*Spin-Orbital physics in transition metal oxides*

Transition metal oxides comprise more fascinating properties and theoretical challenges apart from high-temperature superconductivity. These are connected with the orbital degree of freedom which is present for example in ions like Mn^{3+} and V^{3+}, where the outer d-electrons can choose between different orbitals or linear combination of these. As a consequence of strong local interactions (strong correlations) the undoped parent compounds LaMnO_{3} and LaVO_{3} are Mott insulators and the orbital occupation and also its fluctuations strongly influence the magnetic and other properties of these systems.

The study of orbital degeneracy in transition metal oxides (TMO) is motivated, e.g., by the large variety of spin-charge-orbital ordered phases, the possible control of the orbital degree of freedom at interfaces and hetero structures, the switching from orbital ordered to metallic phases in applied magnetic fields, and the colossal magnetoresistance.

**Figure 1:** Spin-Orbital order of the V^{3+} ion sublattice in YVO_{3}. (a) Low temperature phase: G-AF spin order and C-type alternating orbital occupation of *t _{2g}* orbitals xz and yz; (b) Intermediate phase: C-AF spin order and G-type alternating orbital occupation of

*t*orbitals xz and yz; The red arrow indicates strong orbital singlet fluctuations, which is the origin of strong ferromagnetic interactions between spin neighbors in c-direction. A second

_{2g}*t*electron per V

_{2g}^{3+}occupies the xy orbital and is not shown for transparency. The xy occupation is stabilized by a crystal field and is responsible for the AF correlations in the ab-plane. The different orientation of spins in (a) and (b) follows from the relativistic spin-orbit coupling, and is controlled by a subtle interplay of magnetic correlations and the induced orbital moments.

The rather complex interplay of orbital and spin degrees of freedom can be formulated in terms of spin-orbital models, which are derived from more general multi-band Hubbard models. Central for the theoretical discussion of spin-orbital physics is the superexchange interaction, which for orbital degenerate systems lead to quite complex spin-orbital (or Kugel-Khomskii type) models

_{<i,j>}[J

_{ij}(

**T**

_{i}

*,*

**T**

_{j})

**S**

_{i}·

**S**

_{j}+ K

_{ij}(

**T**

_{i},

**T**

_{j})],

Such systems can also be viewed as Heisenberg spin models H = *J*∑_{ }**S**_{i} · **S**_{j}, where the exchange parameters *J _{ij }*depend on the bond direction and the orbital occupation.

Exchange parameters *J _{ij}* and

*K*are now no longer constants but operators. These are expressed by pseudospin operators

_{ij}**T**

_{i}which represent the orbital degrees of freedom. Thus the size and even the sign of

*J*in a spin-orbital state may fluctuate, i.e. determined by the quantum dynamics of spins and orbitals. Candidates for such strongly entangled spin and orbital degrees of freedom are

_{ij}*t*electron systems, i.e., like the vanadate perovskites, which form a central challenge in our research.

_{2g}In more conventional cases, however, the orbital state is frozen due to strong Jahn-Teller interactions with the lattice, i.e., leading to a quenching of the orbital dynamics. In general the balance of this interplay depends not only on the strength of the electron-lattice interaction but also on the particular state and its spin-orbital correlations. Frequently also the relativistic spin-orbit coupling H_{LS} = Λ∑_{i}(**S**_{i}* ·*** I**_{i})* *contributes. Then the spin part of the Hamiltonian will also turn out anisotropic.

(The logo indicates the orbital order in the *a-b* plane of LaMnO_{3}.)

*Vanadate Perovskites*

A central research topic are the RVO_{3} vanadate perovskites transition metal oxides, [R=Lu, Y, Sm, Pr, La], with orbital degeneracy in the t_{2g} orbital sector. In these compounds the Jahn-Teller coupling is relatively weak in contrast to e_{g} electron systems, therefore the interplay of quantum spin-orbital fluctuations is expected not to be quenched. The investigation and derivation of spin-orbital Hamiltonians for the cubic vanadates went parallel to work in Keimer's group.

A clear manifestation of the strong spin-orbital fluctuations described by the spin-orbital Hamiltonian of the cubic vanadates is the dimensional lowering leading to the magnetic C-phase, e.g., in LaVO_{3} and YVO_{3}, with strong ferromagnetic correlations in combination with strong orbital-singlet fluctuations along the c direction and antiferro correlations in the (a,b)-plane. This is triggered by the larger gain of energy due to quantum fluctuations in 1D as compared to 3D. This mechanism is in striking contrast to the conventional Goodenough-Kanamori mechanism of superexchange interactions.

