We propose a theory for bilayer nickelate materials, where a large tetragonal field - intrinsic or induced by epitaxial strain - lifts the orbital degeneracy and localizes the 3z^2-r^2 orbital states. These states host local spins S=1/2 bound into singlets by strong interlayer coupling, and their dynamics is described by weakly dispersive singlet-triplet excitations ("triplons"). The charge carriers occupy the wide bands of x^2-y^2 symmetry, and their Cooper pairing is mediated by the high-energy triplon excitations. As the x^2-y^2 band filling increases, i.e. moving further away from the Ni^3+ valence state, the indirect Ruderman-Kittel-Kasuya-Yosida interactions between local spins induce spin-density-wave order via triplon condensation. Implications of the model for compressively strained La_3Ni_2O_7 films and electron doped oxychloride Sr_3Ni_2O_5Cl_2 are discussed.

We theoretically study quasi-two-dimensional nickel compounds, where the nickel ions assume Ni^2+ d^8 valence state and feature a low-spin S=0 ground state quasidegenerate with S=1 ionic excitations. Such a level structure is supported by square-planar coordination of nickel ions or a suitable substitution of apical oxygens. We construct the corresponding singlet-triplet exchange model and explore its phase diagram and excitation spectrum. By hole doping, we further introduce mobile Ni^3+ d^7 ionic configurations with effective spins S=1/2, and analyze their interactions with the d^8 singlet-triplet background. The interplay with the triplet excitations in the d^8 sector is found to have a deep impact on the propagation of the doped hole-like charge carriers and is identified as a powerful source of Cooper pairing among them.

Mott insulators based on 4d and 5d transition-metal ions, where spin-orbit interaction plays a key role, can exhibit various forms of unusual magnetism. A particular example is the antiferromagnet Ca2RuO4 containing d4 Ru4+ ions. Here the spin-orbit interaction stabilizes the nonmagnetic J = 0 singlet ionic ground state, which gets dynamically mixed, via exchange interactions, with low-energy J = 1 ionic excitations. Thanks to a sufficient strength of the exchange, these excitations condense and a long-range order emerges. The resulting ordered moments are soft and prone to fluctuations of their effective length. The corresponding amplitude mode appears as a prominent magnetic excitation and complements the conventional magnons involving rotations of the moments. Motivated by this peculiar kind of magnetic order and the specific spectrum of magnetic excitations, we study their influence on the propagation of doped carriers. To this end, we construct a microscopic model including both d4 and d5 degrees of freedom and address the propagation of an injected electron by employing self-consistent Born approximation. We find that the electron shows a combination of both free and a polaronic type of motion, where the mobile carrier strongly interacts with an accompanying cloud of magnetic excitations. Remarkably, in the latter case it is the exotic excitation, the amplitude mode, that is found to dominate over the contribution of magnons. Our soft-spin situation thus largely contrasts with spin polarons widely discussed in the context of doped Heisenberg-type magnets based on rigid spin moments.

Mott insulators based on $4d$ and $5d$ transition-metal ions, where spin-orbit interaction plays a key role, can exhibit various forms of unusual magnetism. A particular example is the antiferromagnet Ca$_2$RuO$_4$ containing $d^4$ Ru$^{4+}$ ions. Here the spin-orbit interaction stabilizes the non-magnetic $J=0$ singlet ionic ground state, which gets dynamically mixed - via exchange interactions - with low-energy $J=1$ ionic excitations. Thanks to a sufficient strength of the exchange, these excitations condense and a long-range order emerges. The resulting ordered moments are soft and prone to fluctuations of their effective length. The corresponding amplitude mode appears as a prominent magnetic excitation and complements the conventional magnons involving rotations of the moments. Motivated by this peculiar kind of magnetic order and the specific spectrum of magnetic excitations, we study their influence on the propagation of doped carriers. To this end, we construct a microscopic model including both $d^4$ and $d^5$ degrees of freedom and address the propagation of an injected electron by employing self-consistent Born approximation. We find that the electron shows a combination of both free and a polaronic type of motion, where the mobile carrier strongly interacts with an accompanying cloud of magnetic excitations. Remarkably, in the latter case it is the exotic excitation - the amplitude mode - that is found to dominate over the contribution of magnons. Our soft-spin situation thus largely contrasts with spin polarons widely discussed in the context of doped Heisenberg-like magnets based on rigid spin moments.

