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

We study the 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, and 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 nontrivial 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.

Figure 1

(Left panel) Schematic view of four representative orbital configurations on a representative cube of the 3D lattice: (a) AO order with $\langle {\tau }_i^{a\left(b\right)}\rangle =1/2$ changing from site to site and $\langle {\tau }_i^c\rangle =-1/4$, obtained for $E_z<0$; (b) AO order with $\langle {\tau }_i^{a\left(b\right)}\rangle =-1/2$ changing from site to site and $\langle {\tau }_i^c\rangle =1/4$, obtained for $E_z>0$; (c) FO order with occupied $z$ orbitals and $\langle {\tau }_i^c\rangle =-1/2$ (cigar-shaped orbitals); and (d) FO order with occupied $x$ orbitals and $\langle {\tau }_i^c\rangle =1/2$ (clover-shaped orbitals). (Right panel) Schematic view of four representative spin configurations (arrows stand for up or down spins) shown on a cubic cluster: (i) $A$-AF configuration, (ii) $C$-AF configuration, (iii) FM configuration, and (iv) $G$-AF configuration.

Figure 2

Phase diagram of the 3D KK model obtained in the single-site MF approximation. Shaded dark gray (green) area indicates phases with AO order while the remaining magnetic phases are accompanied by FO order with fully polarized orbitals, either $x$ (for $E_z>0$) or $z$ (for $E_z<0$). In this approach the $G$-AF$x$ and $C$-AF$x$ phases with FO order are degenerate.

Figure 3

Schematic view of the clusters (solid lines) used in the cluster MF approach of Sec. 3 to the 3D KK model: (a) a plaquette in the $ab$ plane, and (b) a chain along the $c$ axis. Vertices $i=1,2,3,4$ and directions $\gamma =a,b,c$ are marked; dashed lines stand for the outgoing bonds where the spin-orbital interactions are replaced by the MF terms at neighboring sites [see Eq. (3.1)].

Figure 4

Phase diagram of the 3D KK model in the cluster MF approximation. Plaquette valence-bond (PVB) phase with alternating spin singlets in the $ab$ planes, highlighted in light gray (yellow), occurs between the phases with magnetic long-range order; see Fig. 1. Phases with exotic magnetic order are shaded in dark gray (orange).

Figure 5

Classical view of the ortho-$G$-AF spin order realized in the 3D KK model within $ab$ planes, and staggered along the $c$ axis. Vertical and horizontal arrows correspond to two interpenetrating AF states on the sublattices of next nearest neighbor sites. Up (down) arrows stand for $\langle S_i^z\rangle =\pm 1/2$; right (left) arrows stand for $\langle S_i^x\rangle =\pm 1/2$.

Figure 6

Schematic views of the two exotic spin orders realized by the 3D KK model at large Hund's exchange $\eta >0.2$. (a) Striped-AF order in the $ab$ plane, with AF order along the $c$ axis and angle ${\varphi }^a$ between the NN spins along the $a$ axis; (b) spin order realized in the canted-$A$-AF phase with FM order in $ab$ planes and spin canting angle $\theta$ along the $c$ axis. The regions of stability of these phases are shown in Fig. 4 by dark gray (orange) shading.

Figure 7

Evolution of the magnetic order for increasing $\eta$ at $E_z=-0.5J$. (a) Cosine of the canting angle $\theta$ (3.3), and (b) total magnetization $\left|s\right|$ (3.4); both in the $A$-AF, canted-$A$-AF, and FM phases from left to right. Quantum phase transitions are indicated by vertical dotted lines.

Figure 8

Evolution of spin and orbital correlations along the $c$ axis at distance $d$, for $E_z=-0.5J$ and increasing $\eta$. (a) Spin correlation functions $C_s^c\left(d\right)$ (3.5), and (b) on-site $r^{a\left(b\right)}$ (3.6) and bond $R^c\left(d\right)$ (3.7) spin-orbital covariances, as obtained in the $A$-AF, canted-$A$-AF, and FM phase. Quantum phase transitions are indicated by vertical dotted lines.

Figure 9

Evolution from the striped-AF to $G$-AF phase at $\eta =0.22$. (a) Spin $s_{1,3}^{x\left(z\right)}$ and orbital $t^{a\left(b\right)}$ order parameters, and (b) total magnetization $\left|s\right|$ (3.4) and cosines of spin angles ${\varphi }^a$ and ${\varphi }^b$; see Fig. 6a. Quantum phase transition is indicated by vertical dotted lines.

Figure 10

Evolution of covariances from the striped-AF to $G$-AF phase at $\eta =0.22$ under increasing $E_z/J$ in the $ab$ plane. (a) On-site spin-orbital covariances (3.6) for the $x$ spin component, $r_{1\left(3\right)}^{x,a}$ and $r_{1\left(3\right)}^{x,b}$, and (b) on-site spin-orbital covariances (3.6) for the $x$ spin component, $r_{1\left(3\right)}^{z,a}$ and $r_{1\left(3\right)}^{z,b}$ (3.6), together with bond spin-orbital covariances $R^{a,b}$ (3.7). Quantum phase transitions are indicated by vertical dotted lines.

Figure 11

Orbital moment $\left|\tau \right|$ (3.9) found in the cluster MF approximation at the horizontal cut of the phase diagram of Fig. 4 for $\eta =0.13$ and increasing $E_z$ (solid line). Quantum fluctuations reduce $\left|\tau \right|$ from the classical value of $0.5$ found in the single-site MF (dashed line). Quantum phase transitions are indicated by vertical dotted lines.

Figure 12

Schematic view of the order imposed by the Hamiltonian (4.7) in the $ab$ plane. Interactions between NNN spins 1 and 3 (and equivalent) are AF, between 3NN 1 and 4 are FM, but they vanish for NN spins 1 and 2, leaving the angle $\phi$ undetermined.

Figure 13

Schematic view of the $G$-AF spin order on two $ab$ planes of 3D cubic lattice stabilized by the third-order FM NNN Heisenberg interactions between sites $i+\gamma$ and $i+c$ shown by dashed (green) lines [see Eq. (4.22)]. The NN FM interactions between $i$ and $i+c$, shown by vertical (red) line, are frustrated in the $G$-AF phase.

Figure 14

Schematic views of second-order corrections in the effective spin Hamiltonian: (a) effective NNN interactions, and (b) effective 3NN interaction. Red frames stand for Heisenberg bond with $\pm$ sign depending on the bond's direction and solid (magenta) dots are the single-site orbital excitations in the ground state.

Figure 15

Schematic view of third-order corrections to $H_s$ (frames stand for Heisenberg bonds with $\pm$ sign depending on their direction; solid circles indicate orbital flips): (a) term bringing new physics, chain product of three bonds, (b) term doubling the results from second-order expansion, interaction between spins 2 and 3 is squared (bond marked by a thick frame).