We present a novel approach to study the ground state and elementary excitations in compounds where spins and orbitals are entangled by on-site relativistic spin-orbit interaction. The appropriate degrees of freedom are localized states with an effective angular momentum J. We generalize J to arbitrary large values while maintaining the delicate spin-orbital entanglement. After projecting the inter-site exchange interaction to the manifold of effective spins, a systematic 1/J expansion of the effective Hamiltonian is realized using the Holstein-Primakoff transformation. Applications to representative compounds Sr2IrO4 and particularly vanadium spinels AV2O4 are discussed.
Dynamical Jahn-Teller (DJT) effect in a spin-orbital coupled system on a honeycomb lattice is examined, motivated from recently observed spin-liquid behavior in Ba$_3$CuSb$_2$O$_9$. An effective vibronic Hamiltonian, where the superexchange interaction and the DJT effect are taken into account, is derived. We find that the DJT effect induces a spin-orbital resonant state where local spin-singlet states and parallel orbital configurations are entangled with each other. This spin-orbital resonant state is realized in between an orbital ordered state, where spin-singlet pairs are localized, and a…
We extend an approximation earlier developed by us for the single-impurity Anderson model to a full-size impurity solver for models of interacting electrons with multiple orbitals. The approximation is based on parquet equations simplified by separating small and large energy fluctuations justified in the critical region of a pole in the two-particle vertex. We show that an $l$-orbital model with most general interaction is described within this approximation by $4l^{2}\times 4l^{2}$ matrices and is Fermi liquid in the metallic phase. We explicitly calculate properties of a paramagnetic soluti…
We revisit the formulations and simulations of angular distributions in polarization-dependent core-level photoemission spectra of strongly correlated electron systems, in order to explain the recently discovered linear dichroism (LD) in the core-level photoemission of 4f-based rare-earth compounds. Owing to the selection rules for the optical process of core-level excitations, the LD originating from the anisotropic outer localized charge distributions determined by the ground-state orbital symmetry can be observed. Our simulations show that core d-level excitations are essential for the LD i…
One of the most important properties of topological insulators (TIs) is the helical spin texture of the Dirac surface states, which has been theoretically and experimentally argued to be left-handed helical above the Dirac point and right handed helical below. However, since the spin is not a good quantum number in these strongly spin-orbit coupled systems, this can not be a complete statement, and we must consider the total angular momentum J = L + S that is a contribution of the spin and orbital terms. Using a combination of orbital and spin-resolved angle-resolved photoemission spectroscopy…
We present a microscopic magnetic model for the spin-liquid candidate volborthite Cu3V2O7(OH)2(H2O)2. The essentials of this DFT-based model are (i) the orbital ordering of Cu(1) 3d 3z2-r2 and Cu(2) 3d 3x2-y2, (ii) three relevant couplings J_ic, J_1 and J_2, (iii) the ferromagnetic nature of J_1 and (iv) frustration governed by the next-nearest-neighbor exchange interaction J_2. Our model implies magnetism of frustrated coupled chains in contrast to the previously proposed anisotropic kagome model. Exact diagonalization studies reveal agreement with experiments.
Recent angle resolved photoemission spectroscopy measurements on strong spin-orbit coupled materials have shown an in-plane orbital texture switch at their respective Dirac points, regardless of whether they are topological insulators or "trivial" Rashba materials. This feature has also been demonstrated in a few materials ($\text{Bi}_2\text{Se}_3$, $\text{Bi}_2\text{Te}_3$, and $\text{BiTeI}$) though DFT calculations. Here we present a minimal orbital-derived tight binding model to calculate the electron wave-function in a two-dimensional crystal lattice. We show that the orbital components o…
We report on an extended fluctuation regime in the spin dimer system Sr3Cr2O8 based on anomalies in Raman active phonons and magnetic scattering. The compound has two characteristic temperatures, TS = 275 K, related to a Jahn-Teller transition with structural distortions and orbital ordering and a second, T*= 150 K, which is due to further changes in the orbital sector. Below TS quasielastic scattering marks strong fluctuations and in addition phonon anomalies are observed. For temperatures below T* we observe an exponential decrease of one phonon linewidth and determine a gap of the orbital e…
By this Comment we would like to point out [2] that, though such consideration can be quite sophisticated, the consideration of a spin-pseudo-spin Hamiltonian has nothing in common with the reality. It means that such the consideration are useless as far as the properties of real systems are concerned.
