coulomb inter-action between t2g electrons:

Hel−el = U�i�

ni�↑ni�↓ − 2JH�i��

Si� · Si�

+ �U� −JH

2��

i��

ni�ni� + J��i��

di�↑† di�↓

† di�↑di�↓,

�3�

where ni��, Si� are the onsite charge for spin � and the spinoperators for the � orbital, respectively. U �U�� is the intra-�inter� orbital Coulomb repulsion, JH is the Hund coupling,and J� the pair hopping term. Due to the invariance for ro-tations in the orbital space, the following relations hold: U=U�+2JH, J�=JH.

The Hcf part of the Hamiltonian H is the crystalline fieldterm, directly related to the expansive or compressive modeof the RuO6 octahedra:

Hcf = �i

�nixy − nixz − niyz� . �4�

Zero amplitude for the parameter indicates no distor-tions and full local orbital degeneracy. Positive �negative�values of are related to elongated �flat� RuO6 octahedronalong the c axis, and thus the occupation in the d�z �dxy�

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sector is favored, respectively. Within this description, themicroscopic parameter contains the information of thestrong coupling between the extended 4d Ru orbitals and thenearest-neighbor oxygen orbitals, both due to �i� the contri-bution of symmetry lowering from the static Coulomb poten-tial of the surrounding oxygens, and �ii� the energy shift dueto the formation of antibonding molecular orbitals betweenthe TM atom and the neighbors oxygen.

Finally, we would like to point out that, in the presenteffective model, the electron variables of the oxygen havebeen projected out and we have limited ourselves to the puredynamics of the 4d bands.

III. PHASE DIAGRAM: COULOMB INTERACTIONVS C-AXIS OCTAHEDRAL DISTORTIONS

By means of the Lanczos technique, we performed thenumerical simulation on a 2�2 plaquette, the smallest sizethat contains the symmetry features of a planar structure.Due to the large dimension of the Hilbert space, and theconsequent high computational effort, it turns out to be con-siderably demanding to perform a larger size simulation thattakes into account all the quantum configurations of thethree-orbital Hubbard model. However, having the advantageto determine the ground state �GS� in the whole phase spacewithout any bias in the computation, one can explore thenature of the SOC correlations ranging from intermediate tolarge Coulomb coupling under different octahedral distor-tions. The analysis of the ground state GS has been per-formed by evaluating the relevant correlation functionslinked to the SOC pattern and by evaluating the spin gapbetween the GS and the lowest energy configurations in theother subspaces with different total spin projection. For thispurpose, it is useful to introduce the following correlationfunctions in the available momentum space:

S�q� = �i,j

eiq�Ri−Rj��Siz Sjz� ,

P����q� = �i,j

eiq�Ri−Rj��pi�pj��� ,

where S�q� and P����q� are the spin and doublon structurefactors, and q is the characteristic wave vector associatedwith the spin and charge pattern. Here, Siz=��Si�z is the totalz projection on the site i, while pi�=ni�,↑ni�,↓ is the localdouble occupancy operator counting the double occupiedconfigurations in the orbital �. Still, � is label indicating ageneric orbital �xz ,yz ,xy�, while �=z indicates the doubleoccupancy operator in the �z sector, i.e., piz= �pixz+ piyz�.

Hereafter, to label the ground state in each region of thephase diagram, it is useful to adopt the notation Z�l� to indi-cate a phase with a magnetic character Z, whose local distri-bution of double occupied states is such that �pixy�= l and�pixz�= �piyz�=1− l. In particular, the Z character can be anti-ferromagnetic �AF� or ferromagnetic, like Fk with k beingthe average total projection of the spin momentum perplaquette �see Fig. 1�. The F state stands for a GS configu-ration where all the spins are polarized.

A. Degenerate case

We start from the zero crystal field potential case, corre-sponding to the complete local degeneracy of the three orbit-als. Nevertheless, due to the constraint of the two dimension-ality, there occurs a dynamical symmetry breaking related tothe unequal kinetic energy for the xy and �z bands. By ana-lyzing the correlation functions for the spin, orbital, andcharge channel as function of the microscopic parameters, itis possible to extract the following information about thecharacter of the GS and on the related phase boundaries. InFig. 2, we have reported the phase diagram as a function ofthe scaled Coulomb repulsion �U�−JH� versus the Hund cou-pling JH. The main part of the diagram is characterized by aC-AF�1/2� Fig. 1�f� with antiferromagnetic correlations ofC type. Indeed, the system manifests the tendency to haveantiferromagnetic exchange along the x �y� direction, andferromagnetic spin coupling in the y �x�, respectively. Thisbehavior can be inferred by investigating the spin structure

FIG. 1. A schematic view of the representative spin/charge con-figurations that contribute to the ground state on the 2�2 plaquette.The circle with the filled shadow stands for the double occupiedstate. �a� shows the level structure of the t2g sector and theplaquette. From �b� to �e� are reported the different ferromagneticconfigurations. Here the circle on one bond indicates the tendencyto form a singletlike state that quenches the magnetic moments onthe related sites. From �f� to �i� are plotted the possible antiferro-magneticlike states with different charge and orbital occupation.The orbital correlations between the DO states are mainly antiferro-orbital in the C-AF, ferro-orbital in the G-AF, and without anypreferential pattern in the ferromagnetic configurations,respectively.

