Spin-Wave Spectrum of the Quantum Ferromagnet on the Pyrochlore Lattice ${\mathrm{Lu}}_2{\mathrm{V}}_2{\mathrm{O}}_7$

Neutron inelastic scattering has been used to probe the spin dynamics of the quantum ($S=1/2$) ferromagnet on the pyrochlore lattice ${\mathrm{Lu}}_2{\mathrm{V}}_2{\mathrm{O}}_7$. Well-defined spin waves are observed at all energies and wave vectors, allowing us to determine the parameters of the Hamiltonian of the system. The data are found to be in excellent overall agreement with a minimal model that includes a nearest-neighbor Heisenberg exchange $J=8.22(2)\mathrm{meV}$ and a Dzyaloshinskii-Moriya interaction (DMI) $D=1.5(1)\mathrm{meV}$. The large DMI term revealed by our study is broadly consistent with the model originally used to explain the magnon Hall effect in this compound [Onose et al., Science 329, 297 (2010) and Ideue et al., Phys. Rev. B 85, 134411 (2012)]. However, our ratio of $D/J=0.18(1)$ is roughly half of their value, and is much larger than those found in other theoretical studies [Xiang et al., Phys. Rev. B 83, 174402 (2011) and Mook et al., Phys. Rev. B 89,134409 (2014)].

Figure 1

(a) Pyrochlore structure of ${\mathrm{Lu}}_2{\mathrm{V}}_2{\mathrm{O}}_7$. The ${\mathrm{V}}^{4+}$ ions (blue) form a network of vertex-sharing tetrahedra with four tetrahedra in the unit cell. (b) A single tetrahedron. The arrows located between adjacent ${\mathrm{V}}^{4+}$ sites denote the directions of the DMI vectors; e.g., the vector ${\overrightarrow{D}}_{i,j}$ is associated with the interaction between sites $i$ and $j$. Calculated spin-wave dispersion for ${\mathrm{Lu}}_2{\mathrm{V}}_2{\mathrm{O}}_7$ for (c) $D=0$ and (d) $D=J/3$ with the local magnetic field $\overrightarrow{H}$ parallel to either (100) (blue) or (010) (red). $\mathrm{\Gamma }$, $X$, $W$, and $K$ label the high-symmetry points of the conventional fcc Brillouin zone.

Figure 2

Neutron inelastic scattering data ($E_i=50\mathrm{meV}$) from ${\mathrm{Lu}}_2{\mathrm{V}}_2{\mathrm{O}}_7$ plotted as a function of the reduced momentum $t$ defined in the text. The data between 0 and 27 meV energy transfer (below the white gap) are obtained by averaging over data with $|\overrightarrow{Q}|<3\mathrm{r}\mathrm{.}\mathrm{l}\mathrm{.}\mathrm{u}\mathrm{.}$, while between 28 and 40 meV energy transfer (above the white gap) they are obtained by considering only data with $|\overrightarrow{Q}|<5\mathrm{r}\mathrm{.}\mathrm{l}\mathrm{.}\mathrm{u}\mathrm{.}$. The solid red line represents a fit of the dispersion with $J=8.1(1)$ and $D=0\mathrm{meV}$.

Figure 3

Comparison of data (left half) and mcphase simulations (right half). (a) $E_i=25\mathrm{meV}$, averaged with $L$ varying between 0.9 and 1.1 r.l.u. (c) $E_i=50\mathrm{meV}$, averaged between 1.9–2.1 r.l.u. in $L$. (e) $E_i=80\mathrm{meV}$, averaged between 31–35 meV in energy. A background has been subtracted for this data set only. Energy varies from 12 meV at the center to 4 meV at the edges in (a)–(b), 20 to 10 meV in (c)–(d). $L$ varies from 4.5 in the center to 2 at the edges in (e)–(f). (b), (d), and (f) are simulations performed with $J=8.1$ and $D=0\mathrm{meV}$. Dashed boxes represent regions of interest analyzed in cuts shown in Fig. 4.

Figure 4

Analysis of specific regions of ($\overrightarrow{Q}$, $\omega$) to determine the best values of $J$ and $D$. Thick red lines represent the best fit of the mcphase simulations with $J=8.22(2)$ and $D=1.5(1)\mathrm{meV}$, thin blue lines, are $J=8.22(2)$ and $D=0\mathrm{meV}$, dashed black lines are background. (a) and (b) Energy scans at $\sim 4J$ obtained by averaging data over $1.5<t(\overrightarrow{Q})<1.7$ and $0.7<t(\overrightarrow{Q})<0.9$, respectively [22]. (c) Wave vector dependence along H averaged for $0.9\le K\le 1.1$, $0.9\le L\le 1.1$, and $6\le E\le 8$ [box, Figs. 3]. (d) Wave vector dependence along ($HH1$) averaged for $-0.5\le (K\overline{K}1)\le 0.5$, $0\le L\le 6$, and $28\le E\le 35$ [box, Figs. 3].