  States of matter that break time-reversal symmetry are invariably associated
with magnetism or circulating currents. Recently, one of us proposed a phase,
the directional scalar spin chiral order (DSSCO), as an exception: it breaks
time-reversal symmetry via chiral ordering of spins along a particular
direction, but is spin-rotation symmetric. In this work, we prove the existence
of this state via state-of-the-art density matrix renormalization group (DMRG)
analysis on a spin-1 chain with nearest-neighbor bilinear-biquadratic
interactions and additional third-neighbor ferromagnetic Heisenberg exchange.
Despite the large entanglement introduced by the third-neighbor coupling, we
are able to access system sizes up to $L=918$ sites. We find first order phase
transitions from the DSSCO into the famous Haldane phase as well as a
spin-quadrupolar phase where spin nematic correlations dominate. In the Haldane
phase, we propose and demonstrate a method for detecting the topological edge
states using DMRG that could be useful for other topological phases too.