A new scheme of the iterative perturbation theory is proposed for the strongly correlated electron systems with orbital degeneracy. The method is based on the modified self-energy of Yeyati, et al. which interpolates between the weak and the strong correlation limits, but a much simpler scheme is proposed which is useful in the case of the strong correlation with orbital degeneracy. It will be also useful in the study of the electronic structures combined with the band calculations.
A unified approach is presented for investigating coupled spin-orbital fluctuations within a realistic three-orbital model for strongly spin-orbit coupled systems with electron fillings $n=3,4,5$ in the $t_{2g}$ sector of $d_{yz},d_{xz},d_{xy}$ orbitals. A generalized fluctuation propagator is constructed which is consistent with the generalized self-consistent Hartree-Fock approximation where all Coulomb interaction contributions involving orbital diagonal and off-diagonal spin and charge condensates are included. Besides the low-energy magnon, intermediate-energy orbiton and spin-orbiton, an…
The correlated motion of electrons in the presence of strong orbital fluctuations and correlations is investigated with respect to magnetic couplings and excitations in an orbitally degenerate ferromagnet. Introduction of the orbital degree of freedom results in a class of diagrams representing spin-orbital coupling which become particularly important near the orbital ordering instability. Low-energy staggered orbital fluctuation modes, particularly with momentum near (π/2,π/2,0) (corresponding to CE-type orbital correlations), are shown to generically yield strong intrinsically non-Heisenberg…
Based on the generalized valence bond picture, a Schwinger boson mean field theory is applied to the symmetric SU(4) spin-orbital systems. For a two-leg SU(4) ladder, the ground state is a spin-orbital liquid with a finite energy gap, in good agreement with recent numerical calculations. In two-dimensional square and triangle lattices, the SU(4) Schwinger bosons condense at (π/2,π/2) and (π/3,π/3), respectively. Spin, orbital, and coupled spin-orbital static susceptibilities become singular at the wave vectors, twice of which the bose condensation arises at. It is also demonstrated that there …
Doping the distorted-perovskite Mott insulators LaTiO$_3$ and GdTiO$_3$ with a single SrO layer along the [001] direction gives rise to a rich correlated electronic structure. A realistic superlattice study by means of the charge self-consistent combination of density functional theory with dynamical mean-field theory reveals layer- and temperature-dependent multi-orbital metal-insulator transitions. An orbital-selective metallic layer at the interface dissolves via an orbital-polarized doped-Mott state into an orbital-ordered insulating regime beyond the two conducting TiO$_2$ layers. We find…
Theory of dilution effect in orbital ordered system is presented. The $e_g$ orbital model without spin degree of freedom and the spin-orbital coupled model in a three-dimensional simple-cubic lattice are analyzed by the Monte-Carlo simulation and the cluster expansion method. In the $e_g$ orbital model without spin degree of freedom, reduction of the orbital ordering temperature due to dilution is steeper than that in the dilute magnet. This is attributed to a modification of the orbital wave-function around vacant sites. In the spin-orbital coupled model, it is found that magnetic structure i…
Spectra of one-electron and collective excitations in narrow-band ferromagnets with unquenched orbital moments are calculated in various theoretical models. The interaction of spin and orbital excitations with conduction electrons results in the damping of the former which, however, turns out to be rather small; therefore, apart from usual spin waves, well-defined orbitons can exist. Non-quasiparticle states occur in the electron energy spectrum near the Fermi energy due to this interaction. The criteria of stability of the saturated spin and orbital ferromagnetic ordering are considered. Poss…
We study the Kondo effect of a two-orbital vertical quantum dot (QD) coupled to two ferromagnetic leads by employing an equation of motion method. When the ferromagnetic leads are coupled with parallel spin polarization, we find three peaks in the single-particle excitation spectra. The middle one is the Kondo resonance caused by the orbital degrees of freedom. In magnetic fields, the Kondo effect vanishes. However, at a certain magnetic field new two-fold degenerate states arise and the Kondo effect emerges there. In contrast, when the ferromagnetic leads are coupled with antiparallel spin po…
We study the interplay of spin, orbital and lattice degrees of freedom in a one-dimensional Kugel-Khomskii model coupled to phonons. In the vicinity of the dimer point we analyze the excitation spectrum, mapping the spin and orbital degrees of freedom to bond operators. In particular we study the renormalization of the phonon propagator due to coupling to the orbital and magnetic excitations. Considering both, ferro- and antiferro-orbital ordering we will show that the appearance of orbiton shake-up processes in the phonon spectrum is sensitive to the type of Jahn-Teller distortion. The releva…
The local structure of NaTiSi$_{2}$O$_{6}$ is examined across its Ti-dimerization orbital-assisted Peierls transition at 210 K. An atomic pair distribution function approach evidences local symmetry breaking preexisting far above the transition. The analysis unravels that on warming the dimers evolve into a short range orbital degeneracy lifted (ODL) state of dual orbital character, persisting up to at least 490 K. The ODL state is correlated over the length scale spanning $\sim$6 sites of the Ti zigzag chains. Results imply that the ODL phenomenology extends to strongly correlated electron sy…
We report results of the study of the recently discovered magnetic resonance in the orbitally ordered phase of CeB6 (the orbital ordering resonance) in a wide frequency range 44-360 GHz. It is found that the g-factor for this resonance increases with frequency from g(44 GHz)~1.55 to g(>250 GHz)~1.7. In addition to the orbital ordering resonance for the frequencies exceeding 200 GHz a new magnetic resonance with the g-factor 1.2-1.3 is detected.
