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Spin and orbital structure in the singlet phases of the mean-field phase diagram. Solid line indicates AF, dashed line FM spin correlations.
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Motivated by the absence of cooperative Jahn-Teller effect and of magnetic ordering in LiNiO2, a layered oxide with triangular planes, we study a general spin-orbital model on the triangular lattice. A mean-field approach reveals the presence of several singlet phases between the SU(4) symmetric point and a ferromagnetic phase, a conclusion support...
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... we have checked that this orbital structure also re- produces satisfactorily the mean value of T i .T j measured in the mean-field ground state. This turned out to give a clear picture in all phases except A and E. The informa- tion obtained in this way is summarized in Fig. 9. In the following, we describe in more details all these ...
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... the magnetic point of view, this phase consists essentially of weakly coupled, antiferromagnetic chains (see Fig. 9), while the orbital structure turns out to be rather subtle with an antiferro-orbital arrangement of ferro-orbital chains with orbitals which are neither pure |a = |d 3z 2 −r 2 nor |b = |d x 2 −y 2 but alternate between (−|a + |b). The detailed magnetic structure depends a priori on the residual couplings be- tween the chains. If the ...
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... they correspond to d 3z 2 −r 2 , d 3x 2 −r 2 or d 3y 2 −r 2 depend- ing on the orientation of the bond. Note that all these orbitals are Jahn-Teller active, leading in all cases to two long bonds and four short bonds. One might be tempted to conclude that these phases correspond to two types of valence bond solids with the patterns depicted in Fig. 9. The mean-field approach has a very remarkable property however: In addition to the mean-field solutions with lowest energy shown in Fig. 9, there are several other mean-field solutions of the self-consistent equations with energies very close to the lowest energy corresponding to other dimer coverings of the triangular lattice. In ...
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... Jahn-Teller active, leading in all cases to two long bonds and four short bonds. One might be tempted to conclude that these phases correspond to two types of valence bond solids with the patterns depicted in Fig. 9. The mean-field approach has a very remarkable property however: In addition to the mean-field solutions with lowest energy shown in Fig. 9, there are several other mean-field solutions of the self-consistent equations with energies very close to the lowest energy corresponding to other dimer coverings of the triangular lattice. In such circumstances, going beyond mean-field is likely to cou- ple these solutions, and the relevant model would then be a quantum dimer model ...
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... that on a given bond either T i = −T j , or T i + T j is per- pendicular to n z ij . These conditions are satisfied with the collinear orbital order shown in Fig. 10: we choose |a + |b along every second chain with the bond variable n z ij = (0, 0, 1), and |a − |b along the remaining chains (the orbital configuration is the same as in phase B in Fig. 9). There are 6 such configurations, which can be obtained by translations and rotations, with variational ...
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... The ground state is dimerized on the square lattice and breaks the SU(4) symmetry [20]. The fate of the triangular lattice, considered initially as a model for LiNiO 2 in Ref. [21], is still open. Exact diagonalization suggests the importance of the SU(4) singlet plaquettes [22], the uniform mean field produces a parton Fermi sea with closed orbits in the reciprocal space [23], extended honeycomb reduced (8-site u.c.) Brillouin zones Γ′ 2 (a) (b) FIG. 1. ...
... The ground state is dimerized on the square lattice and breaks the SU(4) symmetry [20]. The fate of the triangular lattice, considered initially as a model for LiNiO 2 in Ref. [21], is still open. Exact diagonalization suggests the importance of the SU(4) singlet plaquettes [22], the uniform mean field produces a parton Fermi sea with closed orbits in the reciprocal space [23], ...
We compute the momentum resolved dynamical spin structure factor $S(k,\omega)$ of the SU(4) Heisenberg model on the honeycomb lattice assuming the $\pi$-flux Dirac spin liquid ground state by two methods: (i) variationally using Gutzwiller projected particle-hole excitations of the $\pi$-flux Fermi sea and (ii) in the non-interacting parton mean-field picture. The two approaches produce qualitatively similar results. Based on this analogy, we argue that the energy spectrum of the projected excitations is a gapless continuum of fractional excitations. Quantitatively, the Gutzwiller projection shifts the weight from higher to lower energies, thus emphasizing the lower edge of the continuum. In the mean-field approach, we obtained the $1/\text{distance}^4$ decay of the spin correlation function, and the local correlations show $S^{33}_{\text{MF}}(\omega)\propto \omega^3$ behavior.
... Independently, concrete spin-orbital 15 ) models, which capture single-site spin and orbital degeneracies, have been shown to host a rich spectrum of phenomena 15 , notably, valence bond solids [16][17][18][19][20][21][22][23][24] and orbital liquids (e.g. refs. ...