Mott -insulating transition metal oxides containing t42g ions with strong spin -orbit coupling were recently demonstrated to display unusual magnetism due to a dynamical mixing of the low -energy multiplet states via exchange processes. Here we derive exchange interactions in the situation where a tetragonal or trigonal crystal field selects a single relevant excited state on top of the singlet ionic ground state, producing an effective spin -21. We show that these moments universally obey antiferromagnetic transverse -field Ising model (TFIM) with an intrinsic transverse field generated by the splitting of the two ionic singlets. Using Ru4+ as a example ion, we provide quantitative estimates of the exchange and illustrate the emergent TFIM physics based on phase diagrams and excitation spectra obtained for several 2D lattices-square, honeycomb, as well as for the frustrated triangular lattice.

The density functional plus dynamical mean-field theory is used to study the spin excitation spectra of SrRu_2O_6. A good quantitative agreement with experimental spin excitation spectra is found. Depending on the size of the Hund's coupling J_H the systems chooses either Mott insulator or covalent insulator state when magnetic ordering is not allowed. We find that the nature of the paramagnetic state has negligible influence on the charge and spin excitation spectra. We find that antiferromagnetic correlations hide the covalent insulator state for realistic choices of the interaction parameters.

We present an analytical study of the exchange interactions between pseudospin one-half ${d}^{5}$ ions in honeycomb lattices with edge-shared octahedra. Various exchange channels involving Hubbard $U$, charge-transfer excitations, and cyclic exchange are considered. Hoppings within ${t}_{2g}$ orbitals as well as between ${t}_{2g}$ and ${e}_{g}$ orbitals are included. Special attention is paid to the trigonal crystal field $\mathrm{\ensuremath{\Delta}}$ effects on the exchange parameters. The obtained exchange Hamiltonian is dominated by ferromagnetic Kitaev interaction $K$ within a wide range of $\mathrm{\ensuremath{\Delta}}$. It is found that a parameter region close to the charge-transfer insulator regime and with a small $\mathrm{\ensuremath{\Delta}}$ is most promising to realize the Kitaev spin liquid phase. Two representative honeycomb materials ${\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}$ and $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ are discussed based on our theory. We have found that both materials share dominant ferromagnetic $K$ and positive nondiagonal $\mathrm{\ensuremath{\Gamma}}$ values. However, their Heisenberg $J$ terms have opposite signs: AFM $J>0$ in ${\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}$ and FM $J<0$ in $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$. This brings different magnetic fluctuations and results in their different magnetization behaviors and spin excitation spectra. Proximity to FM state due to the large FM $J$ is emphasized in $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$. The differences between the exchange couplings of these two materials originate from the opposite $\mathrm{\ensuremath{\Delta}}$ values, indicating that the crystal field can serve as an efficient control parameter to tune the magnetic properties of ${d}^{5}$ spin-orbit Mott insulators.

We present an analytical study of the exchange interactions between pseudospin one-half d(5) ions in honeycomb lattices with edge-shared octahedra. Various exchange channels involving Hubbard U, charge-transfer excitations, and cyclic exchange are considered. Hoppings within t(2g) orbitals as well as between t(2g) and e(g) orbitals are included. Special attention is paid to the trigonal crystal field Delta effects on the exchange parameters. The obtained exchange Hamiltonian is dominated by ferromagnetic Kitaev interaction K within a wide range of Delta. It is found that a parameter region close to the charge-transfer insulator regime and with a small Delta is most promising to realize the Kitaev spin liquid phase. Two representative honeycomb materials Na2IrO3 and alpha-RuCl3 are discussed based on our theory. We have found that both materials share dominant ferromagnetic K and positive nondiagonal Gamma values. However, their Heisenberg J terms have opposite signs: AFM J > 0 in Na2IrO3 and FM J < 0 in alpha-RuCl3. This brings different magnetic fluctuations and results in their different magnetization behaviors and spin excitation spectra. Proximity to FM state due to the large FM J is emphasized in alpha-RuCl3. The differences between the exchange couplings of these two materials originate from the opposite Delta values, indicating that the crystal field can serve as an efficient control parameter to tune the magnetic properties of d(5) spin-orbit Mott insulators.