Soft and hard x-ray orbital-resolved photoemission study of a strongly correlated Cd-Ce quasicrystal approximant
Julius-Maximilians-Universität Würzburg（维尔茨堡大学）；The University of Osaka（大阪大学）；RIKEN SPring-8 Center（理化学研究所SPring-8中心）；Institute for Open and Transdisciplinary Research Initiatives, The University of Osaka（大阪大学开放与跨学科研究倡议研究院）
Orbital-lattice coupled excitation dynamics in orbitally degenerate correlated systems are examined. We present a theoretical framework, where both local vibronic excitations and superexchange-type inter-site interactions are dealt with on an equal footing. We generalize the spin-wave approximation so as to take local vibronic states into account. Present method is valid from weak to strong Jahn-Teller coupling magnitudes. Two characteristic excitation modes coexist; a low-energy dispersive mode and high-energy multi-peak mode. These are identified as a collective vibronic mode, and Flanck-Con…
The interplay of charge, spin, lattice, and orbital degrees of freedom leads to a wide range of emergent phenomena in strongly correlated systems. In heterobilayer transition metal dichalcogenide moiré systems, recent observations of Mott insulators and generalized Wigner crystals are well described by triangular lattice single-orbital Hubbard models based on K-valley derived moiré bands. Richer phase diagrams, mapped onto multi-orbital Hubbard models, are possible with hexagonal lattices in $Γ$-valley derived moiré bands and additional layer degrees of freedom. Here we report the tunable inte…
We study the competing phases and the phase transition phenomena in an effective spin-orbital coupled model derived for pyroxene titanium oxides ATiSi2O6 (A=Na, Li). Using the mean-field-type analysis and the numerical quantum transfer matrix method, we show that the model exhibits two different ordered states, the spin-dimer and orbital-ferro state and the spin-ferro and orbital-antiferro state. The transition between two phases is driven by the relative strength of the Hund's-rule coupling to the onsite Coulomb repulsion and/or by the external magnetic field. The ground-state phase diagram i…
The pairing mechanism in A$_3$C$_{60}$ is investigated by studying the properties of a three-orbital Hubbard model with antiferromagnetic Hund coupling in the normal and superconducting phase. Local orbital fluctuations are shown to be substantially enhanced in the superconducting state, with a fluctuation energy scale that matches the low-energy peak in the spectral weight of the order parameter. Our results demonstrate that local orbital fluctuations provide the pairing glue in strongly correlated fulleride superconductors and support the spin/orbital freezing theory of unconventional superc…
We propose an orbital optimized method for unitary coupled cluster theory (OO-UCC) within the variational quantum eigensolver (VQE) framework for quantum computers. OO-UCC variationally determines the coupled cluster amplitudes and also molecular orbital coefficients. Owing to its fully variational nature, first-order properties are readily available. This feature allows the optimization of molecular structures in VQE without solving any additional equations. Furthermore, the method requires smaller active space and shallower quantum circuit than UCC to achieve the same accuracy. We present nu…
This preprint has been withdrawn, as the evidence for persistent currents found with the "infinite-size" DMRG algorithm has not been reproduced with the (presumably more accurate) "finite-size" version of the algorithm (M. Troyer and U. Schollwoeck, private communiation). A much clearer example of long-range order in orbital currents is presented in: J. O. Fjaerestad and J. B. Marston, ``Staggered Orbital Currents in the Half-Filled Two-Leg Ladder,'' cond-mat/0107094.
