Excitonic Magnetism at the intersection of Spin-orbit coupling and crystal-field splitting

Excitonic magnetism involving superpositions of singlet and triplet states is expected to arise for two holes in strongly correlated and spin-orbit coupled $t_{2g}$ orbitals. However, uncontested material examples for its realization are rare. Applying the Variational Cluster Approach to the square lattice, we find conventional spin antiferromagnetism combined with orbital order at weak and excitonic order at strong spin-orbit coupling. We address the specific example of Ca$_2$RuO$_4$ using ab-initio modeling and conclude it to realize excitonic magnetism despite its pronounced orbital polarization.

Introduction.Strong correlations in transition-metal oxides have for decades been a focus of intense scientific activity.While the prominent 3d metals, e.g.iron, nickel and copper, are usually understood to have rather weak spin-orbit coupling (SOC), the interplay of strong SOC and correlations has attracted attention more recently.One important driver of this interest is the search for topologically nontrivial phases like topological Mott insulators [1] or spin liquids.In particular, honeycomb-lattice compounds with a single hole in strongly correlated t 2g orbitals of 4d (e.g.Ru) or 5d (e.g.Ir) elements are candidates for 'Kitaev' spin liquids [2] driven by distinct magnetic anisotropies [3,4], which has led to extensive research efforts [5,6].
The impact of SOC is particularly transparent for a t 2g electron, where it couples effective orbital angular momentum L = 1 and spin S = 1  2 to total angular momentum J = 3  2 or J = 1 2 [7].If sufficiently large, this leads to a band splitting and the reduced width of the resulting subbands effectively enhances correlations.The single-layer square-lattice compound Sr 2 IrO 4 has been established as a prime example for such spin-orbit driven Mott physics [8].With five t 2g electrons, its J = 3  2 levels are completely and the J = 1  2 level is half filled.The resulting effective single-band description focusing on the J = 1  2 states is even robust enough to survive doping, where Fermi-arcs [9] and hints of superconductivity [10] have been reported, as well as photo-doping [11].
While a description in terms of J = 1  2 is thus well established for a single hole in the t 2g levels of Ru or Ir and forms the basis for the proposed spin-liquid scenario, the situation for two holes is less clear.Doubleperovskite iridates [12,13] likely realize a nonmagnetic singlet groundstate and might already be close to the j-jcoupling regime with a doubly occupied J = 1  2 band [14].Theoretical predictions for substantial but not extreme spin-orbit coupling, where L-S coupling is more appropriate, argue that it combines effective angular momentum L = 1 with total spin S = 1 of the two holes into an ionic ground state with J = 0 [15,16], see the left-hand side of Fig. 1(a).Superexchange then determines dynamics of J = 1 excitations as well as their potential condensation into excitonic magnetic order.This unconventional type of magnetism has been predicted to host a bosonic Kitaev-Heisenberg model and topologically nontrivial excitations [17,18], or to mediate triplet superconductivity [19].Ca 2 RuO 4 is a promising candidate for such a scenario, where superexchange mixes J = 0 and J = 1 states so that the superposition acquires a magnetic moment that can order, see the schematic illustration in Fig. 1(c).Features like xy-symmetry breaking (rather than Ising or Heisenberg) can be explained by such an excitonic arXiv:1910.13977v2[cond-mat.str-el]15 Nov 2019 model [20,21].However, a structural phase transition is well established [22,23], which leads to a crystal field (CF) lowering the energy of the xy orbital as shown on the right-hand side of Fig. 1(a).Based on ab initio calculations [24,25], the two holes are distributed mostly in xz and yz orbitals, suggesting more conventional S = 1 spin order, see the sketch of Fig. 1(d).SOC would then be a correction rather than a main driver [24,26].
