111 Orbital Physics
Andrzej M. Ole ´sMarian Smoluchowski Institute of PhysicsJagiellonian UniversityProf. S. Łojasiewicza 11, Krak ´ow, Poland
Contents e g orbitals 10 and K CuF . . . . . . . . . . . . . . . . . 143.3 Spin-orbital superexchange model for LaMnO . . . . . . . . . . . . . . . . . 18 t g orbitals 21 . . . . . . . . . . . . . . . . . . 214.2 Spin-orbital superexchange model for LaVO . . . . . . . . . . . . . . . . . . 22 t - J -like model for ferromagnetic manganites 297 Conclusions and outlook 31 a r X i v : . [ c ond - m a t . s t r- e l ] A ug Strong local Coulomb interactions lead to electron localization in Mott or charge transfer corre-lated insulators. The simplest model of a Mott insulator is the non-degenerate Hubbard modelwhere large intraorbital Coulomb interaction U suppresses charge fluctuations due to the ki-netic energy ∝ t . As a result, the physical properties of a Mott insulator are determined by aninterplay of kinetic exchange ∝ J , with J = 4 t U , (1)derived from the Hubbard model at U (cid:29) t , and the motion of holes in the restricted Hilbertspace without double occupancies, as described by the t - J model [1]. Although this genericmodel captures the essential idea of strong correlations, realistic correlated insulators arise intransition metal oxides (or fluorides) and the degeneracy of partly filled and nearly degenerate d (or d ) strongly correlated states has to be treated explicitly. Quite generally, strong localCoulomb interactions lead then to the multitude of quite complex behavior with often puzzlingtransport and magnetic properties [2]. The theoretical understanding of this class of compounds,including the colossal magneto-resistance (CMR) manganites as a prominent example [3], hasto include not only spins and holes but in addition orbital degrees of freedom which have to betreated on equal footing with the electron spins [4]. For a Mott insulator with transition metalions in d m configurations, charge excitations along the bond (cid:104) ij (cid:105) , d mi d mj (cid:10) d m +1 i d m − j , lead tospin-orbital superexchange which couples two neighboring ions at sites i and j .It is important to realize that modeling of transition metal oxides can be performed on differentlevels of sophistication. We shall present some effective orbital and spin-orbital superexchangemodels for the correlated d -orbitals depicted in Fig. 1 coupled by hopping t between nearestneighbor ions on a perovskite lattice, while the hopping for other lattices may be generated bythe general rules formulated by Slater and Koster [5]. The orbitals have particular shapes andbelong to two irreducible representations of the O h cubic point group:( i ) a two-dimensional (2D) representation of e g -orbitals { z − r , x − y } , and( ii ) a three-dimensional (3D) representation of t g -orbitals { xy, yz, zx } .In case of absence of any tetragonal distortion or crystal-field due to surrounding oxygens, the d -orbitals are degenerate within each irreducible representation of the O h point group andhave typically a large splitting D q ∼ . eV between them. When such degenerate orbitalsare party filled, electrons (or holes) have both spin and orbital degree of freedom. The kineticenergy H t in a perovskite follows from the hybridization between d and p -orbitals. In aneffective d -orbital model the oxygen p -orbitals are not included explicitly and we define thehopping element t as the largest hopping element obtained for two orbitals of the same typewhich belong to the nearest neighbor d ions.We begin with conceptually simpler t g orbitals where finite hopping t results from the d − p hybridization along π -bonds and couples each a pair of identical orbitals active along a givenbond. Each t g orbital is active along two cubic axes and the hopping is forbidden along theone perpendicular to the plane of this orbital, e.g. the hopping between two xy -orbitals is notrbital Physics 11.3 Fig. 1:
Schematic representation of d orbitals: Top — two e g orbitals { z − r , x − y } ;Bottom — three t g orbitals { zx, yz, xy } . Image by courtesy of Yoshinori Tokura. allowed along the c axis (due to the cancelations caused by orbital phases). It is thereforeconvenient to introduce the following short-hand notation for the orbital degree of freedom [6], | a (cid:105) ≡ | yz (cid:105) , | b (cid:105) ≡ | zx (cid:105) , | c (cid:105) ≡ | xy (cid:105) . (2)The labels γ = a, b, c thus refer to the cubic axes where the hopping is absent for orbitals of agiven type, H t ( t g ) = − t (cid:88) α (cid:88) (cid:104) ij (cid:105)(cid:107) γ (cid:54) = α a † iασ a jασ , (3)Here a † iασ is an electron creation operator in a t g -orbital α ∈ { xy, yz, zx } with spin σ = ↑ , ↓ at site i , and the local electron density operator for a spin-orbital state is n iασ = a † iασ a iασ . Notonly spin but also orbital flavor is conserved in the hopping process ∝ t .The hopping Hamiltonian for e g electrons couples two directional e g -orbitals {| iζ α (cid:105) , | iζ α (cid:105)} along a σ -bond (cid:104) ij (cid:105) [7], H t ( e g ) = − t (cid:88) α (cid:88) (cid:104) ij (cid:105)(cid:107) α,σ a † iζ α σ a jζ α σ . (4)Indeed, the hopping with amplitude − t between sites i and j occurs only when an electron withspin σ transfers between two directional orbitals | ζ α (cid:105) oriented along the bond (cid:104) ij (cid:105) direction,i.e., | ζ α (cid:105) ∝ x − r , y − r , and z − r along the cubic axis α = a , b , and c . We willsimilarly denote by | ξ α (cid:105) the orbital which is orthogonal to | ζ α (cid:105) and is oriented perpendicular tothe bond (cid:104) ij (cid:105) direction, i.e., | ξ α (cid:105) ∝ y − z , z − x , and x − y along the axis α = a , b , and c .For a moment we consider only electrons with one spin, σ = ↑ , to focus on the orbital problem.While such a choice of an over-complete basis { ζ a , ζ b , ζ c } is convenient for writing down the1.4 Andrzej M. Ole´skinetic energy, a particular orthogonal basis is needed. The usual choice is to take | z (cid:105) ≡ √ (3 z − r ) , | ¯ z (cid:105) ≡ √ ( x − y ) , (5)called real e g orbitals [7]. However, this basis is the natural one only for the bonds parallel tothe c axis but not for those in the ( a, b ) plane, and for ↑ -spin electrons the hopping reads (herefor clarity we omit spin index σ ) H ↑ t ( e g ) = − t (cid:88) (cid:104) ij (cid:105)(cid:107) ab (cid:104) a † i ¯ z a j ¯ z + a † iz a jz ∓ √ (cid:16) a † i ¯ z a jz + a † iz a j ¯ z (cid:17)(cid:105) − t (cid:88) (cid:104) ij (cid:105)(cid:107) c a † iz a jz , (6)and although this expression is of course cubic invariant, it does not manifest this symmetry buttakes a very different appearance depending on the bond direction. However, the symmetry isbetter visible using the basis of complex e g orbitals at each site j [7], | j + (cid:105) = √ (cid:0) | jz (cid:105) − i | j ¯ z (cid:105) (cid:1) , | j −(cid:105) = √ (cid:0) | jz (cid:105) + i | j ¯ z (cid:105) (cid:1) , (7)corresponding to “up” and“down” pseudospin flavors, with the local pseudospin operators de-fined as τ + i ≡ c † i + c i − , τ − i ≡ c † i − c i + , τ zi ≡ ( c † i + c i + − c † i − c i − ) = ( n i + − n i − ) . (8)The three directional | iζ α (cid:105) and three planar | iξ α (cid:105) orbitals at site i , associated with the three cubicaxes ( α = a , b , c ), are the real orbitals, | iζ α (cid:105) = √ (cid:2) e − iϑ α / | i + (cid:105) + e + iϑ α / | i −(cid:105) (cid:3) = cos( ϑ α / | iz (cid:105) − sin( ϑ α / | i ¯ z (cid:105) , (9) | iξ α (cid:105) = √ (cid:2) e − iϑ α / | i + (cid:105) − e + iϑ α / | i −(cid:105) (cid:3) = sin( ϑ α / | iz (cid:105) + cos( ϑ α / | i ¯ z (cid:105) , (10)with the phase factors ϑ ia = − π/ , ϑ ib = +4 π/ , and ϑ ic = 0 , and thus correspond to thepseudospin lying in the equatorial plane and pointing in one of the three equilateral “cubic”directions defined by the angles { ϑ iα } .Using the above complex-orbital representation (7) we can write the orbital Hubbard model for e g electrons with only one spin flavor σ = ↑ in a form similar to the spin Hubbard model, H ↑ ( e g ) = − t (cid:88) α (cid:88) (cid:104) ij (cid:105)(cid:107) α (cid:104)(cid:16) a † i + a j + + a † i − a j − (cid:17) + γ (cid:16) e − iχ α a † i + a j − + e + iχ α a † i − a j + (cid:17)(cid:105) + ¯ U (cid:88) i n i + n i − , (11)with χ a = +2 π/ , χ b = − π/ , and χ c = 0 , and where the parameter γ , explained below,takes for e g orbitals the value γ = 1 . The appearance of the phase factors e ± iχ α is charac-teristic of the orbital problem — they occur because the orbitals have an actual shape in realspace so that each hopping process depends on the bond direction and may change the orbitalflavor. The interorbital Coulomb interaction ∝ ¯ U couples the electron densities in basis orbitals n iα = a † iµ a iµ , with µ ∈ { + , −} ; its form in invariant under any local basis transformation toa pair of orthogonal orbitals, i.e., it gives an energy ¯ U for a double occupancy either whenrbital Physics 11.5two real orbitals are simultaneously occupied, ¯ U (cid:80) i n iz n i ¯ z , or when two complex orbitals areoccupied, ¯ U (cid:80) i n i + n i − .In general, on-site Coulomb interactions between two interacting electrons in d -orbitals de-pend both on spin and orbital indices and the interaction Hamiltonian takes the form of thedegenerate Hubbard model. Note that the electron interaction parameters in this model areeffective ones, i.e., the p -orbital parameters of O (F) ions renormalize on-site Coulomb inter-actions for d -orbitals. The general form which includes only two-orbital interactions and theanisotropy of Coulomb and exchange elements is [8]: H int = U (cid:88) iα n iα ↑ n iα ↓ + (cid:88) i,α<β (cid:18) U αβ − J αβ (cid:19) n iα n iβ − (cid:88) i,α<β J αβ (cid:126)S iα · (cid:126)S iβ + (cid:88) i,α<β J αβ (cid:16) a † iα ↑ a † iα ↓ a iβ ↓ a iβ ↑ + a † iβ ↑ a † iβ ↓ a iα ↓ a iα ↑ (cid:17) . (12)Here a † iασ is an electron creation operator in any d -orbital α ∈ { xy, yz, zx, z − r , x − y } and ¯ σequiv − σ , with spin states σ = ↑ , ↓ at site i . The parameters { U, U αβ , J αβ } depend inthe general case on the three Racah parameters A , B and C [9] which may be derived fromsomewhat screened atomic values. While the intraorbital Coulomb element is identical for all d -orbitals, U = A + 4 B + 3 C, (13)the interorbital Coulomb U αβ and exchange J αβ elements are anisotropic and depend on theinvolved pair of orbitals; the values of J αβ are given in Table 1. The Coulomb U αβ and ex-change J αβ elements are related to the intraorbital element U by a relation which guarantees theinvariance of interactions in the orbital space, U = U αβ + 2 J αβ . (14)In all situations where only the orbitals belonging to a single irreducible representation of thecubic group ( e g or t g ) are partly filled, as e.g. in the titanates, vanadates, nickelates, or copperfluorides, the filled (empty) orbitals do not contribute, and the relevant exchange elements J αβ Table 1:
On-site interorbital exchange elements J αβ for d orbitals as functions of the Racahparameters B and C (for more details see Ref. [9]). d orbital xy yz zx x − y z − r xy B + C B + C C B + Cyz B + C B + C B + C B + Czx B + C B + C B + C B + Cx − y C B + C B + C B + C z − r B + C B + C B + C B + C t g ( e g ) orbitals, J tH = 3 B + C, (15) J eH = 4 B + C. (16)Then one may use a simplified degenerate Hubbard model with isotropic form of on-site inter-actions (for a given subset of d orbitals) [10], H (0) int = U (cid:88) iα n iα ↑ n iα ↓ + (cid:18) U − J H (cid:19) (cid:88) i,α<β n iα n iβ − J H (cid:88) i,α<β (cid:126)S iα · (cid:126)S iβ + J H (cid:88) i,α<β (cid:16) a † iα ↑ a † iα ↓ a iβ ↓ a iβ ↑ + a † iβ ↑ a † iβ ↓ a iα ↓ a iα ↑ (cid:17) . (17)It has two Kanamori parameters: the Coulomb intraorbital element U (13) and Hund’s exchange J H standing either for J tH (15) or for J eH (16). Now ¯ U ≡ ( U − J H ) in Eq. (11). We emphasizethat in a general case when both types of orbitals are partly filled (as in the CMR manganites)and both thus participate in charge excitations, the above Hamiltonian with a single Hund’sexchange element J H is insufficient and the full anisotropy given in Eq. (17) has to be usedinstead to generate correct charge excitation spectra of a given transition metal ion [9]. If the spin state is ferromagnetic (FM) as e.g. in the ab planes of KCuF (or LaMnO ), chargeexcitations d mi d mj (cid:10) d m +1 i d m − j with m = 9 (or m = 4 ) concern only high-spin (HS) A (or A ) state and the superexchange interactions reduce to an orbital superexchange model [11].Thus we begin with an orbital model for e g -holes in K CuF , with a local basis at site i definedby two real e g -orbitals, see Eq. (5), being a local e g -orbital basis at each site. The basis consistsof a directional orbital | iζ c (cid:105) ≡ | iz (cid:105) and the planar orbital | iξ c (cid:105) ≡ | i ¯ z (cid:105) . Other equivalent orbitalbases are obtained by rotation of the above pair of orbitals by angle ϑ to | iϑ (cid:105) = cos ( ϑ/ | iz (cid:105) − sin ( ϑ/ | i ¯ z (cid:105) , | i ¯ ϑ (cid:105) = sin ( ϑ/ | iz (cid:105) + cos ( ϑ/ | i ¯ z (cid:105) , (18)i.e., to a pair {| iϑ (cid:105) , | i, ϑ + π (cid:105)} . For angles ϑ = ± π/ one finds equivalent pairs of directionaland planar orbitals in a 2D model, {| iζ a (cid:105) , | iξ a (cid:105)} and {| iζ b (cid:105) , | iξ b (cid:105)} , to the usually used e g -orbitalreal basis given by Eq. (5).Consider now a bond (cid:104) ij (cid:105) (cid:107) γ along one of the cubic axes γ = a, b, c , and a charge excitation gen-erated by a hopping process i → j . The hopping t couples two directional orbitals {| iζ γ (cid:105) , | jζ γ (cid:105)} .Local projection operators on these active and the complementary inactive {| iξ γ (cid:105) , | jξ γ (cid:105)} orbitalsare P γiζ = | iζ γ (cid:105)(cid:104) iζ γ | = (cid:18)
12 + τ ( γ ) i (cid:19) , P γiξ = | iξ γ (cid:105)(cid:104) iξ γ | = (cid:18) − τ ( γ ) i (cid:19) , (19)where τ ( γ ) i ≡
12 ( | iζ γ (cid:105)(cid:104) iζ γ | − | iξ γ (cid:105)(cid:104) iξ γ | ) , (20)rbital Physics 11.7(c) Fig. 2:
Virtual charge excitations leading to the e g orbital superexchange model for a stronglycorrelated system with | z (cid:105) and | x (cid:105) ≡ | ¯ z (cid:105) real e g orbitals (5) in the subspace of ↑ -spin states:(a) for a bond along the c axis (cid:104) ij (cid:105) (cid:107) c ; (b) for a bond in the ab plane (cid:104) ij (cid:105) (cid:107) ab . In a FM planeof KCuF (LaMnO ) the superexchange (27) favors (c) AO state of | AO ±(cid:105) orbitals (28).Images (a-b) are reproduced from Ref. [11]; image (c) by courtesy of Krzysztof Bieniasz. and these operators are represented in the fixed {| iz (cid:105) , | i ¯ z (cid:105)} basis as follows: τ ( a ) i = − (cid:16) σ zi − √ σ xi (cid:17) , τ ( b ) i = − (cid:16) σ zi + √ σ xi (cid:17) , τ ( c ) i = 12 σ zi . (21)A charge excitation between two transition metal ions with partly filled e g -orbitals will arise bya hopping process between two active orbitals, | iζ γ (cid:105) and | jζ γ (cid:105) . To capture such processes weintroduce two projection operators on the orbital states for each bond, P ( γ ) (cid:104) ij (cid:105) ≡ (cid:18)
12 + τ ( γ ) i (cid:19) (cid:18) − τ ( γ ) j (cid:19) + (cid:18) − τ ( γ ) i (cid:19) (cid:18)
12 + τ ( γ ) j (cid:19) , (22) Q ( γ ) (cid:104) ij (cid:105) ≡ (cid:18) − τ ( γ ) i (cid:19) (cid:18) − τ ( γ ) j (cid:19) . (23)Unlike for a spin system, the charge excitation d mi d mj (cid:10) d m +1 i d m − j is allowed only in onedirection when one orbital is directional | ζ γ (cid:105) and the other is planar | ξ γ (cid:105) on the bond (cid:104) ij (cid:105) (cid:107) γ ,i.e., (cid:68) P ( γ ) (cid:104) ij (cid:105) (cid:69) = 1 ; such processes generate both HS and low-spin (LS) contributions. On thecontrary, when both orbitals are directional, i.e., one has (cid:68) Q ( γ ) (cid:104) ij (cid:105) (cid:69) = 2 , only LS terms contribute.To write the superexchange model we need the charge excitation energy which for the HSchannel is, ε ≡ E ( d m +1 ) + E ( d m − ) − E ( d m ) = U − J H = ¯ U , (24)where E ( d m ) in the ground state energy for an ion with m electrons. Note that this energyis the same for KCuF and LaMnO [8], so the e g orbital model presented here is universal.Second order perturbation theory shown in Figs. 2(a-b) gives [11], H ↑ ( e g ) = − t ε (cid:88) (cid:104) ij (cid:105)(cid:107) γ P ( γ ) (cid:104) ij (cid:105) . (25)1.8 Andrzej M. Ole´s(a-b) (c) Fig. 3: (a-b) Orbital-wave excitations obtained for different values of the crystal-field splitting E z for a 3D (left) and 2D (right) orbital superexchange model (27), with J r ≡ J . The resultshown for a 3D model at E z = 0 corresponds to the E z → limit. (c) Gap ∆/J in theorbital excitation spectrum and energy quantum correction ∆E/J as functions of the crystal-field splitting E z /J , for the 3D (2D) model shown by full (dashed) lines.Images are reproduced from Ref. [11]. For convenience we define the dimensionless Hund’s exchange parameter η , η ≡ J H U . (26)The value of J defines the superexchange energy scale and is the same as in the t - J model [1],while the parameter η (26) characterizes the multiplet structure when LS states are included aswell, see below. The e g orbital model (25) (for HS states) takes the form, H ↑ ( e g ) = 12 J r (cid:88) (cid:104) ij (cid:105)(cid:107) γ (cid:18) τ ( γ ) i τ ( γ ) j − (cid:19) + E z (cid:88) i τ ( c ) i , (27)where r = U/ε HS = U/ ¯ U = 1 / (1 − η ) . Here we include the crystal-field term ∝ E z whichsplits off the e g orbitals. The same effective model is obtained from the e g Hubbard modelEq. (11) at half-filling in the regime of ¯ U (cid:29) t . It favors consistently with its derivation pairsof orthogonal orbitals along the axis γ , with the energy gain for such a configuration − J r .When both orbitals would be instead selected as directional along the bond, (cid:68) τ ( γ ) i τ ( γ ) j (cid:69) = ,the energy gain vanishes as this orbital configuration corresponds to the situation incompatiblewith the HS excited states considered here and the superexchange is blocked. The ground statein the 2D ab plane has alternating orbital (AO) order between the sublattices i ∈ A and j ∈ B , | i + (cid:105) = √ (cid:0) | iz (cid:105) + | i ¯ z (cid:105) (cid:1) , | j −(cid:105) = √ (cid:0) | jz (cid:105) − | j ¯ z (cid:105) (cid:1) , (28)of orbitals occupied by holes in KCuF and by electrons in LaMnO , see Fig. 2(c).rbital Physics 11.9Here we are interested in the low temperature range T < . J and the 2D (and 3D) e g orbitalmodel orders at finite temperature T < T c [12], i.e., below T c = 0 . J for a 2D model [13],so we assume perfect orbital order given by a classical Ansatz for the ground state, | Φ (cid:105) = (cid:89) i ∈ A | iθ A (cid:105) (cid:89) j ∈ B | jθ B (cid:105) , (29)with the orbital states, | iθ A (cid:105) and | jθ B (cid:105) , characterized by opposite angles ( θ A = − θ B ) andalternating between two sublattices A and B in the ab planes. The orbital state at site i : | iθ (cid:105) = cos ( θ/ | iz (cid:105) + sin ( θ/ | i ¯ z (cid:105) , (30)is here parameterized by an angle θ which defines the amplitudes of the orbital states defined inEq. (5). The AO state specified in Eq. (29) is thus defined by: | iθ A (cid:105) = cos ( θ/ | iz (cid:105) + sin ( θ/ | ix (cid:105) , | jθ B (cid:105) = cos ( θ/ | jz (cid:105) − sin ( θ/ | jx (cid:105) , (31)with θ A = θ and θ B = − θ .The excitations from the ground state of the orbital model (27) are orbital waves (orbitons)which may be obtained in a similar way to magnons in a quantum antiferromagnet. An im-portant difference is that the orbitons have two branches which are in general nondegenerate,see Fig. 3(a-b). In the absence of crystal field ( E z = 0 ) the spectrum for the 2D e g orbitalmodel has a gap and the orbitons have weak dispersion, so the quantum corrections to the orderparameter are rather small. They are much larger in the 3D model but still smaller than in an an-tiferromagnet [11]. The gap closes in the 3D model at E z = 0 , but the quantum corrections aresmaller than in the Heisenberg model. Note that the shape of the occupied orbitals changes atfinite crystal field, and the orbitons have a remarkable evolution, both in the 3D and 2D model,see Figs. 3(a-b). Increasing E z > first increases the gap but when the field overcomes theinteractions and polarizes the orbitals (at E z = 4 J in 2D and E z = 6 J in 3D model), the gapcloses, see Fig. 3(c). This point marks a transition from the AO order to uniform ferro-orbital(FO) order. Note that in agreement with intuition the quantum corrections ∆E/J are maximalwhen the gap closes and low-energy orbitons contribute.To see the relation of the 2D e g orbital model to the compass model [14] we introduce a 2D generalized compass model (GCM) with pseudospin interactions on a square lattice in ab plane( J cm > ) [15], H ( θ ) = − J cm (cid:88) { ij }∈ ab (cid:8) σ aij ( θ ) σ ai +1 ,j ( θ ) + σ bij ( θ ) σ bi,j +1 ( θ ) (cid:9) . (32)The interactions occur along nearest neighbor bonds and are balanced along both lattice di-rections a and b . Here { ij } labels lattice sites in the ab plane and { σ aij ( θ ) , σ bij ( θ ) } are linearcombinations of Pauli matrices describing interactions for T = pseudospins: σ aij ( θ ) = cos( θ/ σ xij + sin( θ/ σ zij ,σ bij ( θ ) = cos( θ/ σ xij − sin( θ/ σ zij . (33)1.10 Andrzej M. Ole´s Fig. 4:
Artist’s view of the evolution of orbital interactions in the generalized compass modelEq. (32) with increasing angle θ . Heavy (blue) lines indicate favored spin direction induced byinteractions along two nonequivalent lattice axes a and b . Different panels show: (a) the Isingmodel at θ = 0 ◦ , (b) the 2D e g orbital model at θ = 60 ◦ , and (c) the OCM at θ = 90 ◦ . Spin orderfollows the interactions in the Ising limit, while it follows one of the equivalent interactions, σ a or σ b , in the OCM. This results in the symmetry breaking quantum phase transition (QPT) whichoccurs between (b) and (c). Image is reproduced from Ref. [15]. The interactions in Eq. (32) include the classical Ising model for σ xij operators at θ = 0 ◦ and become gradually more frustrated with increasing angle θ ∈ (0 ◦ , ◦ ] — they interpolatebetween the Ising model (at θ = 0 ◦ ) and the isotropic compass model (at θ = 90 ◦ ), see Fig. 4.The latter case is equivalent by a standard unitary transformation to the 2D compass model withstandard interactions, σ xij σ xi,j +1 along the a and σ zij σ zi +1 ,j along the b axis [15], H ( π/
2) = − J cm (cid:88) (cid:104) ij (cid:105)(cid:107) a σ xij σ xi +1 ,j − J cm (cid:88) (cid:104) ij (cid:105)(cid:107) b σ zij σ zi,j +1 . (34)The model (32) includes as well the 2D e g orbital model as a special case, i.e., at θ = 60 ◦ .Increasing angle θ between the interacting orbital-like components (33) in Fig. 4 is equivalentto increasing frustration which becomes maximal in the 2D compass model. As a result, asecond order quantum phase transition from Ising to nematic order [16] occurs at θ c (cid:39) . ◦ which is surprisingly close to the compass point θ = 90 ◦ , i.e., only when the interactions aresufficiently strongly frustrated. The ground state has high degeneracy d = 2 L +1 for a 2D cluster L × L of one-dimensional (1D) nematic states which are entirely different from the 2D AOorder in the e g orbital model depicted in Fig. 4(c), yet it is stable in a range of temperaturebelow T c (cid:39) . J cm [17]. e g orbitals We consider the case of partly filled degenerate d -orbitals and large Hund’s exchange J H .In the regime of t (cid:28) U , electrons localize and effective low-energy superexchange interac-tions consist of all the contributions which originate from possible virtual charge excitations,rbital Physics 11.11 Fig. 5:
Energies of charge excitations ε n (35) for selected cubic transition metal oxides, for:(a) e g excitations to Cu ( d ) and Mn ( d ) ions; (b) t g excitations to Ti ( d ) and V ( d ) ions. The splittings between different states are due to Hund’s exchange element J H whichrefers to a pair of e g and t g electrons in (a) and (b). Image is reproduced from Ref. [8]. d mi d mj (cid:10) d m +1 i d m − j — they take a form of a spin-orbital model, see Eq. (37) below. The chargeexcitation n costs the energy ε n = E n ( d m +1 ) + E ( d m − ) − E ( d m ) , (35)where the d m ions are in the initial HS ground states with spins S = m and have the Coulombinteraction energy E ( d m ) = (cid:0) m (cid:1) ( U − J H ) each (if m < , else if m > one has to considerhere m holes instead, while the case of m = 5 is special and will not be considered here as in the t g e g configuration the orbital degree of freedom is quenched). The same formula for groundstate energy applies as well to Mn ions in d configuration with S = 2 spin HS ground state,see Sec. 3.3. By construction also the ion with less electrons (holes) for m < is in the HSstate and E ( d m − ) = (cid:0) m − (cid:1) ( U − J H ) . The excitation energies (35) are thus defined by themultiplet structure of an ion with more electrons (holes) in the configuration d m +1 , see Fig. 5.The lowest energy excitation is given by Eq. (24) — it is obtained from the HS state of the d m +1 ion with total spin S = S + and energy E ( d m +1 ) = (cid:0) m +12 (cid:1) ( U − J H ) . Indeed, onerecovers the lowest excitation energy in the HS subspace, see Eq. (24), with J H being Hund’sexchange element for the electron (hole) involved in the charge excitation, either e g or t g . Weemphasize that this lowest excitation energy ε (24) is universal and is found both in t g and e g systems, i.e., it does not depend on the electron valence m . In contrast, the remaining energies { ε n } for n > are all for LS excitations and are specific to a given valence m of the consideredinsulator with d m ions. They have to be determined from the full local Coulomb interactionHamiltonian (12), in general including also the anisotropy of { U αβ } and { J αβ } elements.Effective interactions in a Mott (or charge transfer) insulator with orbital degeneracy take the1.12 Andrzej M. Ole´sform of spin-orbital superexchange [4, 18]. Its general structure is given by the sum over all thenearest neighbor bonds (cid:104) ij (cid:105) (cid:107) γ connecting two transition metal ions and over the excitations n possible for each of them as, H = − (cid:88) n t ε n (cid:88) (cid:104) ij (cid:105)(cid:107) γ P (cid:104) ij (cid:105) ( S ) O γ (cid:104) ij (cid:105) , (36)where P (cid:104) ij (cid:105) ( S ) is the projection on the total spin S = S ± and O γ (cid:104) ij (cid:105) is the projection operatoron the orbital state at the sites i and j of the bond. Following this general procedure, one findsa spin-orbital model with Heisenberg spin interaction for spins S = m of SU(2) symmetry cou-pled to the orbital operators which have much lower cubic symmetry, with the general structureof spin-orbital superexchange ∝ J (1) [8], H J = J (cid:88) γ (cid:88) (cid:104) ij (cid:105)(cid:107) γ (cid:110) ˆ K ( γ ) ij (cid:16) (cid:126)S i · (cid:126)S j + S (cid:17) + ˆ N ( γ ) ij (cid:111) . (37)It connects ions at sites i and j along the bond (cid:104) ij (cid:105) (cid:107) γ and involves orbital operators, ˆ K ( γ ) ij and ˆ N ( γ ) ij which depend on the bond direction γ = a, b, c for the three a priori equivalent directionsin a cubic crystal. The spin scalar product, (cid:16) (cid:126)S i · (cid:126)S j (cid:17) , is coupled to orbital operators ˆ K ( γ ) ij whichtogether with the other ”decoupled” orbital operators, ˆ N ( γ ) ij , determine the orbital state in a Mottinsulator. The form of these operators depends on the type of orbital degrees of freedom in agiven model. They involve active orbitals on each bond (cid:104) ij (cid:105) (cid:107) γ along direction γ . Thus theorbital interactions are directional and have only the cubic symmetry of a (perovskite) latticeprovided the symmetry in the orbital sector is not broken by other interactions, for instance bycrystal-field or Jahn-Teller terms.The magnetic superexchange constants along each cubic axis J ab and J c in the effective spinmodel, H = J ab (cid:88) (cid:104) ij (cid:105)(cid:107) ab (cid:126)S i · (cid:126)S j + J c (cid:88) (cid:104) ij (cid:105)(cid:107) c (cid:126)S i · (cid:126)S j , (38)are obtained from the spin-orbital model (37) by decoupling spin and orbital operators andnext averaging the orbital operators over a given orbital (ordered or disordered) state. It giveseffective magnetic exchange interactions: J c along the c axis, and J ab within the ab planes. Thelatter J ab ones could in principle still be different between the a and b axes in case of finitelattice distortions due to the Jahn-Teller effect or octahedra tilting, but we limit ourselves toidealized structures with J ab being the same for both planar directions. We show below that thespin-spin correlations along the c axis and within the ab planes, s c = (cid:104) (cid:126)S i · (cid:126)S j (cid:105) c , s ab = (cid:104) (cid:126)S i · (cid:126)S j (cid:105) ab , (39)next to the orbital correlations, play an important role in the intensity distribution in opticalspectroscopy.In the correlated insulators with partly occupied degenerate orbitals not only the structure ofthe superexchange (37) is complex, but also the optical spectra exhibit strong anisotropy andrbital Physics 11.13temperature dependence near the magnetic transitions, as found e.g. in LaMnO [28] or inthe cubic vanadates LaVO and YVO [29]. In such systems several excitations contribute tothe excitation spectra, so one may ask how the spectral weight redistributes between individ-ual subbands originating from these excitations. The spectral weight distribution is in generalanisotropic already when orbital order sets in and breaks the cubic symmetry, but even more sowhen A -type or C -type AF spin order occurs below the N´eel temperature T N .At orbital degeneracy the superexchange consists of the terms H ( γ ) n ( ij ) as a superposition ofindividual contributions on each bond (cid:104) ij (cid:105) due to charge excitation n (35) [19], H = J (cid:88) n (cid:88) (cid:104) ij (cid:105)(cid:107) γ H ( γ ) n ( ij ) , (40)with the energy unit for each individual H ( γ ) n ( ij ) term given by the superexchange constant J (1). It follows from d − d charge excitations with an effective hopping element t betweenneighboring transition metal