The effects of electronic correlations and orbital degeneracy on thermoelectric properties are studied within the context of multi-orbital Hubbard models on different lattices. We use dynamical mean field theory with iterative perturbation theory as a solver to calculate the self-energy of the models in wide range of interaction strengths. The Seebeck coefficient, which measures the voltage drop in response to a temperature gradient across the system, shows a non-monotonic behavior with temperatures in the presence of strong correlations. This anomalous behavior is associated with a crossover …
We study the renormalization of magnons by charge and coupled orbital-lattice fluctuations in colossal magnetoresistance compounds. The model considered is an orbitally degenerate double-exchange system coupled to Jahn-Teller active phonons. The modulation of ferromagnetic bonds by low-energy orbital fluctuations is identified as the main origin of the unusual softening of the zone-boundary magnons observed experimentally in manganites.
Wave functions based on electron-pair states provide inexpensive and reliable models to describe quantum many-body problems containing strongly-correlated electrons, given that broken-pair states have been appropriately accounted for by, for instance, a posteriori corrections. In this article, we analyse the performance of electron-pair methods in predicting orbital-based correlation spectra. We focus on the (orbital-optimized) pair-coupled cluster Doubles (pCCD) ansatz with a linearized coupled-cluster (LCC) correction. Specifically, we scrutinize how orbital-based entanglement and correlatio…
In a two orbital double exchange model we formulate Jahn-Teller coupled charge density wave in one electron per lattice site limit. Softening of Jahn-Teller phonons corresponding to distortion modes $Q_2$ or $Q_3$ associated with perfect nesting of Fermi surface leads to this instability at low temperature. The gap equation for charge density wave state and its dependences on electron-lattice coupling are calculated explicitly when any one of the Jahn-Teller modes is excited cooperatively. We find that the $Q_2$ distortion mode yields lowest free energy. The calculated heat capacity for each m…
Different types of order are discussed in the context of strongly correlated transition metal oxides, involving pure compounds and $3d^{3}-4d^{4}$ and $3d^{2}-4d^{4}$ hybrids. Apart from standard, long-range spin and orbital orders we observe also exotic non-colinear spin patterns. Such patters can arise in presence of atomic spin-orbit coupling, which is a typical case, or due to spin-orbital entanglement at the bonds in its absence, being much less trivial. Within a special interacting one-dimensional spin-orbital model it is also possible to find a rigorous topological magnetic order in a g…
In this work, we study a biquadratic Heisenberg model with coupled orbital degree of freedom using Monte Carlo simulation in order to investigate the phase transitions in iron-based superconductors. The antiferro-quadrupolar state, which may be related to the magnetism of FeSe [Phys. Rev. Lett. 115, 116401 (2015)], is stabilized by the anisotropic biquadratic interaction induced by a ferro-orbital-ordered state. It is revealed that the orbital and nematic transitions occur at the same temperature for all the cases, supporting the mechanism of the orbital-driven nematicity as revealed in most r…
We show that the strongly correlated 4f-orbital symmetry of the ground state is revealed by linear dichroism in core-level photoemission spectra as we have discovered for YbRh2Si2 and YbCu2Si2. Theoretical analysis tells us that the linear dichroism reflects the anisotropic charge distributions resulting from crystalline electric field. We have successfully determined the ground-state 4f symmetry for both compounds from the polarization-dependent angle-resolved core-level spectra at a low temperature well below the first excitation energy. The excited-state symmetry is also probed by temperatu…
Mott insulators with both spin and orbital degeneracy are pertinent to a large number of transition metal oxides. The intertwined spin and orbital fluctuations can lead to rather exotic phases such as quantum spin-orbital liquids. Here we consider two-component (spin 1/2) fermionic atoms with strong repulsive interactions on the $p$-band of the optical square lattice. We derive the spin-orbital exchange for quarter filling of the $p$-band when the density fluctuations are suppressed, and show it frustrates the development of long range spin order. Exact diagonalization indicates a spin-disorde…
A three-orbital-model approach for studying spin wave excitations in the strongly spin-orbit coupled layered perovskite iridates is presented which provides a unified description of magnetic excitations as well as the electronic band structure. The calculated spin wave dispersions with realistic three-band parameters are in excellent agreement with the RIXS data for iridates, including the strong AF zone boundary dispersion in the single-layer compound $\rm Sr_2 Ir O_4$ and the large anisotropy gap in the bilayer compound $\rm Sr_3 Ir_2 O_7$. The RIXS spin wave data is shown to provide evidenc…
