Global phase diagram of competing ordered and quantum spin-liquid phases on the kagome lattice

We study the quantum phase diagram of the spin-1/2 Heisenberg model on the kagome lattice with first-, second-, and third-neighbor interactions J₁, J₂, and J₃ by means of density matrix renormalization group. For small J₂ and J₃, this model sustains a time-reversal invariant quantum spin-liquid phase. With increasing J₂ and J₃, we find in addition a q= (0, 0) N\'eel phase, a chiral spin-liquid phase, an apparent valence-bond crystal phase, and a complex noncoplanar magnetically ordered state with spins forming the vertices of a cuboctahedron known as a cuboc1 phase. Both the chiral spin liquid and cuboc1 phase break time-reversal symmetry in the sense of spontaneous scalar spin chirality. We show that the chiralities in the chiral spin liquid and cuboc1 are distinct, and that these two states are separated by a strong first-order phase transition. The transitions from the chiral spin liquid to both the q= (0, 0) phase and to time-reversal symmetric spin liquid, however, are consistent with continuous quantum phase transitions.