Quantum melting of magnetic long-range order near orbital degeneracy: Classical phases and Gaussian fluctuations

We address the role played by orbital degeneracy in strongly correlated transition-metal compounds. Specifically, we study the effective spin-orbital model derived for the $d^9$ ions in a three-dimensional perovskite lattice, as in ${\mathrm{KCuF}}_3,$ where at each site the doubly degenerate $e_g$ orbitals contain a single hole. The model describes the superexchange interactions that depend on the pattern of orbitals occupied and shows a nontrivial coupling between spin and orbital variables at nearest-neighbor sites. We present the ground-state properties of this model, depending on the splitting between the $e_g$ orbitals $E_z,$ and the Hund’s rule coupling in the excited $d^8$ states, $J_H.$ The classical phase diagram consists of six magnetic phases which all have different orbital ordering: two antiferromagnetic (AF) phases with G-AF order and either $x^2-y^2$ or ${3z}^2-r^2$ orbitals occupied, two phases with mixed orbital (MO) patterns and A-AF order, and two other MO phases with either C-AF or G-AF order. All of them become degenerate at the multicritical point $M\equiv {(E}_z{,J}_H)=(0,0).$ Using a generalization of linear spin-wave theory we study both the transverse excitations which are spin waves and spin-and-orbital waves, as well as the longitudinal (orbital) excitations. The transverse modes couple to each other, providing a possibility of measuring the new spin-and-orbital excitations in inelastic neutron spectroscopy. As the latter excitation turns into a soft mode near the M point, quantum corrections to the long-range-order parameter are drastically increased near the orbital degeneracy, and classical order is suppressed in a crossover regime between the G-AF and A-AF phases in the ${(E}_z{,J}_H)$ plane. This behavior is reminiscent of that found in frustrated spin models, and we conclude that orbital degeneracy provides a different and physically realizable mechanism which stabilizes a spin liquid ground state due to inherent frustration of magnetic interactions. We also point out that such a disordered magnetic phase is likely to be realized at low $J_H$ and low electron-phonon coupling, as in ${\mathrm{LiNiO}}_2.$