Entangled spin-orbital phases in the bilayer Kugel-Khomskii model

Abstract

We derive the Kugel-Khomskii spin-orbital (SO) model for a bilayer and investigate its phase diagram depending on Hund's exchange $J_H$ and the $e_g$ orbital splitting $E_z$. In the (classical) mean-field approach with on-site spin $<S_i^z>$ and orbital $<\tau_i^z>$ order parameters and factorized spin-and-orbital degrees of freedom, we demonstrate a competition between the phases with either $G$-type or $A$-type antiferromagnetic (AF) or ferromagnetic long-range order. Next we develop a Bethe-Peierls-Weiss method with a Lanczos exact diagonalization of a cube coupled to its neighbors in $ab$ planes by the mean-field terms --- this approach captures quantum fluctuations on the bonds which decide about the nature of disordered phases in the highly frustrated regime near the orbital degeneracy. We show that the long-range spin order is unstable in a large part of the phase diagram which contains then six phases, including also the valence-bond phase with interlayer spin singlets stabilized by holes in $3z^2-r^2$ orbitals (VB$z$ phase), a disordered plaquette valence-bond (PVB) phase and a crossover phase between the VB$z$ and the $A$-type AF phase. When on-site SO coupling is also included by the $<S_i^z\tau_i^z>$ order parameter, we discover in addition two entangled phases which compete with $A$-type AF phase and another crossover phase in between the $G$-AF phase with occupied $x^2-y^2$ orbitals and the PVB phase. Thus, the present bilayer model provides an example of SO entanglement which generates novel disordered phases. We analyze the order parameters in all phases and identify situations where SO entanglement is crucial and factorization of the spins and orbitals leads to qualitatively incorrect results. We point out that SO entanglement may play a role in a bilayer fluoride K$_3$Cu$_2$F$_7$ which is a realization of the VB$z$ phase.Comment: 22 pages, 21 figure

Exotic Spin Order due to Orbital Fluctuations

“…Thereby we summarize the results of the earlier studies for a 2D monolayer [47], bilayer [48], and a 3D perovskite (cubic) system [49]. We show below that spin-orbital fluctuations and entanglement [7,15] plays a very important role here and stabilizes exotic types of magnetic order in all these systems.…”

“…Taking different types of spin order (i)-(iv), and assuming the classical average values of the spin projection operators, one finds the MF equations which are next solved for each of the considered three systems: the 2D monolayer, the bilayer, and the 3D perovskite. Solutions of the self-consistency equations and ground state energies in different phases can be obtained analytically, as explained on the example of the bilayer system in [48].…”

“…In a bilayer two ab planes are connected by interlayer bonds along the c axis [48], while a monolayer has only bonds within a single ab plane [47], i.e., γ = a, b. In all these cases spin interactions are described with the help of spin projection operators on a triplet or a singlet configuration on a bond ij ,…”

“…In reality the angles for the AO states vary in a continuous way as functions of the CF parameter E z /J, and therefore an efficient way of solving the one-site MF equations consists of assuming possible magnetic orders and for each of them deriving the effective orbital-only model. Such models are next compared and the phases with the lowest energy is found, as presented below for the case of the 2D monolayer, [47], the KK bilayer [48], and for the 3D cubic model [49]. The simplest approximation to obtain the possible types of order in a spin-orbital model, like the KK models Eq.…”

“…Phase diagrams of the KK models in the single-site MF approximation as obtained for: (a) the 2D monolayer [47], (b) the bilayer [48], and (c) the 3D perovskite system [49]. Shaded dark gray (green) areas indicate phases with AO order: FM and G-AF phase for the monolayer (a), and the A-AF, C-AF and G-AF phases for the bilayer (b) and the 3D perovskite (c) [here the FM phases which coexist also with the AO order are not shadded for clarity].…”