Spin--orbital interaction for face-sharing octahedra: Realization of a highly symmetric SU(4) model
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Spin–orbital interaction for face-sharing octahedra: Realization of a highly symmetricSU(4) model
K. I. Kugel, D. I. Khomskii, A. O. Sboychakov, and S. V. Streltsov
3, 4 Institute for Theoretical and Applied Electrodynamics,Russian Academy of Sciences, Izhorskaya str. 13, 125412 Moscow, Russia II.
Physikalisches Institut, Universit¨at zu K¨oln, Z¨ulpicher Str. 77, 50937 K¨oln, Germany Institute of Metal Physics, Ural Branch, Russian Academy of Sciences,S. Kovalevskaya Str. 18, Ekaterinburg, 620990 Russia Ural Federal University, Mira Str. 19, Ekaterinburg, 620002 Russia (Dated: September 22, 2018)Specific features of orbital and spin structure of transition metal compounds in the case of theface-sharing MO octahedra are analyzed. In this geometry, we consider the form of the spin–orbitalHamiltonian for transition metal ions with double ( e σg ) or triple ( t g ) orbital degeneracy. Trigonaldistortions typical of the structures with face-sharing octahedra lead to splitting of t g orbitals intoan a g singlet and e πg doublet. For both doublets ( e σg and e πg ), in the case of one electron or hole persite, we arrive at a symmetric model with the orbital and spin interaction of the Heisenberg typeand the Hamiltonian of unexpectedly high symmetry: SU(4). Thus, many real materials with thisgeometry can serve as a testing ground for checking the prediction of this interesting theoreticalmodel. We also compare general trends in spin–orbital (“Kugel–Khomskii”) exchange interactionfor three typical situations: those of MO octahedra with common corner, common edge, and thepresent case of common face, which has not been considered yet. PACS numbers: 75.25.Dk, 75.30.Et, 75.47.Lx, 71.27.+a, 71.70.Ej, 75.10.Dg
I. INTRODUCTION
Systems with orbital degeneracy usually exhibit quitediverse properties, often much different from those ofpurely spin systems.
In particular, the coupling be-tween orbital and spin degrees of freedom, besides be-ing of practical importance for many specific materi-als, leads to several interesting theoretical models, suchas spin–orbital model (often called the Kugel–Khomskiimodel) , the popular nowadays compass model , aparticular version of which is the renowned Kitaevmodel. .It turns out that the specific features of one or an-other system with spin and orbital degeneracy stronglydepend on the local geometry. The most typical cases,widely discussed in the literature, are those with MO octahedra (M is a transition metal ion) sharing commonoxygen (or common corner), typical e.g. of perovskiteslike LaMnO or layered systems such as La CuO , andthe situation with two common oxygens for neighboringoctahedra (octahedra with common edge), met in manylayered systems with triangular lattices such as NaCoO and LiNiO . The features of spin–orbital systems in boththese cases were studied in detail, see e.g. Refs. 2,8. How-ever, there exists yet the third typical geometry, which isalso very often met in many real materials – the case ofoctahedra with common face (three common oxygens).Strangely enough, this case has not been actually con-sidered in the literature. To fill this gap and to develop atheoretical description of spin–orbital (Kugel–Khomskii)model for this “third case” is the main goal of the presentpaper.Interestingly enough, after fulfilling this program, we have found out that the resulting model has a very sym-metric form – more symmetric that for the cases of com-mon corner or common edge. The resulting Hamiltonianin the main approximation turned out to have a veryhigh symmetry: SU(4). Actually, SU(4) model appearedalready in the very first treatment of these models for the “artificial” illustrative case, in which for doubly-degenerate orbitals only the diagonal inter-site hoppingexists, that is, t = t = t, t = 0 , (1)where 1 and 2 are the indices denoting two degenerateorbitals. The resulting exchange Hamiltonian, derivedfrom the degenerate Hubbard model in the strong cou-pling limit t/U ≪ U is the on-site Coulomb repul-sion), written in terms of spin s =1/2 and pseudospin τ = 1 / H = t U X h i,j i (cid:18)
12 + 2 s i s j (cid:19) (cid:18)
12 + 2 τ i τ j (cid:19) . (2)This Hamiltonian not only has SU(2) × SU(2) symmetry(it contains scalar products of s and τ vector operators),but it shows even much higher SU(4) symmetry (inter-change of 4 possible states: 1 ↑ , 1 ↓ , 2 ↑ , 2 ↓ ).The SU(4) spin–orbital model was extensively dis-cussed in the literature with the main emphasis on novelquantum states (exact solution of the 1D model ; thepresence of three Goldstone modes ; the gap forma-tion ; spin-orbital singlets on plaquettes in square lat-tice and in two-leg ladders ; spontaneous symmetrybreaking with the formation of dimer columns ; real FIG. 1: (Color online) A chain of face-sharing octahedra.Large (red) and small (blue) circles denote metal and ligandions, respectively. spin-orbital liquid on honeycomb lattice ). There werealso some attempts to apply this model to real ma-terials. Recently the SU(4) model (or more gen-eral SU(N) model with N “colors”) has been appliedalso to cold atoms on a lattice. However, especially asto real transition metal compounds, these applicationswere still rather questionable.
