We assess the ground-state phase diagram of the J(1) - J(2) Heisenberg model on the Kagome lattice by employing Gutzwiller-projected fermionic wave functions. Within this framework, different states can be represented, defined by distinct unprojected fermionic Hamiltonians that include hopping and pairing terms, as well as a coupling to local Zeeman fields to generate magnetic order. For J(2) = 0, the so-called U(1) Dirac state, in which only hopping is present (such as to generate a pi-flux in the hexagons), has been shown to accurately describe the exact ground state [Y. Iqbal, F. Becca, S. Sorella, and D. Poilblanc, Phys. Rev. B 87, 060405(R) (2013); Y.-C. He, M. P. Zaletel, M. Oshikawa, and F. Pollmann, Alys. Rev. X 7, 031020 (2017).]. Here we show that its accuracy improves in the presence of a small antiferromagnetic superexchange J(2), leading to a finite region where the gapless spin liquid is stable; then, for J(2)/J(1) = 0.11(1), a first-order transition to a magnetic phase with pitch vector q = (0, 0) is detected by allowing magnetic order within the fermionic Hamiltonian. Instead, for small ferromagnetic values of vertical bar J(2)vertical bar/J(1), the situation is more contradictory. While the U(1) Dirac state remains stable against several perturbations in the fermionic part (i.e., dimerization patterns or chiral terms), its accuracy clearly deteriorates on small systems, most notably on 36 sites where exact diagonalization is possible. Then, on increasing the ratio vertical bar J(2)vertical bar/J(1), a magnetically ordered state with root 3 x root 3 periodicity eventually overcomes the U(1) Dirac spin liquid. Within the ferromagnetic J(2) regime, evidence is shown in favor of a first-order transition at J(2)/J(1) = -0.065(5).
