  We study ground-state correlation functions in one- and two-dimensional
lattice models of interacting spinful fermions - BCS-like models, which exhibit
continuous quantum phase transitions. The considered models originate from a
two-dimensional model of d-wave superconductivity proposed by Sachdev. Due to
the exact diagonalizability of the considered models in any dimensionality,
exact phase diagrams, with several kinds of quantum-critical points, are
constructed and closed-form analytic expressions for two-point correlation
functions are obtained. In one- and two-dimensional cases we provide analytic
expressions for the asymptotic behavior of those correlation functions at large
distances and in neighborhoods of quantum-critical points. The novelty of our
results is that in two-dimensions explicit expressions for direction-dependent
correlation lengths in terms of model parameters and the values of
direction-dependent universal critical indices $\nu$, that characterize the
divergence of correlation lengths on approaching critical points, are
determined. Moreover, specific scaling properties of correlation functions with
respect to parameters of underlying Hamiltonians are revealed. Besides
enriching the knowledge of properties of lattice fermion systems exhibiting
continuous quantum phase transitions, especially in two dimensions, our results
open new possibilities of testing unconventional methods of studying quantum
phase transitions, as the promising fidelity approach or the entanglement
approach, beyond one-dimension and beyond the realm of paradigmatic XY and
Ising chains in transverse magnetic fields.

Statistical Mechanics