  In the solar atmosphere, the chromospheric and coronal plasmas are much
hotter than the visible photosphere. The heating of the solar atmosphere,
including the partially ionized chromosphere and corona, remains largely
unknown. In this paper we demonstrate that the ions can be substantially heated
by Alfv\'{e}n waves with very low frequencies in partially ionized low beta
plasmas. This differs from other Alfv\'{e}n wave related heating mechanisms
such as ion-neutral collisional damping of Alfv\'{e}n waves and heating
described by previous work on resonant Alfv\'{e}n wave heating. In this paper,
we find that the non-resonant Alfv\'{e}n wave heating is less efficient in
partially ionized plasmas than when there are no ion-neutral collisions, and
the heating efficiency depends on the ratio of the ion-neutral collision
frequency to the ion gyrofrequency.