A new method of computing multipolar exchange interaction in spin-orbit coupled systems is developed using multipolar tensor expansion of the density matrix in LDA+U electronic structure calculation. Within mean-field approximation, exchange constants can be mapped into a series of total energy calculations by pair-flip technique. Application to Uranium dioxide shows an antiferromagnetic superexchange coupling in dipoles but ferromagnetic in quadrupoles which is very different from past studies. Further calculation of spin-lattice interaction indicates it is of the same order with superexchang…
Effects of spontaneous parity breaking by charge, spin, and orbital orders are investigated in a two-band Hubbard model on a honeycomb lattice. This is a minimal model in which the inter-orbital hopping, atomic spin-orbit coupling, and strong electron correlation give rise to fascinating properties, such as the magnetoelectric effects, quantum spin Hall effect, and spin or valley splitting in the band structure. We perform the symmetry analysis of possible broken-parity states and the mean-field analysis of their competition. We find that the model at 1/4 filling exhibits a spin-orbital compos…
We apply grazing-incidence femtosecond x-ray diffraction to investigate the details of the atomic motion connected with a displacively excited coherent optical phonon. We concentrate on the low frequency phonon associated with the charge and orbital order in the mixed valence manganite La0.25Pr0.375Ca0.375MnO3 for T < 210 K. We measure the response of three superlattice reflections that feature different sensitivities to the motion of the unit cell constituents. The results support the assignment to a translational mode of the Mn4+ atoms together with the oxygen atoms connecting adjacent Mn4+ …
We report a ground state with strongly coupled magnetic and charge density wave orders mediated via orbital ordering in the layered compound \tbt. In addition to the commensurate antiferromagnetic (AFM) and charge density wave (CDW) orders, new magnetic peaks are observed whose propagation vector equals the sum of the AFM and CDW propagation vectors, revealing an intricate and highly entwined relationship. This is especially interesting given that the magnetic and charge orders lie in different layers of the crystal structure where the highly localized magnetic moments of the Tb$^{3+}$ ions ar…
In order to numerically study electron correlation effects in multi-orbital systems, we propose a new type of discrete transformation for the exchange (Hund's coupling) and pair-hopping interactions to be used in the dynamical mean field theory + quantum Monte Carlo method. The transformation, which is real and exact, turns out to suppress the sign problem in a wide parameter region including non-half-filled bands. This enables us to obtain the dominant pairing symmetry in the double-orbital Hubbard model, which shows that the spin-triplet, orbital-antisymmetric pairing that exploits Hund's co…
We consider strongly spin-orbit coupled double perovskites A$_2$BB'O$_6$ with B' magnetic ions in either $d^1$ or $d^2$ electronic configuration and non-magnetic B ions. We provide insights into several experimental puzzles, such as the predominance of ferromagnetism in $d^1$ versus antiferromagnetism in $d^2$ systems, the appearance of negative Curie-Weiss temperatures for ferromagnetic materials, and the size of effective magnetic moments. We develop and solve a microscopic model with both spin and orbital degrees of freedom within the Mott insulating regime at finite temperature using mean …
Density functional theory calculations with an on-site Coulomb repulsion term (GGA+U method) reveal competing ground states in (111) oriented (LaAlO$_3$)$_M$/(SrTiO$_3$)$_N$ superlattices with n-type interfaces, ranging from spin, orbital polarized, Dirac point Fermi surface to charge ordered flat band phases. These are steered by the interplay of (i) Hubbard U, (ii) SrTiO$_3$ quantum well thickness and (iii) crystal field spitting tied to in-plane strain. In the honeycomb lattice bilayer case N=2 under tensile strain inversion symmetry breaking drives the system from a ferromagnetic Dirac poi…
Motivated by recent experiments on quasi-1D vanadium oxides, we study quantum phase transitions in a one-dimensional spin-orbital model describing a Haldane chain and a classical Ising chain locally coupled by the relativistic spin-orbit interaction. By employing a field-theoretical approach, we analyze the topology of the ground-state phase diagram and identify the nature of the phase transitions. In the strong coupling limit, a long-range Néel order of entangled spin and orbital angular momentum appears in the ground state. We find that, depending on the relative scales of the spin and orbit…
We study a one-dimensional system composed of three charged bosons confined in an external harmonic potential. More precisely, we investigate the ground-state correlation properties of the system, paying particular attention to the strong-interaction limit. We explain for the first time the nature of the degeneracies appearing in this limit in the spectrum of the reduced density matrix. An explicit representation of the asymptotic natural orbitals and their occupancies is given in terms of some integral equations.