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factor see Fig. 3�a�. It turns out that S�q� has a maximumfor a wave vector amplitude of q= �0,�� q= �� ,0�, show-ing that the spins tend to have a predominant pattern withantiferromagnetic exchange along the x�y� axis. Looking at

the orbital and doublon channel, P����q� see Fig. 3�b� hasthe following behavior. The value of P in the xy sector indi-cates that the double occupied configurations distribute al-most homogeneously without any preferential arrangement,though the value of P�,��� ,�� has an amplitude larger thanthe longitudinal Pxy,xy�0,�� Pxy,xy�� ,0�. This result showsa tendency of the system toward an antiferro-type orbitalpattern for the double occupied states �DO�. Still, the averagedistribution of the DO configurations is such that one hashalf occupation in the xy band and half in the �z sector. Theoccurrence of a C-AF state is a consequence of the one-dimensional character of the hopping connectivity for the �zbands and of the orbital exchange on a given bond when thedouble occupation is not sitting on homologue orbitals. We

will see later that in the intermediate/strong coupling regime,within an effective spin/orbital model able to describe thelow energy dynamics, it naturally emerges, the C-AF state asa ground state.

As expected, the large Hund limit is characterized by aregion with full polarized spin distribution. The ferromag-netic F�1/4� Fig. 1�e� region is marked by a low density perorbital of DO configurations in the xy band compared to the�z sector. There occurs a local interorbital charge redistribu-tion in the transition from the C-AF state to the F one, un-derlining the correlation between the DO density and themagnetic character of the ground state. Thus, the competitionbetween the F and C-AF state manifests directly in connec-tion with the change in the DO orbital population. This spin/charge coupling reflects the tendency of the system to exhibita jump of the magnetization accompanied by a structuralchange under the application of an external field, which isusually observed in the Ca/Sr-based family of ruthenates.5,23

As final consideration, we would like one to notice thatfor any small nonzero scaled Coulomb repulsion or Hundcoupling, the amplitude of the spin structure factor S�q� isalmost constant as a function of q with a little maximum atq= �0,�� q= �� ,0�, indicating the predominant character ofthe C-AF type spin correlations. This result is strictly relatedto the limited size of the system; indeed, we do expect that,for larger size clusters, due to the growing charge fluctua-tions present when �U�−JH� tends to zero, the ground statewould exhibit quantum paramagnetic correlations.

B. c-axis compressive octahedral distortions

To study the change of the GS configurations in presenceof flatten distortions we have first considered one represen-tative case with / t=−0.5 and then analyzed the relevantcorrelators in the parameter space JH , �U�−JH�.

In Fig. 4, the associated phase diagram is reported. As onecan see, the general outcome induced by the lowering of thexy local energy with respect to the �z can be summarized inthe following features: �i� the threshold in the ratio JH / �U�

FIG. 2. Phase diagram as a function of the scaled Coulombrepulsion �U�−JH� vs the Hund coupling energy JH at zero crystal-line field amplitude.

FIG. 3. �Color online� �a� Spin and �b� doublon/orbital correla-tions as a function of the crystalline field potential for a represen-tative value of the Coulomb parameters JH / t=1.5 and JH /U�=1/4.

FIG. 4. Flat octahedral distortions: phase diagram as a functionof the scaled Coulomb repulsion �U�−JH� vs the Hund couplingenergy JH for a value of the crystalline field given by / t=−0.5.

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−JH� for setting the ferromagnetic phase is considerably en-hanced; �ii� a new antiferromagnetic configuration at largevalues of the Coulomb repulsion appears. In case �i�, as ex-pected due to the charge distribution within the F region, thecritical Hund coupling to fully polarize all the spins in-creases, because now the crystal field potential opposes tothe large occupation of DO in the �z sector. On the otherhand, the case �ii� arises as a consequence of the interplaybetween the renormalization of the bandwidth and the gain inthe crystalline field energy. Thus, above a critical Coulombrepulsion the C-AF state undergoes a transition into a G-AFstate. The G-type antiferromagnetic state has an orbital popu-lation given by a complete occupied xy orbital ��pixy�=1� andsingly occupied states for the �z sector ��pi�z�=0�. This or-bital configuration is usually indicated as ferro-orbitally or-dered see Fig. 1�g�. Concerning the antiferromagnetic pat-tern, the spin structure factor gives a dominant peak at awave vector q= �� ,��, indicating a G-type state, namely theeffective exchanges between the local Ru spins are antifer-romagnetic along both the planar directions see Fig. 3�a�.The magnetic character is easily understood in the G-AF�1�configuration, since now the degree of freedom of the xyorbital is completely frozen, and in the half-filled �z sectorthe Coulomb correlations lead to a dominant AF superex-change. Looking at the phase boundary between the C-AFand the G-AF state, one can observe that the critical �U�−JH� value is rather independent from changes of JH, thusindicating a linear relationship between the Coulomb andHund coupling at the interface. Hence, we conclude that theratio between the Coulomb repulsion and the electron band-width is a tunable parameter that, upon a given c-axis com-pressive distortion, can drive the transition from a state withcompeting ferromagnetic and antiferromagnetic exchange toan isotropic AF configuration. It is worth pointing out that,though the average crystalline field energy is not changed,the effective c-axis distortions in these two regions are dif-ferent because in the G-AF�1� the octahedra are completelyflatten, while in the C-AF�1/2� are averagely undistorted.

