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Motivated by the recent theoretical study by Okubo $et \ al$ [Phys. Rev.Lett. ${\bf 108}$, 017206 (2012)] on the possible realization of thefrustration-induced $ symmetric$ skyrmion-lattice state in the $J_1$-$J_2$ (or$J_1$-$J_3$) triangular-lattice Heisenberg model without theDzyaloshinskii-Moriya interaction, we investigate the ordering of the classical$J_1$-$J_2$ honeycomb-lattice Heisenberg antiferromagnet under magnetic fieldsby means of a Monte Carlo simulation, a mean-field analysis and alow-temperature expansion. The model has been known to have an infinitering-like degeneracy in the wavevector space in its ground state for$1/6<J_2/J_1<0.5$, in distinction with the triangular-lattice model. Asreported by Okumura $et \ al$ [J. Phys. Soc. Jpn. ${\bf 79}$, 114705 (2010)],such a ring-like degeneracy gives rise to exotic spin liquid states in zerofield, $e.g$, the "ring-liquid" state and the "pancake-liquid" state. In thispaper, we study the in-field ordering properties of the model paying attentionto the possible appearance of exotic multiple-$q$ states. Main focus is made onthe $J_2/J_1=0.3$ case, where we observe a rich variety of multiple-$q$ statesincluding the single-$q$, double-$q$ and triple-$q$ states. While theskyrmion-lattice triple-$q$ state observed in the triangular-lattice model isnot realized, we instead observe an exotic double-$q$ state consisting ofmeron/antimeron lattice textures.

Multiple-$q$ states of the $J_1$-$J_2$ classical honeycomb-lattice Heisenberg antiferromagnet under magnetic fields