The $J_1$-$J_2$ Kagome Heisenberg Model with Dzyaloshinskii-Moriya Interaction

We study the static and dynamic properties of an $S=1/2$ kagome antiferromagnetic (AFM) Heisenberg model using large-scale exact diagonalization. We map out the phase diagram as functions of the next-nearest-neighbor exchange $J_2$ and the z-axis Dzyaloshinskii-Moriya interaction $D_z$. In particular, the phase boundary between a magnetically disordered state and a ${\bf Q}=0, 120^{\circ}$ AFM long-range order is identified through finite-size extrapolation of the transverse spin-spin correlation function. We also compute the dynamical structure form factors and the cross sections of two-magnon Raman scattering in different regimes of the phase diagram. The implications of our numerical results to the $S=1/2$ kagome compound herbertsmithite ZnCu$_3$(OH)$_6$Cl$_2$ will be discussed.