  Non-reciprocal spin wave can facilitate the realization of spin wave logic
devices. It has been demonstrated that the non-reciprocity can emerge when an
external magnetic field is applied to chiral magnets, of which spin structures
depend crucially on an asymmetric exchange interaction, that is, the
Dzyaloshinskii-Moriya interaction (DMI). Here we demonstrate that the
non-reciprocity can arise even without the DMI. We demonstrate this idea for
the chiral antiferromagnet Ba$_2$NbFe$_3$Si$_2$O$_{14}$, of which DMI is very
small and chiral spin structure arises mainly from the competition between
symmetric exchange interactions. We show that when an external magnetic field
is applied, asymmetric energy gap shift occurs and the spin wave becomes
non-reciprocal from the competition between symmetric exchange interactions and
the external magnetic field.