Tomohiro Takayama, 2 Jǐŕı Chaloupka, 4 Andrew Smerald, Giniyat Khaliullin, and Hidenori Takagi 2, 5 Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany Institute for Functional Matter and Quantum Technologies, University of Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany Department of Condensed Matter Physics, Masaryk University, Brno, Czech Republic Central European Institute of Technology, Masaryk University, Brno, Czech Republic Department of Physics, University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan (Dated: February 5, 2021)

$\alpha$-RuCl$_{3}$ is a major candidate for the realization of the Kitaev quantum spin liquid, but the presence of zigzag antiferromagnetic order at low temperatures indicates deviations from the Kitaev Hamiltonian. To enable a realistic description of the ground state and elementary excitations of $\alpha$-RuCl$_{3}$, we have quantitatively determined the low-energy effective Hamiltonian by carrying out a comprehensive resonant inelastic x-ray scattering (RIXS) study at the Ru $L_{3}$ absorption edge and by comparing the results to detailed model calculations. The high-energy segment of the RIXS spectrum yields the parameters characterizing the electronic structure of individual Ru$^{3+}$ ions and demonstrates that the ground-state wavefunction can be described by a spin-orbit entangled $\widetilde{S}=1/2$ pseudospin. The quasi-elastic RIXS intensity, in conjunction with numerical calculations of the equal-time pseudospin correlation functions, serves as a fingerprint of the exchange interactions between the pseudospins. In the paramagnetic state, it has a broad intensity maximum around the zone center without any local maxima at the zigzag magnetic Bragg wavevectors. This finding implies that the zigzag order is destabilized by pronounced ferromagnetic correlations, and is only slightly lower in energy than other competing phases. Theoretical evaluations based on the RIXS data show that the dominant Kitaev interaction is ferromagnetic and that the magnitudes of Heisenberg and off-diagonal couplings are about half of the Kitaev term. The pseudospin Hamiltonian determined in this way provides a solid foundation for further research on $\alpha$-RuCl$_{3}$, including the quest for a theoretical understanding of its magnetic field-induced spin-liquid behavior.

This paper reviews the current progress on searching the Kitaev spin liquid state in 3[Formula: see text] electron systems. Honeycomb cobaltates were recently proposed as promising candidates to realize the Kitaev spin liquid state, due to the more localized wavefunctions of [Formula: see text] ions compared with that of [Formula: see text] and [Formula: see text] ions, and also the easy tunability of the exchange Hamiltonian in favor of Kitaev interaction. Several key parameters that have large impacts on the exchange constants, such as the charge-transfer gap and the trigonal crystal field, are identified and discussed. Specifically, tuning crystal field effect by means of strain or pressure is emphasized as an efficient phase control method driving the magnetically ordered cobaltates into the spin liquid state. Experimental results suggesting the existence of strong Kitaev interactions in layered honeycomb cobaltates are discussed. Finally, the future research directions are briefly outlined.

A Kitaev quantum spin liquid is a prime example of novel quantum magnetism of spin-orbit entangled pseudospin-1/2 moments in a honeycomb lattice. Most candidate materials such as Na2IrO3 have many competing exchange interactions beyond the minimal Kitaev-Heisenberg model whose small variations in the strength of the interactions produce huge differences in low-energy dynamics. Our incomplete knowledge of dynamic spin correlations hampers identification of a minimal model and quantification of the proximity to the Kitaev quantum spin-liquid phase. Here, we report momentum- and energy-resolved magnetic excitation spectra in a honeycomb lattice Na2IrO3 measured using a resonant inelastic x-ray scattering spectrometer capable of 12 meV resolution. Measured spectra at a low temperature show that the dynamic response lacks resolution-limited coherent spin waves in most parts of the Brillouin zone but has a discernible dispersion and spectral weight distribution within the energy window of 60 meV. A systematic investigation using the exact diagonalization method and direct comparison of high-resolution experimental spectra and theoretical simulations allow us to confine a parameter regime in which the extended Kitaev-Heisenberg model reasonably reproduces the main feature of the observed magnetic excitations. Hidden Kitaev quantum spin-liquid and Heisenberg phases found in the complex parameter space are used as references to propose the picture of renormalized magnons as explaining the incoherent nature of magnetic excitations. Magnetic excitation spectra are taken at elevated temperatures to follow the temperature evolution of the resonant inelastic x-ray scattering dynamic response in the paramagnetic state. Whereas the low-energy excitation progressively diminishes as the zigzag order disappears, the broad high-energy excitation maintains its spectral weight up to a much higher temperature of 160 K. We suggest that the dominant nearest-neighbor interactions keep short-range correlations up to quite high temperatures with a specific short-range dynamics which has a possible connection to a proximate spinliquid phase.

