Spin dynamics of the quasi-two-dimensional spin-$\frac{1}{2}$ quantum magnet ${\mathrm{Cs}}_2{\mathrm{CuCl}}_4$

We study dynamical properties of the anisotropic triangular quantum antiferromagnet ${\mathrm{Cs}}_2{\mathrm{CuCl}}_4$. Inelastic neutron scattering measurements have established that the dynamical spin correlations cannot be understood within a linear spin wave analysis. We go beyond linear spin wave theory by taking interactions between magnons into account in a $1∕S$ expansion. We determine the dynamical structure factor and carry out extensive comparisons with experimental data. We find that compared to linear spin wave theory, a significant fraction of the scattering intensity is shifted to higher energies and strong scattering continua are present. However, the $1∕S$ expansion fails to account for the experimentally observed large quantum renormalization of the exchange energies.

Figure 1

(Color online) The magnetic sites and exchange couplings within a single layer of ${\mathrm{Cs}}_2{\mathrm{CuCl}}_4$. Layers are stacked along the crystallographic $a$ direction with an interlayer spacing $a∕2$ and a relative displacement in the $c$ direction.

Figure 2

(Color online) The reciprocal space diagram of ${\mathrm{Cs}}_2{\mathrm{CuCl}}_4$ projected along the $(0,k,l)$ plane. The $\Gamma$ points refer to the center of the Brillouin zone and $\mathbf{Q}$ is the ordering wave vector. The path of the cut shown in Fig. 5 is depicted as a dashed line.

Figure 3

(Color online) Incommensurate spiral order. The thick arrows represent the spin direction of the spiral order. The crystallographic axes $(a,b,c)$ and the local reference frame of the spins $(x,y,z)$ are indicated. The spins lie in the $bc$ plane and the angle between consecutive spins along the b direction is $\theta =2\pi (0.5+{ϵ}_0)=199°$. In the rotating reference frame, the $z$ axis is the average spin direction, whereas the $x$ and $y$ axes are the transverse directions.

Figure 4

(Color online) The renormalization of the spin wave spectrum. The solid and dashed lines are results of the $1∕S$ expansion ${\tilde{\omega }}_{\mathbf{k}}$ [Eq. (40)] and the linear spin wave dispersion ${\omega }_{\mathbf{k}}$ [Eq. (16)]. The cut in the paramagnetic Brillouin zone runs from (000) to (010) to (011) and back to the center of the Brillouin zone (000).

Figure 5

(Color online) Density plot of the scattering cross section [Eq. (39)] as a function of energy and wave vector. The light (dark) regions represent regions of small (large) scattering intensity. The results have been convolved with the experimental energy resolution of the detectors (the full width at half maximum is $\Delta E=0.016\mathrm{meV}$). The magnetic form factor of copper in Eq. (39) shows very weak wave vector dependence in the regime of interest and, therefore, was taken to be at unity (Ref. 33). The filled circles along the (010) direction are the experimental position of the most intense peaks in the line shapes taken in the spiral phase $(T<0.1\mathrm{K})$ (Ref. 4).

Figure 6

(Color online) Upper panel (a): The integrated spectral weights for the three modes [Eq. (41)] as functions of momentum transfer. The principal ${\omega }^0$ and secondary ${\omega }^{+},{\omega }^{-}$ spin wave modes are defined in the text. Lower panel (b): The ratios of the spectral weights of the single-particle excitations of each mode to their respective integrated intensities [Eq. (42)].

Figure 7

(Color online) The scattering intensity as a function of energy [Eq. (43)] for LSW theory (dashed line) and $\mathrm{LSW}+1∕\mathrm{S}$ expansion (solid line). The contributions of the scattering continua are shown using thin lines.

Figure 8

(Color online) Density plot of the scattering cross section as a function of energy and wave vector along the cut $\left(0k\frac{3}{2}\right)$. The light (dark) regions represent regions of small (large) scattering intensity.

Figure 9

(Color online) The dispersion relation of magnetic excitations. The shaded regions labeled with capital letters $A$ through $K$ indicate scan directions (the line thickness indicates the wave vector averaging). The filled symbols are the main peaks in the line shape as determined experimentally in the ordered phase (from Ref. 4. The dotted area indicates the extent of the scattering continuum. The open circles and squares are, respectively, the upper and lower boundaries of the scattering continuum as determined experimentally. The upper thick dashed line is a guide to the eye. The thin solid line is the experimental fit to the principal mode using effective parameters $(\tilde{J}=0.61\mathrm{meV},{\tilde{J}}^{\prime }=0.107\mathrm{meV})$. The thick solid, dashed, and dash-dotted lines are, respectively, the $1∕S$ results for the principal $\left({\omega }_{\mathbf{k}}^0\right)$ and secondary $({\omega }_{\mathbf{k}}^{+},{\omega }_{\mathbf{k}}^{-})$ modes determined from Eq. (40).

Figure 10

(Color online) Scattering cross section. The numbered panels (1)–(4) correspond to energy scans $B$, $E$, $G$, and $H$, respectively. The data has been convolved with the energy and spatial resolution. $\Delta E=0.016\mathrm{meV}$ for all plots and $\Delta k=\{0.035,0.039,0.085,0.056\}$ for plots (1)–(4). The insets show the results of LSW theory and the $1∕S$ expansion $(\mathrm{LSW}+1∕S)$ for $\Delta k=0,\Delta E=0.002\mathrm{meV}$.

Figure 11

Observed neutron scattering line shape in scan $G$ (1) and $H$ (2) (data from Ref. 4 in the ordered phase $(T<0.1\mathrm{K})$. In scan $G$, the shaded area represents the the $1∕S$ calculation [Eq. (39)] convolved with the experimental resolution.

Figure 12

(Color online) Calculated scattering cross sections in LSW theory and $1∕S$ expansion $(\mathrm{LSW}+1∕S)$ for scans $A$ and $C$. Panels (2) and (4) show the $\mathrm{LSW}+1∕S$ results with instrumental resolutions of $\Delta E=0.016\mathrm{meV}$ and $\Delta k=0.02$ (for scan $A$) and $\Delta k=0.04$ (for scan $C$) taken into account. See Fig. 5 from Ref. 4 to compare to experimental data.