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The many faces of quantum kagome materials: Interplay of further-neighbour exchange and Dzyaloshinskii-Moriya interaction

The field of frustrated magnetism has been enriched significantly by the discovery of various kagome lattice compounds. These materials exhibit a great variety of macroscopic behaviours ranging from magnetic orders to quantum spin liquids. Using large-scale exact diagonalization, we construct the phase diagram of the $S=1/2$ $J_1$-$J_2$ kagome Heisenberg model with $z$-axis Dzyaloshinskii-Moriya interaction $D_z$. We show that this model can systematically account for many of the experimentally observed phases. Small $J_2$ and $D_z$ can stabilize respectively a gapped and a gapless spin liquid. When $J_2$ or $D_z$ is substantial, the ground state develops a $\mathbf{Q}=0$, $120^\textrm{o}$ antiferromagnetic order. The critical strengths for inducing magnetic transition are $D^c_z \sim 0.1\, J_1$ at $J_2=0$, and $J^c_2\sim 0.4\, J_1$ at $D_z=0$. The previously reported values of $D_z$ and $J_2$ for herbertsmithite [ZnCu$_3$(OH)$_6$Cl$_2$] place the compound in close proximity to a quantum critical point.