C. c-axis expansive octahedral deformations

Let us now consider the case of the other distortive mode.Assuming that the octahedra are now elongated and simulat-ing this deformation by a value of the crystal field potentialof / t=0.5, we again perform a systematic analysis of theGS correlations, spanning the complete space of Coulomband Hund couplings. The results are summarized in Fig. 5where the correspondent phase diagram is plotted. As onecan observe, by comparing the diagram of Fig. 2 with thepresent one, the region with a C-AF ground state is com-pletely replaced by different configurations with ferromag-netic character. In this regime of c-axis distortions, the �zorbitals are the two lowest degenerate levels separated by anenergy �� from the single xy orbital. Hence, the gain of thecrystalline field potential would lead to the largest number ofDO configurations in the �z orbital sector. This behavior iscounteracted by the requirement of optimizing the kineticenergy between the correlated electrons in the �z and thosein the xy orbitals, leading to competing ferro or antiferro

patterns between the DO configurations. Furthermore, theconstraint of the hopping connectivity within the �z manifold�only one of the two z orbitals is active for each planar di-rection� acts as a frustration in the dynamics of the DO con-figurations. Let us consider the features of the phase dia-gram. By an inspection of Fig. 5 it is possible to individuatetwo distinct regions: �a� an area, marked by �U�−JH� / t7.5, with shells having a GS characterized by unequal spinpolarized configurations; �b� a complementary part character-ized by an antiferromagnetic G-type ground state where allthe DO are placed in the �z sector.

The �a� case is dominated by GS configurations with anonzero value of the total spin momentum. The F part indi-cates a GS with maximum allowed spin polarization. Theother regions are labeled with Fk indicating configurationswith an incomplete ferromagnetism Figs. 1�b�–1�e�. Whenthe transitions from one to another GS within the phase dia-gram are considered, a few peculiar aspects have to be un-derlined. In the case �a�, assuming a constant difference be-tween the interorbital Coulomb repulsion and the Hundcoupling, and by increasing the amplitude of JH / t startingfrom zero, there occurs a sequence of transitions from theAF-like state to the F configuration going through the F3, F2and again the F3 intermediate polarized GS, respectively.This reentrant behavior in the spin channel reflects the com-petition and the frustration between the ferromagnetic andantiferromagnetic exchange in terms of the DO distribution.Indeed, the F3 region in the range of JH / t�1 is characterizedby the absence of DO in the xy orbital, while the large JH / tF3 configuration has an average of 1 /4 DO in the xy orbital.The occurrence of an intermediate state with a lower value ofthe total magnetization between the two F3 states reveals asubtle competition between the charge and orbital degrees offreedom in this regime of Coulomb repulsion and tetragonaldistortions. In the F3�0� region, it is the gain in the crystalfield potential, due to the completely filled DO distribution inthe �z manifold, that favors an almost complete spin polar-ized configuration. Increasing the Hund coupling and the

FIG. 5. Elongated octahedral configuration: phase diagram as afunction of the scaled Coulomb repulsion �U�−JH� vs the Hundcoupling energy JH for a representative value of the crystalline fieldgiven by / t=0.5.

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Coulomb repulsion, the charge fluctuations get more sup-pressed and, to optimize the orbital exchanges in the pres-ence of a different distribution of DO, first there occurs areduction of the spin magnetization and then the F3 and Fstates are recovered due to the gain in the Hund energy.

Another interesting aspect of the ground state properties isinferred if one fixes the value of the Hund coupling and tunesthe Coulomb repulsion from weak to strong coupling. Thishorizontal scan of the phase diagram contains the informa-tion of the change in the spin/orbital correlations induced bythe suppression of the intra- and interorbital charge fluctua-tions due to a possible reduction in the bandwidth. As onecan see, depending on the relative ratio between the Hundcoupling and the bare hopping amplitude, there is a differentsequence of transitions from the polarized to the antiferro-magnetic configuration. Starting with the F region, the con-nection from the fully polarized to the AF state goes througha gradual change in the magnetization from the maximumvalue to a vanishing value, via all the allowed intermediatepolarization steps �F3, F2�. This behavior reflects a peculiarcompetition between the ferro- and antiferromagnetic ex-change such that the smooth suppression of the ferromag-netic correlations is gradually replaced by the AF-like ex-changes. It is then possible to argue that such change of themagnetization could manifest in a bulk system as a continu-ous �second orderlike� quantum phase transition between aferromagnetic and an antiferromagnetic state. Otherwise, inthe limit of weak Hund coupling, the system can go directlyfrom an almost ferromagnetic to the antiferromagnetic state.