In the present paper,we demonstrate that there exists situation in transitionmetal solids, in which the SU(4) physics might be closeto reality – this is the case of spin–orbital systems withface-sharing MO octahedra. If we include the termsin the effective exchange Hamiltonian, which break thisSU(4) symmetry, see Appendix C, such terms are usu-ally much weaker than the dominant SU(4) exchange, sothat in any case the SU(4) physics would dominate theproperties of a system in a broad temperature interval J ′ < T < J , where J is the scale of SU(4) terms inthe exchange, and J ′ - that of symmetry-breaking terms(typically in systems with 3 d elements J ′ ∼ . J ). Evenat T = 0 strong quantum fluctuations in SU(4) model,especially in one-dimensional systems, may overcome theeffect of symmetry-breaking terms.As far as the actual materials are concerned, in mosttypical and best studied geometries, such as in systemslike perovskites, with corner-sharing MO octahedra andwith ∼ ◦ M–O–M bonds (M is the transition metal),the problem is that conditions (1) required for SU(4)model (2) are not fulfilled. In effect, whereas the spinpart of the spin–orbital exchange is of the Heisenberg s i s j type [SU(2)], the orbital part of the exchange turnsout to be very anisotropic, containing terms of the type τ z τ z , τ x τ x , τ z τ x , and also some linear terms, but not,for example τ y τ y . (The latter terms can appear for com-plex combinations of the basis orbitals, which usually donot lead to static lattice distortions but may be sometime important giving rise to quite exotic types of the groundstate. ) Also for another well-studied case, that withedge-sharing octahedra and with 90 ◦ M–O–M bonds, thesituation is more complicated: sometimes the orbital partof the exchange is anisotropic, and in some cases the lead-ing term in the exchange, ∼ t /U , drops out at all andthe remaining exchange depends on the Hund’s rule ex-change (not included above). The third, much less studied situation, that with theface-sharing MO octahedra (see Fig. 1) is considered be-low. In situation with face-sharing octahedra, one natu-rally obtains for the doubly-degenerate case ( e σg orbitals,or e πg orbitals obtained from triply-degenerate t g or-bitals due to trigonal crystal field, typical for this geom-etry) that the resulting spin–orbital (Kugel–Khomskii)model is of the type of Eq. (2), i.e. it is the SU(4)-symmetric model. Thus, the systems with this geometry,which are in fact quite abundant among transition metalcompounds, represent an actual realization of the high-symmetry SU(4) model, and can provide a natural test-ing ground for it. The experimental study of the systemswith face-sharing arrays may thus allow for verificationof the predictions of this model, such as the strong spin–orbital entanglement, and the presence of three Gold-stone modes.Experimentally, there are many transition metal com-pounds with the face-sharing geometry. Such materialsinclude for example hexagonal crystals like BaCoO ,BaVS or CsCuCl , containing infinite columns offace-sharing ML octahedra (L stands here for ligandsO, Cl, S, ...), as shown in Fig. 1. Many other similarsystems have finite face-sharing blocks, e.g. BaIrO , BaRuO , or Ba Ru O with blocks of threesuch face-sharing octahedra, connected between them-selves by common corners; or blocks of two such octa-hedra as in large series of systems with general formulaA (M1)(M2) O , where A is Ba, Ca, Sr, Li, or Na,and face-sharing M2O octahedra of transition metalsare separated by M1O octahedra (which have commoncorners with M2O ). Such systems have very diverseproperties: some of them are metallic , others are insu-lators or undergo metal–insulator transition ; despitesimilar crystal structures they may have charge ordered or uniform charge states and their magnetic propertiesare also quite different changing from the singlet groundstate , to the situations when part of the magneticmoments turn out be suppressed and to ferro- or an-tiferromagnetic order . However, in any case, thefirst problem to consider for such systems is that of apossible orbital and magnetic exchange in this geome-try. The analysis of this problem is the main task of thepresent paper.In Section II, we formulate a minimal model for theface-sharing geometry, which is in fact the Hubbardmodel taking into account the orbital degrees of free-dom. In Sections III and IV, we consider the chainsof face-sharing octahedra with e g and t g , respectively,and demonstrate that in both cases we arrive at a highly D D s g e g t g a p g e M2 O z x y (a) (b) q M2 M1
FIG. 2: (Color online) (a) Magnetic atom (M) surroundedby trigonally distorted oxygen (O) octahedron in transitionmetal compounds with face-sharing octahedra. The globaltrigonal coordinate is shown. Trigonal distortion is deter-mined by the angle θ ; the value cos θ = 1 / √ z axis. (b) Crystal fieldsplitting of d orbitals of the magnetic atom. The splitting of t g levels (∆ ) is due to both the trigonal distortions of oxy-gen octahedra and contribution from neighboring M atoms tothe crystal field. The sign of ∆ can be different dependingon the type of distortions. symmetrical spin–orbital model. The obtained resultsare discussed in Section V. More technical issues are dis-cussed in appendices. In Appendix A, we show that trig-onal distortions characteristic of the face-sharing geom-etry do not affect the symmetric form of the effectivespin–orbital Hamiltonian. In Appendix B, we derive theexplicit form of the electron hopping integrals via lig-ands as function of an angle characterizing the trigonaldistortion of octahedra. In Appendix C, we present thegeneral form of the exchange Hamiltonian including theterms with the Hund’s rule coupling, going beyond thesymmetric SU(4) form. II. MODEL
