Mixing of t_{2g}-e_{g} orbitals in 4d and 5d transition metal oxides

# Mixing of t2g-eg orbitals in 4d and 5d transition metal oxides

## Abstract

Using exact diagonalization, we study the spin-orbit coupling and interaction-induced mixing between and -orbital states in a cubic crystalline environment, as commonly occurs in transition metal oxides. We make a direct comparison with the widely used only or only model, depending on electronic filling. We consider all electron fillings of the -shell and compute the total magnetic moment, the spin, the occupancy of each orbital, and the effective spin-orbit coupling strength (renormalized through interaction effects) in terms of the bare interaction parameters, spin-orbit coupling, and crystal field splitting, focusing on the parameter ranges relevant to 4d and 5d transition metal oxides. In various limits we provide perturbative results consistent with our numerical calculations. We find that the - mixing can be large, with up to 20% occupation of orbitals that are nominally “empty”, which has experimental implications for the interpretation of the branching ratio in experiments, and can impact the effective local moment Hamiltonian used to study magnetic phases and magnetic excitations in transition metal oxides. Our results can aid the theoretical interpretation of experiments on these materials, which often fall in a regime of intermediate coupling with respect to electron-electron interactions.

## I Introduction

Transition metal oxides have undergone intensive study because of their remarkably rich phase diagrams and sensitivity to external fields, strain, disorder, and doping.Bednorz and Müller (1988); Lee et al. (2006); Tokura and Nagaosa (2000); Gardner et al. (2010) High-temperature superconductors (e.g., cuprates) and colossal magnetoresistance materials (e.g., manganites) are two notable examples, but both of these have light transition elements drawn from the 3d series.Maekawa et al. (2004); Imada et al. (1998) On the other hand, the study of topological insulators in recent yearsQi and Zhang (2011); Hasan and Kane (2010); Moore (2010); Ando (2013) has brought attention to the importance of large spin-orbit coupling, which may induce topological phase transitions in materials. As a result, some focus has shifted to the heavier transition metals from the 4d and 5d series, which have significantly enhanced spin-orbit coupling relative to those in the 3d series.Witczak-Krempa et al. (2014); Rau et al. (2016); Schaffer et al. (2016)

Iridates, in particular, have undergone much theoretical and experimental study.Witczak-Krempa et al. (2014); Rau et al. (2016); Schaffer et al. (2016) An interesting body of theoretical studies has suggested that novel interaction-driven topological states in which the quantum numbers of the electron are fractionalized may appear.Maciejko and Fiete (2015); Stern (2016) However, in some of the iridates even the nature of the conventional order, such as the magnetic order (and the underlying microscopic spin Hamiltonian), is not easy to determine,Chaloupka et al. (2010); Jiang et al. (2011); Kimchi et al. (2015); Takayama et al. (2015); Alpichshev et al. (2015); Williams et al. (2016); Biffin et al. (2014); Chern et al. (2017); Sizyuk et al. (2014) in part due to the large neutron absorption cross-section which makes neutron scattering experiments challenging.Choi et al. (2012) An experimental tool known as resonant inelastic X-ray scattering (RIXS) is particularly well suited to studies of the iridates.Ament et al. (2011); Kotani and Shin (2001); Gretarsson et al. (2016); Lu et al. (2017); Moretti Sala et al. (2015); Kim et al. (2012a); Sala et al. (2014) While there is some understanding of the microscopic details revealed in the RIXS signal, the theory is still under development.Savary and Senthil (2015) Our work will facilitate that development.

A further challenge to understanding the iridates and other 4d/5d transition metal oxides is that the materials fall into a regime of comparable energy scales where it is difficult to argue a priori that a particular term in the Hamiltonian is small compared to the others: The typical kinetic energy, interaction energies, Hund’s coupling, spin-orbit coupling, and crystal field splitting are all on the scale of an electron volt.Witczak-Krempa et al. (2014); Rau et al. (2016); Schaffer et al. (2016) With respect to theoretical analysis, this means it is not clear if one should approach the iridates from a weak-coupling band-like description in which correlations are included within the band description,Hu et al. (2014); Chen et al. (2015); Zhang et al. (2013, 2017); Wan et al. (2011) or from the strong-coupling limit in which a local moment modelHozoi et al. (2014); Katukuri et al. (2012); Mohapatra et al. (2017); Kim et al. (2012b); Perkins et al. (2014); Svoboda et al. (2017); Meetei et al. (2015); Yuan et al. (2017); Laurell and Fiete (2017) is natural to describe the various types of magnetic orders that typically occur in the 4d/5d transition metal oxides (characteristic magnetic transition temperatures are on the order of 100K).Witczak-Krempa et al. (2014); Rau et al. (2016); Schaffer et al. (2016) In this work, we start from an atomic limit of the transition metal ions and treat the interaction effects non-pertubatively using exact diagonalization. In this way, we are able to work within an intermediate regime that reduces to a tight-binding-type Hamiltonian (for multiple ions) in the limit of vanishing interactions and a local moment model in the limit of strong interactions.