**Figure 2:** Theoretical phase diagram of the RVO_{3} vanadate perovskites, [R=Lu, Y, Sm, Pr, La], as function of the ionic radius of the R-ions. The blue line markes the phase transition to the G-type orbital ordered phase, while the red line indicates the Neel transition to a phase with C-type spin and G-type orbital order (experimental data Tokura). The spin-orbital order is not only controlled by the spin-orbital superexchange but also influenced by Jahn-Teller interactions and the GdFeO_{3} distortion. The latter is shown as inset. The GdFeO_{3} distortion consists of a tilting and rotation of the VO octahedra, and increases with decreasing radius of the R-ions (blue circle).

In subsequent work based on using cluster mean-field theory (CMFT) we have analysed the influence of orbital-lattice coupling, i.e., due to the GdFeO_{3}-like rotations of the VO_{6} octahedra, and orthorhombic lattice distortions. In particular we have shown that lattice strain affects the onset of the magnetic and orbital order by partial suppression of orbital fluctuations and influences the phase diagram of the RVO_{3} system (see Fig. 2).

*Spin-Orbital Entanglement*

A particularly challenging system that stimulated many studies is the spin-orbital model in one-dimensions with free coupling parameters x and y:

H = -J Σ_{i} (**S**_{i}**S**_{i+1}+x)(**T**_{i}**T**_{i+1}+y)

On the one hand this model already contains the full complexity of the interplay of the different degrees of freedom, orbital and spin. it also allows to attack the problem with special tools particularly designed for one-dimensions. On the other hand this model reflects to a certain extent aspects of the physics of orbital degenerate Mott insulators in three dimension, namely for phases with quasi-one-dimensional correlations. An example is the C-AF phase in YVO_{3} or in LaVO_{3} that reveals strong orbital fluctuations along the c-direction.

**Figure 3:** Spin-orbital entanglement S_{vN} of the one-dimensional spin-orbital model (1) as function of coupling constants x and y. The calculations for an overall ferromagnetic coupling constant reveals high von Neumann entropy S_{vN} in the ground state of phase III, which is characterized by AF-spin and alternating-orbital correlations, whereas in the other phases spin and orbital degrees of freedom are non-entangled. The phase boundaries (dashed lines) in the phase diagram were obtained by comparison of the fidelity of the corresponding groundstate wave functions.

The one-dimensional models are convenient systems for the study of the dynamics and thermodynamics of spin-orbital systems. Apart from spin-orbital excitations which form a continuum, one finds here collective spin, orbital and spin-orbital modes. In the latter case a bound spin-orbital excitation moves coherently through the system.

We have found that the von Neumann entropy spectral function S_{vN}(ω) is a useful tool to analyze the entanglement of elementary excitations of spin-orbital systems. This also reveals that spin-orbital bound states have maximal entanglement entropy.

*Orbital-Peierls Dimerization*

Another striking outcome of spin-orbital entanglement is the orbital Peierls effect, which is actually believed to occur in the C-phase of cubic vanadates. We have discussed this phenomen theoretically in the context of the 3D systems earlier. However it is natural to study the effect on the example of a pure spin-orbital model (1).

In contrast to the usual Peierls effect where a lattice dimerization is an essential ingredient for the modulation. In the spin-orbital system the two electronic degrees of freedom can modulate each other in such way that the electronic structure results in a dimerization of the spin and orbital correlations.

**Figure 4:** Orbital-Peierls Effect: Phase diagram of spin-orbital Hamiltonian Eq. (1) for J=-1 and x=1 as function of orbital coupling parameter y. The most remarkable feature of the orbital-Peierls phase is that its phase only occurs in a finite temperature intervall, but not at zero temperature. This is another important feature which is distinct from the usual Peierls transition.

Here ferromagnetic spin correlations at finite temperature develop a modulation and induce singlet-pair correlations in the orbital degrees of freedom. This fundamental problem was investigated using finite temperature modified spin-wave theory, perturbation theory and by the transfer matrix renormalization group method.

*Spin-Orbital Polarons*

Motion of holes or single charge carriers in spin-orbital systems is a fundamental topic, similar to hole motion in the high-T_{c} cuprates. Here holes are scattered not only by spinwaves but also by orbital excitations. These problems were analysed for hole motion in t_{2g} systems in several works (see also Physics Viewpoint in Physical Review B) .

**Figure 5:** Schematic picture of a doped hole in the antiferromagnetic spin (and alternating orbital) structure of the ab-plane of LaVO_{3}. The polaron motion of the hole is influenced by the emission and reabsorption of both antiferromagnetic spin waves and orbital waves.