In this Letter, we apply the variational cluster approach (VCA) to a spin-orbit coupled and correlated t 2g three-orbital model in order to address the competition between SOC and CF as illustrated in Fig. 1.We find that without CF splitting, SOC ζ suppresses orbital order, so that the complex spin and orbital pattern of Fig. 1(b) is replaced by the excitonic antiferromagnetic (AFM) state Fig. 1(c).In the presence of CF splitting, its impact will be seen to be more gradual, but it nevertheless induces a transition from spin order, as in Fig. 1(d), to excitonic order.In order to connect to Ca 2 RuO 4 , we derive a realistic model using density-functional theory and subsequent projection to Wannier states.The approach is validated by the magnetic excitation spectrum of a spin-orbital model based on the same parameters, which reproduces neutron-scattering data, and leads to the conclusion that the compound falls into the excitonic regime despite its strong orbital polarization.
Model and Methods.
We discuss four electrons (two holes) in a three-orbital system modeling t 2g orbitals on a square lattice.The kinetic energy Tetragonal CF splitting ∆ > 0 originates from the strained octahedral coordination of the low-temperature phase of Ca 2 RuO 4 and t o from the orthorhombic distortion.
The SOC operator for t 2g orbitals can be written as [27,28] where ε αβγ is the totally antisymmetric Levi-Civita tensor and τ α with α = x, y, z are Pauli matrices.
In addition to these one-particle terms, we use the symmetric Kanamori form of the effective onsite Coulomb interaction [29] with intraorbital Hubbard U , interorbital U ′ , and Hund's coupling J H which are related by U ′ = U − 2J H . Unless specified differently, we fix U = 12.5t and J H = 2.5t, which is consistent with their order of magnitude in Ca 2 RuO 4 [25].
In order to address long-range order and symmetry breaking, we use the VCA [30], which has already been established to explain angle-resolved photo emission for strongly spin-orbit coupled Sr 2 IrO 4 [31].The self energy of a 2 × 2-sites cluster, with three t 2g orbitals per site, is calculated using exact diagonalization (ED) and then inserted into the one-particle Green's function of the thermodynamic limit to obtain the grand potential.According to self-energy functional theory [32], the thermodynamic grand potential Ω[Σ τ ′ ] can be optimized by varying the one-particle parameters τ ′ used to obtain the cluster self energy Σ τ ′ .In particular, the self energy is optimized w.r.t.fictitious symmetry-breaking fields, e.g. a staggered magnetic field inducing AFM order, see supplemental material [33] for details.It should be noted that these fields only act on the reference system, they are not included in the thermodynamic-limit Green's functions.If, however, a symmetry-broken self energy optimizes the grand potential of the fully symmetric Hamiltonian, one can infer spontaneous symmetry breaking.
Excitonic AFM ordering without CF.Strong onsite repulsion and Hund's coupling ensure that exactly one orbital at each site is doubly occupied, which can be described with an effective angular momentum L = 1.Together with the S = 1 formed by the two half-filled orbitals, this gives a nine-fold degeneracy, see Fig. 1(a).Without either CF or spin-orbit coupling, the kinetic energy (here taken as isotropic NN hopping t = t NN yz xz xy , t NNN xy = t o = 0) resolves this degeneracy into a complex pattern with orbital and spin order having orthogonal ordering vectors (0, π) and (π, 0), see Fig. 1(b).Such a state is consistent with Goodenough-Kanamori rules, has also been found by ED [34], and remains stable for ζ ≲ 0.4t.The only role of SOC is here a slight preference for S x over S z .At ζ ≈ 0.4t, the magnetic ordering vector switches to (π, π) and orbital densities become more equal as orbital order is lost, see Fig. 2. At the same time, total onsite angular momentum J becomes more relevant, as can be  inferred from the weight in eigenstates J i , J z i ⟩, where we used the ED ground state φ 0 ⟩ for optimized parameters and averaged over sites i.The J = 0 state quickly gains weight with the onset of checkerboard AFM order, as similarly seen using dynamical mean-field theory [35].In fact, it almost entirely describes the paramagnetic (PM) state already for ζ ≈ 0.6t, see the grey symbols in Fig. 2(a).Magnetic ordering increases weight in J = 1 (see black symbols), exactly as expected for excitonic magnetism, while the J = 2 are nearly irrelevant in the entire checkerboard regime.The small orbital polarization n xy > n xz , n yz , see Fig. 2(b) is due to the layered geometry and favors an ordered moment along z rather than in plane.The onset of checkerboard order together with the shift to the J = 0 state suggests that ordering vector (π, π) is favored as soon as ζ sufficiently lifts orbital degeneracy.
