Three-Magnon Bound State in the Quasi-One-Dimensional Antiferromagnet $\alpha$-${\mathrm{NaMnO}}_2$

Here we report on the formation of a three-magnon bound state in the quasi-one-dimensional antiferromagnet $\alpha$-${\mathrm{NaMnO}}_2$, where the single-ion, uniaxial anisotropy inherent to the ${\mathrm{Mn}}^{3+}$ ions in this material provides a binding mechanism capable of stabilizing higher order magnon bound states. While such states have long remained elusive in studies of antiferromagnetic chains, neutron scattering data presented here demonstrate that higher order $n>2$ composite magnons exist, and, specifically, that a weak three-magnon bound state is detected below the antiferromagnetic ordering transition of ${\mathrm{NaMnO}}_2$. We corroborate our findings with exact numerical simulations of a one-dimensional Heisenberg chain with easy-axis anisotropy using matrix-product state techniques, finding a good quantitative agreement with the experiment. These results establish $\alpha$-${\mathrm{NaMnO}}_2$ as a unique platform for exploring the dynamics of composite magnon states inherent to a classical antiferromagnetic spin chain with Ising-like single ion anisotropy.

Figure 1

Neutron scattering data collected with $E_i=30\mathrm{meV}$ at $T=4\mathrm{K}$. The data were integrated throughout the entire zone in both $H$ and $L$. (a) Intensity of the data on a logarithmic scale, where gray regions indicate no detector coverage. The dashed, white line shows the direction of the energy cut plotted in (b). (b) Energy cut through the intensity map centered at the AF zone center, $K=0.5$, with the $n=1$, $n=2$, and $n=3$ magnon modes.

Figure 2

Constant $K$ cuts parametrizing the dispersions of the $n=2$ and $n=3$ modes. The data were integrated throughout the entire zone in both $H$ and $L$ and represent cuts through the color plot in Fig. 1. (a) Individual $K$ cuts parametrizing the dispersion of the $n=2$ mode. (b) Fit energies of the $n=2$ mode plotted as a function of $K$. (c) Individual $K$ cuts parametrizing the dispersion of the $n=3$ mode. (d) Energies of the $n=3$ mode as a function of $K$. The $K$ cuts in panels (a) and (c) are offset from one another for clarity, and the solid black lines are fits parametrizing the modes’ dispersions as described in the text. The solid orange lines in (b) and (d) are fits to the 1D dispersion described in the text. Error bars represent plus and minus one standard deviation.

Figure 3

$E_i=30\mathrm{meV}$ data showing the temperature dependence of spin excitations determined via an energy cut at the quasi-1D zone center, $\mathbf{Q}=(0,0.5,0)$. The inset shows a closeup in the region of the data, highlighting the temperature dependence of the $n=3$ mode. Data are integrated across a width of 0.01 r.l.u. in $K$.

Figure 4

Spectral function $\mathrm{S}(K,E)$ obtained numerically using the 1D model, for $J_1=5.34\mathrm{meV}$ and $D=0.46\mathrm{meV}$. The intensity is normalized to a log scale. Three modes are clearly seen in the vicinity of $K=0.5$. Blue dashed lines plotted on top correspond to dispersions obtained from experiment. The $n=1$ data is obtained from a $(H,H,0)$ cut, with integration in the $L$ direction from $-0.5<L(\mathrm{r}.\mathrm{l}.\mathrm{u}.)<0.5$ and integration in the $(-H,H,0)$ direction from $-0.01<H(\mathrm{r}.\mathrm{l}.\mathrm{u}.)<0.01$. The $n=2$ and $n=3$ data were obtained via cuts about the AF zone center from Fig. 1 and Fig. 2, respectively.