  We study the interplay of competing interactions in spin-$1/2$ triangular
Heisenberg model through tuning the first- ($J_1$), second- ($J_2$), and
third-neighbor ($J_3$) couplings. Based on large-scale density matrix
renormalization group calculation, we identify a quantum phase diagram of the
system and discover a new {\it gapless} chiral spin liquid (CSL) phase in the
intermediate $J_2$ and $J_3$ regime. This CSL state spontaneously breaks
time-reversal symmetry with finite scalar chiral order, and it has gapless
excitations implied by a vanishing spin triplet gap and a finite central charge
on the cylinder. Moreover, the central charge grows rapidly with the cylinder
circumference, indicating emergent spinon Fermi surfaces. To understand the
numerical results we propose a parton mean-field spin liquid state, the $U(1)$
staggered flux state, which breaks time-reversal symmetry with chiral edge
modes by adding a Chern insulator mass to Dirac spinons in the $U(1)$ Dirac
spin liquid. This state also breaks lattice rotational symmetries and possesses
two spinon Fermi surfaces driven by nonzero $J_2$ and $J_3$, which naturally
explains the numerical results. To our knowledge, this is the first example of
a gapless CSL state with coexisting spinon Fermi surfaces and chiral edge
states, demonstrating the rich family of novel phases emergent from competing
interactions in triangular-lattice magnets.
