insulators characterized by the Cu spin structure shown in Fig. 1a, where the tetragonal or pseudo-tetragonal unit cells have a nearest-neighbor Cu-Cu spacing with $a\approx b\approx 3.8$ Å. In the notation of reciprocal lattice units (rlu) $(2\pi/a,2\pi/b,2\pi/c)$, the AF Bragg peaks occur at the in-plane ordering wave vectors ${\bf Q}_{AF}=(\pm 1/2+m,\pm 1/2+n)$, where $m,n=0,\pm 1, \pm 2, \cdots$ rlu, shown as red circles in Fig. 1e [@vaknin; @tranquada88]. Time-of-flight INS experiments [@coldea; @headings] have mapped out spin waves of the insulating La$_2$CuO$_4$ throughout the Brillouin zone and found no evidence for spin-wave broadening at high energies. The dispersions of spin waves are well described by a Heisenberg Hamiltonian with nearest-neighbor (NN) exchange coupling $J_1=111.8\pm 4$ meV and next-nearest-neighbor (NNN) exchange $J_2=-11.4\pm 3$ meV [@coldea]. Therefore, the dominant magnetic exchange coupling in La$_2$CuO$_4$ is the NN magnetic interaction and the higher-order interactions amount to only $\sim$10% of the total magnetic energy with a bandwidth of $\sim$320 meV (Fig. 1i).

![Antiferromagnetic structure and spin-wave dispersions for the insulating copper oxide La$_2$CuO$_4$ and the parent compounds of iron-based superconductors BaFe$_2$As$_2$, FeTe, and $A_y$Fe$_{1.6+x}$Se$_2$. The chemical unit cells are marked light green. The dark and light brown As/Te/Se atoms indicate their vertical positions above 