Competition of SOC and CF.Orbital degeneracy can alternatively be broken by a CF ∆ favoring the xy orbital so that the two holes occupy xz and yz orbitals, form a total spin 1 and checkerboard AFM order, see the cartoon of Fig. 1(d).6) found in states with total onsite angular momentum J = 0, 1, 2, and (c) shows analogous weights in eigenstates of (S13), i.e., in the levels S0, D 1 2 , and S3 of Fig. 1(a).Grey/black symbols are results obtained without/with AFM order.Background color refers to the optimal order parameter given below the figure, which always lies in the x-y plane, see [33], except for the isotropic limit ζ = 0. Parameters as in Fig. 2 except that ∆ = 1.5t.Orange/red symbols refer to the parameters used to model Ca2RuO4, see text, and the state without/with AFM symmetry breaking, the latter prefers moments along b = 1 √ 2 (x + y) direction.See supplemental material for corresponding grand potentials [33].
of AFM order at ζ ≳ 1.5t already indicates a considerable role for SOC even in this orbitally polarized regime.
While weight ( 6) is moved from J = 1 and J = 2 states towards J = 0 with growing ζ, Fig. 3(a) shows that the J = 2 states remain populated throughout.Nevertheless, total angular momentum shows signatures of excitonic magnetism for ζ ≳ 0.6t: when comparing the grey symbols obtained in the PM phase to the black ones of the AFM state, one notices a weight transfer from J = 0 to J = 1, while weight in J = 2 is hardly affected.The J = 2 states thus carry finite weight, and contribute to orbital polarization, but are not involved in the magnetic ordering.
A clearer picture emerges when weight in eigenstates of J is replaced by weight in eigenstates of the effective onsite Hamiltonian where S and L give the total spin and orbital angular momentum of the two holes, both of length 1, which are coupled by λ = ζ 2 , and where CF ∆ favors the state L z = 0 with doubly occupied xy orbital.The level structure of (S13) is shown in Fig. 1(a).Figure 3(b) shows that in the whole parameter regime, only the three lowest states S 0 and D 1 2 are relevant and fully capture a continuous transition from a spin-one via an excitonic magnet to the PM.For moderate and strong ζ ≳ 0.7t, magnetic ordering can be described in perfect analogy to Fig. 2(a), i.e., using the excitonic picture of weight transfer from the singlet groundstate S 0 to the doublet D 1 2 .
We now address the case of Ca 2 RuO 4 by obtaining realistic hopping parameters from density-functional theory and projection onto Wannier states, see [36] for details.Neglecting very small hoppings, we find t NN xy = 0.2 eV, t NN xz = t NN yz = 0.137 eV, t NNN xy = 0.1 eV and t o ≈ 0.09 eV, as well as CF ∆ ≈ 0.25 eV.Recent X-ray scattering experiments have concluded that U ≈ 2 eV and J H ≈ 0.34 eV and that oxygen covalency reduces SOC from its free-ion value ζ ≈ 0.16 eV to ζ ≈ 0.13 eV [14].Using these parameters in the VCA, we find the magnetic moment to prefer the x-y direction, as in experiment.Orbital occupations, with n h xy ≈ 0.2, are very similar to those found matching values of ζ t xz , see Fig. 3(a) [37].Figure 3(c) also shows that nearly all weight is captured by the onsite singlet S 0 and doublet D 1 2 and Fig. 3(b) reveals that magnetic ordering transfers weight from J = 0 to J = 1, supporting an excitonic-magnetism description.