In a large class of transition metal oxides, the local oxygen environment of the transition metal ions is an octahedral cage (see Fig. 1) that produces a cubic environment that splits the -orbitals into a lower lying triply degenerate set of orbitals and a higher lying doubly-degenerate set of orbitals. A feature that is shared by nearly all weak (aside from ab initio studies) and strong-coupling theoretical studies of the heavy transition metal oxides is that they assume the - mixing is negligible.Hozoi et al. (2014); Katukuri et al. (2012); Mohapatra et al. (2017); Kim et al. (2012b); Perkins et al. (2014); Svoboda et al. (2017); Meetei et al. (2015); Yuan et al. (2017); Laurell and Fiete (2017) In addition, many theoretical studies motivated by the iridates assume the infinite spin-orbit coupling limit which splits the orbitals into a total angular moment and set of states (that do not mix). For iridates with a nominal -shell filling of 5 electrons, this results in a half-filled band, and thus reduces the Hamiltonian to a one-band model that often helps theoretical studies that rely on methods developed in the context of the cuprates.

In this work, we revisit the assumption of negligible - mixing and study the single ion limit in detail using exact diagonalization that allows a non-perturbative treatment of interaction effects. We consider all -shell fillings and find the neglect of - mixing is not in general justified, with the greatest mixing occurring for fillings of 5,6, and 7 electrons. Our work has implications for the interpretation of RIXS and X-ray absorption spectroscopy (XAS) data for the heavier elements with strong spin-orbit coupling, and the spectra of transition metal ions in oxides more generally. Our work can also be used as a more realistic starting point for determining the best form of the magnetic interactions between two nearby ions: Exchange interactions, exchange anistropies, and the size of local moments differ as a consequence of - mixing.

Our paper is organized as follows. In Sec. II we summarize the effects of a local cubic crystal field on the -orbital level structure of a transition metal ion. In Sec. III we provide the details of the Hamiltonian with and without - mixing in the presence of spin-orbit coupling. In Sec. IV and Sec. V we describe the interaction terms and conserved quantities of the full system we study, and in Sec. VI we present the results of our exact diagonalization studies for all electron fillings. We present the main conclusions of the work in Sec. VII.

## Ii Octahedral crystal fields

A transition metal ion in free space has rotational symmetry SO(3) and therefore five-fold degenerate -orbitals. Frequently, transition metal ions in crystals are held inside regular octahedral cages, surrounded by ligands. A common type of these ligands is oxygen, which form the large class of transition metal oxides. When a free ion is placed inside an octahedral cage, the symmetry is reduced from the full rotational SO(3) symmetry of the -orbital states in the free space, to the symmetry group of the octahedron, SO(3). This consists of all the rotations which take the octahedron into itself. Thus, is a subgroup of the rotation group: SO(3). Hence, any representation of SO(3) provides a representation of . However, irreducible representations of SO(3) will become reducible representations of . Thus, the fivefold degeneracy of the -states is lifted by the crystal field and the -levels are split into a higher-lying two-fold degenerate and a lower-lying three-fold degenerate manifold, as seen in Fig.1, where is the energy difference between them. The oxygen ligands are approximated as point charges siting in the corners of the octahedral cages. The -orbital charge distributions point in between the point charges of the oxygens, and the states point towards the point charges, raising their energy relative to the levels, as shown in Fig. 2.

The and orbitals are formed by linear combinationsMaekawa et al. (2004) of the spherical harmonics , with the orbital angular momentum . The magnetic quantum number takes values from to . For these orbitals states are:

 dyz=−1i√2(Y12+Y−12),dzx=−1i√2(Y12−Y−12),dxy=1i√2(Y22−Y−22), (1)

and for they are:

 d3z2−r2=Y02,dx2−y2=1√2(Y22+Y−22). (2)

The crystal field term in the Hamiltonian, , can be written in a diagonal form as (taking the energy of the states as the zero of energy),

 HCF=∑σ=±1/2Δ(|3z2−r2,σ⟩⟨3z2−r2,σ|+|x2−y2,σ⟩⟨x2−y2,σ|), (3)

where refers to the spin of the electron in a given orbital state.

