Ripple state in the frustrated honeycomb-lattice antiferromagnet

We discover a new type of multiple-$q$ state, "ripple state", in a frustrated honeycomb-lattice Heisenberg antiferromagnet under magnetic fields. The ground state has an infinite ring-like degeneracy in the wavevector space, exhibiting a cooperative paramagnetic state, "ring-liquid" state. We elucidate that the system exhibits the ripple state as a new low-temperature thermodynamic phase via a second-order phase transition from the ring-liquid state, keeping the ring-like spin structure factor. The spin texture in real space looks like a "water ripple" and can induce a giant electric polarization vortex. Possible relationship to the honeycomb-lattice compound, ${\rm Bi_{3}Mn_{4}O_{12}(NO_{3})}$, is discussed.