Motivated by recent experiments on Cs2Cu3SnF12 and YCu3(OH)6Cl3, we consider the S=1/2 Heisenberg model on the kagome lattice with nearest-neighbor super-exchange J and (out-of-plane) Dzyaloshinskii-Moriya interaction JD, which favors (in-plane) Q=(0,0) magnetic order. By using both variational Monte Carlo and tensor-network approaches, we show that the ground state develops a finite magnetization for JD/J≳0.03−0.04; instead, for smaller values of the Dzyaloshinskii-Moriya interaction, the ground state has no magnetic order and, according to the fermionic wave function, develops a gap in the spinon spectrum, which vanishes for JD→0. The small value of JD/J for the onset of magnetic order is particularly relevant for the interpretation of low-temperature behaviors of kagome antiferromagnets, including ZnCu3(OH)6Cl2. For this reason, we assess the spin dynamical structure factor and the corresponding low-energy spectrum, by using the variational Monte Carlo technique. The existence of a continuum of excitations above the magnon modes is observed within the magnetically ordered phase, with a broad peak above the lowest-energy magnons, similarly to what has been detected by inelastic neutron scattering on Cs2Cu3SnF12.
