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arXiv:2303.04169 · Physical review. B./Physical review. B

Impact of gapped spin-orbit excitons on low-energy pseudospin exchange interactions

Sreekar Voleti, F. David Wandler, Arun Paramekanti

Published Mar 18, 2024• 3 citations
arXivHTMLPDF

Abstract

The quest for exotic quantum magnetic ground states, including the Kitaev spin liquid and quantum spin ices, has led to the discovery of several quantum materials where low-energy pseudospin-$1/2$ doublets arise from the splitting of spin-orbit entangled multiplets with higher degeneracy. Such systems include $d$-orbital and $f$-orbital Mott insulators. When the gap between the low-energy pseudospin-$1/2$ levels and the excited levels of the multiplet or ``excitons'' is not large, the effective low-energy exchange interactions between the low-energy pseudospin-$1/2$ moments can acquire significant corrections from coupling to the excitons. We extract these corrections using an extended perturbation theory that takes into account the most relevant higher-order perturbations. Such corrections can impact the exchange matrix for the low-energy pseudospin-$1/2$ levels by renormalizing the strength and the sign of Heisenberg exchange or Ising anisotropies, and potentially even inducing bond-anisotropic couplings such as Kitaev-$\mathrm{\ensuremath{\Gamma}}$ exchange interactions. We discuss recent experiments on various cobaltate and osmate materials, which hint at the ubiquity and importance of this physics. Our study suggests a distinct direction in the search for exotic spin liquids---quantum pseudospin-1/2 systems with low-energy spin-orbit excitons.

DOI
10.1103/physrevb.109.094423

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Líquidos spin-orbitais quânticos j = 3/2

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Quantum spin liquids (QSLs) are strongly correlated systems displaying fascinating phenomena like long-range entanglement and fractionalized excitations. The research on these states has since its beginning followed trends generated by the synthesis of new compounds and the construction of new theoretical tools. In coherence with this history, a manifold of new results about QSLs were established during the past decade due to studies on the integrable Kitaev model on the honeycomb lattice. This j = 1/2 model displays bond-dependent and anisotropic exchanges that are essential to stabilize its QSL ground state with Majorana fermion excitations and emergent Z2 gauge field. Even more interestingly, this model is relevant to understand the magnetism of a certain class of 4/5d5 Mott insulators with specific lattice constraints, t2g orbital degeneracy and strong spin-orbit coupling (SOC). This mechanism defining these so-called Kitaev materials can be applied to similar compounds based on transition metal ions in different electronic configurations. In this thesis, I investigate minimal models for two types of 4/5d1 Mott insulators: the ones on the ordered double perovskite structure (ODP) and the ones isostructural to the Kitaev materials. Their effective models generically show bond-dependent and anisotropic interactions involving multipoles of an effective j = 3/2 angular momentum. Such degrees of freedom are conveniently written in terms of pseudospin s and pseudo-orbital τ operators resembling spin and orbital operators of Kugel-Khomskii models with twofold orbital degeneracy. Despite their anisotropy, the two realistic models display continuous global symmetries in the limit of vanishing Hund\'s coupling enhancing quantum fluctuations and possibly stabilizing a QSL phase. Parton mean-field theory was used to propose fermionic QSLs that will be called quantum spin-orbital liquids (QSOLs) due their dependence with s and τ. On ODPs, I studied a chiral QSOL with Majorana fermion excitations and a gapless spectrum characterized by nodal lines along the edges of the Brillouin zone. These nodal lines are topological defects of a non-Abelian Berry connection and the system exhibits dispersing surface states. Several experimental responses of the chiral QSOL within the mean-field approximation are compared with the experimental data available for the spin liquid candidate Ba2YMoO6. Moreover, based on a symmetry analysis, I discuss the operators involved in resonant inelastic X-ray scattering (RIXS) amplitudes for 4/5d1 Mott insulators and show that the RIXS cross sections allow one to selectively probe pseudospin and pseudo-orbital degrees of freedom. For the chiral spin-orbital liquid in particular, these cross sections provide information about the spectrum for different flavors of Majorana fermions. The model for materials isostructural to the Kitaev materials has an emergent SU(4) symmetry that is made explicit by means of a Klein transformation on pseudospin degrees of freedom. The model is known to stabilize a QSOL on the honeycomb lattice and instigated the investigation of QSOLs on a generalization of this lattice to three dimensions. Parton mean-field theory was used once again to propose the liquid states, and a variational Monte Carlo (VMC) method was used to compute the energies of the projected wave functions. The numerical results show that the lowest-energy QSOL corresponds to a zero-flux state with a Fermi surface of four-color fermionic partons. Further VMC computations also revealed that this state is stable against formation of plaquette ordering (tetramerization). The energy of this QSOL is highly competitive even when Hund\'s coupling induced perturbations are included, as shown by comparison with simple ordered states. Extensions and perspectives for future work are discussed in the end of this thesis.