ions and is the same as that obtained in a Mott insulator withnondegenerate orbitals in the regime of U (cid:29) t . The spectral weight in the optical spectroscopyis determined by the kinetic energy, and reflects the onset of magnetic order and/or orbitalorder [19]. In a correlated insulator the electrons are almost localized and the only kineticenergy which is left is associated with the same virtual charge excitations that contribute alsoto the superexchange. Therefore, the individual kinetic energy terms K ( γ ) n may be directlydetermined from the superexchange (40) using the Hellman-Feynman theorem, K ( γ ) n = − J (cid:10) H ( γ ) n ( ij ) (cid:11) . (41)For convenience, we define here the K ( γ ) n as positive quantities. Each term K ( γ ) n (41) originatesfrom a given charge excitation n along a bond (cid:104) ij (cid:105) (cid:107) γ . These terms are directly related to the partial optical sum rule for individual Hubbard subbands, which reads [19] a (cid:126) e (cid:90) ∞ σ ( γ ) n ( ω ) dω = π K ( γ ) n , (42)where σ ( γ ) n ( ω ) is the contribution of band n to the optical conductivity for polarization alongthe γ axis, a is the distance between transition metal ions, and the tight-binding model withnearest neighbor hopping is implied. Using Eq. (41) one finds that the intensity of each band isindeed determined by the underlying orbital order together with the spin-spin correlation alongthe direction corresponding to the polarization.One has to distinguish the above partial sum rule (42) from the full sum rule for the total spectralweight in the optical spectroscopy for polarization along a cubic direction γ , involving K ( γ ) = − J (cid:88) n (cid:10) H ( γ ) n ( ij ) (cid:11) , (43)which stands for the total intensity in the optical d − d excitations. This quantity is usually of lessinterest as it does not allow for a direct insight into the nature of the electronic structure being asum over several excitations with different energies ε n (35) and has a much weaker temperaturedependence. In addition, it might be also more difficult to deduce from experiment.1.14 Andrzej M. Ole´s and K CuF The simplest and seminal spin-orbital model is obtained when a fermion has two flavors, spinand orbital, and both have two components, i.e., spin and pseudospin are S = T = . Thephysical realization is found in cuprates with degenerate e g orbitals, such as KCuF or K CuF [4], where Cu ions are in the d electronic configuration, so charge excitations d i d j (cid:10) d i d j are made by holes. By considering the degenerate Hubbard model for two e g orbitals one findsthat d ions have an equidistant multiplet structure, with three excitation energies which differby J H [here J H stands for J eH in Eq. (16)], see Table 2. We emphasize that the correct spectrumhas a doubly degenerate energy ( U − J H ) and the highest non-degenerate energy is ( U + J H ) ,see Fig. 5(a). Note that this result follows from the diagonalization of the local Coulombinteractions in the relevant subspaces — it reflects the fact that a double occupancy ( | z ↑ z ↓(cid:105) or | ¯ z ↑ ¯ z ↓(cid:105) ) in either orbital state ( | z (cid:105) or | ¯ z (cid:105) ) is not an eigenstate of the degenerate Hubbard in theatomic limit (17), so the excitation energy U is absent in the spectrum, see Table 2.The total spin state on the bond corresponds to S = 1 or 0, so the spin projection operators P (cid:104) ij (cid:105) (1) and P (cid:104) ij (cid:105) (0) are easily deduced, see Table 2. The orbital configuration which corre-sponds to a given bond (cid:104) ij (cid:105) is given by one of the orbital operators in Sec. 2, either P ( γ ) (cid:104) ij (cid:105) forthe doubly occupied states involving different orbitals, or Q ( γ ) (cid:104) ij (cid:105) for a double occupancy in adirectional orbital at site i or j . This gives a rather transparent structure of one HS and three LSexcitations in Table 2. The 3D Kugel-Khomskii (KK) model then follows from Eq. (36) [20,21]: H ( d ) = (cid:88) γ (cid:88) (cid:104) ij (cid:105)(cid:107) γ (cid:26) − t U − J H (cid:18) (cid:126)S i · (cid:126)S j + 34 (cid:19) P ( γ ) (cid:104) ij (cid:105) + t U − J H (cid:18) (cid:126)S i · (cid:126)S j − (cid:19) P ( γ ) (cid:104) ij (cid:105) + (cid:18) t U − J H + t U + J H (cid:19) (cid:18) (cid:126)S i · (cid:126)S j − (cid:19) Q ( γ ) (cid:104) ij (cid:105) (cid:27) + E z (cid:88) i τ ci . (44)The last term ∝ E z is the crystal field which splits off the degenerate e g orbitals when Jahn-Teller lattice distortion occurs, and is together with Hund’s exchange η a second parameter to Table 2:
Elements needed for the construction of the Kugel-Khomskii model from charge exci-tations on the bond (cid:104) ij (cid:105) : excitation n , its type (HS or LS) and energy ε n , total spin state (tripletor singlet) and the spin projection operator P (cid:104) ij (cid:105) ( S ) , and the orbital state and the correspondingorbital projection operator. charge excitation spin state orbital state n type ε n S P (cid:104) ij (cid:105) ( S ) orbitals on (cid:104) ij (cid:105) (cid:107) γ projection1 HS U − J H (cid:16) (cid:126)S i · (cid:126)S j + (cid:17) | iζ γ (cid:105) | jξ γ (cid:105) ( | iξ γ (cid:105) | jζ γ (cid:105) ) P ( γ ) (cid:104) ij (cid:105) U − J H − (cid:16) (cid:126)S i · (cid:126)S j − (cid:17) | iζ γ (cid:105) | jξ γ (cid:105) ( | iξ γ (cid:105) | jζ γ (cid:105) ) P ( γ ) (cid:104) ij (cid:105) U − J H − (cid:16) (cid:126)S i · (cid:126)S j − (cid:17) | iζ γ (cid:105) | jζ γ (cid:105) Q ( γ ) (cid:104) ij (cid:105) U + J H − (cid:16) (cid:126)S i · (cid:126)S j − (cid:17) | iζ γ (cid:105) | jζ γ (cid:105) Q ( γ ) (cid:104) ij (cid:105) rbital Physics 11.15construct phase diagrams, see below. Here it refers to holes, i.e., large E z > favors holeoccupation in | ¯ z (cid:105) ≡ | x − y (cid:105) / √ orbitals, as in La CuO . On the other hand, while E z (cid:39) ,both orbitals have almost equal hole density.Another form of the Hamiltonian (44) is obtained by introducing the coefficients, r = 11 − η , r = r = 11 − η , r = 11 + η , (45)and defining the superexchange constant J in the same way as in the t − J model Eq. (1). Withthe explicit representation of the orbital operators P ( γ ) (cid:104) ij (cid:105) and Q ( γ ) (cid:104) ij (cid:105) in terms of (cid:110) τ ( γ ) i (cid:111) one finds, H ( d ) = 12 J (cid:88) γ (cid:88) (cid:104) ij (cid:105)(cid:107) γ (cid:26)(cid:20) − r (cid:18) (cid:126)S i · (cid:126)S j + 34 (cid:19) + r (cid:18) (cid:126)S i · (cid:126)S j − (cid:19)(cid:21) (cid:18) − τ ( γ ) i τ ( γ ) j (cid:19) + ( r + r ) (cid:18) (cid:126)S i · (cid:126)S j − (cid:19) (cid:18) τ ( γ ) i + 12 (cid:19) (cid:18) τ ( γ ) j + 12 (cid:19)(cid:27) + E z (cid:88) i τ ci . (46)In the FM state spins are integrated out and one finds from the first term just the superexchangein the e g orbital model analyzed before in Sec. 2.The magnetic superexchange constants J ab and J c in the effective spin-orbital model (46) are ob-tained by decoupling spin and orbital operators and next averaging the orbital operators (cid:68) ˆ K ( γ ) ij (cid:69) over the classical state | Φ (cid:105) as given by Eq. (29). The relevant averages are given in Table 3,and they lead to the following expressions for the superexchange constants in Eq. (38), J c = 18 J (cid:110) − r sin θ + ( r + r )(1 + cos θ ) + r (1 + cos θ ) (cid:111) , (47) J ab = 18 J (cid:40) − r (cid:18)
34 + sin θ (cid:19) + ( r + r ) (cid:18) −
12 cos θ (cid:19) + r (cid:18) − cos θ (cid:19) (cid:41) , (48)which depend on two parameters: J (1) and η (26), and on the orbital order (31) specified bythe orbital angle θ . It is clear that the FM term ∝ r competes with all the other AF LS terms.Nevertheless, in the ab planes, where the occupied hole e g orbitals alternate, the larger FMcontribution dominates and makes the magnetic superexchange J ab weakly FM ( J ab (cid:46) ) (when sin θ (cid:39) ), while the stronger AF superexchange along the c axis ( J c (cid:29) | J ab | ) favors quasione-dimensional (1D) spin fluctuations. Thus KCuF exhibits spinon excitations for T > T N .Consider first the 2D KK model on a square lattice, with γ = a, b in Eq. (46), as in K CuF . Inthe absence of Hund’s exchange, interactions between S = spins are AF. However, they arequite different depending on which of the two e g orbitals are occupied by holes: J zab = J for | z (cid:105) and J ¯ zab = J for | ¯ z (cid:105) hole orbitals. As a result, the AF phases with spin order in Fig. 6(iv)and the FO order shown in Figs. 6(c) and 6(d) are degenerate at finite crystal field E z = − J .This defines a quantum critical point Q = ( − . , in the ( E z /J, η ) plane. Actually, at thispoint also one more phase has the same energy — the FM spin phase of Fig. 6(i) with AO orderof |±(cid:105) orbitals (28) shown in Fig. 6(a) [21].To capture the corrections due to quantum fluctuations, one may construct a plaquette meanfield approximation or entanglement renormalization ansatz (ERA) [22]. One finds important1.16 Andrzej M. Ole´s ac b Fig. 6:
Left — schematic view of four simplest orbital configurations on a representative cubeof the 3D lattice: (a) AO order with (cid:104) τ a ( b ) i (cid:105) = ± changing from site to site and (cid:104) τ ci (cid:105) = ,obtained for E z < , (b) AO order with (cid:104) τ a ( b ) i (cid:105) = − changing from site to site and (cid:104) τ ci (cid:105) = − ,obtained for E z > , (c) FO order with occupied z orbitals and (cid:104) τ ci (cid:105) = (cigar-shapedorbitals), and (d) FO order with occupied ¯ z orbitals and (cid:104) τ ci (cid:105) = − (clover-shaped orbitals).Right — schematic view of four spin configurations (arrows stand for up or down spins) inphases with spin order: (i) A -AF, (ii) C -AF, (iii) FM, and (iv) G -AF. Images are reproducedfrom Ref. [24]. corrections to a mean field phase diagram near the quantum critical point Q , and a plaquettevalence bond (PVB) state is stable in between the above three phases with long range order, withspin singlets on the bonds (cid:107) a ( (cid:107) b ) , stabilized by the directional orbitals | ζ a (cid:105) ( | ζ b (cid:105) ) . A novelortho-AF phase appears as well when the magnetic interactions change from AF to FM onesdue to increasing Hund’s exchange η , and for E z /J < − . , see Fig. 7(a). Since the nearestneighbor magnetic interactions are very weak, exotic four-sublattice ortho-AF spin order is Table 3:
Averages of the orbital projection operators standing in the spin-orbital interactionsin the KK model (46) and determine the spin interactions in H s (38) for the C -type orbital orderof occupied e g orbitals which alternate in ab planes, as given by Eqs. (31). Nonequivalent cubicdirections are labeled by γ = ab, c . operator average ab c Q ( γ ) (cid:104) ij (cid:105) (cid:68)(cid:0) − τ ( γ ) i (cid:1)(cid:0) − τ ( γ ) j (cid:1)(cid:69) (cid:0) − cos θ (cid:1) (cid:0) θ (cid:1) P ( γ ) (cid:104) ij (cid:105) (cid:68) − τ ( γ ) i τ ( γ ) j (cid:69) (cid:0) + sin θ (cid:1) sin θ R ( γ ) (cid:104) ij (cid:105) (cid:68)(cid:0) + τ ( γ ) i (cid:1)(cid:0) + τ ( γ ) j (cid:1)(cid:69) (cid:0) + cos θ (cid:1) (cid:0) − cos θ (cid:1) rbital Physics 11.17 (a) (b)(c) Fig. 7:
Spin-orbital phase diagram and entanglement in the 2D KK model:(a) phase diagram in the plaquette mean field (solid lines) and ERA (dashed lines) variationalapproximation, with insets showing representative spin and orbital configurations on a × plaquette — ¯ z -like (cid:0) t c = − (cid:104) τ ci (cid:105) = (cid:1) and z -like (cid:16) t a,c = − (cid:68) τ c ( a ) i (cid:69) = − (cid:17) orbitals are accom-panied either by AF long range order (arrows) or by spin singlets on bonds in the PVB phase;(b) view of an exotic four-sublattice ortho-AF phase near the onset of FM (or FM z ) phase;(c) artist’s view of the ortho-AF phase — spin singlets (ovals) are entangled with either one ortwo orbital excitations | z (cid:105) → | ¯ z (cid:105) (clovers). Images are reproduced from Ref. [22]. stabilized by second and third nearest neighbor interactions, shown in Fig. 7(b). Such furtherneighbor interactions follow from spin-orbital excitations shown in Fig. 7(c). Note that bothapproximate methods employed in Ref. [22] (plaquette mean field approximation and ERA)give very similar range of stability of ortho-AF phase.In the 3D KK model the exchange interaction in the ab planes (48) and along the c axis (47) areexactly balanced at the orbital degeneracy ( E z = 0 ) and the quantum critical point where severalclassical phases meet in mean field approximation is Q = (0 , , see Fig. 8(a). While finite E z favors one or the other G -AF phase, finite Hund’s exchange η favors AO order stabilizing A -AFspin order, see Fig. 6(i). This phase is indeed found in KCuF at low temperature T < T N and isalso obtained from the electronic structure calculations [23]. We remark that for unrealisticallylarge η > . , spin order changes to FM.Large qualitative changes in the phase diagram are found when spin correlations on bondsare treated in cluster mean field approximation (using plaquettes or linear clusters [24]), seeFig. 8(b). Phases with long range spin order ( G -AF, A -AF, and FM) are again separated byexotic types of magnetic order which arise by a similar mechanism to that described above for1.18 Andrzej M. Ole´s(a) -3 -2 -1 0 1 2 3 E z /J η A-AFz C-AFFMA-AFG-AFz G-AFx(C-AFx)G-AF ortho-G-AF (b)
Fig. 8:
Phase diagram of the 3D KK model obtained in two mean field methods: (a) the single-site mean field, and (b) the cluster mean field. Shaded (green) area indicates phases with AOorder while the remaining magnetic phases are accompanied by FO order with fully polarizedorbitals, either ¯ z ( x ) (for E z > ) or z (for E z < ). In this approach plaquette valence-bond(PVB) phase with alternating spin singlets in the ab planes (yellow) separates the phases withmagnetic long range order, see Fig. 6. Phases with exotic magnetic order are shown in orange.Note different ranges of E z /J shown. Images are reproduced from Ref. [24]. an ab monolayer, i.e., nearest neighbor exchange changes sign along one cubic direction. Nearthe QCP Q one finds again PVB phase, as in the 2D KK model. In addition to the phasediagram of Fig. 7(a), the transitions between G -AF and PVB phases are continuous and mixedPVB-AF phases arise. Electronic structure calculations give A -AF spin order, in agreement with experiment. It followsfrom the spin-orbital superexchange for spins S = 2 in LaMnO , H e , due to the excitationsinvolving e g electrons. The energies of the five possible excited states [9] shown in Fig. 5(a)are: ( i ) the HS ( S = ) A state, and ( ii ) the LS ( S = ) states: A , E ( E (cid:15) , E θ ), and A ,will be parameterized again by the intraorbital Coulomb element U and by Hund’s exchange J eH between a pair of e g electrons in a Mn ( d ) ion, defined in Eq. (16). The Racah parameters B = 0 . eV and C = 0 . eV justify an approximate relation C (cid:39) B , and we find the LSexcitation spectrum: ε ( A ) = U + J H , ε ( E ) = U + J H (twice), and ε ( A ) = U + J H .Using the spin algebra (Clebsch-Gordan coefficients) and considering again two possible e g orbital configurations, see Eqs. (22) and (23), and charge excitations by t g electrons, one findsrbital Physics 11.19(A) (B) Fig. 9:
Kinetic energies per bond K ( γ ) n (41) for increasing temperature T obtained from therespective spin-orbital models for FM (top) and AF (bottom) bonds along the axis γ :(A) LaMnO (with J = 150 meV, η (cid:39) . [8], end experimental points [28]);(B) LaVO with η = 0 . [19] and experimental points [29].The kinetic energies in HS states ( n = 1 , red lines) and compared with the experiment (filledcircles). Vertical dotted lines indicate the value of T N . Images are reproduced from Ref. [8]. a compact expression [25], H e = 116 (cid:88) γ (cid:88) (cid:104) ij (cid:105)(cid:107) γ (cid:26) − t ε ( A ) (cid:16) (cid:126)S i · (cid:126)S j + 6 (cid:17) P ( γ ) (cid:104) ij (cid:105) + (cid:20) t ε ( E ) + 35 t ε ( A ) (cid:21) (cid:16) (cid:126)S i · (cid:126)S j − (cid:17) P ( γ ) (cid:104) ij (cid:105) + (cid:20) t ε ( E ) + t ε ( A ) (cid:21) (cid:16) (cid:126)S i · (cid:126)S j − (cid:17) Q ( γ ) (cid:104) ij (cid:105) (cid:27) + E z (cid:88) i τ ci . (49) H t = 18 J βr t (cid:16) (cid:126)S i · (cid:126)S j − (cid:17) . (50)Here β = ( t π /t ) follows from the difference between the effective d − d hopping elementsalong the σ and π bonds, i.e., β (cid:39) , while the coefficient r t stands for a superposition of all t g excitations involved in the t g superexchange [8]. Note that spin-projection operators forhigh (low) total spin S = 2 ( S = 1 ) cannot be used, but again the HS term stands for a FMcontribution which dominates over the other LS terms when (cid:68) P ( γ ) (cid:104) ij (cid:105) (cid:69) (cid:39) . Charge excitations by1.20 Andrzej M. Ole´s(a) (b) Fig. 10:
Band structure along the high symmetry directions in: (a) G -AF phase at x = 0 and(b) C -AF phase at x = 0 . . Spin majority (minority) bands are shown by solid (dashed) lines.Parameters: t = 0 . eV, J H = 0 . eV, g = 3 eV. Insets shows the Fermi surfaces at low doping.The special points: Γ = (0 , , , X = ( π, , , M = ( π, π, , R = ( π, π, π ) , Z = (0 , , π ) .Images are reproduced from Ref. [30]. t g electrons give double occupancies in active t g orbitals, so H t is AF but this term is small —as a result FM interactions may dominate but again only along two spatial directions. Indeed,this happens for the realistic parameters of LaMnO for the ab planes where spin order is FMand coexists with AO order, while along the c axis spin order is AF accompanied by FO order,i.e., spin-orbital order is A -AF/ C -AF. Indeed, this type of order is found both from the theoryfor realistic parameters and from the electronic structure calculations [26]. One concludes thatJahn-Teller orbital interactions are responsible for the enhanced value of the orbital transitiontemperature [27].The optical spectral weight due to HS states in LaMnO may be easily derived from the presentmodel (49), following the general theory, see Eq. (41). One finds a very satisfactory agree-ment between the present theory and the experimental results of [28], as shown in Fig. 9(A).We emphasize, that no fit is made here, i.e., the kinetic energies (41) are calculated using thesame parameters as those used for the magnetic exchange constants [8]. Therefore, such agood agreement with experiment suggests that indeed the spin-orbital superexchange may bedisentangled, as also verified later [27].To give an example of a phase transition triggered by e g electron doping of Sr − x La x MnO weshow the results obtained with double exchange model for degenerate e g electrons extended bythe coupling to the lattice [30], H = − (cid:88) ij,αβ,σ t ijαβ a † iασ a jβσ − J H (cid:88) i (cid:126)S i · (cid:126)s i + J (cid:88) (cid:104) ij (cid:105) (cid:126)S i · (cid:126)S j − gu (cid:88) i ( n iz − n i ¯ z ) + 12 N Ku . (51)It includes the hopping of e g electrons between orbitals α = z, ¯ z as in Eq. (6). The tetragonaldistortion u is finite only in the C -AF phase. Here we define it as proportional to a differencebetween two lattice constants a and c along the respective axis, u ≡ c − a ) / ( c + a ) , and N is the number of lattice sites. The microscopic model that explains the mechanism of themagnetic transition in electron doped manganites from canted G -AF to collinear C -AF phaserbital Physics 11.21at low doping x (cid:39) . . The double exchange supported by the cooperative Jahn-Teller effectleads then to dimensional reduction from an isotropic 3D G -AF phase to a quasi-1D orderof partly occupied z − r orbitals in the C -AF phase [30]. We emphasize that this theoryprediction relies on the shape of the Fermi surface which is radically different in the G -AF and C -AF phase. Due to the Fermi surface topology, spin canting is suppressed in the C -AF phase,in agreement with the experiment. t g orbitals LaTiO would be electron-hole symmetric compound to KCuF , if not the orbital degree offreedom which t g here. This changes the nature of orbital operators from the projections foreach bond to scalar products of pseudospin T = operators. The superexchange spin-orbitalmodel (37) in the perovskite titanates couples