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... Two key considerations make the search for quantumdisordered phases in spin-orbital systems particularly promising [21][22][23][24][25][26]: On one hand [1], some spin-orbital models may have SU(4)-symmetric points in their parameter space [7,11,23,27,28]. It is expected that near such high-symmetry points, quantum fluctuations become enhanced, as magnetic order in generalized Heisenberg antiferromagnets with SU(N ) or Sp(N ) symmetry have been shown to become increasingly unstable upon enlarging the symmetry group, even on unfrustrated lattices [29,30]. ...
Spin-orbital liquids are quantum disordered states in systems with entangled spin and orbital degrees of freedom. We study exactly solvable spin-orbital models in two dimensions with selected Heisenberg-, Kitaev-, and $\Gamma$-type interactions, as well as external magnetic fields. These models realize a variety of spin-orbital-liquid phases featuring dispersing Majorana fermions with Fermi surfaces, nodal Dirac or quadratic band touching points, or full gaps. In particular, we show that Zeeman magnetic fields can stabilize nontrivial flux patterns and induce metamagnetic transitions between states with different topological character. Solvable nearest-neighbor biquadratic spin-orbital perturbations can be tuned to stabilize zero-energy flat bands. We discuss in detail the examples of $\mathrm{SO}(2)$- and $\mathrm{SO}(3)$-symmetric spin-orbital models on the square and honeycomb lattices, and use group-theoretical arguments to generalize to $\mathrm{SO}(\nu)$-symmetric models with arbitrary integer $\nu > 1$. These results extend the list of exactly solvable models with spin-orbital-liquid ground states and highlight the intriguing general features of such exotic phases. Our models are thus excellent starting points for more realistic modellings of candidate materials.
... Another quantum spin liquid state on a hexagonal lattice is realized in Ba 3 CuSb 2 O 9 with Cu 2+ (S=1/2) [10,11]. The spin liquid phase in Ba 3 CuSb 2 O 9 can be associated with fluctuation in the orbital sector [12][13][14][15] in contrast to the Kitaev state where the orbital degeneracy is lifted due to the strong spin-orbit interaction. ...
Both the Jahn-Teller distortion of Cu$^{2+}$O$_6$ octahedra and magnetic ordering are absent in hexagonal Ba$_3$CuSb$_2$O$_9$ suggesting a Cu 3$d$ spin-orbital liquid state. Here, by means of resonant x-ray scattering and absorption experiment, we show that oxygen 2$p$ holes play crucial role in stabilizing this spin-orbital liquid state. These oxygen holes appear due to the "reaction" Sb$^{5+}$$\rightarrow$Sb$^{3+}$ $+$ two oxygen holes, with these holes being able to attach to Cu ions. The hexagonal phase with oxygen 2$p$ holes exhibits also a novel charge-orbital dynamics which is absent in the orthorhombic phase of Ba$_3$CuSb$_2$O$_9$ with Jahn-Teller distortion and Cu 3$d$ orbital order. The present work opens up a new avenue towards spin-charge-orbital entangled liquid state in transition-metal oxides with small or negative charge transfer energy.
... Considering Sb atom does not have any spin electrons, thus no magnetic frustration can be observed. [30] Similar with LiNiO 2 , [31] NaNiO 2 can be considered as a prominent example of strong geometrically frustrated material as it shows neither spin nor orbital ordering, of which the frustration parameter value is estimated to be 1.8. However in www.advmat.de ...
... Mater. 2019,31, 1903483 ...
Sodium‐based layered oxides are among the leading cathode candidates for sodium‐ion batteries, toward potential grid energy storage, having large specific capacity, good ionic conductivity, and feasible synthesis. Despite their excellent prospects, the performance of layered intercalation materials is affected by both a phase transition induced by the gliding of the transition metal slabs and air‐exposure degradation within the Na layers. Here, this problem is significantly mitigated by selecting two ions with very different MO bond energies to construct a highly ordered Ni6‐ring superstructure within the transition metal layers in a model compound (NaNi2/3Sb1/3O2). By virtue of substitution of 1/3 nickel with antimony in NaNiO2, the existence of these ordered Ni6‐rings with super‐exchange interaction to form a symmetric atomic configuration and degenerate electronic orbital in layered oxides can not only largely enhance their air stability and thermal stability, but also increase the redox potential and simplify the phase‐transition process during battery cycling. The findings reveal that the ordered Ni6‐ring superstructure is beneficial for constructing highly stable layered cathodes and calls for new paradigms for better design of layered materials.
... The most well-studied examples of KK models display two-orbital degeneracy and can be implemented in three distinct solid-state platforms. Historically, the first one arises in Mott insulators with e g orbitals [1], where the orbital Hilbert space is spanned by d 3z 2 −r 2 and d x 2 −y 2 orbitals [10][11][12][13][14]. The second platform comprises t 2g Mott insulators with 4/5d 1 magnetic species, in which the strong spin-orbit coupling (SOC) favors a low-energy j = 3/2 multiplet [15]. ...