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arXiv:cond-mat/9809013•Journal of Applied Physics•Apr 15, 1999•1 citation
The quest for exotic quantum magnetic ground states, including the Kitaev spin liquid and quantum spin ices, has led to the discovery of several quantum materials where low-energy pseudospin-1/21/21/2 doublets arise from the splitting of spin-orbit entangled multiplets with higher degeneracy. Such systems include ddd-orbital and fff-orbital Mott insulators. When the gap between the low-energy pseudospin-1/21/21/2 levels and the excited levels of the multiplet or ``excitons'' is not large, the effective low-energy exchange interactions between the low-energy pseudospin-1/21/21/2 moments can acquire significant corrections from coupling to the excitons. We extract these corrections using an extended perturbation theory that takes into account the most relevant higher-order perturbations. Such corrections can impact the exchange matrix for the low-energy pseudospin-1/21/21/2 levels by renormalizing the strength and the sign of Heisenberg exchange or Ising anisotropies, and potentially even inducing bond-anisotropic couplings such as Kitaev-\ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ exchange interactions. We discuss recent experiments on various cobaltate and osmate materials, which hint at the ubiquity and importance of this physics. Our study suggests a distinct direction in the search for exotic spin liquids---quantum pseudospin-1/2 systems with low-energy spin-orbit excitons.
In the search for quantum spin liquids, candidate materials for the Kitaev model and its extensions have been intensively explored during the past decade, as the models realize the exact quantum spin liquids in the ground state. Thus far, insulating magnets in the low-spin d5{d}^{5}d5 electron configuration under the strong spin-orbit coupling have been studied for realizing the Kitaev-type bond-dependent anisotropic interactions between the spin-orbital entangled Kramers doublets. To extend the candidates, here we investigate the systems in a high-spin d7{d}^{7}d7 electron configuration, whose ground state is described by the spin-orbital entangled Kramers doublet. By the second-order perturbation in terms of the ${t}_{2g}\ensuremath{-}{t}_{2g}$ and ${t}_{2g}\ensuremath{-}{e}_{g}$ hoppings, we show that the effective spin model possesses the anisotropic Kitaev interactions as well as the isotropic Heisenberg ones. While the Kitaev interaction is always ferromagnetic, the Heisenberg interaction can become either ferromagnetic or antiferromagnetic depending on the Coulomb interactions and the crystalline electric fields. We also derive the effective model for the low-spin d5{d}^{5}d5 electron configuration within the same perturbation scheme, in which the Kitaev interaction becomes both ferromagnetic and antiferromagnetic, while the Heisenberg one is always ferromagnetic. Referring to the previous study for the Kitaev-Heisenberg model, we find that the quantum spin liquid phase exists in the reasonable parameter region in both d7{d}^{7}d7 and d5{d}^{5}d5 cases, while the former has a richer structure of the phase diagram. We discuss the advantages of the d7{d}^{7}d7 case in comparison with the d5{d}^{5}d5 case. Our results indicate that the high-spin d7{d}^{7}d7 state provides another platform for the Kitaev-type quantum spin liquid.
Can we realize Kitaev quantum spin liquids in the Jeff=1/2J_\text{eff}=1/2Jeff​=1/2 honeycomb cobaltates?
The quest for Kitaev quantum spin liquids has led to great interest in honeycomb quantum magnets with strong spin-orbit coupling. It has been recently proposed that even Mott insulators with 3d3d3d transition metal ions, having nominally weak spin-orbit coupling, can realize such exotic physics. Motivated by this, we study the rhombohedral honeycomb cobaltates CoTiO3_33​, BaCo2_22​(PO4_44​)2_22​, and BaCo2_22​(AsO4_44​)2_22​, using ab initio\textit{ab initio}ab initio density functional theory, which takes into account realistic crystal field distortions and chemical information, in conjunction with exact diagonalization numerics. We show that these Co2+^{2+}2+ magnets host jeff=1/2j_\text{eff}=1/2jeff​=1/2 local moments with highly anisotropic ggg-factors, and we extract their full spin Hamiltonians including longer-range and