## Iii Spin-orbit coupling in a crystal field

The spin-orbit coupling strength is comparable to other energy scales in heavy transition metal oxides.Witczak-Krempa et al. (2014); Rau et al. (2016); Schaffer et al. (2016) In its presence the orbital angular momentum and spin angular momentum are no longer independently conserved quantities. Moreover, the spin-orbit coupling can also induce mixing between the and manifolds.

The matrix elements of orbital angular momentum for a single electron in the basis of the , Eq. (1), and , Eq.(2), states: , and that of a single electron in atomic -orbitals in the basis are:Sugano et al. (1970)

 lx=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣000−√3i−i00i000−i000√3i0000i0000⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦,l′x=⎛⎜⎝00000−i0i0⎞⎟⎠, (4)
 ly=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣00−i00000√3i−ii00000−√3i0000i000⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦,l′y=⎛⎜⎝00i000−i00⎞⎟⎠, (5)
 Unknown environment 'array% (6)

By comparing the matrix elements of in the states with those in the -states in free atoms, one can map the former = 2 -states onto the latter -states with = 1 using the relation:

 l(t2g)=−l(p). (7)

This relation is called the T-P equivalence, Sugano et al. (1970); Fazekas (1999) according to which the orbital angular momentum in states is partially quenched from to . When the cubic crystal field splitting is large, one can neglect the off-diagonal elements between and manifolds and the T-P equivalence can be conveniently used. Note, however, that the spin-orbit coupling generally mixes the and states so if the spin-orbit coupling is large enough compared to the crystal field splitting (and we will see it can be enhanced by electron-electron interactions) then the mixing may have non-negligible effects.

Using the expression of the orbital angular momentum of Eqs.(4)-(6) and the Pauli matrices, we can construct the spin-orbit interaction matrix. Written in the basis it becomes,

 HSOC=ζ2Ψ†AΨ, (8)

where is a row vector, and is the complex conjugate column vector, and

 \resizebox469.755pt$A=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝0−ii√3−1i0−1−i√3−i−i−100−2i√3i√3000−1i2i00000ii−√31−i01−i√3−i−i1002i−√3i√30001i−2i00⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠$, (9)

expresses the spin-orbit coupling in the full 10 states of the and manifolds, including spin. The matrix elements are split into terms that act only on the -subspace, , terms that acts only one the subspace, , and terms that have matrix elements between and states, . The angular momentum matrix elements in the states are zero. Thus, the matrix elements of the are zero as well.

The full Hamiltonian of the one-electron states is

 H=HSOC+HCF. (10)

In the T-P equivalence one neglects the off-diagonal matrix elements of the angular momentum, that connect the - subspaces,

 HTP=Ht2gSOC+HegSOC+HCF, (11)

which is given from the expressions above without the mixing. Diagonalizing Eq.(11), the states evolve as shown in Fig. 3 via the green lines. In particular, the states are not affected by the spin-orbit coupling, and are separated from the states by an energy difference . On the other hand, the states are split into eigenstates of energy :

 |Jeff=12,m=−12⟩=1√3|dyz↑⟩−i√3|dxz↑⟩−1√3|dxy↓⟩,|Jeff=12,m=12⟩=1√3|dyz↓⟩+i√3|dxz↓⟩+1√3|dxy↑⟩, (12)

and eigenstates of energy :

 |Jeff=32,m=−32⟩=1√2|dyz↓⟩−i√2|dxz↓⟩,|Jeff=32,m=32⟩=−1√2|dyz↑⟩−i√2|dxz,↑⟩,|Jeff=32,m=−12⟩=1√6|dyz↑⟩−i√6|dxz↑⟩+√23|dxy↓,⟩|Jeff=32,m=12⟩=−1√6|dyz↓⟩−i√6|dxz↓⟩+√23|dxy↑⟩. (13)

The results in Eq.(12) and Eq.(13) are commonly used in the literature. Beyond the T-P equivalence one needs to consider the neglected mixing of the - subspaces of the spin-orbit coupling . Here, we consider it as a perturbation to the T-P equivalence terms of Eq. (11).