Let us suppose that we have a linear chain of 3 d mag-netic ions. Each of them is located at the center of anoctahedron of anions with face-sharing geometry. In con-trast to the case of corner-sharing octahedra, where the z direction is usually chosen along the fourfold symmetryaxis connecting the transition metal ion with one of theapexes of the ligand octahedron (tetragonal coordinatesystem), here it is convenient to choose trigonal systemwith the z axis along the chain and the x and y axes inthe plane perpendicular to the chain (see Fig. 2a). Insuch geometry, two nearest-neighbor ions, M1 and M2,are non-equivalent: a pair ligand triangles surroundingone metal ion can be considered as rotated by 180 ◦ withrespect to that surrounding another ion.To formulate a minimal model for the chain, we startfrom the well-known Hamiltonian in the second quantiza-tion that corresponds to a general problem of interacting electrons H = X ij X γγ ′ σ t γγ ′ ij c † iγσ c jγ ′ σ + (3)12 X i X γβγ ′ β ′ X σσ ′ U γβ ; γ ′ β ′ c † iγσ c † iβσ ′ c iβ ′ σ ′ c iγ ′ σ . Here, i and j denote lattice sites, where the magnetic ionis located, γ , γ ′ , β , β ′ run over the active orbitals on eachsite, σ , σ ′ denote spin up or spin down and c iγσ , ( c † iγσ )are the annihilation (creation) operators for an electronat site i with the quantum numbers γ and σ . The firstterm describes the kinetic energy and the second one cor-responds to the on-site Coulomb repulsion, where U γβ ; γ ′ β ′ = Z Z d r d r ′ φ ⋆γ ( r ) φ ⋆β ( r ′ ) V ( r , r ′ ) φ γ ′ ( r ) φ β ′ ( r ′ ) . Here, φ ( r ) are one-particle wave functions and V ( r , r ′ )describes the interparticle interactions. The crystal fieldfelt by the magnetic ions has an important component ofcubic O h symmetry due to octahedra of anions. It splitsthe one-electron d levels into a triply degenerate level( t g ) and a doubly degenerate level ( e g ). In the caseof the face-sharing octahedra, actual symmetry is usu-ally lower than O h due to, e.g., axial order of the metalions, which in such a geometry often form chains, dimers,trimers, etc. This type of low-dimensional packing in itsturn results in drastic distortions of the ligand octahe-dra by itself so that octahedra appear to be trigonallydistorted (elongation or compression along the vertical z direction in Fig. 2a). Such local distortions of D d sym-metry lead to splitting of t g orbitals into an a g singletand e πg doublet; the original e g ( e σg ) doublet by that re-mains unsplit (see below Fig. 2b).The model treatment will be performed separately fortwo situations, when e g and t g orbitals are active, takinginto account trigonal distortions. III. e g LEVELS
We first consider the case of one hole (electron) at thedegenerate e g level, which corresponds e.g. to the orbitalfilling of Cu ions in CsCuCl . It has been established(see, e.g. Ref. 44) that both the trigonal field and thespin-orbit coupling do not split the e g levels.In the case of ideal MO octahedra, one may use thetrigonal coordinate system. The e g doublet for two neigh-boring magnetic ions along the chain can be written as | d i = 1 √ | x − y i − r | xz i , | e i = − √ | xy i − r | yz i (4)for an ion M1, and | d i = 1 √ | x − y i + r | xz i , | e i = − √ | xy i + r | yz i (5)for the nearest-neighbor ion M2 (the corresponding struc-ture is illustrated in Fig. 2a).We start from the two-band 1D Hubbard Hamiltonianof the form of Eq. (3), where orbital indices γ take thevalues d , e for the M1 sites (sites with e.g. odd i ) or d , e for the M2 sites (sites with even i ). We restrictourselves by the consideration of the nearest neighborhopping amplitudes along the chain, t γγ ′ ≡ t γγ ′ ii +1 . Thesehopping amplitudes have two contributions, which, forthis particular geometry, could be of the same order ofmagnitude; direct hopping between two magnetic ionsalong the chain, t d − dγγ ′ , and the indirect hopping via neigh-boring anions, t viaAγγ ′ . We consider both these situationsseparately.We begin by calculating the direct hopping terms. Wechoose the z direction (trigonal axis) parallel to the chain.In this situation, the only monzero d – d Slater–Kosterparameters are t xy,xy = t x − y ,x − y = V ddδ ,t yz,yz = t xz,xz = V ddπ . (6)Therefore, we have for the direct case t d − d ≡ t d − dd ,d = t d − de ,e = 13 V ddδ − V ddπ , (7)and t d − de ,d = t d − dd ,e = 0 . (8)The calculation of effective hoppings via ligands t viaAγγ ′ is more complicated. The direct derivation is performedin Appendix B. Here, we only show that t viaAγγ ′ ∝ δ γγ ′ us-ing simple considerations. Assume that we know hoppingintegrals along a superexchange path between two neigh-boring cations involving an anion (A1) located at one ofthe apexes of the octahedron. In general, we have threenonzero hopping integrals t viaA d ,d = t , t viaA e ,e = t , and t viaA d ,e = t viaA e ,d = t between M1 and M2 ions. Then, thehopping integrals for other two superexchange paths (viaA2 and A3) could be found by rotating the xy plane by ± π about the trigonal axis. Denoting by primes the axisin the coordinate system rotated by π , ( x ′ , y ′ , z ′ ), z ′ = z ,we can write taking into account that | xy i ∝ xy/r and | x − y i ∝ ( x − y ) / (2 r ) | d i i = | d ′ i i cos 2 π − | e ′ i i sin 2 π , | e i i = | d ′ i i sin 2 π | e ′ i i cos 2 π , (9) where i = 1 ,
2. Therefore | d ′ i i = | d i i cos 2 π | e i i sin 2 π , | e ′ i i = −| d i i sin 2 π | e i i cos 2 π . (10)After the rotation the path M1–A2–M2 becomes thepath M1–A1–M2. Thus, we can express the hopping in-tegrals via A2, t viaA d ,d = t ′ , t viaA e ,e = t ′ , and t viaA d ,e = t viaA e ,d = t ′ in terms of those via A1 according to t viaA µν = t viaA µ ′ ν ′ . Using Eq. (10), we obtain t ′ = t cos π t sin π t sin 4 π ,t ′ = t sin π t cos π − t sin 4 π , (11) t ′ = t − t π t cos 4 π . The hopping integrals via A3 are found by substituting π for − π : t ′′ = t cos π t sin π − t sin 4 π ,t ′′ = t sin π t cos π t sin 4 π , (12) t ′′ = t − t π t cos 4 π . The total hopping integrals are t viaAi = t i + t ′ i + t ′′ i ( i =1 , , t viaAγγ ′ = t δ γγ ′ , t = 32 ( t + t ) . (13)The value of t as a function of the p – d Slater–Kosterparameters V pdσ and V pdπ , and the p – d charge trans-fer energy ∆ is calculated in Appendix B. Here, we seethat the situation is again similar to the direct exchange,for which we have equal hopping integrals between thesame orbitals, and hopping between different orbitals isabsent. This is a rather general result based only onthe existence of the threefold trigonal axis and it doesnot depend on the specific features of the superexchangepaths. Therefore, the results (7), (8), (13) show thatthe parameters for the hopping part of the Hamiltonianare t d ,d = t e ,e = t and t e ,d = t d ,e = 0 with t = t d − d + t .For the Coulomb part of Hamiltonian (3), we canuse the standard parametrization: the on-site Coulomb(Hubbard) repulsion on the same orbital U ee,ee = U dd,dd = U , and that on different