Concerning the orbital correlations, both the ferromag-netic and antiferromagnetic configurations with full or par-tially full DO in the �z sector, do not manifest any specifictendency in the orbital arrangement of the DO within themanifold, namely there is an almost equal probability offerro- or antiferro-orbital correlations between the xz and yzorbitals.

D. From flatten to elongated octahedra in different regimesof Coulomb/Hund ratio

In the previous sections we have investigated the spin/orbital patterns in the space of the Hund coupling against thedifference �U�−JH�, for three representative cases of crystal-line field potential, corresponding to the case of undistorted,flatten, and elongated octahedra, respectively. As a summaryof the previous analysis, we have seen that the tendency ofcompressive c-axis distortions is to stabilize a ground statewith preferential antiferromagnetic correlations of C- orG-type, depending on the DO orbital distribution. On thecontrary, the expansive distortions generally lead to a ferro-magneticlike configuration characterized by different pos-sible values of the total magnetization, ranging from zero tothe maximum allowed value. Still, a G-AF state is stabilizedin the limit of large intra-atomic Coulomb repulsion.

Now we look at the evolution of the different GS configu-rations by simulating an inversion of the c-axis distortions,tuning the amplitude of the crystalline field potential fromnegative to positive values. To get insight into the competi-tion between the charge fluctuations induced by the c-axis

distortions, and those due to the Coulomb repulsion, we fixthe ratio JH /U� and vary JH vs . This parametrization al-lows for scanning the parameter phase space of the JH ,U�along a specific direction in various regimes of couplings. Inparticular, we consider two representative cases, correspond-ing to the situation of small and comparable Hund couplingwith respect to the Coulomb repulsion, i.e., �a� JH /U�=4/5and �b� JH /U�=1/4.

The phase diagram for the case �a� is reported in Fig. 6.One can clearly notice that the region at negative crystal fieldamplitudes �flatten octahedra� is entirely antiferromagnetic,while the part for the elongated octahedra is dominated byconfigurations with pronounced ferromagnetic correlations.Concerning the negative side of the phase diagram, few com-ments are worth pointing out. The critical value of for thetransition from the C-AF state to the G-AF state is of theorder of the bare hopping amplitude in the range of t ,2t,even for large Hund coupling and correspondent Coulombrepulsions. This aspect indicates a weak renormalization ofthe effective bandwidth via the Coulomb interaction, as theenergy required to quench the DO configurations within thexy orbital is given by the direct competition between the gainin the crystalline field potential and the loss of kinetic en-ergy. Another interesting feature is represented by the factthat the transition between the C-AF�1/2� configuration andthe G-AF�1� state is separated by an intermediate phase char-acterized by a noninteger distribution of DO between the �zand the xy sector. This state, indicated as G-AF�3/4�, exhibitsG-type antiferromagnetic correlations with weak ferromag-netic correlations Fig. 1�i�. Moving to the other side of thephase diagram, the interface of the C-AF�1/2� region sepa-rates the antiferromagnetic GS from different ferromagneticconfigurations, depending on the amplitude of the Hund cou-pling. The change, due to c-axis expansive distortions, tendsto weaken the antiferromagnetic correlations within theC-AF state and enhances the tendency to have an incompleteferromagnetism. The latter is an evidence of some residualantiferromagnetic correlations within the GS. The transitionfrom the C-AF to an F-like state is always accompanied by amodification of the charge distribution due to the different

FIG. 6. Evolution of the phase boundaries vs the CF amplitudefrom flat to elongated configuration for a ratio between the Hundand the Coulomb repulsion given by JH /U�=4/5.

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density of DO per orbital. From a general point of view, theferromagnetic states are characterized by a small density ofDO in the xy band.

Let us now discuss the case �b�. When JH /U�=1/4, thephase diagram presents some features that are different ifcompared to the previous case. As one can see in Fig. 7, thescenario for the compressive octahedral distortions is quali-tative similar to the case �a�. We do notice that the transitionfrom the C-AF configuration to the G-AF one does not gothrough the G-AF�3/4� ground state, thus revealing a charac-teristic that manifests when the Hund and Coulomb couplingare comparable. Furthermore, the critical crystalline fieldamplitude to drive the transition from the C-AF to the G-AFstate has an opposite trend, when compared to the previoussituation, as the Hund coupling is increased from small tolarge values. In the case �a� the upturn of the critical line wasmoving toward a larger amplitude of , as the Hund cou-pling is varied. In the present condition, the G-AF region ismore easily stabilized as in the limit of JH / t�2 the value forcrossing from one to another AF state is renormalized downto / t�0.25. The place where the boundary starts to bendindicates the crossover between the weak and theintermediate/strong coupling, followed by a renormalizationin the relevant scale that controls the orbital exchange. Con-cerning the region corresponding to elongated octahedra, anew portion emerges that is characterized by a G-AF statewith all the DO sitting in the �z band, G-AF�0�. This con-figuration is somehow symmetric with respect to theG-AF�1� obtained in the extreme flat situation. Now, at theinterface of the C-AF�1/2� state there are two configurationswith incomplete ferromagnetism. It is not possible in thisregime of the Hund/Coulomb relation to stabilize the fullferromagnetic state in the c-axis elongated distortions. Thisaspect reflects the subtle competition between the magneticand orbital exchange in different limits of Coulomb andHund coupling.

