We investigate the effects of Dzyaloshinsky-Moriya (DM) interactions on the frustrated $J_1$-$J_2$ kagome-Heisenberg model using the pseudo-fermion functional-renormalization-group (PFFRG) technique. In order to treat the off-diagonal nature of DM interactions, we develop an extended PFFRG scheme. We benchmark this approach in parameter regimes that have previously been studied with other methods and find good agreement of the magnetic phase diagram. Particularly, finite DM interactions are found to stabilize all types of non-collinear magnetic orders of the $J_1$-$J_2$ Heisenberg model ($mathbf{q}=0$, $sqrt{3}timessqrt{3}$, and cuboc orders) and shrink the extents of magnetically disordered phases. We discuss our results in the light of the mineral {it herbertsmithite} which has been experimentally predicted to host a quantum spin liquid at low temperatures. Our PFFRG data indicates that this material lies in close proximity to a quantum critical point. In parts of the experimentally relevant parameter regime for {it herbertsmithite}, the spin-correlation profile is found to be in good qualitative agreement with recent inelastic-neutron-scattering data.
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