Pseudo spin orbital

As it follows from (12), ground-state averages of orbital pseudo spin operators determine the numeric values of Q (i) and, thereby, the respective symmetry-adapted distortion of every elementary cell of the crystal. Instead of ferromagnetic , antiferromagnetic , etc., the appropriate name to use is ferrodistortive , antiferrodistortive , etc. 

The greatest advantage of the OOA is its combined description of orbital and magnetic ordering patterns. In this case, besides orbital pseudo spin system and nuclear distortions (phonons), we include real spins, the third participant in the nontrivial coupling. 

It is interesting to follow the role of the IT effect in this picture. Dressed with phonons, orbital states transform into the so-called IT polaron states. The orbital pseudo spin becomes vibronic pseudo spin (a detailed discussion of this side of the story is given below in Sects. 5.2 and 5.3). Instead of free rotations of the orbital pseudo spin we come to hindered rotations of the vibronic pseudo spin. At each metal site, the equipotential continuum of different orientations of the orbital spin is transformed into alternating bumps and wells of the vibronic pseudo spin. The latter ones correspond to favorable directions of the vibronic pseudo spin along metal-ligand chemical bonds. 

Behind the OOA, the basic concept is the so-called orbital pseudo spin. For one-center JT effect, in general the JT coupling does not change transformation... 

As distinguished from the orbital pseudo spin, for the vibronic pseudo spin, the mechanism of intercell coupling is different. It is dominated by the respective terms of elastic intercell interaction. 

Fig. 3 Schematic views of the degenerate states in the classical ground state. Arrows represente the orbital pseudo-spins in the plane, (a) and (b) correspond to the type-1 and 11 degenerate states (see text), respectively...

The second term is —3JN/16, when Tj is a two-dimensional classical spin, and is —3JN/% in the quantum-spin case. A total number of sites is N. This model is proposed as a orbital state for the layered iron oxide, [5, 6, 26], and is also recently proposed in study of the optical lattice [27-29]. A similar orbital model in a honeycomb lattice termed the Kitaev model is recently well examined. [30,31] Let us introduce the Fourier transformation for the orbital pseudo-spin operator. The Hamiltonian (15) is represented in the momentum space, [5,27]... 

We assume that each nucleon has a pseudo-spin i and pseudo-orbital angular momentum k. These couple to form the single particle angular momenta J,J (in [j]) of the two interacting nucleons. The wavefunction of a pair of nucleons coupled to a total angular momentum L (and z component p) is then given by ... 

Teichteil et al.41 fit a spin-orbit pseudo-operator such that its action on a pseudo-orbital optimally reproduces the effect of the true spin-orbit operator on the corresponding all-electron orbital. Ermler, Ross, Christiansen, and co-workers42 9 and Titov and Mosyagin50 define a spin-orbit operator as the difference between the and j dependent relativistic effective pseudopotentials (REPs)51... 

The electron Hamiltonian (15) describes the so-called orbital exchange coupling in a three-dimensional (3D) crystal lattice. The Pauli matrices, cr O ), have the same properties as the z-component spin operator with S = As a i) represents not a real spin but orbital motion of electrons, it is called pseudo spin. For the respective solid-state 3D-exchange problem, basic concepts and approximations were well developed in physics of magnetic phase transitions. The key approach is the mean-fleld approximation. Similar to (8), it is based on the assumption that fluctuations, s(i) = terms quadratic in s i) can be neglected. We do not go into details here because the respective solution is well-known and discussed in many basic texts of solid state physics (e.g., see [15]). 

The OOA was not designed for and does not apply to temperature dependencies of any kind in JT crystals. In particular, one cannot expect a reasonable estimate of the temperature of phase transitions in crystal lattice (structural), electron orbital, and/or spin system. This follows from the partitioning procedure that includes averaging over vibrational degrees of freedom. One can see the same reason from another perspective. The pseudo spin of a JT site, as the basic concept used in the OOA, operates in the basis of degenerate ground state wave functions. Excited vibronic states are beyond the pseudo spin setup. Therefore, in the OOA, by its very definition, temperature population of excited states does not make sense. 

We start from the model Hamiltonian to describe the strongly correlated electron system with the eg orbital degree of freedom. A system of the present interest is a Mott insulator where one electron occupies one of the doubly degenerate orbitals at each site in a simple cubic lattice. The doubly degenerate orbital degree of freedom is represented by the pseudo-spin operator with an amplitude of 1/2 defined by... 

Because of this symmetry, it is proven that the conventional long-range order is not realized. In spite of this, a kind of orbital order, termed the directional order [20,23-25] is realized at finite temperature. A schematic picture of the directional order is presented in Fig. 8. In a one-dimensional chain along the z direction, the z component of the pseudo-spin is aligned ferromagnetically. However, there is no correlation between the chains. A similar order is possible along the x direction. This directional order is identified by the following order parameter. 

In this section, we introduce another interesting orbital model with the frustration effect, termed the honeycomb lattice orbital model. We consider the model where the doubly degenerate orbitals, e.g. the d i y7. and d y orbitals, which are described by the pseudo-spin operator T,, are located at each site in a two dimensional honeycomb lattice. The explicit form of the model Hamiltonian is given by... 

Orbital structure in the classical ground state is obtained from the Hamiltonian in (17). The ground state energy is —37/16, when the pseudo-spins satisfy the following condition in all NN bonds [5,28,29]... 

This relation implies that the projection components of pseudo-spins are equal with each other for all NN bonds. There is a macroscopic number of orbital structures which satisfy this condition. Two of them are shown in Fig. 13. 

Instead of the conventional orbital order, there is a remarkable enhancement of a physical parameter q for the pseudo-spin angle 9i defined by... 

The projectors ljmj) ljmj and their smns P/,/ i/2 are then reduced to combinations of usual spin-free projectors P/ onto spherical harmonics. Due to the spin projectors oc) (oc, 1)3) (j31, oc) (/3 [ and /3) (a, the computation of spin-orbit integrals is carried out for separate (oc, a) and (a,)3) spin-orbit matrices on the basis of atomic pseudo-spin-orbitals it can be shown that (/3, )3) = (oc, oc) and ()3, oc) = — (oc, )3). Of coiirse both approaches give the same results. 

The scalar-relativistic effects can be easily absorbed into the effective potential by taking the all-electron (AE) calculation results of the same order of relativistic approximation as the references to parametrize the potentials. Taking the two-component (or even four-component) form of the pseudo-valence orbitals, the spin-orbit coupling effect can also be absorbed into the ECR Because the pseudovalence orbitals are energetically the lowest-eigenvalue eigenvectors of the Fock... 

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