  We propose a method to construct universal order parameters for quantum phase
transitions in many-body lattice systems. The method exploits the
$H$-orthogonality of a few near-degenerate lowest states of the Hamiltonian
describing a given finite-size system, which makes it possible to perform
finite-size scaling and take full advantage of currently available numerical
algorithms. An explicit connection is established between the fidelity per site
between two $H$-orthogonal states and the energy gap between the ground state
and low-lying excited states in the finite-size system. The physical
information encoded in this gap arising from finite-size fluctuations clarifies
the origin of the universal order parameter.We demonstrate the procedure for
the one-dimensional quantum formulation of the $q$-state Potts model, for
$q=2,3,4$ and 5, as prototypical examples, using finite-size data obtained from
the density matrix renormalization group (DMRG) algorithm.
