P. Horsch, G. Khaliullin, and A.M. Ole´s Large Coulomb interactions play a crucial role
in transition metal oxides, and are responsible for the collective behavior of strongly correlated d electrons which localize in Mott-Hubbard in-sulators. Such localized electrons may occupy degenerate orbital states which makes it nec-essary to consider orbital degrees of freedom on equal footing with electron spins, and leads to effective (superexchange) spin-orbital mod-els to describe the low-energy physics. A re-markable feature of these models is that the su-perexchange interaction is highly frustrated on a cubic lattice, which was recognized as the ori-gin of novel quantum effects in transition metal oxides. In the case of eg systems, however, quantum fluctuations of orbitals are largely sup-pressed by the Jahn-Teller (JT) effect, which to-gether with superexchange often leads to struc-tural phase transitions accompanied by a classi-cal ordering of occupied orbitals.
The transition metal oxides with partly filled t2g orbitals, like the titanates and vanadates, exhibit different and more interesting phenomena due to the stronger quantum fluctuations among or-bitals. This occurs because of the relative weak-ness of the JT coupling in this case and due to the higher degeneracy of t2gstates.
We derived the spin-orbital superexchange model for cubic vanadates, and investigated the low-temperature phases of LaVO3 and YVO3. The magnetic order in LaVO3is C-type (ferro-magnetic chains along c-axis which stagger within (ab) planes), with a N´eel temperature TN= 143 K, whereas the magnetic order is stag-gered in all three directions (G-type) in YVO3 for T77 K and C-type at higher tempera-tures 77T118 K (Fig. 38). The C-phase is particularly surprising in the practically undis-torted structure of LaVO3at TTN.
Figure 38: Magnetic G- (a) and C-structures (b) with S = 1 spins indicated by black arrows. For the C-phase a red arrow indicates the fluctuating orbital occupation between a = yz and b = xz orbitals along the c-direction, while the secondt2gelectron occupies the xy orbital (nc= 1) at each vanadium ion, which is not shown.
In order to understand the microscopic ori-gin of the competition of C- and G-phases we address the following questions: (i) can the superexchange interactions alone explain why the ferromagnetic (FM) and antiferromagnetic (AF) interactions coexist in LaVO3in spite of a practically ideal cubic structure at TTNwith almost equal V–V bonds; (ii) why does the structural transition in LaVO3 occur only be-low the magnetic transition; and (iii) why is the G-type AF order stable in the low-temperature phase of YVO3, while the C-type order wins at higher temperatures?
We start with a Mott-insulator picture of cu-bic vanadites. Due to the large Hund cou-pling JH the V3 ions are in a triplet configu-ration. The t2g superexchange interactions be-tween S = 1 spins arise from the virtual excita-tions d2id2j d3id1j on a given bondij, with the hopping t allowed only between two out of three t2g orbitals: yz, zx, and xy, depending on the bond direction. These orbitals are perpen-dicular to three cubic directions, and will be labeled byγ= a, b, and c, respectively.
It is instructive to consider first the spin-orbital Hamiltonian in the absence of Hund and Jahn-Teller coupling: where the first factor represents the Heisenberg interaction between spins, while the second fac-tor is an operafac-tor which acts in the pseudospin space of orbitals. A remarkable feature of the t2gsuperexchange in Eq. (13) is that every bond is represented by two equivalent orbitals giving a SU(2) symmetric structureτiτj1
4ninjγof the orbital part. Hereτiare Pauli-matrices in the space of the (two) active orbitals, whose selec-tion depends on the direcselec-tionγ. Depending on the type of orbital correlations this may result in a spin coupling constant of either sign. This important property resembles that of the one-dimensional (1D) SU(4) model. The present problem is however more involved since there are three t2gflavors in a cubic crystal, and SU(2)
orbital correlations among two of them along a particular direction will necessarily frustrate those correlations in the other directions. We ar-gue that orbital singlets (with nia+ nib= 1) may form on the bonds parallel to c axis, thereby ex-ploiting fully the SU(2) symmetry of the orbital interactions in one direction. The second elec-tron occupies the third t2g orbital (nic= 1), and controls the spin interactions in the (ab) planes.
