  Quantum phase transitions (QPTs), including symmetry breaking and topological
types, always associated with gap closing and opening. We analyze the
topological features of the quantum phase boundary of the XY model in a
transverse magnetic field. Based on the results from graphs in the auxiliary
space, we find that gapless ground states at boundary have different
topological characters. On the other hand, In the framework of Majorana
representation, the Majorana lattice is shown to be two coupled SSH chains. The
analysis of the quantum fidelity for the Majorana eigen vector indicates the
signature of QPT for the gapless state. Furthermore numerical computation shows
that the transition between two types of gapless phases associates with
divergence of second-order derivative of groundstate energy density, which
obeys scaling behavior. It indicates that a continues QPT can occur among
gapless phases. The underlying mechanism of the gapless QPT is also discussed.
The gap closing and opening are not necessary for a QPT.