orbitals U de,de = U ′ = U − J . Here J is the Hund’s rule coupling constant. Notehere that the latter relationship is valid only for the un-screened Coulomb potential and can be violated in realtransition metal compounds since U is usually screenedmore by surrounding ligands than the purely intra-atomicparameter J . In the general case, other Slater integrals,not only U and J , may enter ; we use below this, the socalled Kanamori parametrization, which in most cases issufficient.Assuming that t ≪ ( U, J ), we can change over to aneffective Hamiltonian that acts on the subspace of func-tions with singly occupied sites. The calculation is stan-dard (see, e.g., Refs. 4,5). In the first approximation( J = 0), the result is the symmetrical SU(4) model H eff = t U X h i,j i ( 12 + 2 s i s j )( 12 + 2 τ i τ j ) , (14)where s i is the spin operator of e g electron at site i de-fined as s i = P γαβ c † iγα σ αβ c iγβ and τ i is the pseu-dospin operator for the orbital degree of freedom atsite i defined as τ i = P αγγ ′ c † iγα σ γγ ′ c iγ ′ α ( σ are thePauli matrices). Notice that the same τ operators corre-sponds to different orbitals at the neighboring sites (sincethe neighboring face-sharing anion octahedra are rotatedwith respect to each other). A more general form of thespin-orbital Hamiltonian with the finite Hund’s rule cou-pling J is presented in Appendix C.Thus, the transition metal compounds with face-sharing octahedra could provide the closest realizationof the high-symmetry spin–orbital model. The leadingterm of the exchange ∼ t /U has the high SU(4) symme-try, but the terms of higher order containing the Hund’srule coupling constant would have a more complicatedform, see Appendix C. The ground state of this generalHamiltonian including terms ∼ J/U , in the the mean-field approximation is well known to be ferromagnetic inspin and antiferromagnetic in pseudospin.
In general,however, quantum effects related to the SU(4) symmetrymay favor other types of states, and the total resultingtype of the ground state requires a special analysis.The value of the effective electron–electron hopping t depends on the details of the crystal structure, in partic-ular, on the M1–O–M2 angle. Note that at some valuesof this angle, the contribution of the M1–O–M2 exchangevia oxygens can vanish (see Appendix B), and such caseshould be treated separately. IV. t g LEVELS
There are many materials, which have the orbital fillingcorresponding to the present case. These are not onlywell-known V O and BaVS , but also many other 3 d and especially 4 d and 5 d transition metal compounds,such as Ba Ru O and BaRuO . As was mentionedabove, even in the case of ideal MO octahedra, thereexists the trigonal symmetry, which is inherent to face-sharing geometry.The trigonal crystal field acts on the triplet t g levelfurther splitting it into a doublet ( e πg ) and a singlet ( a g ).The corresponding part of the Hamiltonian due to a trig- onal field can be written as H t = δ ( L z − I ) , (15)where I is the unit operator, L z is the angular momen-tum operator in the basis of trigonal axes, and parameter δ can be positive or negative. We now analyze the sign ofthe possible contributions to δ . The trigonal field due toa distortion of the octahedra can have both signs, positivefor an elongation and negative for a compression of theoctahedra along the trigonal axis. The trigonal field dueto the neighboring magnetic cations forming 1D struc-tures is always positive ( δ > δ > δ < a g sin-glet, | a i = | z − r i , (16)and a doublet e πg , | b i = − √ | xy i + 1 √ | yz i , | c i = 2 √ | x − y i + 1 √ | xz i , (17)for an ion M1, and the same singlet | a i = | z − r i , (18)and a doublet, | b i = − √ | xy i − √ | yz i , | c i = 2 √ | x − y i − √ | xz i , (19)for the nearest neighbor ion M2.It has to be mentioned that these expressions for thewave functions [and Eqs. (4) – (5)] are given for the case ofthe ideal MO octahedra, where M–O–M angle is about70.5 ◦ . The trigonal distortions will mix e πg and e σg or-bitals. More detailed calculations, which take into ac-count such modification of the wave function due to trig-onal distortions are presented in Appendix A. This mix-ing, however, only changes some numerical coefficientsand does not change the main conclusion that there ex-ist only equal diagonal hoppings, the hopping betweendifferent orbitals being zero – the conditions importantfor getting SU(4) model (14).Here, we consider the electronic configuration as shownin Fig. 2b: the a g level has energy lower than that forthe e πg level. The conditions for the existence of such aconfiguration are discussed in Appendix A. Further on,we assume that the a g level is fully occupied, there isone electron at the doubly degenerate e πg level, and theupper e σg levels are empty. In this case, we can use 1Dtwo-band Hubbard Hamiltonian in the form of Eq. (3),but now orbital indices γ take the values b , c for theM1 sites (odd i ) or b , c for the M2 sites (even i ). Thehopping amplitudes t γγ ′ are the sum of the direct d – d and indirect (via ligands) hopping amplitudes. Note thatour analysis is relevant also for the case of negative butlarge in absolute value ∆ , when the empty a g level liesfar above the e πg level with one electron.For the direct d – d hopping, we have now t d − d ≡ t d − db ,b = t d − dc ,c = 23 V ddδ − V ddπ , (20) t d − db ,c = t d − dc ,b = 0 . To find the relations for the hopping via ligands, we canuse the consideration similar to that used in the previousSection, but with the replacement | d i i → | b i i , | e i i → | c i i , i = 1 , . (21)Repeating after this substitution all calculations as de-scribed above, we obtain that the hopping amplitudeshave again the symmetric form t γγ ′ = ( t d − d + t ) δ γγ ′ , (22)where direct hopping amplitude t d − d is given by Eq. (20),while the hopping amplitude via ligands, t , is obtainedin Appendix B [Eq. (B7)].Thus, the same arguments as those presented in theprevious section show that for one electron (or hole)at e πg levels (neglecting the contribution of a g states),the effective spin–orbital Hamiltonian for a chain offace-sharing octahedra would have the same form ofEq. (14) as for “real” e g orbitals, including the SU(4)part, Eq. (14), and if necessary the extra terms ∼ J/U (see Appendix C). This form of the effective Hamiltonianis, in fact, a consequence of the lattice symmetry: e πg and e σg are similar representations of the same point group.Moreover, taking into account trigonal distortions of themetal–ligand octahedra and the Coulomb interaction be-tween cations in the chain does not change the symmetryof the Hamiltonian (see Appendix A below). V. CONCLUSIONS