In order to understand why orbital fluctuations support the C-AF type spin order, it is instruc-tive to consider first a single bond along c-axis.
A crucial observation is that the lowest energy of –J2 is obtained when the spins are ferro-magnetic, and the orbitals a and b form a sin-glet, withτiτjc= –34. Thus, one finds a novel mechanism of ferromagnetic interactions which operates due to local fluctuations of a and b orbitals. At the same time, the orbital reso-nance on the bonds in (ab) planes is blocked, as nic= njc= 1, leading to antiferromagnetic in-teractions along these directions. Such an elec-tron distribution and the formation of quasi one-dimensional orbital pseudospin chains supports FM spin order along c-axis, and breaks the cu-bic symmetry in the orbital space as well. This explains why the magnetic transition into the C-phase in LaVO3 is accompanied by a weak structural distortion.
Based on the full Hamiltonian HH0HηHJT, which accounts in addition for the Hund inter-action η= JHU and the Jahn-Teller coupling, we calculated the magnetic exchange constants in the different phases. Taking realistic pa-rameters for the model, i.e., t = 0.2, U = 4.5 and JH= 0.68 eV which results in J = 35.6 meV, we obtain for the exchange constants in the C-phase Jab7.1 and Jc–9.3 meV. We em-phasize that the orbital quantum fluctuations play here a dominant role and the well known Hund’s mechanism due to JH alone would not suffice to obtainJc Jab as suggested by ex-periment (Keimer et al.), and would give in-stead Jc–4.4 meV.
On the other hand we find Jab5.9 and Jc6.9 meV for the G-phase of YVO3. This demonstrates that the magnetic structure de-pends sensitively on the orbital state – the ex-change constant Jc which is FM in LaVO3 changes into the strongest AF bond in the G-phase of YVO3.
Next we consider the reasons for the stability of the G-phase in YVO3. Unlike LaVO3 with al-most equal V–V bonds, this compound crystal-izes in a distorted structure, indicating that the JT effect plays a significant role. It was sug-gested that energy may be gained due to C-type orbital ordering, with a and b orbitals staggered in (ab) planes and repeated along c-axis, while nic= 1. Such ordering can be promoted by
HJT 2V
∑
ijc
τziτzj V
∑
ijab
τziτzj (14) and competes with the orbital disorder. This be-havior is remarkably different from the eg sys-tems, where the JT effect and superexchange support each other, inducing orbital ordering.
While V0 causes orbital splitting by 4 V and thus lowers the energy of the G-phase, it has lit-tle effect on the energy of the C-phase.
Orbital excitations are quite different in both AF phases: while the orbital excitation gap is small and grows∝ ηin C-phase, a large gap
4V opens in the orbital wave spectrum of G-phase. Thus, both the larger quantum fluctua-tions and additional (classical) energy gain due to finite JH in the C-phase have to be overbal-anced by the JT energy ∝V in order to stabi-lize the G-AF order at T 0. However, the G-phase may be destabilized at finite T by larger orbital entropy of the C-phase. Indeed, taking V = 0.65 J andη= 0.15, the free energy gives a transition from G- to C-phase around T0.8 J
(Fig. 39). While this behavior reproduces quali-tatively the first order phase transition observed in YVO3, its quantitative description requires a careful consideration of lattice and spin entropy contributions to the free energy. These effects are expected to reduce the transition tempera-ture Tdown to experimental values.
Figure 39: Free energies for YVO3 (in units of J) as a function of temperature T/J for the G-phase (V = 0.65 J) which is stable at low temperature, and for the C-phase calculated for different Hund cou-pling parametersη=JHU.
Summarizing, strong t2g orbital fluctuations in cubic vanadites lead to a new mechanism of ferromagnetic superexchange which stabilizes the C-phase in first undistorted LaVO3, and the structural transition follows. The JT ef-fect opposes the superexchange and can stabi-lize the G-phase with orbital ordering but only at low temperatures, as the fluctuations of t2g or-bitals release high entropy, and are thus respon-sible for the transition from the orbital ordered G-phase into the orbital disordered C-phase in YVO3.