Exploring the spin-orbital ground state of ${\mathrm{Ba}}_3{\mathrm{CuSb}}_2{\mathrm{O}}_9$

Motivated by the absence of both spin freezing and a cooperative Jahn-Teller effect at the lowest measured temperatures, we study the ground state of ${\mathrm{Ba}}_3{\mathrm{CuSb}}_2{\mathrm{O}}_9$. We solve a general spin-orbital model on both the honeycomb and the decorated honeycomb lattice, revealing rich phase diagrams. The spin-orbital model on the honeycomb lattice contains an SU(4) point, where previous studies have shown the existence of a spin-orbital liquid with algebraically decaying correlations. For realistic parameters on the decorated honeycomb lattice, we find a phase that consists of clusters of nearest-neighbor spin singlets, which can be understood in terms of dimer coverings of an emergent square lattice. While the experimental situation is complicated by structural disorder, we show qualitative agreement between our theory and a range of experiments.

Figure 1

The crystal structure of ${\mathrm{Ba}}_3{\mathrm{CuSb}}_2{\mathrm{O}}_9$ [5]. ${\mathrm{Cu}}^{2+}\text{−}{\mathrm{Sb}}^{5+}$ dumbbells are surrounded by an oxygen bioctahedra, and form a triangular lattice. An electric-dipole interaction between the dumbbells favors antiparallel nearest-neighbor alignment, and this results in a short-range ordered honeycomb lattice of ${\mathrm{Cu}}^{2+}$ ions. ${\mathrm{Cu}}^{2+}$ has the electron configuration $3d^9$, and, therefore, there is on average one hole per Cu site. The dominant interaction is superexchange via Cu-O-O-Cu pathways (shown by yellow dots). Due to the bonding angles, the superexchange interaction is comparable on paths 1 and 2, but considerably weaker on path 3.

Figure 2

Mapping $t/t^{\prime }\to -t/t^{\prime }$, illustrated on a 6-site cluster with periodic boundary conditions. Bond labels A, B, and C correspond to different ${\mathbf{n}}_{ij}$ vectors [see Eq. (12)]. Sites are numbered $1,\cdots ,6$, and are circled to show the periodic boundary conditions. The $t/t^{\prime }\to -t/t^{\prime }$ mapping proceeds in two steps. First an on-site orbital rotation is performed, given in Table 1; and second a lattice restructuring is performed, which involves swapping sites 3 and 5. An equivalent mapping can be made on the 18-site cluster.

Figure 3

Phase diagram for ${\mathcal{H}}_{ST}$ [Eq. (8)] on the honeycomb lattice, as calculated from (a) exact diagonalization and (b) spin-orbital decoupling on a 6-site cluster. Illustrations of phases show spins in blue, with ellipsoids on bonds representing spin singlets, and orbitals in red. At $t/t^{\prime }=1$ and $J/U=0,{\mathcal{H}}_{ST}$ is SU(4) symmetric, and the two regions continuously connected to this point are labeled as SU(4) phases. The gold region has $S_{\mathrm{tot}}=1,T_{\mathrm{tot}}\approx 0$ and the white region $S_{\mathrm{tot}}=0,T_{\mathrm{tot}}\approx 1$. The light blue region has $S_{\mathrm{tot}}=0$, noncollinear ordering of nearest-neighbor spin dimers (NCD) and 3-sublattice orbital order of $d^{x^2-y^2}$ type. The darker blue region has $S_{\mathrm{tot}}=0$, collinear ordering of nearest-neighbor spin dimers (CD) and ferro-orbital order of $d^{x^2-y^2}$ type. The red region is ferromagnetic, with antiferro-orbital ordering for $t/t^{\prime }>0$ and 6-sublattice orbital order for $t/t^{\prime }<0$. The orbitals alternate between $d^{x^2-y^2}$ and $d^{3z^2-r^2}$ type on neighboring sites. Finally, the small yellow region close to $t/t^{\prime }=0$ is an intermediary spin state with $S_{\mathrm{tot}}=1$.

Figure 4

Phase diagram for ${\mathcal{H}}_{ST}$ [Eq. (8)] on the honeycomb lattice, as calculated within the spin-orbital decoupling scheme on an 18-site cluster. Illustrations of phases show spins in blue, with ellipsoids on bonds representing spin-singlets, and orbitals in red. The SU(4) phase contains the SU(4) point at $t=t^{\prime }$ and $J/U=0$. The rotated SU(4) phase is related by the orbital mapping described in Sec. 2c. The noncollinear dimer phase (NCD) is an $S_{\mathrm{tot}}=0$ phase with nearest-neighbor spin singlets crystallized in a noncollinear pattern and $d^{x^2-y^2}$ type orbital order. The collinear dimer (CD) phase involves crystallization of spin dimers in a collinear pattern, with $d^{x^2-y^2}$ type ferro-orbital order. At large $J/U$ the spin configuration is ferromagnetic. For $t/t^{\prime }>0$ there is antiferro-orbital order with an alternation of $d^{x^2-y^2}$ and $d^{3z^2-r^2}$ type orbitals, while for $t/t^{\prime }<0$ the orbital order has a 6-sublattice structure. Finally there is an intermediate spin region with $S_{\mathrm{tot}}=3$.