In the present paper, we considered the effective spin–orbital exchange for the “third case” (as compared tothe first two well-known cases of MO octahedra withcommon corner and common edge), namely, the case oflocal geometry with face-sharing MO octahedra. Thetrigonal distortions are inherent to such systems. Theydetermine the symmetry of the problem, splitting the t g levels to those with a g and e πg orbitals and reduce it toappropriate spin–orbital model with pseudospin-1/2. Weshow that resulting effective spin–orbital Hamiltonian inthis situation is a well known symmetric Kugel–Khomskiimodel, Eq. (2), or, in a more complete form, Eq. (C1),both for the e σg and e πg orbitals. The leading terms of themodel have the SU(4) symmetry. In that sense, the sit-uation with face-sharing geometry is very different from the usually considered cases of MO octahedra with acommon corner (M1–O–M2 angle ∼ ◦ ) and with acommon edge (M1–O–M2 angle ∼ ◦ ).This result is important in several respects. First of all,it points out a class of real physical systems, for whichthe spin–orbital model of SU(4) symmetry can be ap-plied. This opens the possibility to experimentally checksome nontrivial predictions of this model, such as strongspin–orbital entanglement and crucial role of quantumeffects. Second, it is instructive to compare the generaltendencies existing for three typical geometries: thoseof MO octahedra with a common corner (one commonoxygen for two neighboring MO octahedra), commonedge (two common oxygens), and a common face (threecommon oxygens). The general conclusions in the bet-ter known first and second cases are rather different.For the common-corner geometry, the typical well-knownrule is that the ferro-orbital ordering gives antiferromag-netic spin alignment, and vice versa . However, thisis not true for the case of common edges, with ∼ ◦ M1–O–M2 bonds: in this situation, often one has fer-romagnetic spin ordering irrespective of orbital occupa-tion.
In that sense, the situation with face-sharingoctahedra leading, e.g., to Hamiltonian (14) is more sim-ilar to that with a common corner than to the situationwith a common edge: ferro-spins coexist with antiferro-orbitals and vice versa . On the other hand, as stressedin Appendix B, for the superexchange via ligands (butnot for direct d − d hopping!) the leading terms in theexchange ∼ t /U ∼ [ t pd / ∆] /U can drop out for cer-tain values of the M1–O–M2 angle, similar to the caseof common-edge geometry. Thus, the systems with face-sharing geometry represent a class of their own, and theyhave to be considered as such. Our treatment is focusedon the specific features related to such geometry, and theresulting picture turns out to be quite interesting.Turning to real systems, several factors not consid-ered in the present paper may become important, whichcould decrease the symmetry of the resulting model. Oneis the electron-lattice (Jahn–Teller) interaction, which,in principle, could lead to orbital ordering independentof the spin one; in systems like CsCuCl , for example,it could result in helicoidal superstructures (see Ref. 47and references therein). The second one, considered indetail in the Appendix B, is the strong trigonal distor-tion of MO octahedra, which for particular situationscan strongly reduce the M–O–M contribution to the su-perexchange, so that for certain M–O–M angle, only thedirect d − d contribution remains. In this case, one mayneed to take into account higher-order terms ∼ J/U inthe superexchange Hamiltonian. These terms, writtendown in the general expression (C1) presented in Ap-pendix C, have less symmetric form in orbital variables τ ; i.e., they can also violate the SU(4) symmetry. Never-theless, pronounced quantum effects typical of the SU(4)model with its intrinsic strong spin-orbital entanglementcan still can still be dominant and determine the typeof the ground state of the system. However, even if thetype of the ground state at T = 0 would be determinedby these symmetry-breaking terms (with the energy scale J ′ ∼ t U JU , which is are typically about 10% of the mainSU(4) term of the order of t U ), there would exist a broadtemperature range J ′ . T . t U , in which the behaviorwould be determined by the SU(4) physics. However,the situation taking place in each particular real systemrequires a special treatment.As far as real materials are concerned, one more issueis worth discussing. Whereas the situation with commoncorner and common edge geometry is met in all cases,3D, 2D, and 1D, common-face geometry in this senseis more “choosy”: it is typical for one-dimensional sys-tems (CsCuCl , BaCoO ), and often such face-sharingoctahedra exist for just dimers or linear trimers (e.g. inBaIrO ). We are not aware of any real substances withthe 2D or 3D face-sharing geometry, although we can-not exclude such cases in principle. As to the exchangein such hypothetical situations, we can give some argu-ments that in this case for real e g systems the result-ing Hamiltonian would also be in a first approximationSU(4)-symmetric, but for the t g levels it would not bethe case, because the choice of relevant a g and e πg or-bital would depend on the direction and be different fordifferent nearest neighbors. The 1D systems, however,should not necessarily involve straight chains; there maybe zigzag or even spiral chains. In all such cases, theSU(4) physics would be preserved for e g electrons to afirst approximation. In some sense, it might be even ad-vantageous, because such 1D model is exactly soluble –although it would be very interesting (if at all possible)to have similar 2D or 3D systems. Acknowledgments
This work is supported by the Russian Foundationfor Basic Research (projects 14-02-00276-a, 14-02-0058-a,13-02-00050-a, 13-02-00909-a, and 13-02-00374-a), by theRussian Science Support Foundation, by the Ministry ofEducation and Science of Russia (grant MK 3443.2013.2),by the Ural Branch of Russian Academy of Sciences, bythe German projects DFG GR 1484/2-1 and FOR 1346,and by K¨oln University via the German Excellence Ini-tiative.