Magnetic excitations in Ca 2 RuO 4 .Unfortunately, while the VCA provides one-particle spectral functions, two-particle quantities like the dynamic magnetic structure factor where are not readily available.We thus resort to ED on a small cluster.In order to go beyond the 2×2 cluster, we restrict the Hilbert space to the low-energy states of Fig. 1, i.e., we keep only states with exactly two holes per site and L = 1 and S = 1.The effective Hamiltonian then consists of Eq. (S13) as well as superexchange obtained in second-order perturbation theory of the kinetic energy, which is given in the supplemental material [33].
Figure 4(a) shows the spectrum (8) obtained for a tetragonally symmetric variant of the parameters derived from density-functional theory (i.e.neglecting interorbital hopping t o = 0 for technical reasons).The spectrum agrees well with neutron-scattering data for Ca 2 RuO 4 [20,26] and reproduces the salient feature of a maximum at (0, 0) and ω ≈ 50 meV.We then introduce magnetic anisotropy δ(S b ) 2 = δ 1 2 (S x − S y ) 2 , with S a b = 1 √ 2 (S x ± S y ) and δ = −10 meV, to mimic the impact of t o , see above, and to reproduce the experimentally observed gap of ω ≈ 15 meV at (π, π) [38].The resulting spectrum is given in Fig. 4(b) and while excitation energies are slightly too large by ≈ 10 %, it nevertheless agrees well with the data reported in [20]: In addition to the out-of-plane transverse mode M c , the broken inplane symmetry splits the amplitude mode M b off from the in-plane transverse mode M a .
Conclusions.We have used the VCA to investigate excitonic magnetism stabilized by the interplay of superexchange and SOC in strongly correlated t 2g orbitals.Without CF splitting, the main role of weak to moderate SOC is to reduce orbital degeneracy, which in turn changes the magnetic ordering vector from (0, π) to (π, π).A similar transition from a state with complex orbital order to an excitonic antiferromagnet has been found in one dimension [39].When orbital degeneracy is from the outset reduced by a CF, SOC drives a transition from an orbitally polarized S = 1 AFM state, where the excitonic picture is not applicable, via excitonic order coexisting with strong orbital polarization, to a paramagnet where SOC suppresses superexchange.
As an example, we focus on Ca 2 RuO 4 , which we model using hoppings and CF strength derived from densityfunctional theory [36] as well as interaction and SOC parameters inferred from experiment [14].Magnetic excitations obtained using ED for the corresponding superexchange model are found to closely match neutronscattering data, in particular showing the tell-tale maximum at momentum (0, 0), and thus validate our microscopic model.Strong orbital polarization might suggest a description as an orbitally polarized spin-one system, in line with the interpretation of angle-resolved photonemission-spectroscopy data [25,40].However, magnetic ordering transfers weight from a singlet onsite ground state S 0 to doublet D 1 2 , as expected for an excitonic magnet.SOC and the remaining small xy-hole density thus play a decisive role in the AFM state of Ca 2 RuO 4 , and we conclude Ca 2 RuO 4 to realize excitonic magnetism despite its pronounced orbital polarization.
Supplemental Material to "Excitonic Magnetism at the intersection of Spin-orbit coupling and crystal-field splitting" Here, we present (i) details of the VCA calculations, e.g., the grand potentials obtained in the various ordered states and (ii) the Kugel-Khomskii-type Hamiltonian obtained in second-order perturbation theory and used to calculate excitations spectra.

DETAILS OF THE VCA CALCULATIONS
We tried a variety of potential ordering parameters, which can generally be expressed as one-particle terms of the form where Λ i can be any one-particle operator acting on site i at position R i , e.g.spin S α i along direction α, total angular momentum S α i +L α i , magnetization 2S α i −L α i or any other linear combination of S and L, as well as orbital density n i,β in orbital β.Ordering vectors Q accessible to our cluster are (0, 0), (π, π), (π, 0) and (0, π).We also included superpositions of two or three order parameters, e.g., orbital and magnetic order, with identical or different (e.g. in the stripy phase) ordering vectors.