Writing in the diagonal basis of , we have in the basis ,

 H0+H1=ζ2Φ†BΦ (14)

where is a row vector, and is a complex conjugate column vector,

 \resizebox469.755pt$B=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝200000−1i√6000−i√6δ00000−1−i√6000i√6δ00200000−1−i√6000i√6δ00000−1i√6000−i√6δ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠$, (15)

where . Note that are the diagonal matrix elements, and are the non-diagonal ones, of the -matrix, Eq.(15). One sees that there are no matrix elements involving states. Thus they remain unaffected. However, the and subspaces are mixed. Thus, going beyond the T-P equivalence involves mixing the upper and the lower states as seen in Fig. 3 indicated with red lines. Hence the evolution of the and states in the presence of spin-orbit coupling is more complex than the commonly used T-P equivalence assumes.

To first order in the wavefunctions, the lower quartet is modified by

 Unknown environment '%' (16)

and to second order in energy we find a shift by . The upper quartet is modified by

 |d3z2−r2,∓12⟩±i√32ζΔ+ζ/2|Jeff=32,m=±32⟩,|dx2−y2,±12⟩±i√32ζΔ+ζ/2|Jeff=32,m=±12⟩, (17)

with shifts in energies of . Note for typical values for 5d systems, the mixing is , a 20% effect.

## Iv Inclusion of Electron-electron interaction

Having treated the octahedral crystal field in Sec. II and the spin-orbit interaction in Sec. III, we are now ready to add the electron-electron interactions, . We are especially interested in how electron-electron interactions will interplay with the - mixing highlighted in the previous section. This mixing is often ignored in the literature.

### iv.1 T-P equivalence in 3d systems

In the presence of electron-electron interactions, the Hamiltonian of the ion is

 H=HCF+HSOC+He−e, (18)

which contains the crystal field part , the spin-orbit part , and the interacting part . Within the crystal field approximation several different cases arise: weak, intermediate, and strong crystal field.Fazekas (1999); Zeiger and Pratt (1973) The simplest is the weak crystal field case,

 He−e>>HCF>HSOC,

where the interacting part is much larger than the crystal field terms , and the spin-orbit coupling is smaller still. The intermediate crystal field case is

 He−e>HCF>HSOC,

which follows the same order, but the crystal fields are no longer much weaker than the electron-electron interactions.

In 3d systems, the on-site Coulomb interaction is on the order of U= 3-10 eV, crystal fields are =1.5-2 eV, Hund’s coupling is =0.8-0.9 eV, and the spin-orbit coupling is in the order of 0.01eV-0.1eV (=0.02 eV for Ti, and = 0.07 eV for heavier Co).Maekawa et al. (2004) Thus, 3d systems fall into the weak and intermediate crystal field regimes.

Following the above scheme from the most dominant term to the weakest, we have the interacting Hamiltonian, which is rotationally invariant with spin independent (Coulomb) interactions. Thus, the orbital angular momentum and spin are conserved quantum numbers and can be used to label the states. The next important term, the crystal field, is not rotationally invariant and mixes different terms. Because the energy difference of different terms is 3-10 eV, and the crystal field is 1.5-2 eV, as a first approximation we neglect the mixing of different values, and we consider the effect of crystal field splitting within the ground state manifold of the term, following the conventions of the field. The smallest term in the hierarchy, the spin-orbit coupling, mixes states of different crystal field levels ( and in our case), and terms of different levels as well, but we neglect those and only include the splitting within the ground state multiplet of crystal field split levels.

Since the electron-electron interaction is the most dominant term in the above hierarchy and the crystal field mixes states within a given term, Hund’s first and second rule are valid even in the presence of crystal fields. This means that 3d ions can form high spin structures, where the 4 and 5 electrons go into the orbitals, as indicated from Hund’s first rule of maximal spin. The condition for the low-spin to high-spin transition where the 4 electron prefers to go into the orbitals is approximately (larger favors a high-spin configuration, smaller a low-spin configuration). Since =1.5-2 eV and =0.8-0.9eV, this condition is satisfied. However, since crystal fields dominate over the spin-orbit coupling, Hund’s third rule ceases to apply. This means that though and remain valid quantum numbers, and their values are still given by Hund’s first and second rule, the total angular momentum is no longer a good quantum number.