Appendix A: Face-sharing octahedra with trigonaldistortions
Let us now consider a more general case, namely, thatwith the crystal field of trigonal symmetry correspond-ing to the stretching or compression of the chain of face-sharing octahedra. In the main text, we considered ex-change interaction for e g and t g orbitals taking for thecorresponding wave functions those of pure e g and t g or-bitals for cubic symmetry. However, trigonal distortion can modify these wave functions, leading, in particular,to a mixing of e σg and e πg orbitals. In this Appendix, weconsider these effects; as a result, we find that their inclu-sion does not qualitatively modify our main conclusions,and can lead only to some change in certain numericalcoefficients.An elementary building block of transition metal com-pounds with face-sharing octahedra is shown in Fig. 2a.Magnetic atoms form a quasi-one-dimensional chain di-rected along the z axis. Each magnetic atom is sur-rounded by the distorted oxygen octahedron. Distortionsare described by a single parameter θ , which is the an-gle between the z axis and the line connecting M and Oatoms (see Fig. 2a). For undistorted octahedron, we have θ = θ = arccos(1 / √ d electron levels of the transition metal atominto two doubly degenerate e σg , e πg levels, and a g level, asit is shown in Fig. 2b. The energy difference ∆ between e πg and a g levels can be positive or negative dependingon the type of trigonal distortions. Stretching of oxygenoctahedron ( θ < θ ) increases the energy of the a g levelwith respect to e πg one, leading to ∆ <
0. However,the contribution to the crystal field from a neighboringmagnetic cations acts in opposite direction, and, in gen-eral, we can have ∆ > r can be represented as asum of Coulomb terms V ( r ) = v X i r | r − r i | , (A1)where r i are the positions of ligand ions. For d states,the existence of the threefold symmetry axis leads to asignificant simplification of the expression for the crystalfield, which can be, approximately, written in the follow-ing form V ( r ) = V ( r ) + v ( r ) X s =1 P (cos θ s ) + v ( r ) X s =1 P (cos θ s ) , (A2)where P and P are the Legendre polynomials, P ( x ) = (3 x −
1) and P ( x ) = (34 x − x + 3). Here, wetook into account the symmetry in the arrangement oftwo opposite edges of the ligand octahedron and as aresult, we havecos θ s = cos θ ′ cos θ + sin θ ′ sin θ cos (cid:18) φ ′ − πs (cid:19) , (A3)where θ ′ and φ ′ describe the direction of r , that is, r = r { sin θ ′ cos φ ′ , sin θ ′ sin φ ′ , cos θ ′ } .Now, we should find the matrix elements of the crystalfield for the complete set of d functions | xy i = R d ( r ) sin θ ′ sin 2 φ ′ , | xz i = R d ( r ) sin θ ′ cos θ ′ cos φ ′ , | yz i = R d ( r ) sin θ ′ cos θ ′ sin φ ′ , (A4) | x − y i = R d ( r ) sin θ ′ cos 2 φ ′ , | z − x − y i = R d ( r ) 3 cos θ ′ − . Straightforward, but rather cumbersome calculations,lead us to the following matrixˆ V αβ = E × − a − a − b a +5 a b − b a +5 a b − a − a
00 0 0 0 − a +10 a , where E = 10 Dq is the splitting between e g and t g levels, and a = − (cid:18)
52 cos θ −
157 cos θ + 314 (cid:19) ,a = 2735 κ (cid:0) θ − (cid:1) , (A5) b = 3 sin θ cos θ . Here, parameter κ is defined as κ = r R ∞ r R d ( r ) r dr R ∞ r R d ( r ) r dr = r h r ih r i = k (cid:18) r a B (cid:19) , (A6)where r is the cation–ligand distance and a B is theBohr radius. A rough estimate for the factor k canbe found by using the hydrogen-like form for the ra-dial part R d ( r ) of the wave function in metal ions, R d ( r ) ∼ r n ∗ − exp ( − z ∗ r/a B ), where n ∗ and z ∗ , are theeffective values of the principal quantum number and ofthe nuclear charge, respectively. According to Ref. 48,we have n ∗ = 3, 3 .
