In contrast to iron pnictide $A$Fe$_2$As$_2$ [@diallo09; @jzhao; @raewings; @lharriger] and iron chalcogenide Fe$_{1+y}$Te [@lipscombe; @zaliznyak], spin waves in insulating $A$Fe$_{1.6+x}$Se$_2$ can be well-described by a Heisenberg Hamiltonian with NN, NNN, and NNNN exchange couplings [@mywang11]. Comparing effective exchange couplings for different iron-based materials (Table I), it is clear that the NN exchange couplings are quite different, but the NNN exchange couplings are AF and rather similar. In addition, spin waves for iron-based materials are much broader at high energies. This is different from the insulating copper oxides, where the NN exchange coupling dominates the magnetic interactions and spin waves are instrumental resolution limited throughout the Brillouin zone [@coldea; @headings]. These results suggest that itinerant electrons play a role in spin waves of metallic iron-based materials.

  Parent compounds                                 $T_N$ (K)    $SJ_{1a}$ (meV)   $SJ_{1b}$ (meV)   $SJ_{2a}$ (meV)   $SJ_{2b}$ (meV)   $SJ_{3}$ (meV)
  ------------------------------------------------ ------------ ----------------- ----------------- ----------------- ----------------- ----------------
  La$_2$CuO$_4$, Ref. [@coldea]                    $317\pm 3$   $111.8\pm4$       $111.8\pm4$       $-11.4\pm 3$      $-11.4\pm 3$      0
  CaFe$_2$As$_2$, Ref. [@jzhao]                    $\sim$170    $49.9\pm 9.9$     $-5.7\pm4.5$      $18.9\pm 3.4$     $18.9\pm 3.4$     0
  BaFe$_2$As$_2$, Ref. [@lharriger]                $\sim$138    $59.2\pm2.0$      $-9.2\pm 1.2$     $13.6\pm 1$       $13.6\pm 1$       0
  Fe$_{1.05}$Te, Ref. [@lipscombe]                 $\sim$68     $-17.5\pm 5.7$    $-51.0\pm 3.4$    $21.7\pm 3.5$     $21.7\pm 3.5$     $6.8\pm 2.8$
  Rb$_{0.89}$Fe$_{1.58}$Se$_2$, Ref. [@mywang11]   