Collective excitations in chiral spin liquid: chiral roton and long-wavelength nematic mode

Chiral spin liquid (CSL) is a magnetic analogue of the fractional quantum Hall (FQH) liquid. Collective excitations play a vital role in shaping our understanding of these exotic quantum phases of matter and their quantum phase transitions. While the magneto-roton and long-wavelength chiral graviton modes in the FQH liquids have been extensively explored, the collective excitations of CSLs remain elusive. Here we explore the collective excitations in the SU(2) symmetric CSL phase of the spin-1/2 square-lattice $J_1-J_2-J_\chi$ model, where an intriguing quantum phase diagram was recently revealed. Combining exact diagonalization and time-dependent variational principle calculations, we observe two spin-singlet collective modes: a chiral p-wave (low-energy) roton mode at finite momentum and a elliptically polarized d-wave (higher-energy) nematic mode at zero momentum, both of which are prominent across the CSL phase. Such exotic modes exhibit fingerprints distinct from those of FQH liquids, and to the best of our knowledge, are reported for the first time. By tuning $J_2$, we find the nematic mode to be pronouncedly soft, together with the spin-triplet two-spinon bound states, potentially promoting strong nematic and spin stripe instabilities. Our work paves the way for further understanding CSL from the dynamical perspective and provides new spectroscopic signatures for future experiments of CSL candidates.