  At strong repulsion, the triangular-lattice Hubbard model is described by
$s=1/2$ spins with nearest-neighbor antiferromagnetic Heisenberg interactions
and exhibits conventional 120$^\circ$ order. Using the infinite density matrix
renormalization group and exact diagonalization, we study the effect of the
additional four-spin interactions naturally generated from the underlying
Mott-insulator physics of electrons as the repulsion decreases. Although these
interactions have historically been connected with a gapless ground state with
emergent spinon Fermi surface, we find that at physically relevant parameters,
they stabilize a chiral spin-liquid (CSL) of Kalmeyer-Laughlin (KL) type,
clarifying observations in recent studies of the Hubbard model. We then present
a self-consistent solution based on mean-field rewriting of the interaction to
obtain a Hamiltonian with similarities to the parent Hamiltonian of the KL
state, providing a physical understanding for the origin of the CSL.