Compounds of transition metal ions with strong spin-orbit coupling recently attracted attention due to the possibility to host frustrated bond-dependent anisotropic magnetic interactions. In general, such interactions lead to complex phase diagrams that may include exotic phases, e.g. the Kitaev spin liquid. Here we report on our comprehensive analysis of the global phase diagram of the extended Kitaev-Heisenberg model relevant to honeycomb lattice compounds Na2IrO3 and alpha-RuCl3. We have utilized recently developed method based on spin coherent states that enabled us to resolve arbitrary spin patterns in the cluster ground states obtained by exact diagonalization. Global trends in the phase diagram are understood in combination with the analytical mappings of the Hamiltonian that uncover peculiar links to known models - Heisenberg, Ising, Kitaev, or compass models on the honeycomb lattice - or reveal entire manifolds of exact fluctuation-free ground states. Finally, our study can serve as a methodological example that can be applied to other spin models with complex bond-dependent non-Heisenberg interactions.

We study the orbitally frustrated singlet-triplet models that emerge in the context of spin-orbit coupled Mott insulators with t(2g)(4) electronic configuration. In these compounds, low-energy magnetic degrees of freedom can be cast in terms of three-flavor "triplon" operators describing the transitions between spin-orbit entangled J = 0 ionic ground state and excited J = 1 levels. In contrast to a conventional, flavor-isotropic O(3) singlet-triplet models, spin-orbit entangled triplon interactions are flavor-and-bond selective and thus highly frustrated. In a honeycomb lattice, we find close analogies with the Kitaev spin model-an infinite number of conserved quantities, no magnetic condensation, and spin correlations being strictly short-ranged. However, due to the bosonic nature of triplons, there are no emergent gapless excitations within the spin gap, and the ground state is a strongly correlated paramagnet of dense triplon pairs with no long-range entanglement. Using exact diagonalization, we study the bosonic Kitaev model and its various extensions, which break exact symmetries of the model and allow magnetic condensation of triplons. Possible implications for magnetism of ruthenium oxides are discussed.

Motivated by similarities between cuprate superconductors and two-leg ladder copper-oxide compounds and in order to obtain a better understanding of optical properties of cuprate superconductors we have studied the c-axis (along the ladder) optical conductivity [Formula: see text] of a doped [Formula: see text] two-leg ladder. Using exact diagonalization, we have calculated the conductivity and related quantities for cyclic ladders of up to 13 rungs. In agreement with results of an early study by Hayward and coworkers (Hayward et al 1996 Phys. Rev. B 53 8863) we find that [Formula: see text] consists of a Drude peak at zero frequency and an absorption band in the infrared region that is separated from the former by a pseudogap. The width of the pseudogap [Formula: see text] increases with increasing J/t, in parallel with an increase of the magnitude [Formula: see text] of the gap in the quasiparticle excitation spectra. Our central finding is that [Formula: see text], where [Formula: see text] is the magnitude of the gap in the spin excitation spectra. We demonstrate that this approximate relation can be understood in terms of a phenomenological model involving a superconducting ladder and a coupling between charged quasiparticles and spin excitations. The relation is remarkably similar to the one between experimental values of the energy scale of a dip in the in-plane optical conductivity, the superconducting gap [Formula: see text] and the energy of the spin-resonance in cuprate superconductors (for a recent discussion of the optical data, see Šopík et al 2015 New J. Phys. 17 053022). Our findings support the point of view that low energy infrared active excited states of cuprate superconductors can be viewed as consisting of two charged quasiparticles connected with pair-breaking and a spin excitation.

