Low energy spin dynamics of rare-earth orthoferrites YFeO$_{\mathrm{3}}$ and LaFeO$_{\mathrm{3}}$

YFeO$_{\mathrm{3}}$ and LaFeO$_{\mathrm{3\thinspace }}$are members of the rare-earth orthoferrites (RFeO$_{\mathrm{3}})$ family with \textit{Pbnm} space group. With the strong superexchange interaction between Fe3$+$ ions, both compounds exhibit the room temperature antiferromagnetic order (T$_{\mathrm{N}}$ \textgreater 600 K) with a slight spin canting. Here we report low-energy magnetic excitation of YFeO$_{\mathrm{3}}$ and LaFeO$_{\mathrm{3}}$ using inelastic neutron scattering measurements, showing evidence of magnon mode splitting and a spin anisotropy gap at the zone center. Spin wave calculations with the spin Hamiltonian including both Dzyaloshinsky-Moriya interaction and single-ion anisotropy accounts for the observed features well. Our results offer insight into the underlying physics of other RFeO$_{\mathrm{3\thinspace }}$with magnetic rare-earth ions or related Fe$^{\mathrm{3+}}$-based multiferroic perovskites such as BiFeO$_{\mathrm{3}}$.