Chiral spin liquid with spinon Fermi surfaces in the spin-1/2 triangular Heisenberg model

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 a large-scale density-matrix renormalization group calculation, we identify a quantum phase diagram of the system and discover a 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. This realizes an example of a gapless CSL state with coexisting spinon Fermi surfaces and chiral edge states, demonstrating the rich family of interesting quantum phases emergent from competing interactions in triangular-lattice magnets.