We examine a system of doubly degenerate $p$-orbital polarized fermions on a two-dimensional square lattice with a strong on-site interaction. We consider the system density at the half filling limit and tackle the strong attractive interaction using a perturbation theory. We treat the four-site square plaquette interaction term generated from the directional tunneling dependence of $p$-orbitals using the fourth order in perturbation theory. We map the strong coupling particle Hamiltonian into an effective spin-Hamiltonian and then use a variational mean field approach and a linear spin-wave t…
The description of quantized collective excitations stands as a landmark in the quantum theory of condensed matter. A prominent example occurs in conventional magnets, which support bosonic magnons - quantized harmonic fluctuations of the ordered spins. In striking contrast is the recent discovery that strongly spin-orbital coupled magnets, such as $α$-RuCl$_3$, may display a broad excitation continuum inconsistent with conventional magnons. Due to incomplete knowledge of the underlying interactions unraveling the nature of this continuum remains challenging. The most discussed explanation ref…
We present a Linearized Coupled Cluster (LCC) correction based on an Antisymmetric Product of 1 reference orbital Geminals (AP1roG) reference state. In our LCC ansatz, the cluster operator is restricted to double and to single and double excitations as in standard single-reference CC theory. The performance of the AP1roG-LCC models is tested for the dissociation of diatomic molecules (C$_2$ and F$_2$), spectroscopic constants of the uranyl cation (UO$_2^{2+}$), and the symmetric dissociation of the H$_{50}$ hydrogen chain. Our study indicates that an LCC correction based on an AP1roG reference…
In spite of flourishing studies on the topology of spin waves, a generic framework to classify and compute magnon band topology in non-collinear magnets is still missing. In this work we provide such a theory framework, by mapping an arbitrary linear spin wave into a local free-fermion Hamiltonian with exactly the same spectrum, symmetry implementation and band topology, which allows for a full classification and calculation on any topological properties of magnon bands. We apply this fermionization approach to honeycomb Kitaev magnet $α$-RuCl$_3$, and show the existence of topologically prote…
We calculated the polarization-dependent Raman scattering cross section spectra of LaMnO3 below the A-type magnetic ordering temperature. Two strong peaks appear around the MnO6 octahedra stretching phonon frequency. One mode shows Ag symmetry, while the other mode shows Bg symmetry. We found that the Ag symmetry peak is a Jahn-Teller phonon coupled to the orbital wave and the Bg symmetry peak is an orbital wave coupled to a Q2 phonon mode via the Jahn-Teller electron phonon coupling.
Strong correlation effects in a capacitively coupled double quantum-dot setup were previously shown to provide the possibility of both entangling spin-charge degrees of freedom and realizing efficient spin-filtering operations by static gate-voltage manipulations. Motivated by the use of such a device for quantum computing, we study the influence of electromagnetic noise on a general spin-orbital Kondo model, and investigate the conditions for observing coherent, unitary transport, crucial to warrant efficient spin manipulations. We find a rich phase diagram, where low-energy properties sensit…
Transition metal compounds with the $(t_{2g})^4$ electronic configuration are expected to be nonmagnetic atomic singlets both in the weakly interacting regime due to spin-orbit coupling, as well as in the Coulomb dominated regime with oppositely aligned $L=1$ and $S=1$ angular momenta. However, starting with the full multi-orbital electronic Hamiltonian, we show the low energy effective magnetic Hamiltonian contains isotropic superexchange spin interactions but anisotropic orbital interactions. By tuning the ratio of superexchange to spin-orbit coupling $J_\mathrm{SE}/λ$, we obtain a phase tra…
