  In this paper we analyze the ground state phase diagram of a class of
fermionic Hamiltonians by looking at the fidelity of ground states
corresponding to slightly different Hamiltonian parameters. The Hamiltonians
under investigation can be considered as the variable range generalization of
the fermionic Hamiltonian obtained by the Jordan-Wigner transformation of the
XY spin-chain in a transverse magnetic field. Under periodic boundary
conditions, the matrices of the problem become circulant and the models are
exactly solvable. Their free-ends counterparts are instead analyzed
numerically. In particular, we focus on the long range model corresponding to a
fully connected directed graph, providing asymptotic results in the
thermodynamic limit, as well as the finite-size scaling analysis of the second
order quantum phase transitions of the system. A strict relation between
fidelity and single particle spectrum is demonstrated, and a peculiar gapful
transition due to the long range nature of the coupling is found. A comparison
between fidelity and another transition marker borrowed from quantum
information i.e., single site entanglement, is also considered.
