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We study the quantum phase diagram of the spin-$1/2$ Heisenberg model on thekagom\'e lattice with first-, second-, and third-neighbor interactions $J_1$,$J_2$, and $J_3$ by means of density matrix renormalization group. For small$J_2$ and $J_3$, this model sustains a time-reversal invariant quantum spinliquid phase. With increasing $J_2$ and $J_3$, we find in addition a $q=(0,0)$N\'{e}el phase, a chiral spin liquid phase, a valence-bond crystal phase, and acomplex non-coplanar magnetically ordered state with spins forming the verticesof a cuboctahedron known as a cuboc1 phase. Both the chiral spin liquid andcuboc1 phase break time reversal symmetry in the sense of spontaneous scalarspin chirality. We show that the chiralities in the chiral spin liquid andcuboc1 are distinct, and that these two states are separated by a strong firstorder phase transition. The transitions from the chiral spin liquid to both the$q=(0,0)$ phase and to time-reversal symmetric spin liquid, however, areconsistent with continuous quantum phase transitions.

Global Phase Diagram of Competing Ordered and Quantum Spin Liquid Phases on the Kagom\'e Lattice