*Defects in orbital-degenerate Mott insulators*

As in the high-T_{c} cuprates the description of doped system poses a particular challenge to theory. Here we need to understand the properties of defects in strongly correlated systems with orbital degeneracy. Starting from a multiband Hubbard model and using an unrestricted Hartree-Fock approach, we have shown that Ca defects in the representative compound Y_{1-x}Ca_{x}VO_{3} induce defect states inside the Mott-Hubbard gap with 1 eV binding energy at low doping, i.e., consistent with experiment. These defect states are split off the lower Hubbard band. They are partially filled; thus they form the states at the chemical potential and are therefore relevant for transport, and determine the physical properties of the materials.

**Figure 6:** Energy levelstructure of t_{2g} electron states in the vicinity of a charged defect in Y_{1-x}Ca_{x}VO_{3}. The unperturbed level structure (left) shows the stabilization of the c (xy) orbital by a crystal field, and the degeneracy a and b orbitals. The Coulomb potential H_{imp} of the charged impurity shifts all vanadium states in the vicinty of the defect up by an energy V_{1}. In addition the occupied orbitals rotate due to the orbital polarization interaction H_{D} in order to lower their energy. The two electrons per V-ion fill the lowest two levels (1 and 2) in a spin-1 configuration. With each defect comes a doped hole which goes into the topmost filled orbital (2). The inset shows orbitals 2 on a V-cube surrounding a Ca^{2+}-defect (D) that replaces an Y^{3+}-ion. Thus the relative charge of the defect is negative. Therefore all electron states are pushed upward by the defect potential.

Moreover we studied the effect of charged defects on the magnetic structure of Y_{1-x}Ca_{x}VO_{3} and we have shown that the holes bound to the charged defects lead to an instability of the low-temperature G-phase at about 1% doping where the system enters the C-phase. This phase has in the undoped case a tendency towards orbital-Peierls distortion. We have shown that in the moderate doping regime defects support dimerization of the C-phase. Further research will focus on: (a) the role of disorder, (b) the dielectric response, and (c) the metal-insulator transition in such systems.

*Local experts:*

*External collaborators:*

- Enrico Arrigoni (Graz, Austria)
- Maria Daghofer (Dresden)
- Louis Felix Feiner (Eindhofen, Netherlands)
- Jesko Sirker (Kaiserslautern)
- Krzysztof Wohlfeld (Standford)

*Key publications:*

- Adolfo Avella, Peter Horsch, and Andrzej M. Oles
**Defect states and excitations in a Mott insulator with orbital degrees of freedom: Mott-Hubbard gap versus optical transport gaps in doped system**

Phys. Rev. B**87**, 045132 (2013) cond-mat/1302.0020 - Wen-Long You, Andrzej M. Oles, and Peter Horsch

**von Neumann entropy spectra and entangled excitations in spin-orbital models**

Phys. Rev. B**86**, 094412 (2012) cond-mat/1206.1062 - Peter Horsch and Andrzej M. Oles
**Defect states and spin-orbital physics in doped vanadates Y(1-x)Ca(x)VO(3)**

Phys. Rev. B**84**, 064429 (2011)

- Alexander Herzog, Peter Horsch, Andrzej M. Oles, and Jesko Sirker

**Magnetic excitations in one-dimensional spin-orbital models**

Phys. Rev. B**83**, 245130 (2011) cond-mat/1101.5301 - Krzysztof Wohlfeld, Andrzej M. Oles, and Peter Horsch

**Orbitally induced string formation in the spin-orbital polarons**

Phys. Rev. B**79**, 224433 (2009) cond-mat/0905.2380 - Krzysztof Wohlfeld, Maria Daghofer, Andrzej M. Oles, and Peter Horsch

**Spectral properties of orbital polarons in Mott insulators**

Phys. Rev. B**78**, 214423 (2008) cond-mat/0811.3763 - Jesko Sirker, Alexander Herzog, Andrzej M. Oles, and Peter Horsch

**Thermally Activated Peierls Dimerization in Ferromagnetic Spin Chains**

Phys. Rev. Lett.**101**, 157204 (2008) cond-mat/0807.4893 - Peter Horsch, Andrzej M. Oles, Louis Felix Feiner, and Giniyat Khaliullin

**Evolution of Spin-Orbital-Lattice Coupling in the RVO(3) Perovskites**

Phys. Rev. Lett.**100**, 167205 (2008) cond-mat/0805.1545 - Maria Daghofer, Krzysztof Wohlfeld, Andrzej M. Oles, Enrico Arrigoni, and Peter Horsch

**Absence of Hole Confinement in Transition-Metal Oxides with Orbital Degeneracy**

Phys. Rev. Lett.**100**, 066403 (2008) cond-mat/0802.2012

Last modified: July 31, 2013