S = spins and T = pseudospins arising fromthe t g orbital degrees of freedom at nearest neighbor Ti ions, e.g. in LaTiO or YTiO [6].Due to large intraorbital Coulomb element U electrons localize and the densities satisfy thelocal constraint at each site i , n ia + n ib + n ic = 1 . (52)The charge excitations lead to one of four different excited states [9], shown in Fig. 5(b):( i ) the high-spin T state at energy U − J H , and( ii ) three low-spin states — degenerate T and E states at energy ( U − J H ) , and( iii ) an A state at energy ( U + 2 J H ) .As before, the excitation energies are parameterized by η , defined by Eq. (26), and we introducethe coefficients r = 11 − η , r = 11 − η , r = 11 + 2 η . (53)One finds the following compact expressions for the terms contributing to superexchange H J ( d ) Eq. (40) [6]: H ( γ )1 = 12 J r (cid:18) (cid:126)S i · (cid:126)S j + 34 (cid:19) (cid:18) A ( γ ) ij − n ( γ ) ij (cid:19) , (54) H ( γ )2 = 12 J r (cid:18) (cid:126)S i · (cid:126)S j − (cid:19) (cid:18) A ( γ ) ij − B ( γ ) ij + 12 n ( γ ) ij (cid:19) , (55) H ( γ )3 = 13 J r (cid:18) (cid:126)S i · (cid:126)S j − (cid:19) B ( γ ) ij , (56)where A ( γ ) ij = 2 (cid:18) (cid:126)τ i · (cid:126)τ j + 14 n i n j (cid:19) ( γ ) , B ( γ ) ij = 2 (cid:18) (cid:126)τ i ⊗ (cid:126)τ j + 14 n i n j (cid:19) ( γ ) , n ( γ ) ij = n ( γ ) i + n ( γ ) j . (57)As in Sec. 3.2, the orbital (pseudospin) operators (cid:110) A ( γ ) ij , B ( γ ) ij , n ( γ ) ij (cid:111) depend on the directionof the (cid:104) ij (cid:105) (cid:107) γ bond. Their form follows from two active t g orbitals (flavors) along the cubic1.22 Andrzej M. Ole´saxis γ , e.g. for γ = c the active orbitals are a and b , and they give two components of thepseudospin T = operator (cid:126)τ i . The operators (cid:110) A ( γ ) ij , B ( γ ) ij (cid:111) describe the interactions betweenthese two active orbitals, which include the quantum fluctuations, and take either the form of ascalar product (cid:126)τ i · (cid:126)τ j in A ( γ ) ij , or lead to a similar expression, (cid:126)τ i ⊗ (cid:126)τ j = τ xi τ xj − τ yi τ yj + τ zi τ zi , (58)in B ( γ ) ij . These latter terms enhance orbital fluctuations by double excitations due to the τ + i τ + j and τ − i τ − j terms. The interactions along the axis γ are tuned by the number of electrons occu-pying active orbitals, n ( γ ) i = 1 − n iγ , which is fixed by the number of electrons in the inactiveorbital n iγ by the constraint (52). The cubic titanates are known to have particularly pronouncedquantum spin-orbital fluctuations [18], and their proper treatment requires a rather sophisticatedapproach. Therefore, in contrast to AF long range order found in e g -orbital systems, spin-orbitaldisordered state may occur in titanium perovskites, as suggested for LaTiO [6]. As the last cubic system we present the spin-orbital model for V ions in d configurations inthe vanadium perovskite R VO ( R =La, . . . ,Lu). Due to Hund’s exchange one has S = 1 spinsand three ( n = 1 , , ) charge excitations ε n arising from the transitions to [see Fig. 5(b)]:( i ) a high-spin state A at energy ( U − J H ) ,( ii ) two degenerate low-spin states T and E at U , and( iii ) T low-spin state at ( U + 2 J H ) [31].Using η (26) we parameterize this multiplet structure by r = 11 − η , r = 11 + 2 η . (59)The cubic symmetry is broken and the crystal field induces orbital splitting in R VO , hence (cid:104) n ic (cid:105) = 1 and the orbital degrees of freedom are given by the doublet { a, b } which defines thepseudospin operators (cid:126)τ i at site i . One derives a HS contribution H ( c )1 ( ij ) for a bond (cid:104) ij (cid:105) alongthe c axis, and H ( ab )1 ( ij ) for a bond in the ab plane: H ( c )1 ( ij ) = − J r (cid:16) (cid:126)S i · (cid:126)S j + 2 (cid:17) (cid:0) − (cid:126)τ i · (cid:126)τ j (cid:1) , (60) H ( ab )1 ( ij ) = − J r (cid:16) (cid:126)S i · (cid:126)S j + 2 (cid:17) (cid:0) − τ zi τ zj (cid:1) . (61)In Eq. (60) pseudospin operators (cid:126)τ i describe low-energy dynamics of (initially degenerate) { xz, yz } orbital doublet at site i ; this dynamics is quenched in H ( ab )1 (61). Here ( (cid:126)S i · (cid:126)S j + 2) isthe projection operator on the HS state for S = 1 spins. The terms H ( c ) n ( ij ) for LS excitations( n = 2 , ) contain instead the spin operator (1 − (cid:126)S i · (cid:126)S j ) (which guarantees that these termscannot contribute for fully polarized spins (cid:104) (cid:126)S i · (cid:126)S j (cid:105) = 1 ): H ( c )2 ( ij ) = − J (cid:16) − (cid:126)S i · (cid:126)S j (cid:17) (cid:0) − τ zi τ zj − τ xi τ xj + 5 τ yi τ yj (cid:1) ,H ( c )3 ( ij ) = − J r (cid:16) − (cid:126)S i · (cid:126)S j (cid:17) (cid:0) + τ zi τ zj + τ xi τ xj − τ yi τ yj (cid:1) , (62)rbital Physics 11.23while again the terms H ( ab ) n ( ij ) differ from H ( c ) n ( ij ) only by orbital operators: H ( ab )2 ( ij ) = − J (cid:16) − (cid:126)S i · (cid:126)S j (cid:17) (cid:0) ∓ τ zi ∓ τ zj − τ zi τ zj (cid:1) ,H ( ab )3 ( ij ) = − J r (cid:16) − (cid:126)S i · (cid:126)S j (cid:17) (cid:0) ∓ τ zi ∓ τ zj + τ zi τ zj (cid:1) , (63)where upper (lower) sign corresponds to bonds along the a ( b ) axis.First we present a mean field approximation for the spin and orbital bond correlations whichare determined self-consistently after decoupling them from each other in H J (37). Spin inter-actions in Eq. (38) are given by two exchange constants: J c = 12 J (cid:8) ηr − ( r − ηr − ηr )( + (cid:104) (cid:126)τ i · (cid:126)τ j (cid:105) ) − ηr (cid:104) τ yi τ yj (cid:105) (cid:9) ,J ab = 14 J (cid:8) − ηr − ηr + ( r − ηr − ηr )( + (cid:104) τ zi τ zj (cid:105) ) (cid:9) , (64)determined by orbital correlations (cid:104) (cid:126)τ i · (cid:126)τ j (cid:105) and (cid:104) τ αi τ αj (cid:105) . By evaluating them one finds J c < and J ab > supporting C -AF spin order. In the orbital sector one finds H τ = (cid:88) (cid:104) ij (cid:105) c (cid:2) J τc (cid:126)τ i · (cid:126)τ j − J (1 − s c ) ηr τ yi τ yj (cid:3) + J τab (cid:88) (cid:104) ij (cid:105) ab τ zi τ zj , (65)with: J τc = 12 J [(1 + s c ) r + (1 − s c ) η ( r + r )] ,J τab = 14 J [(1 − s ab ) r + (1 + s ab ) η ( r + r )] , (66)depending on spin correlations: s c = (cid:104) (cid:126)S i · (cid:126)S j (cid:105) c and s ab = −(cid:104) (cid:126)S i · (cid:126)S j (cid:105) ab . In a classical C -AFstate ( s c = s ab = 1 ) this mean field procedure becomes exact, and the orbital problem maps toHeisenberg pseudospin chains along the c axis, weakly coupled (as η (cid:28) ) along a and b bonds, H (0) τ = J r (cid:88) (cid:104) ij (cid:105) c (cid:126)τ i · (cid:126)τ j + 12 η (cid:18) r r (cid:19) (cid:88) (cid:104) ij (cid:105) ab τ zi τ zj , (67)releasing large zero-point energy. Thus, spin C -AF and G -AO order with quasi-1D orbitalquantum fluctuations support each other in R VO . Orbital fluctuations play here a prominentrole and amplify the FM exchange J c , making it even stronger that the AF exchange J ab [31].Having the individual terms H ( γ ) n of the spin-orbital model, one may derive the spectral weightsof optical spectra (41). The HS excitations have remarkable temperature dependence and thespectral weight decreases in the vicinity of the magnetic transition at T N , see Fig. 9(B). Theobserved behavior is reproduced in the theory only when spin-orbital interactions are treated ina cluster approach, i.e. they cannot be disentangled, see Sec. 5.2.Unlike in LaMnO where the spin and orbital phase transitions are well separated, in the R VO ( R =Lu,Yb, . . . ,La) the two transitions are close to each other [33]. It is not easy to reproducethe observed dependence of the transition temperatures T OO and N´eel T N on the ionic radius1.24 Andrzej M. Ole´s(a) LaPrYLu Sm r R (A) T N , T OO T OO T N1 ac ϕ b ϑ ( K ) (b) Fig. 11:
Phase transitions in the vanadium perovskites R VO : (a) phase diagram with theorbital T OO and N´eel T N transition temperature obtained from the theory with and withoutorbital-lattice coupling (solid and dashed lines) [32], and from experiment (circles) [33];(b) spin (cid:104) S zi (cid:105) (solid) and G -type orbital (cid:104) τ zi (cid:105) G (dashed) order parameters, vanishing at T OO and T N , and the transverse orbital polarization (cid:104) τ xi (cid:105) (dashed-dotted lines) for LaVO and SmVO (thin and heavy lines). Images are reproduced from Ref. [32]. r R (in the R VO compounds with small r R there is also another magnetic transition at T N [34]which is not discussed here). The spin-orbital model was extended by the coupling to thelattice to unravel a nontrivial interplay between superexchange, the orbital-lattice coupling dueto the GdFeO -like rotations of the VO octahedra, and orthorhombic lattice distortions [32].One finds that the lattice strain affects the onset of the magnetic and orbital order by partialsuppression of orbital fluctuations, and the dependence of T OO is non-monotonous in Fig. 11(a).Thereby the orbital polarization ∝ (cid:104) τ x (cid:105) increases with decreasing ionic radius r R , and the valueof T N is reduced, see Fig. 11(b). The theoretical approach demonstrates that orbital-latticecoupling is very important and reduces both T OO and N´eel T N for small ionic radii. While rather advanced many-body treatment of the quantum physics characteristic for spin-orbital models is required in general, we want to present here certain simple principles whichhelp to understand the heart of the problem and to give simple guidelines for interpreting ex-periments and finding relevant physical parameters of the spin-orbital models of undoped cubicinsulators. We will argue that such an approach based upon classical orbital order is well justi-fied in many known cases, as quantum phenomena are often quenched by the Jahn-Teller (JT)coupling between orbitals and the lattice distortions, which are present below structural phasetransitions and induce orbital order both in spin-disordered and in spin-ordered or spin-liquidrbital Physics 11.25
Fig. 12:
Artist’s view of the GKR [35] for: (a) FO z and AF spin order and (b) AO z and FMspin order in a system with orbital flavor conserving hopping as is alkali R O hyperoxides( R =K,Rb,Cs) [36]. The charge excitations generated by interorbital hopping fully violate theGKR and support the states with the same spin-orbital order: (c) FO z and FM spin order and(d) AO z and AF spin order. Image is reproduced from Ref. [36]. phases.From the derivation of the Kugel-Khomskii model in Sec. 3.2, we have seen that pairs of direc-tional orbitals on neighboring ions {| iζ γ (cid:105) , | jζ γ (cid:105)} favor AF spin order while pairs of orthogonalorbitals such as {| iζ γ (cid:105) , | jξ γ (cid:105)} favor FM spin order. This is generalized to classical Goodenough-Kanamori rules (GKR) [35] that state that AF spin order is accompanied by FO order, while FMspin order is accompanied by AO order, see Figs. 12(a) and 12(b). Indeed, these rules empha-sizing the complementarity of spin-orbital correlations are frequently employed to explain theobserved spin-orbital order in several systems, particularly in those where spins are large, likein CMR manganites [3]. They agree with the general structure of spin-orbital superexchangein the Kugel-Khomskii model where it is sufficient to consider the flavor-conserving hoppingbetween pairs of directional orbitals {| iζ γ (cid:105) , | jζ γ (cid:105)} . The excited states are then double occupan-cies in one of the directional orbitals while no effective interaction arises for two parallel spins(in triplet states), so the superexchange is AF. In contrast, for a pair of orthogonal orbitals, e.g. {| iζ γ (cid:105) , | jξ γ (cid:105)} , two different orbitals are singly occupied and the FM term is stronger than theAF one as the excitation energy is lower. Therefore, configurations with AO order support FMspin order.The above complementarity of spin-orbital order is frustrated by interorbital hopping, or maybe modified by spin-orbital entanglement, see below. In such cases the order in both channelscould be the same, either FM/FO, see Fig. 12(c), or AF/AO, see Fig. 12(d). Again, whendifferent orbitals are occupied in the excited state, the spin superexchange is weak FM and1.26 Andrzej M. Ole´swhen the same orbital is doubly occupied, the spin superexchange is stronger and AF. Thelatter AF exchange coupling dominates because antiferromagnetism, which is due to the Pauliprinciple, does not have to compete here with ferromagnetism. On the contrary, FM exchange iscaused by the energy difference ∝ η between triplet and singlet excited states with two differentorbitals occupied.The presented modification of the GKR is of importance in alkali R O hyperoxides ( R =K,Rb,Cs)[36]. The JT effect is crucial for this generalization of the GKR — without it large interorbitalhopping orders the T x -orbital-mixing pseudospin component instead of the T z component in asingle plane. More generally, such generalized GKR can arise whenever the orbital order on abond is not solely stabilized by the same spin-orbital superexchange interaction that determinesthe spin exchange. On a geometrically frustrated lattice, another route to this behavior can occurwhen the ordered orbital component preferred by superexchange depends on the direction andthe relative strengths fulfill certain criteria. A quantum state consisting of two different parts of the Hilbert space is entangled if it cannotbe written as a product state. Similar to it, two operators are entangled if they give entangledstates, i.e., they cannot be factorized into parts belonging to different subspaces. This happensprecisely in spin-orbital models and is the source of spin-orbital entanglement [37].To verify whether entanglement occurs it suffices to compute and analyze the spin, orbital andspin-orbital (four-operator) correlation functions for a bond (cid:104) ij (cid:105) along γ axis, given respectivelyby S ij ≡ d (cid:88) n (cid:68) n (cid:12)(cid:12) (cid:126)S i · (cid:126)S j (cid:12)(cid:12) n (cid:69) , (68) T ij ≡ d (cid:88) n (cid:68) n (cid:12)(cid:12) ( (cid:126)T i · (cid:126)T j ) ( γ ) (cid:12)(cid:12) n (cid:69) , (69) C ij ≡ d (cid:88) n (cid:68) n | ( (cid:126)S i · (cid:126)S j − S ij )( (cid:126)T i · (cid:126)T j − T ij ) ( γ ) | n (cid:69) (70) = 1 d (cid:88) n (cid:68) n (cid:12)(cid:12) ( (cid:126)S i · (cid:126)S j )( (cid:126)T i · (cid:126)T j ) ( γ ) (cid:12)(cid:12) n (cid:69) − d (cid:88) n (cid:68) n (cid:12)(cid:12) (cid:126)S i · (cid:126)S j (cid:12)(cid:12) n (cid:69) d (cid:88) m (cid:68) m (cid:12)(cid:12) ( (cid:126)T i · (cid:126)T j ) ( γ ) | m (cid:69) , where d is the ground state degeneracy, and the pseudospin scalar product in Eqs. (69) and (70)is relevant for a model with active t g orbital degrees of freedom. As a representative examplewe evaluate here such correlations for a 2D spin-orbital model derived for NaTiO plane [39],with the local constraint (52) as in LaTiO ; other situations with spin-orbital entanglement arediscussed in Ref. [37].To explain the physical origin of the spin-orbital model for NaTiO [39] we consider a rep-resentative bond along the c axis shown in Fig. 13. For the realistic parameters of NaTiO the t g electrons are almost localized in d configurations of Ti ions, hence their interactionswith neighboring sites can be described by the effective superexchange and kinetic exchangerbital Physics 11.27 Fig. 13:
Left — (a) Hopping processes between t g orbitals along a bond parallel to the c axisin NaTiO : (i) t pd between Ti( t g ) and O( p z ) orbitals — two t pd transitions define an effectivehopping t , and (ii) direct d − d hopping t (cid:48) . The t g orbitals shown by different color are labeledas a , b , and c , see Eq. (2). The bottom part gives the hopping processes along γ = a, b, c axesin the triangular lattice that contribute to Eq. (71): (b) superexchange and (c) direct exchange.Right — Ground state for a free hexagon as a function of α (71): (a) bond correlations — spin S ij Eq. (68) (circles), orbital T ij Eq. (69) (squares), and spin–orbital C ij Eq. (70) (triangles);(b) orbital electron densities n γ at a representative site i = 1 (left-most site): n a (circles), n b (squares), n c (triangles). The insets indicate the orbital configurations favored by thesuperexchange ( α = 0 ), by mixed . < α < . , and by the direct exchange ( α = 1 ).The vertical lines indicate an exact range due to the degeneracy. Images are reproduced fromRef. [40]. processes. Virtual charge excitations between the neighboring sites, d i d j (cid:10) d i d j , generatemagnetic interactions which arise from two different hopping processes for active t g orbitals:( i ) the effective hopping t = t pd /∆ which occurs via oxygen p z orbitals with the charge trans-fer excitation energy ∆ , in the present case along the 90 ◦ bonds, and ( ii ) direct hopping t (cid:48) whichcouples the t g orbitals along the bond and give kinetic exchange interaction, as in the Hubbardmodel (1). Note that the latter processes couple orbitals with the same flavor, while the formerones couple different orbitals (for this geometry) so the occupied orbitals may be interchangedas a result of a virtual charge excitation — these processes are shown in Fig. 13.The effective spin-orbital model considered here reads [39], H = J (cid:110) (1 − α ) H s + (cid:112) (1 − α ) α H m + α H d (cid:111) . (71)The parameter α in Eq. (71) is given by the hopping elements as follows, α = t (cid:48) t + t (cid:48) , (72)and interpolates between the superexchange H s ( α = 0 ) and kinetic exchange H d ( α = 1 ),while in between mixed exchange contributes as well; these terms are explained in Ref. [39].1.28 Andrzej M. Ole´sThis model is considered here in the absence of Hund’s exchange η (26), i.e., at η = 0 . Onefinds that all the orbitals contribute equally in the entire range of α , and each orbital stateis occupied at two out of six sites in the entire regime of α , see Fig. 13. The orbital statechanges under increasing α and one finds four distinct regimes, with abrupt transitions betweenthem. In the superexchange model ( α = 0 ) there is precisely one orbital at each site whichcontributes, e.g. n c = 1 as the c orbital is active along both bonds. Having a frozen orbitalconfiguration, the orbitals decouple from spins and the ground is disentangled, with C ij = 0 ,and one finds that the spin correlations S ij = − . , as for the AF Heisenberg ring of L = 6 sites. Orbital fluctuations increase gradually with increasing α and this results in finite spin-orbital entanglement C ij (cid:39) − . for . < α < . ; simultaneously spin correlationsweaken to S ij (cid:39) − . .In agreement with intuition, when α = 0 . and all interorbital transitions shown in Fig. 13have equal amplitude, there is large orbital mixing which is the most prominent feature in theintermediate regime of . < α < . . Although spins are coupled by AF exchange, theorbitals fluctuate here strongly and reduce further spin correlations to S ij (cid:39) − . . The orbitalcorrelations are negative, T ij < , the spin-orbital entanglement is finite, C ij (cid:39) − . , andthe ground state is unique ( d = 1 ). Here all the orbitals contribute equally and n γ = 1 / which may be seen as a precursor of the spin-orbital liquid state which dominates the behaviorof the triangular lattice. The regime of larger values of α > . is dominated by the kineticexchange in Eq. (71), and the ground state is degenerate with d = 2 [40], with strong scatteringof possible electron densities { b iγ } , see Fig. 13. Weak entanglement is found for α > . ,where C ij (cid:39)(cid:54) = 0 . Summarizing, except for the regimes of α < . and α > . the groundstate of a single hexagon is strongly entangled, i.e., C ij < − . , see Fig. 13. As a rule, even when spin and orbital operators disentangle in the ground state, some of theexcited states are characterized by spin-orbital entanglement. It is therefore even more subtle toseparate spin-orbital degrees of freedom to introduce orbitons as independent orbital excitations,in analogy to the purely orbital model and the result presented in Fig. 3 [41]. This problemis not yet completely understood and we show here that in a 1D spin-orbital model the orbitalexcitation fractionalizes into freely propagating spinon and orbiton, in analogy to charge-spinonseparation in the 1D t - J model.While a hole doped to the FM chain propagates freely, it creates a spinon and a holon in an AFbackground described by the t - J model. A similar situation occurs for an orbital excitation inAF/FO spin-orbital chain [41]. An orbital excitation may propagate through the system onlyafter creating a spinon in the first step, see Figs. 14(a) and 14(b). The spinon itself moves viaspin flips ∝ J > t , faster than the orbiton, and the two excitations get well separated, see Fig.14(c). The orbital-wave picture of Sec. 2, on the other hand, would require the orbital excitationto move without creating the spinon in the first step. Note that this would be only possible forimperfect N´eel AF spin order. Thus one concludes that the symmetry between spin and orbitalrbital Physics 11.29 (c)(b)(a)