7, and 4 for 3 d , 4 d , and 5 d shells, re-spectively. For d electrons, there is the following simplerule: the charge of all filled shells inside the d shell issubtracted from the nuclear charge and the charge of all d electrons except the given one is multiplied by 0.35 andalso subtracted. For example, for Co with the nuclearcharge z = 27, we find z ∗ = 7 .
6. In this case, we have k = ( z ∗ ) / ≈ .
07. More accurate estimates using thelinearized muffin-tin orbitals (LMTO) give k = 0 . − . e σg , e πg , and a g energy levels, which depend on the trigonal distor-tions. Choosing the reference frame like shown in Fig. 2a,we obtain for the wave functions expressions having theforms similar to those obtained above for the case of ( d e g r ee s ) degrees ) FIG. 3: (Color online) Angle α versus M1–O–M2 angle β = π − θ ; k = 0 . r = 2 ˚A. undistorted octahedra. Thus, for e g levels ( e σg orbitals)we have [cf. Eqs. (4) and (5)] | d , i = sin α | x − y i ∓ cos α | xz i , | e , i = − sin α | xy i ∓ cos α | yz i . (A7)For t g orbitals, we have the same a g singlet, Eqs. (16)and (18), and the e πg doublet [cf. Eqs. (17) – (19)] | b , i = − cos α | xy i ± sin α | yz i , | c , i = cos α | x − y i ± sin α | xz i . (A8)The ∓ and ± signs in the above expressions for cationwave functions for neighboring M atoms occur since theoxygen octahedra surrounding neighboring metal atomsare transformed to each other by the 180 ◦ rotation about z axis. Parameter α in Eqs. (A7) and (A8) depends onthe trigonal distortions as well on the contribution to thecrystal field from magnetic atoms. Neglecting the lattereffect, we findcos α = a √ a + b , a = a + a . (A9)For the ideal octahedron, we have α = α ≡ π − θ =arccos(1 / α on the M1–O–M2 angle β = π − θ is illustrated in Fig. 3. Parameter α decreases monoton-ically when β increases, and it changes faster for β closeto the value β = π − θ corresponding to the ideal octa-hedron. This decrease becomes sharper for larger valuesof κ .These results were obtained neglecting the effect ofneighboring metal atoms in the chain. Taking into ac-count the contribution to the crystal field from theseatoms modifies the parameter a in the following manner a → a − κ Z ∗
12 cos θ , (A10)where Z ∗ is the effective charge (in units of e ) of themetal ion. Note that Z ∗ can be different from z ∗ men-tioned above. Parameters a and b , as well as the rela-tions (A7)–(A9) remain the same. Stretching of oxygenoctahedra ( θ < θ ) tends to make α < α , while theeffect of neighboring metal atoms acts in the opposite di-rection. For α > α , the energy of the a g level is lowerthan that of the e πg one (see Fig. 2b), leading to ∆ > > with the chains of face-sharing Co O octahe-dra. Here, Co with the d configuration has one holeat the e πg level.The wave functions (A7) and (A8) are the generaliza-tion of those considered in the Sec. III – IV to the case ofarbitrary trigonal distortion characterized by an angle α .In other words, these distortions mix the e πg and e σg wavefunctions for ideal octahedra MO given in Eqs. (4) and(5) and Eqs.(17) – (19).It is quite straightforward to demonstrate that orbitals(A7) and (A8) provide the structure of the spin–orbitalHamiltonian of the same form of Eq. (14) at any given α (taking into account both direct and ligand-assistanthoppings). Thus, our main conclusions remain the sameeven with the e σg − e πg mixing taken into account. Appendix B: Electron hopping via ligands
Here, we analyze a possible dependence of the hoppingintegrals between metal ions via ligand ions (let them calloxygens for brevity) on the M-O-M bond angle. In thechain of face sharing MO octahedra, we chose a unit cellconsisting of two oxygen triangles forming an octahedronand two metal ions, M1 and M2 (see Fig. 2). Then, thetight-binding Hamiltonian describing the charge trans-fer between metal ions via oxygen can be written in thefollowing form (spin indices are omitted for simplicity) H pd = − X nµA h t µ ;1 A d † n µ p n A + t µ ;1 A d † n µ p n A + t µ ;2 A d † n µ p n A + t µ ;2 A d † n µ p n − A + H.c. i +∆ X nA (cid:16) p † n A p n A + p † n A p n A (cid:17) , (B1)where n enumerates unit cells, p † ( p ) and d † ( d ) are cre-ation (annihilation) operators for p and d electrons, re-spectively, numbers 1 and 2 correspond to metal ionsM1 and M2, respectively, and to the oxygen triangleabove each of them, µ is the set of basis d functions[Eqs. (A7) for e σg orbitals or Eqs. (A8) for e πg orbitals],and A = { s, η } , where s = 1 , , η = p x , p y , p z , isa set of subscripts numbering the atoms in each oxygentriangle and denoting the oxygen p orbitals. For eachdoublet ( e σg or e πg ), we have four d nlµ ( l = 1 ,
2) opera-tors and eighteen p njA ( j = 1 ,
