Spin-orbital resonating valence bond liquid on a triangular lattice: Evidence from finite-cluster diagonalization

We investigate the ground state of the $d^1$ spin-orbital model for triply degenerate $t_{2g}$ orbitals on a triangular lattice that unifies intrinsic frustration of spin and orbital interactions with geometrical frustration. Using full or Lanczos exact diagonalization of finite clusters, we establish that the ground state of the spin-orbital model that interpolates between superexchange and direct exchange interactions on the bonds is characterized by valence-bond correlations. In the absence of Hund’s exchange the model describes a competition between various possible valence-bond states. By considering the clusters with open boundary conditions we demonstrate that orbital interactions are always frustrated, but this frustration is removed by pronounced spin singlet correlations that coexist with dimer orbital correlations supporting them. Such local configurations contribute to the disordered ground states found for the clusters with periodic boundary conditions that interpolate between a highly resonating, dimer-based, entangled spin-orbital liquid phase and a valence-bond state with completely static spin-singlet states. We argue that these states are also realized for the infinite lattice and anticipate that pronounced transitions between different regimes found for particular geometries will turn out to smooth crossovers in the properties of the spin-orbital liquid in the thermodynamic limit. Finally, we provide evidence that the resonating spin-orbital liquid phase involves entangled states on the bonds. In such a phase classical considerations based on the mean-field theory cannot be used, spin exchange interactions do not determine spin bond correlations, and quantum fluctuations play a crucial role in the ground states and magnetic transitions.

Figure 1

(a) Schematic view of the hopping processes between $t_{2g}$ orbitals along a bond parallel to the $c$ axis in NaTiO${}_2$: (i) $t_{\mathit{pd}}$ between Ti($t_{2g}$) orbitals and O($2p_z$) orbitals, with two $t_{\mathit{pd}}$ transitions contributing to an effective hopping $t$; and (ii) direct $d$-$d$ hopping, $t^{\prime }$. The hopping $t$ interchanges two orbital flavors on two sites and contributes to the effective superexchange interactions on a bond in the triangular lattice, while the latter (diagonal) hopping element $t^{\prime }$ contributes to the direct (kinetic) exchange. The $t_{2g}$ orbitals are shown by different gray scales (color) and are labeled $a$, $b$, and $c$; see Eq. (2.1). Hopping processes contributing to (b) superexchange and (c) direct exchange are shown for the bonds along $\gamma =a,b,c$ axes in the triangular lattice.

Figure 2

Clusters (shown in gray) used to investigate the spin-orbital model, Eq. (2.3), by full or Lanczos exact diagonalization: (a) triangular clusters with open boundary conditions N3, N6, and N10-here nonequivalent sites and bonds used later are indicated by labels 1, 2, and 3, respectively; and (b) clusters with periodic boundary conditions—rhombic N4, hexagonal N7, and rhombic N9. In the latter case positions of neighboring clusters which cover the triangular lattice are indicated and all the sites and bonds are equivalent (label 1 is used for equivalent sites and bonds in N7 and N9 clusters).

Figure 3

Evolution of the g.s. for a hexagon H6 with OBC as a function of $\alpha$ in the absence of Hund’s exchange ($\eta =0$). (a) Bond correlations: spin $S_{\mathit{ij}}$, Eq. (2.20), circles; orbital $T_{\mathit{ij}}$, Eq. (2.21), squares; and spin-orbital $C_{\mathit{ij}}$, Eq. (2.22), triangles. (b) Orbital electron densities $n_{1\gamma }$ at site $i=1$ (left-most site): $n_{1a}$ (circles), $n_{1b}$ (squares), and $n_{1c}$ (triangles). Insets: Orbital configurations realized in the superexchange limit ($\alpha =0$), for $0.44<\alpha <0.63$, and in the direct exchange limit ($\alpha =1$). Vertical lines indicate the range of possible values of the particular quantity (bond correlation or electron density). They should not be confused with any kind of numerical error, as they indicate an exactly determined range due to the g.s. degeneracy.

Figure 4

Bond correlations for triangular clusters for increasing $\alpha$ in the absence of Hund’s exchange ($\eta =0$), as obtained for topologically equivalent bonds $n=1,2$ in triangular clusters N6 (top) and N10 (bottom) with OBC [see Fig. 2a]: spin (${\mathcal{S}}_n$; circles), orbital (${\mathcal{T}}_n$; squares), and spin-orbital (${\mathcal{C}}_n$; triangles) correlations.

Figure 5

Orbital occupations $n_{1\gamma }$ in triangular clusters for increasing $\alpha$ in the absence of Hund’s exchange ($\eta =0$): at sites $i=1,2$ for the N6 cluster (top) and at sites $i=1,2,3$ for the N10 cluster (bottom) [see Fig. 2a]. Insets: Two representative orbital occupancy distributions for the superexchange (left) and direct exchange (right) regimes.

