  The laws of quantum-critical scaling theory of quantum fidelity, dependent on
the underlying system dimensionality $D$, have so far been verified in exactly
solvable $1D$ models, belonging to or equivalent to interacting, quadratic
(quasifree), spinless or spinfull, lattice-fermion models. The obtained results
are so appealing that in quest for correlation lengths and associated universal
critical indices $\nu$, which characterize the divergence of correlation
lengths on approaching critical points, one might be inclined to substitute the
hard task of determining an asymptotic behavior of a two-point correlation
function by an easier one, of determining the quantum-critical scaling of the
quantum fidelity. However, the role of system's dimensionality has been left as
an open problem. Our aim in this paper is to fill up this gap, at least
partially, by verifying the laws of quantum-critical scaling theory of quantum
fidelity in a $2D$ case. To this end, we study correlation functions and
quantum fidelity of $2D$ exactly solvable models, which are interacting,
quasifree, spinfull, lattice-fermion models. The considered $2D$ models exhibit
new, as compared with $1D$ ones, features:at a given quantum-critical point
there exists a multitude of correlation lengths and multiple universal critical
indices $\nu$, since these quantities depend on spatial directions, moreover,
the indices $\nu$ may assume larger values. These facts follow from the
obtained by us analytical asymptotic formulae for two-point correlation
functions. In such new circumstances we discuss the behavior of quantum
fidelity from the perspective of quantum-critical scaling theory. In
particular, we are interested in finding out to what extent the quantum
fidelity approach may be an alternative to the correlation-function approach in
studies of quantum-critical points beyond 1D.
