Topological and magnetic properties of a noncollinear spin state on a honeycomb lattice in a magnetic field

This paper studies the topological and magnetic properties of a noncollinear spin state on a honeycomb lattice that evolves from coplanar to ferromagnetic with a magnetic field applied along the $z$ axis. The coplanar state is stabilized by nearest-neighbor ferromagnetic interactions, single-ion anisotropy along $z$, and Dzyaloshinskii-Moriya interactions between next-nearest-neighbor sites. Below the critical field ${H}_{c}$ that aligns the spins, the magnetic unit cell contains six sites and the spin dynamics contains six magnon modes. Although the classical energy is degenerate with respect to the twist angle $\ensuremath{\phi}$ between nearest-neighbor spins, the dependence of the free energy on $\ensuremath{\phi}$ at low temperatures is dominated by the magnon zero-point energy, which contains extremum at $\ensuremath{\phi}=\ensuremath{\pi}l/3$ for integer $l$. The only unique ground states $\mathrm{GS}(\ensuremath{\phi})$ have $l=0$ or 1. For $H<{H}_{c}^{\ensuremath{'}}$, the zero-point energy has minima at even $l$ and the ground state is GS(0); for ${H}_{c}^{\ensuremath{'}}<H<{H}_{c}$, the zero-point energy has minima at odd $l$ and the ground state is $\mathrm{GS}(\ensuremath{\pi}/3$). In GS(0), the magnon density of states exhibits five distinct topological phases with increasing field associated with the opening and closing of energy gaps between two or three magnonic bands. While the Berry curvature vanishes for the coplanar $\ensuremath{\phi}=0$ phase in zero field, the Berry curvature and Chern numbers exhibit signatures of the five topological phases below ${H}_{c}^{\ensuremath{'}}$. Whereas the Berry curvature and Chern number are sensitive to changes in the magnon density of states within $\mathrm{GS}(\ensuremath{\pi}/3$), the inelastic spectrum $S(\mathbf{k},\ensuremath{\omega})$ is sensitive to changes in the intensity of the magnon modes in the different magnetic phases GS(0) and $\mathrm{GS}(\ensuremath{\pi}/3$) rather than the five topological phases within $\mathrm{GS}(\ensuremath{\pi}/3$).