Spin waves of 2D $J_1$-$J_2$ model and single hole dynamics via nearest neighbor hole hopping

We study the two dimensional $J_1$-$J_2$ model using the spin wave approximation and find the spin wave modes at different values of parameter $\lambda$=$J_2/J_1$. Competing antiferromagnetic ($\pi,\pi$) and ($\pi,0$) (or, ($0,\pi$)) orders preferred by the $J_1$ and $J_2$ terms respectively creates frustration in such a system and the spin wave excitations about different vacuum states are observed for different values of $\lambda$. We investigate the stability of different zero spin deviation states as $\lambda$ is varied from 0 to 1.5. We discuss the band folding of the super-lattice structure that appears with the emergence of the $J_2$ term and study the different spin wave modes that the system inherits. Dynamics of a foreign hole put in such a system is studied using a $t$-$J_1$-$J_2$ model and the hole spectra obtained by solving the Dyson's equation self-consistently within the non-crossing approximation are compared with the ARPES spectra from the newly discovered FeAs superconducting compounds.