Magnon dispersion and anisotropies in $\mathrm{Sr}{\mathrm{Cu}}_2{\left(\mathrm{B}{\mathrm{O}}_3\right)}_2$

We study the dispersion of the magnons (triplet states) in $\mathrm{Sr}{\mathrm{Cu}}_2{\left(\mathrm{B}{\mathrm{O}}_3\right)}_2$ including all symmetry-allowed Dzyaloshinskii-Moriya interactions [J. Phys. Chem. Solids 4, 241 (1958); Phys. Rev. 120, 91 (1960)]. We can reduce the complexity of the general Hamiltonian to a simpler form by appropriate rotations of the spin operators. The resulting Hamiltonian is studied by both perturbation theory and exact numerical diagonalization on a 32-site cluster. We argue that the dispersion is dominated by Dzyaloshinskii-Moriya interactions. We point out which combinations of these anisotropies affect the dispersion to linear order, and extract their magnitudes.

Figure 1

(Color online) Dispersion relation of the first-triplet excitation of the Heisenberg Shastry-Sutherland model (diamonds), as calculated by exact numerical diagonalization on a 32-site cluster, for $J^{\prime }∕J=0.62$. The vertical axis shows the excitation energy of those states with respect to the ground state. The degeneracy of the two triplets at $(\pi ∕2,\pi ∕2)$ is slightly lifted, reflecting the two dimers per unit cell. Singlet states nearby are shown as open circles.

Figure 2

(Color online) Shastry-Sutherland lattice with Dzyaloshinskii-Moriya interactions allowed by the symmetries of $\mathrm{Sr}{\mathrm{Cu}}_2{\left(\mathrm{B}{\mathrm{O}}_3\right)}_2$. The colored (short gray) arrows are the components of the Dzyaloshinskii-Moriya vectors for each bond [the long arrows show the prescription for the order of the operators, a black arrow from $i$ to $j$ indicates that we should take $\mathbf{D}\bullet ({\mathbf{S}}_i\times {\mathbf{S}}_j)$ with the given $\mathbf{D}$]. $m_1$ and $m_2$ are mirror planes, but the $\left(ab\right)$ plane is not a mirror plane in the low-temperature phase. This allows for all in-plane components of $\mathbf{D}$.

Figure 3

$(J^{\prime }∕J\to 0)$ Dispersion relation and dynamical structure factor (gray scale) of the first- triplet excitations in the limit of small $J^{\prime }∕J$. Note that the overall dimer form factor $M_q$ is not included in the picture. For $(\pi ,0)$, the middle mode has no intensity. For the clarity of the picture, ${\tilde{D}}_{\perp }=0.1J$ and ${\tilde{D}}_{\Vert }=0.05J$ were used.

Figure 4

(Color online) Same as Fig. 1 but for Hamiltonian (11) with $J^{\prime }∕J=0.62$, ${\tilde{D}}_{\perp }=0.02J$ and ${\tilde{D}}_{\Vert }=0.005J$. The lines are guides for the eyes and join together states which would correspond to $S^z=+1$ (triangles), $-1$ (squares), and 0 (diamonds) in the limit ${\tilde{D}}_{\Vert }\to 0$. Note that some of them are indistinguishable for the upper mode.

Figure 5

(Color online) (a) Effect of ${\tilde{D}}_{\perp }$ on the $\mathbf{q}=\mathbf{0}$ triplet (diamonds) and singlet energies (open circles) when ${\tilde{D}}_{\Vert }=0$ and (b) effect of ${\tilde{D}}_{\Vert }$ on the $\mathbf{q}=(\pi ,0)$ states when ${\tilde{D}}_{\perp }=0.02J$. Numbers are calculated using exact diagonalization on the 32-site cluster. The lines are linear fittings to triplet energies.

Figure 6

(Color online) Magnon dispersion: exact diagonalization with Dzyaloshinskii-Moriya interactions (diamonds) and experimental data from Ref. 12 (circles). The lines are guides for the eyes. Parameters are ${\tilde{D}}_{\perp }=0.18\mathrm{meV}$ and ${\tilde{D}}_{\Vert }=0.07\mathrm{meV}$. $J^{\prime }∕J$ was kept equal to 0.62. This is responsible for the shift in the overall scale of the calculated energies.