Figure 6

Evolution of the frustrated g.s. for the N7 cluster with PBC shown in Fig. 2b with increasing $\alpha$ in the absence of Hund’s exchange: (a) intersite bond correlations—spin ($S_{\mathit{ij}}$; circles), orbital ($T_{\mathit{ij}}$; squares), and spin-orbital ($C_{\mathit{ij}}$; triangles); (b) orbital occupations $n_{1\gamma }$ per site (at a representative site $i=1$).

Figure 7

Evolution of the frustrated ground state for the N9 cluster with PBC shown in Fig. 2b with increasing $\alpha$ in the absence of Hund’s exchange: (a) intersite bond correlations—spin ($S_{\mathit{ij}}$, circles), orbital ($T_{\mathit{ij}}$, squares), and spin-orbital ($C_{\mathit{ij}}$, triangles); (b) orbital occupations $n_{1\gamma }$ per site (at a representative site $i=1$). Insets: typical orbital patterns in the superexchange ($\alpha =0$) limit and direct exchange ($\alpha =1$) limit.

Figure 8

Ground-state energy $E_0$ per bond for clusters N4 (filled squares), N7 (diamonds), and N9 (circles) with with PBC for increasing $\alpha$. For comparison the result obtained for cluster N7 with OBC is shown by open squares. Parameter: $\eta =0$.

Figure 9

Evolution of the orbital state in the FM ground state of the hexagonal H6 cluster with OBC, as found at large $\eta =0.2$ for increasing $\alpha$: (a) spin $S\equiv S_{\mathit{ij}}$, orbital $T\equiv T_{\mathit{ij}}$, and spin-orbital $C\equiv C_{\mathit{ij}}$ bond correlations; (b) orbital bond projection operators $P\equiv P_{\mathit{ij}}^{(\gamma )}$ (circles), $Q\equiv Q_{\mathit{ij}}^{(\gamma )}$ (squares), and $R\equiv R_{\mathit{ij}}^{(\gamma )}$ (triangles); (c) orbital electron densities $n_{1\gamma }$ on the leftmost cluster site $i=1$. Insets in (c): Two equivalent states in the superexchange ($\alpha =0$) and direct exchange ($\alpha =1$) limit.

Figure 10

Evolution of the orbital state in the FM g.s. of the N7 cluster for increasing $\alpha$, found with PBC at large $\eta =0.2$: (a) bond orbital correlations $T\equiv T_{\mathit{ij}}$; (b) orbital occupation correlations $P\equiv P_{\mathit{ij}}^{(\gamma )}$ (circles), $Q\equiv Q_{\mathit{ij}}^{(\gamma )}$ (squares), and $R\equiv R_{\mathit{ij}}^{(\gamma )}$ (triangles); (c) orbital electron densities $n_{1\gamma }$.

Figure 11

Phase diagram in the $(\alpha ,\eta )$ plane as obtained for the hexagonal H6 cluster with OBC. The total spin of the g.s. is indicated in the transition regime from the singlet phase ($\mathcal{S}=0$) to the high-spin ($\mathcal{S}=3$) phase.

Figure 12

Phase diagram in the $(\alpha ,\eta )$ plane as obtained for triangular clusters with OBC: N3 (squares),[35] N6 (diamonds), and N10 (circles). The phase diagrams of N6 and N10 clusters also contain intermediate spin phases and the corresponding phase boundaries are shown as dashed lines (for the N6 cluster only near $\alpha =1$).

Figure 13

Phase diagram in the $(\alpha ,\eta )$ plane as obtained for clusters with PBC: N4 (squares),[36] N7 (diamonds), and N9 (circles). Ranges of stability of intermediate spin phases for N7 and N9 clusters are shown by thin lines and arrows.

Figure 14

Phase diagrams obtained for disentangled spin-orbital interactions following Eq. (5.2), obtained for the N4, N7, and N9 clusters with PBC. No intermediate phases were found between low-spin ($\mathcal{S}={\mathcal{S}}_{\min }$) and high-spin ($\mathcal{S}={\mathcal{S}}_{\min }$) phases.

Figure 15

Contour plots of the effective exchange constant $J_{\mathrm{MF}}$ as obtained for the N7 cluster with PBC from Eq. (5.4): (a) within the MF calculation, which includes orbital fluctuations, and (b) using the exact g.s. found in exact diagonalization. (a) The transition from low-spin to high-spin phase occurs when the exchange constant $J_{\mathrm{MF}}$ changes sign and becomes negative. (b) Thick lines indicate the phase boundaries obtained between phases with increasing total spin value $\mathcal{S}=1/2$, 3/2, 5/2, and 7/2 for increasing $\eta$.