Thermal Evolution of Dirac Magnons in the Honeycomb Ferromagnet ${\mathrm{CrBr}}_3$

${\mathrm{CrBr}}_3$ is an excellent realization of the two-dimensional honeycomb ferromagnet, which offers a bosonic equivalent of graphene with Dirac magnons and topological character. We perform inelastic neutron scattering measurements using state-of-the-art instrumentation to update 50-year-old data, thereby enabling a definitive comparison both with recent experimental claims of a significant gap at the Dirac point and with theoretical predictions for thermal magnon renormalization. We demonstrate that ${\mathrm{CrBr}}_3$ has next-neighbor $J_2$ and $J_3$ interactions approximately 5% of $J_1$, an ideal Dirac magnon dispersion at the $K$ point, and the associated signature of isospin winding. The magnon lifetime and the thermal band renormalization show the universal $T^2$ evolution expected from an interacting spin-wave treatment, but the measured dispersion lacks the predicted van Hove features, pointing to the need for more sophisticated theoretical analysis.

Figure 1

(a) Honeycomb layer of ${\mathrm{CrBr}}_3$. The ${\mathrm{Cr}}^{3+}$ ions (blue) host $S=3/2$ spins with FM interactions. (b),(c) Schematic spin-wave spectra in the vicinity of the $K$ points. When inversion symmetry is preserved, the dispersion forms a Dirac cone (b); otherwise, a gap opens to form separate acoustic and optical magnon branches (c). (d)–(g) Scattered intensity obtained by integrating the PANTHER $E_i=30\mathrm{meV}$ dataset over four constant-energy windows (indicated) and compared with linear spin-wave theory (LSWT); all panels have the same intensity scale. White lines indicate the boundaries of the crystallographic BZ. A $\mathbf{Q}$-independent background was subtracted from each spectrum to aid visual comparison. (h) Intensity obtained on winding around the $K$ point, showing a cosinusoidal modulation with inverted phase for energies above and below the Dirac point. Solid lines show the corresponding fits [Supplemental Material Eq. (S1) [19] ] with an additive background term.

Figure 2

Low-temperature magnon dispersion measured along several high-symmetry directions. (a) Reciprocal-space representation of the honeycomb lattice in the $(hk0)$ plane showing the crystallographic BZ in blue and the unfolded zone in orange. Green, red, and violet arrows indicate the paths for the spectra shown, respectively, in panels (b)–(d); the gray arrow indicates the complete path used in Fig. 3. (b)–(d) Spin-wave spectra collected using PANTHER with $E_i=30\mathrm{meV}$ at $T=1.7\mathrm{K}$. The data were integrated by $\pm 0.03$ r.l.u. along the orthogonal in-plane direction and $\pm 5$ r.l.u. in the out-of-plane $l$ direction. Red and green points show, respectively, the dispersions of modes 1 and 2 modeled by LSWT and their sizes represent the calculated intensities.

Figure 3

Thermal magnon renormalization in ${\mathrm{CrBr}}_3$. (a),(b) Magnon spectra for the high-symmetry directions taken from the PANTHER $E_i=30\mathrm{meV}$ dataset at $T=1.7\mathrm{K}$ (a) and $T=30\mathrm{K}$ (b). The data were integrated over $\pm 0.015{\AA }^{-1}$ in-plane and $\pm 2{\AA }^{-1}$ for $l$. (c) Magnon branches 1 and 2 extracted for the four experimental temperatures; circles were taken from the $E_i=30\mathrm{meV}$ dataset and squares from $E_i=15\mathrm{meV}$. The mode intensities vanish in some regions of the unfolded BZ. (d) Temperature-induced renormalization of the measured magnon dispersions for modes 1 (up) and 2 (down), shown in the reduced form of Eq. (2). Solid lines show the real part of the self-energy, $\mathrm{Re}\mathrm{\Sigma }(\mathbf{q})$, obtained by adapting the analytical framework of Ref. [4], in which calculations are performed for the upper (red) and lower (blue) bands in the crystallographic BZ; dashed lines show the Hartree term, ${\mathrm{\Sigma }}_1(\mathbf{q})$. (e) Reduced thermal renormalization of the measured magnon linewidths for modes 1 (up) and 2 (down). Solid lines show $-\mathrm{Im}\mathrm{\Sigma }(\mathbf{q})$ obtained following Ref. [4].

Figure 4

Thermal magnon renormalization at the $M$ point. (a) Constant-$\mathbf{Q}$ cuts measured on EIGER at the $M$ point, $\mathbf{Q}=(1.500)$. Spectra for the different temperatures are offset vertically by $+15$ units for clarity. Red lines show complete fits to the data composed of a magnon contribution (shaded areas) and a temperature-independent background. The horizontal bar indicates the FWHM resolution of EIGER at 10 meV. (b),(c) Dependence of the peak position (b) and linewidth (c) on temperature; solid lines show quadratic fits. Squares and circles correspond, respectively, to results obtained by fitting the PANTHER and EIGER datasets. (d) $\mathbf{Q}$ dependence of the quadratic fitting coefficients $b$ through the $M$ point.