Exact ground states of a spin-$\frac{1}{2}$ Ising-Heisenberg model on the Shastry-Sutherland lattice in a magnetic field

Exact ground states of a spin-1/2 Ising-Heisenberg model on the Shastry-Sutherland lattice with Heisenberg intradimer and Ising interdimer couplings are found by two independent rigorous procedures. The first method uses a unitary transformation to establish a mapping correspondence with an effective classical spin model, while the second method relies on the derivation of an effective hard-core boson model by continuous unitary transformations. Both methods lead to equivalent effective Hamiltonians providing a convincing proof that the spin-1/2 Ising-Heisenberg model on the Shastry-Sutherland lattice exhibits a zero-temperature magnetization curve with just two intermediate plateaus at one-third and one-half of the saturation magnetization, which correspond to stripe and checkerboard orderings of singlets and polarized triplets, respectively. The nature of the remarkable stripe order relevant to the one-third plateau is thoroughly investigated with the help of the corresponding exact eigenvector. The rigorous results for the spin-1/2 Ising-Heisenberg model on the Shastry-Sutherland lattice are compared with the analogous results for the purely classical Ising and fully quantum Heisenberg models. Finally, we discuss to what extent the critical fields of ${\mathrm{SrCu}}_2{(\mathrm{BO}}_3){}_2$ and ${\left(\mathrm{CuCl}\right)\mathrm{Ca}}_2{\mathrm{Nb}}_3{\mathrm{O}}_{10}$ can be described within the suggested Ising-Heisenberg model.

Figure 1

Schematic illustration of the spin-1/2 Ising-Heisenberg model on the Shastry-Sutherland lattice with the $XXZ$ Heisenberg intradimer interaction $J\left(\Delta \right)$ and the Ising interdimer interaction $J^{\prime }$. Thin (blue) lines show the Ising coupling and thick (red) lines denote the Heisenberg coupling. Spins inside the Heisenberg dimers are enumerated from left to right and from bottom to top, respectively (see also Fig. 2).

Figure 2

Schematic representation of the local cluster Hamiltonian $H_{[i-1:i+1],j}$ formed by three consecutive dimers in the horizontal direction and of its reduction to the Ising-like form.

Figure 3

Schematic representation of the effective classical spin model on the Shastry-Sutherland lattice obtained by applying the local unitary transformation (4) to each Heisenberg dimer.

Figure 4

Schematic representation of exact ground states of the spin-1/2 Ising-Heisenberg model on the Shastry-Sutherland lattice. The left and right panels show spin arrangements relevant to the classical effective spin model given by the Hamiltonian (19) and the original quantum spin model defined through the Hamiltonian (2), respectively. The rows from top to bottom correspond to the singlet-dimer phase, the antiferromagnetic phase, the stripe 1/3-plateau phase and the checkerboard 1/2-plateau phase. Shaded (transparent) dimers on the left panel denote the polarized triplet (singlet) states. On the right panel, the ellipse denotes a singlet-dimer state, filled circles denote spins oriented in a direction of the external magnetic field, empty circles denote spins oriented in opposite to the external magnetic field, whereas the reduced diameter of both kinds of circles corresponds to the quantum reduction of the local magnetization (25).

Figure 5

Ground-state phase diagram of the spin-1/2 Ising-Heisenberg model on the Shastry-Sutherland model in the $J^{\prime }\text{−}h$ plane for several values of the exchange anisotropy: (a) $\Delta =0.0$, (b) $0.25$, (c) $1.0$, and (d) $2.0$.

Figure 6

Comparison between the ground-state phase diagrams of the spin-1/2 Ising-Heisenberg and Heisenberg models on the Shastry-Sutherland model. Solid (black) lines represent the ground-state phase boundaries for the Ising-Heisenberg model (2) with the isotropic Heisenberg intradimer coupling $\Delta =1$. Filled circles are exact diagonalization data of the pure quantum Heisenberg model for $N=36$ spins and the broken curve represents the CORE results for the same model adapted from Ref. [12]. Empty squares and circles show the boundaries of the 1/3 and 1/2 plateaux obtained with iPEPS in Ref. [23].