The SU(4)-symmetric spin-orbital model on the honeycomb lattice was recently studied in connection to correlated insulators such as the $e_{g}$ Mott insulator Ba$_{3}$CuSb$_{2}$O$_{9}$ and the insulating phase of magic-angle twisted bilayer graphene at quarter filling. Here we provide a unified discussion of these systems by investigating an extended model that includes the effects of Hund's coupling and anisotropic, orbital-dependent exchange interactions. Using a combination of mean-field theory, linear flavor-wave theory, and variational Monte Carlo, we show that this model harbors a quantum spin-orbital liquid over a wide parameter regime around the SU(4)-symmetric point. For large Hund's coupling, a ferromagnetic antiferro-orbital ordered state appears, while a valence-bond crystal combined with a vortex orbital state is stabilized by dominant orbital-dependent exchange interactions.
... In their pioneering work Kugel and Khomskii [2] have shown that when degenerate orbitals are partly filled, spin-orbital superexchange couples spin and orbital degrees of freedom. It leads to phases with spin-orbital superexchange in two-dimensional (2D) [3][4][5][6][7][8] or in three-dimensional (3D) [9][10][11][12][13][14][15][16][17][18][19][20] systems. When both spin and orbital degrees of freedom are active joint spin-orbital quantum fluctuations arise and may even destabilize long-range order [21]. ...
We show that even when spins and orbitals disentangle in the ground state, spin excitations are renormalized by the local tuning of e g orbitals in ferromagnetic planes of K 2 CuF 4 and LaMnO 3 . As a result, dressed spin excitations (magnons) obtained within the electronic model propagate as quasiparticles and their energy renormalization depends on momentum . Therefore magnons in spin-orbital systems go beyond the paradigm of the effective Heisenberg model with nearest neighbor spin exchange derived from the ground state - spin-orbital entanglement in excited states predicts large magnon softening at the Brillouin zone boundary, and in case of LaMnO 3 the magnon energy at the M = (π, π) point may be reduced by ∼45%. In contrast, simultaneously the stiffness constant near the Goldstone mode is almost unaffected. We elucidate physics behind magnon renormalization in spin-orbital systems and explain why long wavelength magnons are unrenormalized while simultaneously energies of short wavelength magnons are reduced by orbital fluctuations. In fact, the -dependence of the magnon energy is modified mainly by dispersion which originates from spin exchange between second neighbors along the cubic axes a and b. © 2019 The Author(s). Published by IOP Publishing Ltd on behalf of Deutsche Physikalische Gesellschaft.
... In their pioneering work Kugel and Khomskii [2] have shown that when degenerate orbitals are partly filled, spin-orbital superexchange couples spin and orbital degrees of freedom. It leads to phases with spin-orbital superexchange in two-dimensional (2D) [3][4][5][6][7][8] or in threedimensional (3D) [9][10][11][12][13][14][15][16][17][18][19][20] systems. When both spin and orbital degrees of freedom are active joint spin-orbital quantum fluctuations arise and may even destabilize long-range order [21]. ...
... The fit may be refined by taking into account the next-nearest and third neighbor interactions, J 2 and J 3 , in the effective spin model. One finds that J 2 = 0.25 × 10 −3 J is rather small but J 3 1.35 × 10 −3 J is significant and plays an important role. Both fits are shown in Fig. 3(b). ...
We show that even when spins and orbitals disentangle in the ground state, spin excitations are renormalized by the local tuning of $e_g$ orbitals in ferromagnetic planes of K$_2$CuF$_4$ and LaMnO$_3$. As a result, dressed spin excitations (magnons) obtained within the electronic model propagate as quasiparticles and their energy renormalization depends on momentum ${\vec k}$. Therefore magnons in spin-orbital systems go beyond the paradigm of the effective Heisenberg model with nearest neighbor spin exchange derived from the ground state --- spin-orbital entanglement in excited states predicts large magnon softening at the Brillouin zone boundary, and in case of LaMnO$_3$ the magnon energy at the $M=(\pi,\pi)$ point may be reduced by $\sim 45$\%. In contrast, simultaneously the stiffness constant near the Goldstone mode is almost unaffected. We elucidate physics behind magnon renormalization in spin-orbital systems and explain why long wavelength magnons are unrenormalized while simultaneously energies of short wavelength magnons are reduced by orbital fluctuations. In fact, the ${\vec k}$-dependence of the magnon energy is modified mainly by dispersion which originates from spin exchange between second neighbors along the cubic axes $a$ and $b$.
