  To obtain a light mode in two-dimensional staggered fermions, we introduce
four new local operators keeping the rotational invariance for a staggered
Dirac operator. To split masses of tastes, three cases are considered. The mass
matrix and the propagator for free theories are analyzed. We find that one of
three cases is a good candidate for taking a single mode by the mass splitting.
In the case, it is possible that a heavy mode obtains infinite mass on even
sites or odd sites.