In addition to these fictitious fields, we had two more variational parameters, namely the physical chemical potential µ and the fictitious µ ′ .The chemical potential µ should ensure a particle density of 2 holes (4 electrons) per site and translates to the requirement The term −µN target with N target = 2 (in hole notation) comes in addition to the chemical-potential term contained in the Hamiltonian, where the sum runs over over site i, orbital α and spin σ.However, the overall density ⟨N ⟩ µ * obtained for the saddle-point chemical potential µ * does not necessarily fulfill ⟨N ⟩ µ * = N target , and we indeed found it to differ slightly.To ensure thermodynamic consistency [S2], we thus additionally optimized the fictitious chemical potential µ ′ .Like the symmetrybreaking fields h ′ , it acts only on the reference system.Fortunately, the robust Mott gap present in all our calculations ensured a very weak dependence of µ and µ ′ on the various h ′ , so that we fixed µ and µ ′ to the values found for h ′ = 0 and then kept them constant during optimization of the h ′ .
Figure S1 shows the VCA grand potential Ω obtained for the simplest model with varying spin-orbit coupling and t NN xy = t NN xz = t NN yz = t, t NNN xy = t o = 0, ∆ = 0, U = 12.5 t, and J H = 2.5 t.The ground-state ordering, which was subsequently used to obtain Fig. 2 of the main text, is the one giving the lowest Ω.Note that the z-component of angular momentum is here clearly preferred over x, but that differences between various linear combinations of S and L can be small -the crucial aspect of the order parameter is the symmetry broken rather than the exact form of te order parameter.
Similarly, Fig. S2 gives the grand potential in various candidate states for the model with NN hopping and crystal field ∆ = 1.5t, i.e., the corresponding to Fig. 3 of the main text.Again, various linear combinations of S and L can have quite similar grand potentials as long as the ordered moment lies in the x-y plane, while outof-plane moment S z is clearly disfavored by spin-orbit coupling.The dashed line is a guide to the eye connecting the lowest-Ω values, whose color is used for the back ground.

PERTURBATION THEORY AND KUGEL-KHOMSKII-TYPE MODEL
In the low-energy manifold discussed in Fig. 1 of the main text, the two holes on each site reside in two different orbitals, i.e. we conveniently label the site by the remaining doubly occupied orbital, which gives us the effective orbital angular momentum L = 1.Additionally, Hund's-rule coupling enforces a total spin S = 1, leading to a total of nine low-energy states.The VCA results show that these nine states indeed capture nearly all the weight of the cluster ground state, which justifies projecting out charge excitations as well as states with onsite spin S = 0.The restricted Hilbert space then allows us to reach 8 rather than 4 site in exact diagonalization.
The effective Hamiltonian acting on this low-energy manifold is obtained by treating hopping t in secondorder perturbation theory, with the large energy scale being onsite Coulomb repulsion of creating a charge excitation.We give here the various terms of the effective Kugel-Khomskii-type Hamiltonian, restricting ourselves to orbital-conserving hoppings for simplicity.The first possible process is a spin-spin coupling H S⋅S = J i,j (S i S j − 1) ⊗ T i ; T j ⟩⟨T i ; T j (S4) that leaves orbital occupations T i and T j unchanged.Its coupling strength depends on whether the same orbital γ is doubly occupied on both sites, or whether it is orbital α on one site and orbital β on the other, i.e., and accordingly

(S7)
The first of these would be the only superexchange term surviving in a strongly orbitally polarized case for crystal field ∆ ≫ ζ.Additional terms involve the orbital degree of freedom.Again discussing first the case of identical orbital character on both sites,finite Hund's-rule coupling J H > 0 (or rather the associated pair-hopping term) allows the doubly occupied orbital to change on both sites simultaneously, as long as not all spins are parallel, yielding a 'pair-flip' term H p = (S i S j − 1) α≠β −t α t β J H U (U + 2J H ) α; α⟩⟨β; β .(S8)

FIG. 1 .
