Spontaneous Chiral-Spin Ordering in Spin-Orbit Coupled Honeycomb Magnets

Frustrated magnets with highly degenerate ground states are at the heart of hunting exotic states of matter. Recent studies in spin-orbit coupled honeycomb magnets have generated immense interest in bond-dependent interactions, appreciating a symmetric off-diagonal $\Gamma$ interaction which exhibits a macroscopic degeneracy in the classical limit. Here, we study a generic spin model and discover a novel chiral-spin ordering with spontaneously broken time-reversal symmetry near the dominant $\Gamma$ region. The chiral-spin phase is demonstrated to possess a staggered chirality relation in different sublattices, and it exhibits gapless excitations as revealed by the vanishing energy gap and the finite central charge on cylinders. Although there is a vestige of a tiny peak in the corner of the second Brillouin zone, the magnetic order is likely to vanish as the system size increases. Finally, we also attempt to gain insight into the possible topological signature of the chiral-spin phase by calculating the dynamic structure factor and the modular $\mathcal{S}$ matrix.