Effect of Dzyaloshinskii-Moriya interactions on the phase diagram and magnetic excitations of SrCu${}_2$(BO${}_3$)${}_2$

The orthogonal dimer structure of the SrCu${}_2$(BO${}_3$)${}_2$ spin-1/2 magnet provides a realization of the Shastry–Sutherland model. Using a dimer-product variational wave function, we map out the phase diagram of the Shastry–Sutherland model including anisotropies. Based on the variational solution, we construct a bond-wave approach to obtain the excitation spectra as a function of the magnetic field. The characteristic features of the experimentally measured neutron and ESR spectra are reproduced, like the anisotropy-induced zero-field splittings and the persistent gap at higher fields.

Figure 1

(The Shastry–Sutherland lattice in SrCu${}_2$(BO${}_3$)${}_2$. Orthogonal dimers (thick lines) are denoted by A and B, while Cu ions on the dimer are numbered 1 and 2. Interdimer bonds (with Heisenberg exchange $J^{\prime }$) are shown by thin lines. We also show the ${\mathcal{S}}_4$ and ${\mathcal{C}}_{2v}$ symmetry point groups of the buckled CuBO${}_3$ layer. Open and filled circles indicate Cu ions that are below or above the layer, respectively.

Figure 2

Components of symmetry allowed (a) intradimer and (b) interdimer Dzyaloshinskii-Moriya (DM) vectors shown in a unit cell. Arrows running from $i$ to $j$ on bonds indicate the ordering of the spin operators ${\mathbf{S}}_i\times {\mathbf{S}}_j$ in the DM interaction term.

Figure 3

Dispersions of quasi-triplet excitations in a zero magnetic field: ${\omega }_{3,4}^{-}$ (bottom), ${\omega }_{1,2}$ (middle), and ${\omega }_{3,4}^{+}$ (top). We chose $D=D_{\perp }^{\prime }=0.1J$ and $J^{\prime }=0.6J$.

Figure 4

(a) Phase diagram in the $h_z$-$D_{\perp }^{\prime }$ plane for $D=0$ and $J^{\prime }/J=0.6$. DS is the dimer-singlet phase that remains a variational ground state even for finite values of $D_{\perp }^{\prime }$. The spin configurations in the different phases are shown in Fig. 5. $m=1/2$ plat. denotes the half-magnetization plateau phase, with a singlet and a magnetized triplet in each unit cell. (b) Magnetization curves for different values of $\left|D_{\perp }^{\prime }\right|$ as a function of the magnetic field.

Figure 5

Schematic representation of the $O(2)$ symmetric phases in the high-symmetry ($D=0$) case with a magnetic field perpendicular to the CuBO${}_3$ layer (i.e., $h\Vert z$). We have plotted the expectation values of the spin component in the $\mathit{xy}$ plane. Since in this case $S^z$ is conserved, the spins can be arbitrarily rotated by a global $O(2)$ rotation in the plane. Blue (open arrows) and red (filled arrows) represent inequivalent spins.

Figure 6

(a) Phase diagram in the $h_z$-$D_{\perp }^{\prime }$ plane for $J^{\prime }/J=0.6$ and $D/J=0.1$ (low-symmetry case). In comparison to the $D=0$ case in Fig. 4, in a large region of the phase space the dimer singlet and the $O(2)[S_4]$ essentially merged to create the $Z_1[D_{2d}]$ phase, a small part of the $O(2)[S_4]$ phase became a twofold degenerate $Z_2[S_4]$, and the $O(2)[C_4]$ merged with the $m=1/2$ magnetization plateau phase into the $Z_2[C_{2v}]$ phase. (b) Magnetization curves for a few selected values of $D_{\perp }^{\prime }$.

Figure 7

Bond-wave spectrum at the $\Gamma$ point when we keep the singlets and the $S^z=1$ triplets only. The magnetic field is parallel to the $z$ axis, $J^{\prime }=0.6J$, and $D_{\perp }^{\prime }=0.1J$. The transition into the $O(2)$ phase happens at $h_z\approx 0.77J$, and that into the plateau phase, at $h_z\approx 1.1J$. For $0.77\lesssim h_z/J\lesssim 1.03$, one of the excitations becomes identically 0; this is the Goldstone mode of the $O(2)[{\mathcal{S}}_4]$ phase. The Goldstone mode of the $O(2)\times Z_2$ phase is 0 for $1.03\lesssim h_z/J\lesssim 1.1$. The two-dimer variational solution is unstable in the gray region; the dispersion goes to 0 at some wave vector away from the $\Gamma$ point at $h_z=0.96$ and $1.08$. The filled area above the dispersion line shows the strength of the spin structure factor $S^{\mathit{xx}}+S^{\mathit{yy}}$. The dashed line is the approximation from Ref.37.

Figure 8

Excitation spectrum in a magnetic field parallel to the $z$ axis when $D_{\perp }^{\prime }=0$, $J^{\prime }/J=0.6$. The instabilities toward helical states are at $h_z/J=0.9421$ and $1.1002$, while the $\mathbf{k}=0$ instability into the $Z_2$ phase is at 0.9515.

Figure 9

Excitation spectrum in a magnetic field parallel to the $z$ axis in the low-symmetry case, for $D_{\perp }^{\prime }>0$, $D/J=0.1$, and $J^{\prime }/J=0.6$. Notation is the same as in Fig. 7.

Figure 10

Excitation spectrum in a magnetic field parallel to the $z$ axis for $D_{\perp }^{\prime }<0$. $J^{\prime }=0.6J$.

Figure 11

Phase diagram for the magnetic field in the $x$ direction. $J^{\prime }/J=0.6$ and $D/J=0.1$. The spin configurations in the different phases are shown in Fig. 12.

Figure 12

Schematic representation of the spin configurations in the phases $Z_1[{\mathcal{C}}_{2v}]$, $Z_2[{\mathcal{C}}_s^{(x)}]$, and $Z_2[{\mathcal{C}}_s^{(y)}]$ that appear when the field is along the $x$ direction. The darker and lighter arrows represent the two degenerate states.

Figure 13

Excitation spectrum in a magnetic field parallel to the $x$ axis. $J^{\prime }=0.6J$.

Figure 14

(a) Qualitative comparison of the excitation spectrum with the $h\Vert c$ ESR spectrum shown in Fig. 4a of Nojiri et al.[22] (b) Diamonds are the far-infrared data from Ref.43, and squares are the ESR data from Ref.22.

Figure 15

(a) The lowest-lying branch of the excitation spectrum at $\mathbf{k}=0$ is shown when keeping two (thick lines) and two bosons (thin lines) per dimer for $J^{\prime }=0.6J$, $D_{\perp }^{\prime }=0.1J$, and different values of $\mathrm{D̃}$. (b) The bond-wave spectrum has a dip at the $h_{c1}=0.8J$ critical field. The dotted line is the approximation from Ref.37; the dashed line and the circle are the approximations given by Eqs. (B17) and (B19), respectively.