Static and dynamical signatures of Dzyaloshinskii-Moriya interactions in the Heisenberg model on the kagome lattice

Motivated by recent experiments on Cs$_2$Cu$_3$SnF$_{12}$ and YCu$_{3}$(OH)$_{6}$Cl$_{3}$, 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 $J_D$, which favors (in-plane) ${\bf Q}=(0,0)$ magnetic order. By using both variational Monte Carlo (based upon Gutzwiller-projected fermionic wave functions) and tensor-network approaches (built from infinite projected-entangled pair/simplex states), we show that the ground state develops a finite magnetization for $J_D/J \gtrsim 0.03 - 0.04$, while the gapless spin liquid remains stable for smaller values of the Dzyaloshinskii-Moriya interaction. The relatively small value of $J_D/J$ for which magnetic order sets in is particularly relevant for the interpretation of low-temperature behaviors of kagome antiferromagnets, including ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$. In addition, we assess the spin dynamical structure factors 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 reported within the magnetically ordered phase, similarly to what has been detected by inelastic neutron scattering on Cs$_{2}$Cu$_{3}$SnF$_{12}$.