Topological and Magnetic Properties of a Non-collinear Spin State on a Honeycomb Lattice in a Magnetic Field

Abstract

We study the Berry curvature and Chern number of a non-collinear 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 Dzyalloshinskii-Moriya interactions between next-nearest neighbor sites. Below the critical field $H_c$ that aligns the spins, the magnetic unit cell contains $M=6$ sites and the spin dynamics contains six magnon subbands. Although the classical energy is degenerate wrt the twist angle $ϕ$ between nearest-neighbor spins, the dependence of the free energy on $ϕ$ at low temperatures is dominated by the magnon zero-point energy, which contains extremum at $ϕ=πl/3$ for integer $l$. The only unique ground states GS($ϕ)$ have $l=0$ or 1. For $H < H_c'$, the zero-point energy has minima at even $l$ and the ground state is GS(0). For $H_c' < H < H_c$, the zero-point energy has minima at odd $l$ and the ground state is GS($π/3$). In GS(0), the magnon density-of-states exhibits five distinct phases with increasing field associated with the opening and closing of energy gaps between the two or three magnonic bands, each containing between 1 and 4 four magnon subbands. While the Berry curvature vanishes for the coplanar $ϕ=0$ phase in zero field, the Berry curvature and Chern numbers exhibit signatures of the five phases at nonzero fields below $H_c'$. If $ϕ\ne πl/3$, the Chern numbers of the two or three magnonic bands are non-integer. We also evaluate the inelastic neutron-scattering spectrum $S(\vk ,ω)$ produced by the six magnon subbands in all five phases of GS(0) and in GS($π/3$).