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 $S=1\mathrm{/}2$ 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) $\mathbf{Q}=(0,0)$ 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 $J_D\mathrm{/}J\gtrsim 0.03-0.04$, while the gapless spin liquid remains stable for smaller values of the Dzyaloshinskii-Moriya interaction. The relatively small value of $J_D\mathrm{/}J$ 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${}_{12}$.