Quantum-critical scaling of fidelity in 2D pairing models

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.