E. Effective spin-orbital model in the large Coulomb limit

To get more insight into the features of the phase diagramextracted in the previous sections, we introduce and analyze

an effective superexchange Hamiltonian for S=1 spins, thatis suitable to describe the low energy physics in the limit oflarge Coulomb coupling. Such a Hamiltonian is the generali-zation of the one obtained for a Mott insulator having non-degenerate orbitals, for value of U such that U� t. In thepresent case, dealing with an atomic t2g

4 configuration, onehas to consider the possible contributions which originatefrom all the virtual excitations on a bond �i , j� of the typedi

4dj4→di

5dj3 �Ref. 24� with the hopping t allowed only be-

tween two out of three of the t2g orbitals. To get the result,we profit from a particle-hole transformation within the t2gsector, in a way to reduce the dynamics of four electrons tothat of two holes, the double occupied configuration beingmapped into an empty state. Within this frame, the interact-ing part of the Hamiltonian Eq. �1� is not changed and mapsto itself, while the hopping matrix and the part of the crystalfield potential changes the sign. Hence, by means of the anal-ogy with the case of the cubic vanadates,3 after having con-sidered the second order processes and projected on the rel-evant configurations, one obtains the following spin-orbitalHamiltonian �Ref. 3�:

Hs/o = J��

��ij���

�S� i · S� j + 1�Jij��� + Kij

��� �5�

with J=4t2 /U, S� the spin operator, � � �ij� the given bondparallel to the axis direction, relabeled now as �= �a ,b ,c�,respectively. Still, Jij

��� and Kij��� are operators acting only in

the orbital space. They have the following form in terms of apseudospin representation defined in the orbital subspacespanned by the two orbital flavors active along a given di-rection �:

Jij��� =

1

2 �1 + 2�R����i · �� j +

1

4ninj�

− �r��iz� j

z +1

4ninj�

−1

2�R�ni + nj�����

Kij��� = �R���i · �� j +

1

4ninj� + �r��i

z� jz +

1

4ninj�

−1

4�1 + �R��ni + nj�����

.

The coefficients R=1/ �1−3�� and r=1/ �1+2�� are relatedto the energy of the t2g intermediate states with �=JH /U. Tosimplify the notation, let us rename the basis of the t2g mani-

fold as a=yz, b=xz, c=xy and introduce the related operators��i

�= ��xi� ,�yi

� ,�zi� �. Hence, the ��i

� acts in the subspace definedby the two orbitals which can be effectively connected on thegiven bond along the � direction. For example, if the bond isalong the c axis, the pseudospin coupling is between the a

and b orbitals, and it can be expressed via the Schwingerrepresentation: �+i

c =ai†bi, �−i

c =bi†ai, �zi

c = 12 �nia−nib�, and ni

c

=nia+nib. Similar relations with the exchange of a bond in-dex hold for the other axis directions.

FIG. 7. As in Fig. 6 assuming a ratio between the Hund and theCoulomb repulsion JH /U�=1/4.

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In this representation, the crystal field potential for theholes reads as follows:

H = �i

�nia + nib − nic� ,

where a constant energy per Ru ion has been dropped.From an inspection of this Hamiltonian term, one can

deduce that the flatten octahedral distortions tend to favor an

occupation of two holes in the a and b orbitals. Otherwise,the elongated octahedra configuration would favor a state

with one hole in the c and another in a or b orbitals. As wecan see, on each bond there are two out of three orbitals thatcontrol the orbital correlations and in turn the character andthe sign of the spin coupling. The complexity of the problemis due to the fact that there are three flavors and two of themare SU�2� spin/orbital correlated along the correspondent di-rection, thus leading to an orbital frustration in the other one.As pointed out in Ref. 3, this subtle interplay can lead todifferent magnetic and orbital configurations. It is worthpointing out that our problem contains extra elements withrespect to the case of the vanadates,3 since in that case one istreating a cubic system where one electron occupies the corbital and is frozen while the other can fluctuate between

the a and b configurations. The present situation has the con-straint of a layered structure for the electron dynamics, thatquenches a priori the spin-orbital coupling along the c axis.Moreover, due to the possible modification of the octahedraldistortions, as we have seen in the previous sections, thethree orbitals areall involved into the dynamics.

To further investigate the general properties of the spin/orbital correlations for the case of the ruthenates, first we doconsider the competing configurations on one bond and thenwe analyze the full planar problem within a mean-fieldscheme of approximation.

As a starting consideration, one can observe that there arenine possible configurations in the orbital sector. Hence, it ispossible to group the configurations on a generic bond �ij� inthe following way: singlet and triplet along the a axis �sa�=1/�2ai

†aj†�bi

†cj†−ci

†bj†� �v�, �ta�=1/�2ai

†aj†�bi

†cj†+ci

†bj†� �v�

�and the correspondent along the other directions� togetherwith the following states �ab�=ai

†aj†bi

†bj† �v�, �ac�

=ai†aj

†ci†cj

† �v�, �bc�=bi†bj

†ci†cj

† �v�, where �v� is the vacuumstate.