anisotropic exchange couplings. For CoTiO3_33​, we find a nearest-neighbor easy-plane ferromagnetic XXZXXZXXZ model with additional bond-dependent anisotropies and interlayer exchange, which supports three-dimensional (3D) Dirac nodal line magnons. In contrast, for BaCo2_22​(PO4_44​)2_22​ and BaCo2_22​(AsO4_44​)2_22​, we find a strongly suppressed interlayer coupling, and significant frustration from additional third-neighbor antiferromagnetic exchange mediated by P/As. Such bond-anisotropic J1J_1J1​-J3J_3J3​ spin models can support collinear zig-zag or coplanar spiral ground states; we discuss their dynamical spin correlations which reveal a gapped Goldstone mode, and argue that the effective parameters of these pseudospin-1/21/21/2 models may be strongly renormalized by coupling to a low energy spin-exciton. Our results call for re-examining proposals for realizing Kitaev spin liquids in the honeycomb cobaltates.
The current efforts to find the materials hosting Kitaev model physics have been focused on Mott insulators of d5{d}^{5}d5 pseudospin-1/2 ions Ir4+{\mathrm{Ir}}^{4+}Ir4+ and Ru3+{\mathrm{Ru}}^{3+}Ru3+ with t2g5(S=1/2,L=1){t}_{2g}^{5}(S=1/2,L=1)t2g5​(S=1/2,L=1) electronic configuration. Here we propose that the Kitaev model can be realized in materials based on d7{d}^{7}d7 ions with t2g5eg2(S=3/2,L=1){t}_{2g}^{5}{e}_{g}^{2}(S=3/2,L=1)t2g5​eg2​(S=3/2,L=1) configuration such as Co2+{\mathrm{Co}}^{2+}Co2+, which also host the pseudospin-1/2 magnetism. Considering possible exchange processes, we have derived the d7{d}^{7}d7 pseudospin-1/2 interactions in ${90}^{\ensuremath{\circ}}$ bonding geometry. The obtained Hamiltonian comprises the bond-directional Kitaev KKK and isotropic Heisenberg JJJ interactions as in the case of d5{d}^{5}d5 ions. However, we find that the presence of additional spin-active eg{e}_{g}eg​ electrons radically changes the balance between Kitaev and Heisenberg couplings. Most remarkably, we show that the exchange processes involving eg{e}_{g}eg​ spins are highly sensitive to whether the system is in Mott ($U<\mathrm{\ensuremath{\Delta}}$) or charge-transfer ($U>\mathrm{\ensuremath{\Delta}}$) insulating regime. In the latter case, to which many cobalt compounds do actually belong, the antiferromagnetic Heisenberg coupling JJJ is strongly suppressed and spin-liquid phase can be stabilized. The results suggest cobalt-based materials as promising candidates for the realization of the Kitaev model.
The microscopic modeling of spin-orbit entangled j=1/2j=1/2j=1/2 Mott insulators such as the layered hexagonal iridates Na2IrO3{\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}Na2​IrO3​ and Li2IrO3{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}Li2​IrO3​ has spurred an interest in the physics of Heisenberg-Kitaev models. Here we explore the effect of lattice distortions on the formation of the collective spin-orbital states that include not only conventionally ordered phases but also gapped and gapless spin-orbital liquids. In particular, we demonstrate that in the presence of distortions, i.e., spatial anisotropies of the exchange couplings, conventionally ordered states are formed through an order-by-disorder selection, which is not only sensitive to the type of exchange anisotropy but also to the relative strength of the Heisenberg and Kitaev couplings. The spin-orbital liquid phases of the Kitaev limit---a gapless phase in the vicinity of spatially isotropic couplings and a gapped Z2{Z}_{2}Z2​ phase for a dominant spatial anisotropy of the exchange couplings---show vastly different sensitivities to the inclusion of a Heisenberg exchange. While the gapless phase is remarkably stable, the gapped Z2{Z}_{2}Z2​ phase quickly breaks down in what might be a rather unconventional phase transition driven by the simultaneous condensation of its elementary excitations.
