  The fidelity between two infinitesimally close states or the fidelity
susceptibility of a system are known to detect quantum phase transitions. Here
we show that the k-space fidelity between two states far from each other and
taken deep inside (bulk) of two phase s, generically vanishes at the k-points
where there are gapless points in the energy spectrum that give origin to the
lines (edges) separating the phases in the phase diagram. We consider a general
case of two-band models and present a sufficient condition for the existence of
gapless points, given there are pairs of parameter points for which the
fidelity between the corresponding states is zero. By presenting an explicit
counter-example, we showed that the sufficient condition is not necessary.
Further, we showed that, unless the set of parameter points is suitably
constrained, the existence of gapless points generically imply the accompanied
pairs of parameter points with vanishing fidelity. Also, we showed the
connection between the vanishing fidelity and gapless points on a number of
concrete examples (topological triplet superconductor, topological insulator,
1d Kitaev model of spinless fermions, BCS superconductor, Ising model in a
transverse field, graphene and Haldane Chern insulator), as well as for the
more general case of Dirac-like Hamiltonians. We also briefly discuss the
relation between the vanishing fidelity and gapless points at finite
temperatures.