Since we are interested in determining the character of thespin coupling, one has to compare the orbital correlations inthe lowest possible configuration in the different regimes ofstructural distortions, so to extract the nature of the spin cor-relations in an indirect way. By considering for instance abond along the a axis, the expectation values for the operator

Jija given by E�

Ja= �� � Jij

a ���, with ��� being one of the statesintroduced above, are the following:

EabJa

= EacJ =

1

4�1 − �r� ,

EbcJa

=1

2�1 − �r� ,

Esa

Ja= −

1

4�1 + 4�R� ,

EtaJa

=1

4,

Esb

Ja=

1

4�1 − �R − �r� ,

EtbJa

= Esc

Ja= Etc

Ja= Esb

Ja.

The expressions along the other directions are obtained by apermutation of the bond indices. Referring to the contribu-tion related to the crystal field energy, one can easily get thefollowing expressions:

Eab = 4 ,

Esa

= Eta = Esb

= Etb = 2 ,

Ebc = Eac

= Esc

= Etc = 0.

The above relations yield the energy hierarchy of the pos-sible orbital configurations on a given bond. Of course, theresult is equivalent for any bond direction as the crystal fieldterm is an intra-atomic operator.

Before considering the interplay between the Coulomband the crystalline field potential, it is useful to make a fewconsiderations about the sign of the orbital term in front of

the spin interaction as due to the expressions E�Ja

. Since �r is

a quantity always smaller than 1, EabJa

, EbcJa

, and EacJa

occur aspositive quantities. Furthermore, the energy associated withthe singlet and triplet orbital configurations, along the bondperpendicular to a, has a sign that depends on the amplitudeof �. Indeed, the amplitude �1−�R−�r� changes from posi-tive to negative when � �c with �c�0.24. Finally, as �and R are positive quantities, the term associated with theorbital singlet along the assigned bond is always negative.Hence, if the orbital configuration along the a bond is in thestate �ab�, �ac�, �bc�, and �ta� the spins will prefer an antifer-romagnetic exchange. When the singlet along a is preferred,the spin coupling is negative and hence there occurs a ferro-magnetic alignment between the bond spins. Still, the con-figurations associated with the �sb�, �tb�, �sc�, and �tc� wouldyield a antiferromagnetic �ferromagnetic� spin exchange if���c�� �c�, respectively.

At this point we can address some of the features obtainedfrom the analysis in the previous sections, when the compe-tition between the crystal field potential and the Coulombcorrelations, at least in the limit of large Coulomb coupling,is concerned. Starting from the extreme flatten octahedraconfigurations, one observes that when � � �J the energy ofthe �ab� state is stabilized with respect to the other possibili-ties, due to the greater gain in crystal field energy. As aconsequence, on each bond either along the a- or the b-

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direction, the spin coupling is positive, as observed before,thus leading to a G- type antiferromagnet where the a and borbitals are singly occupied.

At lower values of / t, there are other configurations thatcome into play. Concerning the crystal field energy contrib-utes, the first states that enter in competition are the orbitalsinglet �triplet� along a or b, as the gain is −2. Betweenthese states, the orbital correlations, originating from theCoulomb and Hund interactions, lower the energy of the or-bital singlet homologue of the given bond, that in turn leadsto a ferromagnetic coupling between the neighbor spins.Once within the bond along a are set the singlet orbital con-figuration, due to the anisotropic structure of the spin/orbitalHamiltonian, no exchange can occur along the b directionbetween the two singlets. If one forms a state with two or-bital singlets along the a direction in a plaquette, the con-figurations along the b axis are of the type �ac�, �bc�, or �sb�that induce an antiferromagnetic exchange. The formation oforbital singlets stabilize a ferromagnetic exchange within thegiven bond and an antiferromagnetic one between the singletbonds thus leading to the C-AF configuration. Due to thedegeneracy between a and b, the configurations with orbitalsinglets along the two planar directions have the same en-ergy. Hence, when the energy of the crystal field potentialcompetes with that one for the spin/orbital correlations, a

state with partial filled c and a, b orbitals occurs with respectto the previous one, where all the holes where placed in the

a, b orbitals. This configuration reminds that one analyzed inthe vanadates, where the orbital resonance was occurring

along the c axis between the a and b orbitals. In the presentcase, due to two dimensionality, there is still a degeneracydue to the formation of bond singlets both along the a and bdirection.

