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

Functional-renormalization-group analysis of Dzyaloshinsky-Moriya and Heisenberg spin interactions on the kagome lattice