Figure 5

The decorated honeycomb lattice. A honeycomb lattice of Cu ions is decorated by the addition of ${\mathrm{Cu}}^{\prime }$ ions, forming Cu-${\mathrm{Cu}}^{\prime }$-Cu isoceles triangles. Bonds on the honeycomb lattice are labeled A, B, and C, as in Fig. 2, while Cu-${\mathrm{Cu}}^{\prime }$ bonds are labeled ${\text{B}}^{\prime }$ and ${\text{C}}^{\prime }$.

Figure 6

Phase diagram for ${\mathcal{H}}_{ST}+{\mathcal{H}}_{ST}^{\prime }$ [Eqs. (8) and (18)] on the decorated honeycomb lattice, calculated using (a) Lanczos diagonalization and (b) spin-orbital decoupling on a 12-site cluster. Parameters $t_a/t^{\prime }=2/3,t_b/t^{\prime }=0$ and $t_{ab}/t^{\prime }=-1/\sqrt{3}$ are used. At small $J/U$ the phase diagram is dominated by an $S=0$ phase (yellow, ED) in which three nearest neighbor spin singlets (blue ellipses) form on 6-site clusters. This cluster of three singlet bonds and associated orbital order (shown in red) can be thought of as a dimer (black ellipse) on an emergent square lattice (see Figs. 7 and 8), and we label it the emergent dimer phase (ED). The 12-site cluster is too small to determine the ordering pattern of these dimers in the thermodynamic limit. The red $S=0$ phase has the same spin-singlet pattern, but a different orbital state. Other phases include a pair of $S=2$ phases (green), which consist of ferromagnetic spins on the honeycomb lattice aligned antiparallel to the spins on the ${\mathrm{Cu}}^{\prime }$ sites. Orbital order on the honeycomb lattice is of $d^{x^2-y^2}$ type, while orbitals on the ${\mathrm{Cu}}^{\prime }$ sites are $d^{3z^2-r^2}$ type. For large $J/U$ the spins order ferromagnetically (orange, FM), and the orbital ordering is likely incommensurate. There are also intermediate phases (light and dark blue), which do not match between the exact diagonalization and the spin-orbital decoupling. The $S=0$ phase (dark blue) in the spin-orbital decoupling phase diagram involves ferromagnetically aligned chains that order antiferromagnetically with neighboring chains. This has a significant overlap with the first excited state of the $S=4$ (dark blue) phase calculated via exact diagonalization.

Figure 7

6-site cluster composed of four Cu and two ${\mathrm{Cu}}^{\prime }$ sites, useful for understanding the ground state of ${\mathcal{H}}_{ST}+{\mathcal{H}}_{ST}^{\prime }$ [Eqs. (8) and (18)] at small $J/U$. Spins form nearest-neighbor singlets (shown in blue) on the 6-site cluster. An angle $\theta =arctan[\langle T^z\rangle /\langle T^x\rangle ]$ is used to specify the orbital degree of freedom (shown in red). Angles are calculated for $t/t^{\prime }=-1/3$ and $J/U=0$, but are representative of the entire phase.

Figure 8

Emergent square lattice useful for understanding the ground state of ${\mathcal{H}}_{ST}+{\mathcal{H}}_{ST}^{\prime }$ [Eqs. (8) and (18)] at small $J/U$. The decorated honeycomb lattice (see Fig. 5) is shown projected onto a plane (green). Spin singlets (blue ellipses) form on this lattice in groups of three, with associated orbital state, as shown in Fig. 7. These can be thought of as dimers (grey ellipses) on an emergent square lattice (grey sites, black dashed bonds). How the dimers are arranged on this lattice remains an open question.

Figure 9

Overlap between the ground state wave function determined by Lanczos diagonalization and a linear superposition of the eight orientations of the emergent dimer (ED) state found using the spin-orbital decoupling scheme. The coefficients of the superposition are adjusted to maximise the overlap. The overlap is plotted as a function of $J/U$, with $t/t^{\prime }=-1/3$. The overlap is significant throughout the ED phase.