In the case of strong crystal fields,

 HCF≥He−e>HSOC,

the crystal fields are comparable to (or larger than) the electron-electron interaction giving rise to Hund’s first and second rule. Thus, they even mix states belonging to different terms. It is quite usual to find strong crystal fields in 4d and 5d transition metal compounds. On the other hand, there are only rare instances of insulating solids where 3d ions are subject to such strong crystal fields that even Hund’s first rule is put out of action. In next section we will more extensively discuss the case of 4d and 5d systems.

Regardless of the particular energy hierarchy that is relevant, one has

 [He−e+HCF,S2]=0,[He−e+HCF,Sz]=0, (19)

so that and commute with and since they are spin independent. As a consequence, has a ground state with well defined spin quantum number. This holds for arbitrary strength of the Coulomb interaction (including none at all).

Summarizing, the ground state multiplet of is only (for up to 6 electrons) if the ion is in the low spin configuration. For finite spin-orbit coupling, and are no longer good quantum numbers. As discussed in Sec. III, splits into (). Since in 3d systems the spin-orbit coupling is on the order of 0.02-0.07 eV and crystal fields eV, the mixing of and states in the low-spin configuration will be on the order of and can be neglected to first order. Consequently, it is a good approximation in 3d systems to neglect the off-diagonal matrix elements of angular momentum in systems and use the T-P equivalence. This is no longer the case for the heavier transition elements.

### iv.2 Limitations of the T-P equivalence in 4d and 5d systems

As one moves from 3d to 4d to 5d transition metals the outermost electronic wavefunctions become more and more extended, and thus scale of the typical Hubbard becomes smaller, reaching down to =0.5-3eV in 5d elements. The Hund’s coupling is reduced as well, to =0.6-0.7 eV in 4d elements and to =0.5eV in 5d elements. Similarly, the larger spatial extent of the outermost electronic states increase the crystal field splitting to =1-5eV in 5d elements. Heavier elements have larger spin-orbit coupling, and its value is increased to eV in 5d elements. These values bring the 4d/5d elements into the strong crystal field scenario mentioned in the previous section, where the energy scale of the crystal fields is greater than or comparable to the electron interactions.

Since there is mixing of terms. Due to stronger crystal fields and smaller Hund’s coupling , even Hund’s first rule of maximal spin is violated in 4d and 5d systems. Since (the approximate criterion with eV,=0.5eV) is not satisfied, a low-spin ground state configurations are preferred. However, a crucial difference of 4d/5d systems relative to their 3d counterparts is the strong spin-orbit coupling.

To help understand the relevant physics, it is useful to briefly consider 4f systems where,

 He−e>HSOC>HCF,

since the spin-orbit coupling is greater than crystal fields, Hund’s third rule, takes precedence over lattice effects. Crystal field mixing of different -manifolds are dropped in a first approximation and crystal field effects are considered only within a given -manifold.

Returning to 5d systems, we have the following hierarchy:

 HCF≈He−e⪆HSOC.

In this scenario, which occurs mainly in 5d systems and is intermediate to 3d systems and 4f systems, all energy scales are comparable, with spin-orbit coupling smaller, but still the same order of magnitude as the others. None of the approximations used in 3d and 4f systems work in this regime. Therefore, in order to study this regime in detail we turn to an exact diagonalization study.

As mentioned in Sec.III, the off-diagonal elements of spin-orbit coupling mix the and states. In 5d systems spin-orbit coupling is an order of magnitude greater than 3d systems, and although crystal fields are larger as well, they remain of the same order of magnitude. Thus, the first order correction in perturbation theory of the wavefunction due to - mixing is of the order of . Therefore, it is not as small as in 3d systems and neglecting the states by using the T-P equivalence will result in more dramatic differences from the full - space of states.

## V Model and calculations

To study the mixing between and orbitals, we use a five-orbital model, taking in account all the -orbitals. Depending on the electron filling, we compare the five-orbital model with a three-orbital -only model, or to a two-orbital -only model. We compute various observables as a function of the mixing parameter (of and states), which is the bare spin-orbit coupling strength, . We do this for every electron filling, from one electron to nine electrons.

