Excitation spectra and ground state properties of the layered spin-$\frac{1}{2}$ frustrated antiferromagnets ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_4$ and ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Br}}_4$

We use series expansion methods to study ground- and excited-state properties in the helically ordered phase of spin-$1∕2$ frustrated antiferromagnets on an anisotropic triangular lattice. We calculate the ground state energy, ordering wave vector, sublattice magnetization and one-magnon excitation spectrum for parameters relevant to ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_4$ and ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Br}}_4$. Both materials are modeled in terms of a Heisenberg model with spatially anisotropic exchange constants; for ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_4$ we also take into account the additional Dzyaloshinskii-Moriya (DM) interaction. We compare our results for ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_4$ with unpolarized neutron scattering experiments and find good agreement. In particular, the large quantum renormalizations of the one-magnon dispersion are well accounted for in our analysis, and inclusion of the DM interaction brings the theoretical predictions for the ordering wave vector and the magnon dispersion closer to the experimental results.

Figure 1

Exchange constants $J$, $J^{\prime }$ and lattice basis vectors ${\mathbf{\delta }}_1$, ${\mathbf{\delta }}_2$ on a triangular lattice. The standard notation for axes in ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_4$ is also indicated.

Figure 2

The ground state energy per site, as a function of the angle $q$ between nearest-neighbor spins along $J^{\prime }$ bonds, obtained from the series expansion around the Ising limit. (a) $J∕J^{\prime }=2$, relevant for ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Br}}_4$. Here the left and right vertical dashed lines denote the value of $q$ obtained from dimer expansions [Eq. (16) in Ref. 17], and the classical ground state, respectively. (b) $J∕J^{\prime }=2.92$, the ratio appropriate for ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_4$. The DM interaction $(D=0.16J^{\prime })$ lowers the ground state energy and shifts the ordering angle from $q\approx 1.59$ to $q\approx 1.64$.

Figure 3

(Color online) Reciprocal space diagram for the triangular lattice. The path $ABCOAD$ [$A=(\pi ,0)$, $B=(\pi ,\pi ∕\sqrt{3})$, $C=(2\pi ∕3,2\pi ∕\sqrt{3})$, $O=(0,0)$, and $D=(2\pi ,0)$] and the three sets of points denoted (1)–(3) are cuts along which magnon dispersions are plotted in subsequent figures in the paper.

Figure 4

(Color online) Magnon dispersion ${\omega }_{\mathit{k}}$, as calculated from series expansions, for parameters relevant to ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_4$, plotted along the path $ABCOAD$ shown in Fig. 3. When the DM interaction is included the magnon energy is significantly enhanced along $AB$ and near the point $D$, and a gap opens up at the ordering wave vector $\mathit{k}=\mathit{Q}$.

Figure 5

(Color online) Magnon dispersion for $J∕J^{\prime }=2.92$ and $D∕J^{\prime }=0.16$ (solid points from series, black solid line from spin-wave theory with $1∕S$ correction) compared to experimental dispersion in ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_4$ (squares from Ref. 3, dashed line is experimental parametrization) along the path $ABCOAD$ in Fig. 3. Compared to the dispersion from spin-wave theory, the series dispersion is enhanced along the $J$ bonds and decreased perpendicular to them. Note that theoretical and experimental dispersions cannot be directly compared near the ordering wave vector $\mathit{k}=\mathit{Q}$, as the gap in ${\omega }_{\mathit{k}}$ expected at this wave vector cannot be resolved in the unpolarized neutron scattering experiments (see text about how the experimental dispersion is extracted).

Figure 6

(Color online) The magnon modes ${\omega }_{\mathit{k}}$ (principal mode) and ${\omega }_{\mathit{k}\pm \mathit{Q}}\equiv {\omega }_{\mathit{k}}^{\pm }$ (secondary modes), as calculated from spin-wave theory with $1∕S$ corrections,[9] for ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_4$ ($J∕J^{\prime }=2.92$ and $D∕J^{\prime }=0.16$), plotted along the path ABCOAD shown in Fig. 3.

Figure 7

(Color online) Comparison of the experimentally measured dispersion relation in ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_4$ (red squares) and series expansion results for $J=0.385\mathrm{meV}$, $J∕J^{\prime }=2.92$, and $D∕J^{\prime }=0.16$ (black circles), along paths (1), (2), and (3) in Fig. 3. Calculations were made at the same $(k_x,k_y)$ where experimental dispersion points were measured.

Figure 8

(Color online) Magnon dispersion for ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Br}}_4$ [$J=2J^{\prime }$ and $D=0$], calculated from series expansions. Compared to the dispersion from linear spin-wave theory [red dashed line, Eq. (12)] the series dispersion is enhanced along the $J$ bonds and decreased perpendicular to them. The dispersion looks qualitatively rather similar to the case $J∕J^{\prime }=2.92$, $D=0$ in Fig. 4 although in that case the energy is practically flat along $AB$ (i.e., perpendicular to the chains) while here the energy varies somewhat along $AB$, giving a less one-dimensional dispersion, as expected. The horizontal black dashed line is at $\omega ∕J=\pi ∕2$ which is the maximum energy in the one-dimensional limit $J∕J^{\prime }=\infty$.