We present and analyze Raman spectra of the Mott insulator Ca$_2$RuO$_4$, whose quasi-two-dimensional antiferromagnetic order has been described as a condensate of low-lying spin-orbit excitons with angular momentum $J_{eff}=1$. In the $A_g$ polarization geometry, the amplitude (Higgs) mode of the spin-orbit condensate is directly probed in the scalar channel, thus avoiding infrared-singular magnon contributions. In the $B_{1g}$ geometry, we observe a single-magnon peak as well as two-magnon and two-Higgs excitations. Model calculations using exact diagonalization quantitatively agree with the observations. Together with recent neutron scattering data, our study provides strong evidence for excitonic magnetism in Ca$_2$RuO$_4$ and points out new perspectives for research on the Higgs mode in two dimensions.

We explore the phase diagram of the Kitaev-Heisenberg model with nearest neighbor interactions on the honeycomb lattice using the exact diagonalization of finite systems combined with the cluster mean field approximation, and supplemented by the insights from analytic approaches: the linear spin-wave and second-order perturbation theories. This study confirms that by varying the balance between the Heisenberg and Kitaev term, frustrated exchange interactions stabilize in this model either one of four phases with magnetic long range order: Neel phase, ferromagnetic phase, and two other phases with coexisting antiferromagnetic and ferromagnetic bonds, zigzag and stripy phase, or one of two distinct spin-liquid phases. Out of these latter disordered phases, the one with ferromagnetic Kitaev interactions has a substantially broader range of stability as the neighboring competing ordered phases, ferromagnetic and stripy, have very weak quantum fluctuations. Focusing on the quantum spin-liquid phases, we study spatial spin correlations and dynamic spin structure factor of the model by the exact diagonalization technique, and discuss the evolution of gapped low-energy spin response across the quantum phase transitions between the disordered spin liquid and phases with long range magnetic order.

We study the effects of electron doping in Mott insulators containing d(4) ions such as Ru(4+), Os(4+), Rh(5+), and Ir(5+) with J=0 singlet ground state. Depending on the strength of the spin-orbit coupling, the undoped systems are either nonmagnetic or host an unusual, excitonic magnetism arising from a condensation of the excited J=1 triplet states of t(2g)(4). We find that the interaction between J excitons and doped carriers strongly supports ferromagnetism, converting both the nonmagnetic and antiferromagnetic phases of the parent insulator into a ferromagnetic metal, and further to a nonmagnetic metal. Close to the ferromagnetic phase, the low-energy spin response is dominated by intense paramagnon excitations that may act as mediators of a triplet pairing.

We study the ordered moment direction in the extended Kitaev-Heisenberg model relevant to honeycomb lattice magnets with strong spin-orbit coupling. We utilize numerical diagonalization and analyze the exact cluster ground states using a particular set of spin-coherent states, obtaining thereby quantum corrections to the magnetic anisotropy beyond conventional perturbative methods. It is found that the quantum fluctuations strongly modify the moment direction obtained at a classical level and are thus crucial for a precise quantification of the interactions. The results show that the moment direction is a sensitive probe of the model parameters in real materials. Focusing on the experimentally relevant zigzag phases of the model, we analyze the currently available neutron-diffraction and resonant x-ray-diffraction data on Na2IrO3 and RuCl3 and discuss the parameter regimes plausible in these Kitaev-Heisenberg model systems.

We investigate the electronic dispersion of the high-T-c cuprate superconductors using the fully self-consistent version of the phenomenological model, where charge planar quasiparticles are coupled to spin fluctuations. The inputs we use, the underlying (bare) band structure and the spin susceptibility chi, are extracted from fits of angle-resolved photoemission and inelastic neutron scattering data of underdoped YBa2Cu3O6.6 by T. Dahm and coworkers [Nat. Phys. 5, 217 (2009)]. Our main results are as follows: (i) We have confirmed the finding by Dahm and coworkers that the main nodal kink is, for the present values of the input parameters, determined by the upper branch of the hourglass of chi. We demonstrate that the properties of the kink depend qualitatively on the strength of the charge-spin coupling. (ii) The effect of the resonance mode of chi on the electronic dispersion strongly depends on its kurtosis in the quasimomentum space. A low (high) kurtosis implies a negligible (considerable) effect of the mode on the dispersion in the near-nodal region. (iii) The energy of the kink decreases as a function of the angle theta between the Fermi surface cut and the nodal direction, in qualitative agreement with recent experimental observations. We clarify the trend and make a specific prediction concerning the angular dependence of the kink energy in underdoped YBa2Cu3O6.6.