Theory of Magnetization Plateaux in the Shastry-Sutherland Model

Motivated by the remarkable properties of ${\mathrm{SrCu}}_2({\mathrm{BO}}_3{)}_2$ in a magnetic field, we use perturbative continuous unitary transformations to determine the magnetization plateaux of the Shastry-Sutherland model, unveiling an unexpected sequence of plateaux progressively appearing at $2/9$, $1/6$, $1/9$, and $2/15$ upon increasing the interdimer coupling. We predict that a $1/6$ plateau should be present in ${\mathrm{SrCu}}_2({\mathrm{BO}}_3{)}_2$, even if residual interactions beyond the Shastry-Sutherland are strong enough to modify the other plateaux below $1/3$. The method is extended to calculate the magnetization profile within the plateaux, leading to a local structure around triplons that agrees with NMR results on ${\mathrm{SrCu}}_2({\mathrm{BO}}_3{)}_2$.

Figure 1

Shastry-Sutherland lattice and definition of the two-body interactions. $V_n$ is the coefficient of the two-body interactions between the thick dimer and the dimer labeled $V_n$.

Figure 2

Largest process of most relevant types in the Hamiltonian $H_{\mathrm{eff}}$ and its amplitude $C_{r_1,\cdots ,r_n}$ in units of $J$ at $J^{\prime }/J=0.5$.

Figure 3

Coefficients of the extrapolated two-body interactions as a function of $J^{\prime }/J$. Inset: Different extrapolants (solid lines) as well as the bare series (dashed lines) for $V_3^{\prime }$ and $V_5$.

Figure 4

Magnetization plateaux as a function of $\mu$ and $J^{\prime }/J$. The boson density $n$ is equal to the magnetization in units of the saturation value, and the chemical potential $\mu$ is equal to the magnetic field $B$. Solid line denote results that are fully converged with respect to the terms kept in the Hamiltonian (see Fig. 6). Well-converged results are then connected by dashed lines.

Figure 5

Spin density ($S^z$) profile of the main plateaux at $J^{\prime }/J=0.5$. Full (empty) circles corresponds to magnetization along (opposite to) the magnetic field. The radius of the circles is proportional to the magnetization amplitude. The blue line shows the unit cell compatible with the periodicity of the state. For the $2/15$ plateau, two structures have the same energy within the error bars of the method.

Figure 6

Magnetization curve at $J^{\prime }/J=0.5$: comparison of the results obtained by keeping terms with up to 2 creators (dotted blue curve), 3 creators (solid blue curve) and 4 creators (black curve). Well-converged plateaux used in Fig. 4 are indicated explicitly. Inset: Energy of the $1/9$, rhomboid $1/8$, $2/15$, and $1/6$ plateaux as a function of $\mu$ at $J^{\prime }/J=0.5$. The error bars from the $D\log $ Padé extrapolation are smaller than the line width.