  Traditionally, the transverse spin-1/2 XY model is mapped to a fermionic
"c-cycle" problem, where the prior periodic boundary condition is applied to
the fermionic chain and the additional boundary term has been neglected.
However, the "a-cycle" problem (the original problem without any approximation)
has not been treated seriously up to now. In this paper, we consider the XY
model with ring frustration and diagonalize it without any approximation with
the help of parity constraint. Then two peculiar gapless phases have been
found.