If one changes the sign of the crystal field potential, mov-ing to the regime of expansive c-axis octahedral deforma-tions, the energy relations between the bond configurationsare drastically modified. For smaller values of / t, it is stillconvenient to accommodate one orbital singlet on one bondand it turns out to be favorable introducing on the other bondconfigurations of the type �bc�, �ac�, �sc�, or �tc�, that have noloss in the crystalline field potential. This arrangement yieldsan incomplete ferromagnetism, due to a frustration in thespin/orbital correlations, if on one bond one has an F ex-change and on the other an AF, and they are linked via AFcorrelations. The attempt to optimize the magnetic exchangeleaves uncompensated some of the antiferromagnetic ex-changes. Otherwise, by increasing the value of �, it is pos-sible to modify the sign of the �sc� or �tc� orbital correlationsboth along the a or b directions, thus stabilizing the fullalignment of the spins. By further increasing / t, the systemtends to completely disfavor the orbital singlets. In this re-gime, the sign of the magnetic exchange is always uniformand positive, except in the limit of � �c. The situation ofuniform positive orbital correlations yields a G-type antifer-romagnetism that reproduces the G-AF�0� configuration ob-tained in Sec. III C. Otherwise, for strong Hund coupling, theoccurrence of configurations of the type �sc� or �tc�, withrelated negative orbital correlations, leads to a ferromagneticground state. Concerning the orbital correlations, due to the

degeneracy between the a, b orbitals, there is no tendency toa specific ordering in between the empty configurations.

In order to be more quantitative, we have determined thephase diagram � versus /J, in a mean field approach wherethe spin and the orbital parts have been decoupled. Theground state phase diagram is calculated by comparing ener-gies of different spin/orbital patterns as reported in Fig. 8. Inthis figure, the orbital configurations adopted for the calcu-lation are also represented insets from �b� to �f�, while themeaning of the orbitals is depicted in the box �a�. Indeed, interms of the Schwinger notation, the local states �a�=bi

†ci† �v� , �b�=ai

†ci† �v� , �c�=ai

†bi† �v� stand for the orbitals

that are empty in the hole representation, respectively. Al-though a direct comparison with the phase diagrams plottedin Figs. 6 and 7 is not possible, we may infer that somefeatures of these curves are recovered in Fig. 8. We refer tothe G-AF�1� G-AF�2� region at negative �positive� crystalfield potential, for low values of � �see Fig. 7�, as well as thetransition from C-AF�1/2� to F2�1/4� for intermediate � val-ues �see Fig. 6�. Nevertheless, a striking difference is thepresence of a partial ferromagnetic region, for intermediate-large �, which extends also to negative values of . Theexistence of this solution, at negative crystal field potential,is a consequence of the mean field calculations used whichtend to overestimate the ferromagnetic correlations towardthe antiferromagnetic ones due to the frustration of the or-bital exchange.

IV. DISCUSSION AND CONCLUSIONS

We have presented a detailed analysis of the spin/orbitalcorrelations that emerge in the dynamics of a t2g

4 system uponthe competition of the Coulomb correlations and the c-axisdistortions. We have shown that the control of the doubleoccupation distribution among the different orbitals is a keyaspect in characterizing the spin and orbital correlations ofthe ground state. When the system is in the extreme flattenconfiguration, the DO is quenched in the xy sector and thespin coupling turns out be isotropically antiferromagnetic.Our study has shown that the onset of this state, as a functionof the crystalline field potential, has a different behavior de-pending on the relative ratio between the Hund’s rule and theCoulomb repulsion. By increasing the Hund’s coupling andmore generally any mechanisms that would reinforce the fer-romagnetic correlations, the system stabilizes the C-AFground state with respect to the G-AF one. Indeed, thethreshold of the crystal field potential for the crossing be-tween the two configurations shifts at values of the order ofthe bare xy half-bandwidth.

The G-AF configuration with ferro-orbital order has beenalready proposed as the candidate for the low temperatureground state in the Ca2RuO4 compound.6,12 Nevertheless, aspointed out in the Introduction, there have been experimentsshowing that the orbital order configuration is more complexand may contain contributions of antiferro-orbital character.Our analysis suggests that in the region �see Fig. 7� of inter-mediate Coulomb coupling, small variations in the structuralparameters may lead to significative changes in the magneticand orbital correlations. Assuming that the compressive oc-

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tahedral mode stabilizes the xy ferro-orbital pattern, the clos-est configuration which can be activated has a weaker anti-ferromagnetism and enhanced antiferro-orbital correlations,as it occurs in the C-AF�1/2� state. It is worth reminding onethat, due to the degeneracy of the C-AF state with respect tothe x and y axis, there is no a priori symmetry breakingwithin this analysis. Nevertheless, a small perturbation dueto the orthorhombic distortions would symmetrize such aconfiguration in a form where the antiferromagnetic correla-tions are more isotropic along the x and y directions. Hence,few possible scenarios can be considered to account for theconcomitant occurrence of ferro- and antiferro-orbital pat-terns. �i� The simplest picture would be that of an inhomo-geneous distribution of distorted octahedra in a way that thesystem separates into regions with large/small compressiveoctahedral distortions. �ii� Since the experimental analysis isperformed by varying the temperature, one may expect that,within the thermal evolution, the C-AF-like configurationsare activated, so that the global response contains features asan average of the two states. A preliminary study of the spinand orbital correlations performed within the Lanczos tech-nique extended at finite temperature, on the boundary be-tween the G-AF and the C-AF region, indicates that, whenthe spin starts to be disordered, the homogeneous ferro-orbital and the antiferro-orbital pattern come with almostequal weight in the structure factor. �iii� Another source of

hybrid orbital pattern may be originated by the spin-orbitcoupling. Indeed, this interaction would tend to give a non-null overlap between the G-AF and the C-AF configuration,by removing the constraint of local orbital quenching in theangular momentum, thus leading to a complex ground statethat, to the lowest order, can be seen as a quantum superpo-sition of the two antiferromagnetic configurations.14