... On the other hand, electronic orbital degree of freedom, which responds to 'the crystal electric field', is the minimal unit of degrees of freedom related to 'the shape' of localized valence electron inside a material, and is an important element that determines, based on the Neumann principle [4], the relation between anisotropic physical properties and crystallographic point group, and the dimensionality of the system responding to the external field. 'Orbital liquid' material, which does not show any orbital freezing at low temperatures, has been predicted theoretically since 1990s and its experimental verification in some transition metal oxides have been carried out [5][6][7][8][9][10][11][12]. However, their realization has remained hypothetical to date. ...
... Hence, relevant choice of system, precise physical properties measurements and in-depth theoretical understanding are necessary for the search of the candidate materials for orbital liquids. Among such attempts to realize the novel state, the quasi two dimensional (2D) triangular lattice magnet Li 1−x Ni 1+x O 2 with active e g orbitals was once thought to be a promising candidate [9,11,13,14]. Extensive research has been made to reveal the origin of the observed behaviors hinted a possible orbital liquid state. In contrast with the general wisdom that the intersite orbital exchange coupling should be highly anisotropic due to the Hund's coupling as well as anisotropic character of the wave function, an interesting limit of the spin-orbital lattice (Kugel-Khomskii) model with highly isotropic SU(4) symmetry was discovered and found to stabilize a spin-orbital liquid state [7]. ...
... In contrast with the general wisdom that the intersite orbital exchange coupling should be highly anisotropic due to the Hund's coupling as well as anisotropic character of the wave function, an interesting limit of the spin-orbital lattice (Kugel-Khomskii) model with highly isotropic SU(4) symmetry was discovered and found to stabilize a spin-orbital liquid state [7]. More realistic models have been proposed to understand experimental observations revealing intrinsic and extrinsic ferromagnetic interactions supported by orbital ordered and/or disordered states [9,11,13,14]. Experimentally, however, it was later found that Li 1−x Ni 1+x O 2 exhibits orbital freezing [10]. ...
Structure with orbital degeneracy is unstable toward spontaneous distortion. Such orbital correlation usually has a much higher energy scale than spins, and therefore, magnetic transition takes place at a much lower temperature, almost independently from orbital ordering. However, when the energy scales of orbitals and spins meet, there is a possibility of spin-orbital entanglement that would stabilize novel ground state such as spin-orbital liquid and random singlet state. Here we review on such a novel spin-orbital magnetism found in the hexagonal perovskite oxide Ba<sub>3</sub>CuSb<sub>2</sub>O<sub>9</sub>, which hosts a self-organized honeycomblike short-range structural order of a strong Jahn-Teller ion Cu<sup>2+</sup>. Comprehensive structural and magnetic measurements have revealed that the system has neither magnetic nor Jahn-Teller transition down to the lowest temperatures, and Cu spins and orbitals retain the hexagonal symmetry and paramagnetic state. Various macroscopic and microscopic measurements all indicate that spins and orbitals remain fluctuating down to low temperatures without freezing, forming a spin-orbital entangled liquid state.
... Also, geometrical frustration may cause a heavy fermion state in LiV 2 O 4 (Kondo et al., 1997;Urano et al., 2000) and other materials (Stewart, 1984), or an anomalous Hall effect induced by the chiral order (Tatara & Kawamura, 2002), a resonating valence bond state predicted in LiNiO 3 (Vernay et al., 2004), and so on. ...
The spinel oxide AlV 2 O 4 is a unique material, in which the formation of clusters is accompanied by atomic, charge and orbital ordering and a rhombohedral lattice distortion. In this work a theory of the structural phase transition in AlV 2 O 4 is proposed. This theory is based on the study of the order-parameter symmetry, thermodynamics, electron density distribution, crystal chemistry and mechanisms of formation of the atomic and orbital structures of the rhombohedral phase. It is established that the critical order parameter is transformed according to irreducible representation k9 (τ 4 ) (in Kovalev notation) of the Fd \bar{3}m space group. Knowledge of the order-parameter symmetry allows us to show that the derived AlV 2 O 4 rhombohedral structure is a result of displacements of all atom types and the ordering of Al atoms (1:1 order type in tetrahedral spinel sites), V atoms (1:1:6 order type in octahedral sites) and O atoms (1:1:3:3 order type), and the ordering of d xy , d xz and d yz orbitals. Application of the density functional theory showed that V atoms in the Kagomé sublattice formed separate trimers. Also, no sign of metallic bonding between separate vanadium trimers in the heptamer structure was found. The density functional theory study and the crystal chemical analysis of V—O bond lengths allowed us to assume the existence of dimers and trimers as main clusters in the structure of the AlV 2 O 4 rhombohedral modification. The trimer model of the low-symmetry AlV 2 O 4 structure is proposed. Within the Landau theory of phase transitions, typical diagrams of possible phase states are built. It is shown that phase states can be changed as a first-order phase transition close to the second order in the vicinity of tricritical points of the phase diagrams.
