Possible chiral spin liquid state in the $S=1/2$ kagome Heisenberg model

The nature of the ground state for the $S = 1/2$ kagome Heisenberg antiferromagnet (KHAF) has been elusive. We revisit this challenging problem and provide numerical evidence that its ground state might be a chiral spin liquid. Combining the density matrix renormalization group method and analytical analyses, we demonstrate that the previously observed chiral spin liquid phase in the KHAF with longer-range couplings is stable in a broader region. We characterize the nature of the ground state by computing energy derivatives, revealing ground-state degeneracy arising from spontaneous breaking of time-reversal symmetry, and targeting the semion sector. We further investigate the phase diagram in the vicinity of the KHAF and observe a $\sqrt{3}\times\sqrt{3}$ magnetically ordered phase and two valence-bond crystal phases.