FIG. 1.(a) Level structure starting from nine-fold degeneracy (far left and right) and interpolating between spin-orbit coupled (left, α = 0) and crystal field (CF) split (right, α = 1) regimes.(b) Stripy state arising out of nine-fold degeneracy via Goodenough-Kanamori rules.(c) Excitonic magnetism for dominant SOC, one of the J = 1 states condenses to yield a magnetic moment.(d) Orbitally ordered state with doubly occupied xy orbital, where holes in xz and yz form S = 1.In (b) and (d), the doubly occupied orbital is drawn.
) annihilates (creates) an electron with spin σ =↑, ↓ in orbital α = xy, xz, yz and momentum k, has non-zero contributions xy,xy (k) = −2t NN xy (cos k x + cos k y ) − 4t NNN xy cos k x cos k y − ∆ , Figure 3 explores the transition between such a spin-AFM and the excitonic AFM state discussed above.Setting CF to ∆ = 1.5t and starting from ζ = 0 with full orbital polarization n h xy = 0, hole density n h xy in xy increases with SOC, see Fig. 3(c), but remains rather low with n h xy ≲ 0.4 vs. n h xz yz ≳ 0.8.Nevertheless, the loss

Ca 2 FIG. 3 .
FIG. 3. Spin and spin-orbital order in the presence of a CF.(a) gives orbitally resolved hole occupation numbers n h yz xz xy , (b) weights (6) found in states with total onsite angular momentum J = 0, 1, 2, and (c) shows analogous weights in eigenstates of (S13), i.e., in the levels S0, D 1 2 , and S3 of Fig.1(a).Grey/black symbols are results obtained without/with AFM order.Background color refers to the optimal order parameter given below the figure, which always lies in the x-y plane, see [33], except for the isotropic limit ζ = 0. Parameters as in Fig.2except that ∆ = 1.5t.Orange/red symbols refer to the parameters used to model Ca2RuO4, see text, and the state without/with AFM symmetry breaking, the latter prefers moments along b = 1

FIG. S1 .
FIG.S1.Grand canonical potentials obtained with VCA for various order parameters, corresponding to Fig.2of the main text.'AFM' refers to checkerboard pattern, i.e.Q = (π, π) in Eq. (S1), 'stripy' to Q = (0, π), and 'FO' to ferrorobital order [Q = (0, 0)] favoring the xy orbital, 'TO' is the complex threeorbital-order of Ref.[S1].'Stripy orb.' denotes the pattern of Fig.1(a) of the main text, with Q orb = (π, 0) orthogonal to Qspin = (0, π) of the concurrent stripy spin pattern.Magnetic operators are x and z components of spin S, orbital angular momentum L, magnetization M = 2S − L, total angular momentum J = S + L, as well as linear combination 2S + L. The dashed line is a guide to the eye connecting the lowest-Ω values, whose color is used for the back ground.
FIG.S2.Grand canonical potentials obtained with VCA corresponding as in Fig.S1, but corresponding to Fig.3of the main text, i.e., with crystal field ∆ = 1.5t.All magnetic ordering vectors are Q orb = (π, π) and the substantial CF suppresses any orbital order, so that corresponding Weiss fields are all 0.
Black symbols indicate the same observables, but obtained for the ordered state, i.e. with optimized symmetry-breaking cluster parameters.Background color labels the ordered phase yielding the lowest grand potential: orange -stripy [see Fig.1(b), but with in-plane spins], blue -checkerboard AFM order with out-ofplane moment (light blue: Sz plus a small xy polarization; dark blue: 2Sz + Lz), and grey -PM.See supplemental material for corresponding grand potentials[33].Parameters are t NN xy = t NN xz = t NN yz = t, ∆ = 0, U = 12.5t and J H = 2.5t.