Quantum spin liquids (QSLs) are strongly correlated systems displaying fascinating phenomena like long-range entanglement and fractionalized excitations. The research on these states has since its beginning followed trends generated by the synthesis of new compounds and the construction of new theoretical tools. In coherence with this history, a manifold of new results about QSLs were established during the past decade due to studies on the integrable Kitaev model on the honeycomb lattice. This j = 1/2 model displays bond-dependent and anisotropic exchanges that are essential to stabilize its QSL ground state with Majorana fermion excitations and emergent Z2 gauge field. Even more interestingly, this model is relevant to understand the magnetism of a certain class of 4/5d5 Mott insulators with specific lattice constraints, t2g orbital degeneracy and strong spin-orbit coupling (SOC). This mechanism defining these so-called Kitaev materials can be applied to similar compounds based on transition metal ions in different electronic configurations. In this thesis, I investigate minimal models for two types of 4/5d1 Mott insulators: the ones on the ordered double perovskite structure (ODP) and the ones isostructural to the Kitaev materials. Their effective models generically show bond-dependent and anisotropic interactions involving multipoles of an effective j = 3/2 angular momentum. Such degrees of freedom are conveniently written in terms of pseudospin s and pseudo-orbital τ operators resembling spin and orbital operators of Kugel-Khomskii models with twofold orbital degeneracy. Despite their anisotropy, the two realistic models display continuous global symmetries in the limit of vanishing Hund\'s coupling enhancing quantum fluctuations and possibly stabilizing a QSL phase. Parton mean-field theory was used to propose fermionic QSLs that will be called quantum spin-orbital liquids (QSOLs) due their dependence with s and τ. On ODPs, I studied a chiral QSOL with Majorana fermion excitations and a gapless spectrum characterized by nodal lines along the edges of the Brillouin zone. These nodal lines are topological defects of a non-Abelian Berry connection and the system exhibits dispersing surface states. Several experimental responses of the chiral QSOL within the mean-field approximation are compared with the experimental data available for the spin liquid candidate Ba2YMoO6. Moreover, based on a symmetry analysis, I discuss the operators involved in resonant inelastic X-ray scattering (RIXS) amplitudes for 4/5d1 Mott insulators and show that the RIXS cross sections allow one to selectively probe pseudospin and pseudo-orbital degrees of freedom. For the chiral spin-orbital liquid in particular, these cross sections provide information about the spectrum for different flavors of Majorana fermions. The model for materials isostructural to the Kitaev materials has an emergent SU(4) symmetry that is made explicit by means of a Klein transformation on pseudospin degrees of freedom. The model is known to stabilize a QSOL on the honeycomb lattice and instigated the investigation of QSOLs on a generalization of this lattice to three dimensions. Parton mean-field theory was used once again to propose the liquid states, and a variational Monte Carlo (VMC) method was used to compute the energies of the projected wave functions. The numerical results show that the lowest-energy QSOL corresponds to a zero-flux state with a Fermi surface of four-color fermionic partons. Further VMC computations also revealed that this state is stable against formation of plaquette ordering (tetramerization). The energy of this QSOL is highly competitive even when Hund\'s coupling induced perturbations are included, as shown by comparison with simple ordered states. Extensions and perspectives for future work are discussed in the end of this thesis.
We consider the exchanged Hamiltonian HST=−J∑〈rr′〉(2Sr⋅Sr′−12)(2Tr⋅Tr′−12), describing two isotropic spin-1/2 Heisenberg antiferromagnets coupled by a quartic term on equivalent bonds. The model is relevant for systems with orbital degeneracy and strong electron-vibron coupling in the large Hubbard repulsion limit. To investigate the ground state properties we use a Green’s Function Monte Carlo, calculating energy gaps and correlation functions, the latter through the forward walking technique. In one dimension we find that the ground state is a “crystal” of valence bond dimers. In two dimensions, the spin gap appears to remain finite in the thermodynamic limit, and, consistently, the staggered magnetization—signal of Néel long range order—seems to vanish. From the analysis of dimer–dimer correlation functions, however, we find no sign of a valance bond crystal. A spin liquid appears as a plausible scenario compatible with our findings.