When the energy associated with the c-axis compressivedistortions gets smaller and comparable to the bare hopping,the system prefers to have a partial filling of DO in the xyand the �z sector, with equal probability of occupying thetwo subsectors of the t2g manifold. In this limit, the antiferro-orbital exchange, between the xz�yz� and xy orbitals, inducesa ferromagnetic spin coupling along one direction but ittends to be frustrated in the perpendicular one, leading to aC-type antiferromagnetic pattern. Switching to the other oc-tahedral configuration, one notices that the antiferromagneticcorrelations get frustrated or in general are reduced, in a waythat, in the region of coupling where there was a C-AF statefor flatten octahedral distortions, there occur ferromagnetic-like ground states with incomplete spin polarization. TheseGS are characterized by an asymmetric filling of DO be-tween the xy and the �z sectors. In general, more chargetends to fill the �z orbitals. The elongation tends in fact tosuppress the occupation of the xy orbital and, as a conse-quence, to destroy the AF order. As a result, we see that the

FIG. 8. �Color online� Ground state phase dia-gram for the effective spin orbital model in the� /J, �� plane with J=4t2 /U and �=JH /U. Thephase diagram has been obtained by comparingthe lowest energies for the schematic configura-tions reported in the boxes �b�–�f�, whereas inbox �a� the doubly occupied orbitals used in thecomputation are represented.

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energy gain of the AF state with respect to the F-like onesreduces, being consistent with the reduced TN temperatureexperimentally observed for example by increasing the Srdoping concentration.17,18

Still, the large intra-atomic Coulomb repulsion restores anantiferromagnetic G-type state with an integer distribution ofelectrons in the two subsectors. The evolution from the fer-romagnetic to the antiferromagnetic state can be smooth sothat the systems pass gradually all the intermediate spin po-larized configurations, from the maximum to the unpolarizedAF state. This feature, that occurs in a specific range of Cou-lomb, Hund, and elongated octahedra, reveals the subtlecompetition between the ferromagnetic and antiferromag-netic exchange, as tuned by the different orbital correlationswithin the ground state. The ferromagnetism that emerges inthis region has a novel character being related to the occur-rence of competing antiferromagnetic/ferromagnetic cou-pling and the presence of isotropic in plane orbital reso-nance. This may be interesting in connection with the debateon the nature of the ferromagnetic correlations induced in theCa1−xSrxRuO4, by doping or pressure, in proximity of x=0.5, where the octahedra are in the regime of expansivedistortions and the experiments18 suggest the existence oflarge ferromagnetic correlations, at least at short range.

Furthermore, as a synthesis and simple view of the com-peting mechanisms that come into play in the present prob-lem, we have introduced an effective spin/orbital model todescribe the low energy features in the limit of large Cou-lomb interaction. We have seen that the character of theground state, obtained via the numerical analysis, can becaptured in the description of a spin/orbital model where allthe orbital degrees of freedom are taken into account. Thesimple analysis of the orbital correlations on a bond and on aplaquette, as a function of the different octahedral distor-

tions, permits one to correlate the character of the spin cou-pling to the charge and orbital patterns. In particular, theorbital resonance between the xz�yz� and xy orbitals gener-ally competes with orbital configurations where no exchangecan occur. The interplay of those states together with theconstraint of the low dimensionality gives rise to a differentvariety of spin/orbital patterns. The feature of the ferromag-netic state out of an orbital and spin frustration naturallyemerges into the effective picture of the spin/orbital model.

Finally, we would like it noted that there have been sys-tematic Hartree-Fock studies on essentially the same Hamil-tonian model adopted here to determine the most favorablephases with respect to the interaction parameters.25–27 Com-pared to our calculations, these studies adopt a specific elec-tronic structure for t2g electrons, using a tight-binding ap-proximation for their band structure. Moreover, looking atthe role played by the splitting between the xy orbital and theother two orbitals �xz, yz�, a striking difference with ourresults is the observation that a ferromagnetic stable solutionis obtained also when is negative. Indeed, our exact solu-tion on small cluster presented here shows that only for posi-tive value of a stable FM configuration is realized. This ofcourse is related to the mean field treatment of the modelHamiltonian which gives rise to an FM configuration, as re-ported in Ref. 25. When the role played by U is investigated,the mean field calculations26 show that, at zero crystal fieldamplitude , a stable G-type solution is obtained with the xyorbital occupied by two electrons and the other two electronssit on the xz and yz orbital, forming a half-filled band gappedby the AF order. We notice that this region is only stable forsmall value of JH /U. On the other hand, this configuration inour calculations occurs only as a combined effect of flattenoctahedral distortions together with Coulomb correlations.

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