Fig. 14:
Schematic representation of the orbital motion and the spin-orbital separation in a1D spin-orbital model. The first hop of the excited state (a) → (b) creates a spinon (wavy line)that moves via spin exchange ∝ J . The next hop (b) → (c) gives an orbiton freely propagatingas a holon with an effective hopping t ∼ J/ . Image is reproduced from Ref. [41]. sector is broken also for this reason and orbitals are so strongly coupled to spin excitations inrealistic spin-orbital models with AF/FO order that the mean field picture separating these twosectors of the Hilbert space breaks down. t - J -like model for ferromagnetic manganites Even more complex situations arise when charge degrees of freedom are added to spin-orbitalmodels. The spectral properties of such models are beyond the scope of this discussion butwe shall only point out that macroscopic doping changes radically spin-orbital superexchangeby adding to it ferromagnetic exchange triggered by e g orbital liquid realized in hole dopedmanganites. As a result, the CMR effect is observed and the spin order changes to FM [3].Similar to the spin case, the orbital Hubbard model Eq. (11) gives at large ¯ U (cid:29) t the e g t - J model [42], i.e., e g electrons may hop only in the restricted space without doubly occupied e g sites. The kinetic energy is gradually released with increasing doping x in doped manganeseoxides La − x A x MnO , with A = Sr,Ca,Pb, which is a driving mechanism for effective FMinteraction generated by the kinetic energy ∝ ˜ H ↑ t ( e g ) in the double exchange [3]. It competeswith AF exchange which eventually becomes frustrated in FM metallic phase, arising typicallyat sufficient hole doping x (cid:39) . . The evolution of magnetic order with increasing dopingresults from the above frustration: at low doping x ∼ . AF spin order becomes stable andfirst changes to FM insulating phase, see Fig. 15(a). Only at larger doping x , an insulator-to-metal transition takes place which explains the CMR effect [3].1.30 Andrzej M. Ole´s SE AF FI FM X ΓΓ M (a) (b) Fig. 15:
Theoretical predictions for magnon spectra in the FM metallic phase in manganites:(a) spin-wave stiffness D (solid line) as a function of hole doping x given by double exchange(dashed) reduced by superexchange (SE) for: A -AF, FM insulating (FI), and FM metallic (FM)phases, and experimental points for La − x Sr x MnO (diamonds) and La . Pb . MnO (circle);empty circles for the hypothetical AO |±(cid:105) state unstable against the e g orbital liquid;(b) magnon dispersion ω (cid:126)q obtained at x = 0 . (solid line) and the experimental points forLa . Pb . MnO [43] (circles and dashed line).Parameters: U = 5 . , J eH = 0 . , t = 0 . , all in eV. Images are reproduced from Ref. [42]. In the FM metallic phase the magnon excitation energy is derived from manganite t - J model andconsists of two terms [42]: ( i ) superexchange being AF for the orbital liquid and ( ii ) FM doubleexchange J DE , proportional to the kinetic energy of e g electrons (6), J DE = 12 z S (cid:12)(cid:12)(cid:12)(cid:68) ˜ H ↑ t ( e g ) (cid:69)(cid:12)(cid:12)(cid:12) . (73)Here z is the number of neighbors ( z = 6 for the cubic lattice), and S = 4 − x is the averagespin in a doped manganese oxide. The kinetic energy (cid:12)(cid:12)(cid:12)(cid:68) ˜ H ↑ t ( e g ) (cid:69)(cid:12)(cid:12)(cid:12) measures directly the bandnarrowing due to the strong correlations in the e g orbital liquid. This explains why the spin-wave stiffness D is: ( i ) reduced by the frustrating AF superexchange J SE but ( ii ) increasesproportionally to the hole doping x in the FM metallic phase, see Fig. 15(a). As a result, themagnon dispersion in the FM metallic phase is given by, ω (cid:126)q = ( J DE − J SE ) z S (1 − γ (cid:126)q ) , (74)where γ (cid:126)q = z (cid:80) (cid:126)δ e i(cid:126)q · (cid:126)δ , and (cid:126)δ is a vector which connects the nearest neighbors.An experimental proof that indeed the e g orbital liquid is responsible for isotropic spin excita-tions in the FM metallic phase of doped manganites we show the magnon spectrum observedin La . Pb . MnO , with rather large stiffness constant D = 7 . meV, see Fig. 15(b). Notethat D would be much smaller in the phase with AO order of |±(cid:105) orbitals (28). Summarizing,the isotropy of the spin excitations in the simplest manganese oxides with FM metallic phase isnaturally explained by the orbital liquid state of disordered e g orbitals.rbital Physics 11.31 Spin-orbital physics is a very challenging field in which only certain and mainly classical as-pects have been understood so far. We have explained the simplest principles of spin-orbitalmodels deciding about the physical properties of strongly correlated transition metal oxideswith active orbital degrees of freedom. In the correlated insulators exchange interactions areusually frustrated and this frustration is released by certain type of spin-orbital order, with thecomplementarity of spin and orbital correlations at AF/FO or FM/AO bonds, as explained bythe Goodenough-Kanamori rules [35].One of the challenges is spin-orbital entanglement which becomes visible both in the groundand excited states. The coherent excitations such as magnons or orbitons are frequently notindependent and also composite spin-orbital excitations are possible. Such excitations are notyet understood, except for some simplest cases as e.g. the 1D spin-orbital model with SU(4)symmetry where all these excitations are on equal footing and contribute to the entropy in thesame way [44]. Such a perfect symmetry does not occur in nature however, and the orbitalexcitations are more complex due to finite Hund’s exchange interaction and, at least in somesystems, orbital-lattice couplings. They may be a decisive factor explaining why spin-orbitalliquids do not occur in certain models. For the same reason in the absence of geometricalfrustration, the orbital liquid seems easier to obtain than the spin liquid.Doping of spin-orbital systems leads to very rich physics with phase transitions induced bymoving charge carriers, as for instance in the well known example of the CMR manganites.Yet, the holes doped to the correlated insulators with spin-orbital order may be of quite differentnature. Charge defects may prevent the holes from coherent propagation [45] and as a result thespin-orbital order will persist to rather high doping level.Recently doping by transition metal ions with different valence was explored [46] — in such t g systems local or global changes of spin-orbital order result from the complex interplay ofspin-orbital degrees of freedom at orbital dilution , see Fig. 16(a). In general, the observed orderin the doped system will then depend on the coupling between the ions with different valencecompared with that within the host J imp /J host , and on Hund’s exchange at doped ions η imp . Notonly a crossover between AF and FM spin correlations is expected with increasing η imp , but alsothe orbital state will change from inactive orbitals to orbital polarons on the hybrid bonds withincreasing J imp , see Fig. 16(c). Quite a different case is given when double occupancies arereplaced by empty orbitals in charge doping as shown in Fig. 16(b). Here orbital fluctuationsare remarkably enhanced by the novel double excitation ∝ T + i T + j terms, see Figs. 16(d-e). Onthe one hand, large spin-orbital entanglement is expected in such cases when Hund’s exchangeis weak, while on the other hand the superexchange will reduce to the orbital model in the FMregime. By mapping of this latter model to fermions one may expect interesting topologicalstates in low dimension that are under investigation at present.1.32 Andrzej M. Ole´s J host spin-orbital order orbitaldilution host+ d impurity J imp J host spin-orbital order chargedilution host+ d impurity (a) (b) η i m p J imp /J host AF' FM AF (c) (d) J imp abc T i T j+ + (e) abc abcabc + abcabc T i T j+ - T i T j z z abcabc Fig. 16:
Top — Doping by transition metal ions in an ab plane with C -AF/ G -AO order of { a, c } orbitals found in d Mott insulators (ruthenates) with: (a) orbital dilution by the d impurity with S = 3 / spin, and (b) charge dilution by the d impurity with S = 1 spin. Host S = 1 spins (red/black arrows) interact by J host and doublons in a ( c ) orbitals shown by greensymbols. Here doping occurs at a doublon site and spins are coupled by J imp along hybrid (red)bonds.Bottom — (c) phase diagram for a single d impurity replacing a doublon in c orbital in the C -AF host [46], with changes in the orbital order indicated by dashed boxes (note a → b orbitalflips); (d-e) orbital fluctuations promoted on d – d hybrid bonds with (d) AF and (e) FM spincorrelations. In the latter case (e) the doublons at two orbitals are coupled in excited states(doublon and hole in ovals), and one obtains orbital flips ∝ T − i T + j accompanied by Ising terms ∝ T zi T zj , while double excitations ∝ T + i T + j occur on AF bonds (d) even in the absence ofHund’s exchange and are amplified by finite η . Image is reproduced from Ref. [47]. Acknowledgments
We kindly acknowledge support by Narodowe Centrum Nauki (NCN, National Science Centre,Poland), under Project MAESTRO No. 2012/04/A/ST3/00331.rbital Physics 11.33
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