2) operators, for which weshould take into account all possible electron hoppings. In the second order of the perturbation theory on t pd / ∆, we can derive a Hamiltonian describing the ef-fective hoppings of electrons between the states of d doublets under study via the oxygen p orbitals. To dothis, we first proceed to the momentum representationfor electronic operators d klµ = P n e − ikz n d nlµ / √ N , p kjA = P n e − ikz n p njA / √ N , where N is the numberof unit cells in the chain and z = 2 r cos θ is the dis-tance between neighboring M1 and M2 atoms ( r is theM–O distance). Following then the standard procedure,we obtain for the effective Hamiltonian H eff = − X kµν h t µν ( k ) d † k µ d k ν + H.c. i , (B2)where ( t lµ ; jA are assumed to be real) t µν ( k ) = 1∆ X A (cid:2) t µ ;1 A t ν ;1 A + t µ ;2 A t ν ;2 A e − ikz (cid:3) . (B3)According to Ref. 46, the hopping amplitudes t lµ ; jA can be expressed via two Slater–Koster parameters V pdσ and V pdπ and directing cosines of the radius vector r con-necting the corresponding oxygen and metal ions . If wechoose the reference frame as shown in Fig. 2, the radiusvector r l ; js directed from the oxygen atom s (= 1 , ,
3) inthe j th ( j = 1 ,
2) group of oxygens to the neighboringmetal ion l (= 1 ,
2) is r l ; js = r ( − j { sin θ cos ϕ s , sin θ sin ϕ s , ( − l − cos θ } , (B4)where ϕ s = 2 π ( s − /
3. Using these relations, Table Iof Ref. 46, and Eqs. (A7) for e σg orbitals or Eqs. (A8) for e πg orbitals, we calculate the hopping amplitudes t lµ ; jA as functions of Slater–Koster parameters V pdσ and V pdπ ,the angle θ , and the parameter α describing the or-bital states. Substituting then the obtained t lµ ; jA intoEq. (B3) and performing the summation, we arrive fi-nally to the following relation for the effective d – d hop-ping amplitudes t µν ( k ) = δ µν t (cid:0) e − ikz (cid:1) . (B5)This relation is valid both for e σg and e πg orbitals. For e σg orbitals, the parameter t is t = − V pdσ ∆ sin θ cos 2 θ (cid:16) θ cos α − sin θ sin α (cid:17) +32 V pdπ ∆ (cid:20)(cid:16) sin θ sin α θ cos α (cid:17) +cos 2 θ (cid:16) sin θ cos θ sin α − cos 2 θ cos α (cid:17) (cid:21) +3 √ V pdσ V pdπ ∆ sin θ sin 2 θ h sin θ cos θ sin α − sin θ (cid:0) θ − sin θ (cid:1) sin α α θ cos 2 θ cos α i . (B6)0 t ( a r b . un it s ) (degrees) FIG. 4: (Color online) Hopping integral for e σg orbitals versusM1–O–M2 angle β = π − θ ; k = 0 . r = 2 ˚A. In the case of e πg electrons, the hopping integral t reads t = − V pdσ ∆ sin θ cos 2 θ (cid:16) θ sin α − sin θ cos α (cid:17) +32 V pdπ ∆ (cid:20)(cid:16) sin θ cos α θ sin α (cid:17) +cos 2 θ (cid:16) sin θ cos θ cos α − cos 2 θ sin α (cid:17) (cid:21) +3 √ V pdσ V pdπ ∆ sin θ sin 2 θ h sin θ cos θ cos α − sin θ (cid:0) θ − sin θ (cid:1) sin α α θ cos 2 θ sin α i . (B7)Note, that the effective Hamiltonian (B2) with hoppingamplitudes of the form of Eq. (B5) is equivalent to thesimple tight-binding Hamiltonian of the form H eff = − t X mµ h d † mµ d m +1 µ + H.c. i . (B8)This can be easily checked by using the transformationfor electronic operators d n µ → d mµ , d n µ → d m +1 µ , m ∈ Z . (B9)Thus, from viewpoint of the electronic properties, themagnetic sites M1 and M2 are equivalent to each othereven though crystallographically they are different. Oneshould keep in mind, however, that d -orbitals wave func-tions of neighboring magnetic sites are different.The dependence of the hopping integral t for e σg and e πg orbitals on the M–O–M bond angle β = π − θ is illustrated in Figs. 4 and 5, respectively. For theratio V pdσ /V pdπ , we took the commonly used value equal to 2 .
16. We see that at some value of β thehopping integral via oxygens either changes sign (for e πg orbitals) or becomes close to zero (for e σg orbitals). t ( a r b . un it s ) (degrees) FIG. 5: (Color online) Hopping integral for e πg orbitals versusM1–O–M2 angle β = π − θ ; k = 0 . r = 2 ˚A. For e σg orbitals this happens for M–O–M bond angleclose to that characteristic of an undistorted octahedron β = π − / √ ∼ = 70 . ◦ , while for e πg orbitals t changes the sign at a bit smaller value of β (com-pressed octahedron). The total d – d hopping amplitudeis t = t + t d − d . Thus, for e σg orbitals t is always positive,while for e πg orbitals it can change sign. Usually, the di-rect d – d hopping amplitude t d − d is assumed to be smallerthan the characteristic value of the effective hopping viaoxygens t ∼ V pdσ / ∆. Our calculations show however,that for some M–O–M bond angles the direct hoppingbecomes dominant. Moreover, for e σg orbitals this canbe the case of the ideal octahedron. When the hoppingis suppressed, the higher-order corrections to the SU(4)model, containing the terms ∼ J/U , which have less sym-metric form in orbital τ variables (see e.g. Refs. 4 and 5)may have to be included. Appendix C: General form of the exchangeHamiltonian
Let us now present the full form of the exchange Hamil-tonian, including terms containing the Hund’s rule cou-pling constant J . These terms appear when we considernot the virtual hopping between occupied orbitals, butthe hopping to an empty orbital of the neighbor, withthe consecutive effect of Hund’s coupling. The deriva-tion of these terms is straightforward , although a bitcumbersome. In our case of face-sharing octahedra, theresulting spin–orbital Hamiltonian has the form H eff = t U X h i,j i (cid:26) ( 12 + 2 s i s j )( 12 + 2 τ i τ j )+ (C1) JUU − J [2( τ i τ j − τ zi τ zj ) − ( 12 + 2 s i s j )( 12 − τ zi τ zj )] + J U − J [(2 τ zi τ zj −
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