Exotic spin orders driven by orbital fluctuations in the Kugel-Khomskii model

Abstract

We study zero temperature phase diagram of the three-dimensional Kugel-Khomskii model on a cubic lattice using the cluster mean field theory and different perturbative expansions in the orbital sector. The phase diagram is rich, goes beyond the single-site mean field theory due to spin-orbital entanglement. In addition to the antiferromagnetic (AF) and ferromagnetic (FM) phases, one finds also a plaquette valence-bond phase with singlets ordered either on horizontal or vertical bonds. More importantly, for increasing Hund's exchange we identify three phases with exotic magnetic order stabilized by orbital fluctuations in between the AF and FM order: (i) an AF phase with two mutually orthogonal antiferromagnets on two sublattices in each ab plane and AF order along the c axis (ortho-G-type phase), (ii) a canted-A-type AF phase with a non-trivial canting angle between nearest neighbor FM layers along the c axis, and (iii) a striped-AF phase with anisotropic AF order in the ab planes. We elucidate the mechanism responsible for each of the above phases by deriving effective spin models which involve second and third neighbor Heisenberg interactions as well as four-site spin interactions going beyond Heisenberg physics, and explain how the entangled nearest neighbor spin-orbital superexchange generates spin interactions between more distant spins. Published in Physical Review B 87, 064407 (2013).

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.…”

“…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]. The remaining G-AF phases (G-AFx and G-AFz) are accompanied by FO order with fully polarized orbitals, either x (for Ez > E [47], (b) the bilayer [48], and (c) the 3D perovskite system [49]. Plaquette valence-bond (PVB) phase with alternating spin singlets in the ab planes is highlighted in light gray (yellow) -it occurs between the phases with long-range magnetic order, and the QCP is hidden.…”

“…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].…”

“…We developed this so-called cluster MF approximation for the KK models by embedding a foursite cluster in the plaquette MF (PMF) used in the 2D model [47], and a cube in the bilayer case [48]. The most natural choice for the 3D perovskite is a cube as well, but here we limit ourselves to four-site clusters [49] (a plaquette or a linear cluster) to avoid tedious numerical analysis.…”