We model the electron-electron interaction with the Kanamori Hamiltonian,Maekawa et al. (2004)

 H(Kanamori)=U∑m^nm↑^nm↓+U′∑m≠m′^nm↑^nm′↓+(U′−JH)∑m

where is the electron annihilation(creation) operator, are associated with labels respectively, and . For the three orbital -only model and for the two orbital -only model . We assume that the relation is satisfied, which is a good approximation for many materials.Maekawa et al. (2004) We take eV in all calculations, leaving only one free parameter, the Hund’s coupling . For the five-orbital model, Eq.(20) is supplemented by , which is given in the Eq. (3). The full Hamiltonian we consider is then

 H=H(Kanamori)+HCF+HSOC, (21)

with 1-5. For the three-orbital -only model with , and for the two-orbital -only model with . Using exact diagonalization we will compare the results of the full Hamiltonian in Eq.(21) with the -only model and the -only model.

We calculate expectation values of different operators , where is the ground state of the many-electron system. We compute the expectation value of the total spin angular momentum , the total orbital angular momentum , the zero, the single, and the double occupancies of different orbitals defined byMatsuura and Miyake (2013)

 ^Zi≡1−ni↑−ni↓+ni↑ni↓, (22) ^Si≡ni↑+ni↓−2ni↑ni↓, (23) ^Di≡ni↑ni↓, (24)

where stands for the orbital index. The amplitudes of the spin, orbital, and total angular magnetic moments, respectively, are defined by , , and , where and are the z components of the spin and orbital angular momenta of the orbital respectively, and the effective spin-orbit interaction is

 ¯¯¯ζ=−1ζHSOC, (25) ¯¯¯ζt2g=−1ζHt2gSOC, (26) ¯¯¯ζt2g−eg=−1ζHt2g−egSOC, (27)

where is in units of .

We note that the effective spin-orbit coupling can be probed experimentally through X-ray absorption spectroscopy (XAS) measurements.Thole and van der Laan (1988a, b); van der Laan and Thole (1988) Core electrons from the occupied states and are excited to the unoccupied states and , respectively, since these are allowed from the selection rules . These absorption processes are referred to as the intensity peaks and , respectively. Van de Laan and TholeThole and van der Laan (1988a, b); van der Laan and Thole (1988) have shown that the ratio of the integrated intensities (area) of the peaks, [called the branching ratio (BR)] is directly related to the ground state expectation value of the spin-orbit coupling (which we call ), through the relation , where , and is the average number of holes in the unoccupied -states (including the full five orbitals).

When the spin-orbit coupling is zero, the =3/2 and =5/2 -states are degenerate (see right side of Fig. 3), and the ratio of the intensities is equal to the ratio of the occupied states and which is 2:1. This yields a branching ratio of . A deviation from this value is a clear indication of strong spin-orbit coupling, and can give information on the nature of the ground state.

Since the effective spin-orbit coupling is a local property of the ion, a single-site calculation is expected to capture the essential physics of the experimental measurements. In our exact diagonalization (ED) calculations, we place an infinitesimal magnetic field in the z-direction, of the order of eV, in order to lift the degeneracy of the ground state, and obtain a unique expression for the eigenvectors of the ground state. We have verified this small value does not numerically change the expectation values we compute.

## Vi Exact Diagonalization Results

### vi.1 Comparison of t2g-eg model with t2g only model

For electron filling from one to six electrons, we will compare the results of the full - model with the only model.

#### 1 electron

In the -only model, we have for the orbital angular momentum, and s=1/2. Thus, there is no magnetic moment M=-l+2s=0, since due to spin-orbit coupling, orbital angular momentum and spin angular momentum favor an antiparallel alignment. This is what we see in Fig.4(a). However, the quenching of the orbital angular momentum is overestimated in the -only model. As we see in the 5-orbital model (for which ), the restoration of orbital angular momentum due to spin-orbit coupling becomes significant. We compute the total magnetic moment for crystal field energy eV and find it is reduced as the crystal field splitting is increased. A significant moment remains, for example, for eV and eV.

As shown in Sec. III using perturbation theory for a single electron, the off diagonal - matrix elements of the spin-orbit coupling creates a small occupancy of -orbitals in the ground state. This is seen in Fig.4(b), with the single, zero, and double -occupancy of the -orbital, for three different crystal field energies eV (the single, zero, and double -occupancy of the -orbital are zero). As expected, the occupancies are reduced as the crystal field energy is increased, and they are increased as the spin-orbit coupling strength is increased. In Fig.4(c) we see for the only model , coming from in the ground state. In the 5-orbital model, by using Eq.(16) in calculating the extra contribution from of the off-diagonal matrix elements of matrix B in Eq.(15), we get , thus which gives the correct trend shown in Fig.4(c), explaining the missing part not captured from the -only model.

#### 2 electrons

In the -only model, for zero spin-orbit coupling () and . Thus, a non-zero magnetic moment is achieved. However, for the 5-orbital model gives because the crystal field mixes different terms (with the same as the -only model, following Hund’s first rule) as discussed in Sec. IV.2. At one has the same total magnetic moment as with the -only model, .

However, when the spin-orbit coupling is turned on, and , so the magnetic moment abruptly plunges to zero, consistent with the approximate rule , , . In Fig.5(a) we see for the -only model with eV the magnetic moment is reduced as the spin-orbit coupling is increased. This can be understood as a competition with the Hund’s coupling aligning the spins of the electrons, while the spin-orbit coupling “unaligns” them as it tries to align the spin with the orbital motion. Thus, for eV where Hund’s coupling is stronger, the effect of the spin-orbit coupling is weaker.

In Fig.5(b) we see the spin quantum number , for eV for the -only and for the 5-orbital model as a function of the spin-orbit coupling. We see that for the smaller Hund’s coupling the reduction of the spin is greater, due to the same explanation given for the magnetic moment. The two models match for small spin-orbit coupling, but for eV a deviation between them appears for eV. In Fig. 5(c) we see the single, zero and double occupancy per orbital, for crystal field energy eV and eV is increased as the spin-orbit coupling is increased. While the curves are similar to the one-electron case, the total result is roughly doubled since it is per -orbital.

In Fig.5(d) the effective spin-orbit coupling is shown for eV for the -only model and for the 5-orbital model. As the Hund’s coupling is increased, the effective spin-orbit coupling is decreased. As the crystal field is increased, the results from the two models approach each other. However, is quite robust even for eV, eV, and eV where the -only model gives and the 5-orbital model gives .

We can understand these results qualitatively using a single particle analysis. By taking the ground state to be a tensor product of the single-particle eigenstates given in Sec. III for the -only model and the 5-orbital model, we get for two electrons, . The weaker the electronic correlations (i.e. eV), the closer one gets to this single electron result. Using this result for the -only model gives and the 5-orbital model gives an extra contribution , which for reasonable values in the 5d elements (i.e eV), gives for the 5-orbital model close to what is observed in Fig.5(d). We also see that the two models match at . Thus, for 3d systems the T-P equivalence is a good approximation even for the most dramatically different expectation value, the effective spin-orbit coupling.

#### 3 electrons

For zero spin-orbit coupling for the -only model we have , and , while for the 5-orbital model and , as predicted from Hund’s first rule for maximal spin. With this in mind, we turn our attention first to the total magnetic moment, which we expect to reduce with increasing spin-orbit coupling because the spin-orbit coupling tends to “unalign” the spins. This will be true for both models. However, comparing our results for the total magnetic moment with Ref. [Matsuura and Miyake, 2013] where a -only model was used, we find a significant difference using a 5-orbital model, as seen in Fig.6(a). Thus, the quenching of orbital angular momentum is underestimated in the -only model. There is an increased and decreased in the 5-orbital model compared to the -only model. When () the magnetic moment is reduced rapidly with spin-orbit coupling. For , when becomes greater than () spin-orbit coupling overcomes the aligning of the spins caused from Hund’s coupling. For eV there is a transition at eV, and for eV at eV. The transitions can be seen from the discontinuity in the occupancies where some small electron occupancy is transferred from one orbital to the other (the average -occupancy remains constant). There is also some transfer of double occupancy from two orbitals to the third one, where the average -occupancy remains constant as well.

As one increases the spin-orbit coupling strength, the total spin is more affected compared to the two-electron system, because it is tightly connected to the orbital angular momentum. The of the and 5-orbital models begin to deviate with increasing strength of the spin-orbit coupling, as seen in the Fig.9(b). For small Hund’s coupling this deviation is small, and for larger Hund’s coupling this deviation is larger.

For the effective spin-orbit coupling, there is a more dramatic difference between the two models compared to the two-electron system, where for eV and eV we have =1.5 for the -only model, while for the 5-orbital model =2.8. Using a single particle analysis similar to that of two-electron filling, we get , which is very close to what we observe in Fig.6(e) for eV, while for eV a significant decrease occurs in the effective spin-orbit coupling.

#### 4 electrons

For four electrons the total magnetic moment is zero in both models: . In the -only model, and as indicated from the law of the T-P equivalence. In the five-orbital model there is a low-spin to high-spin transition. For eV at zero spin-orbit coupling and