  The effect of boundaries on the bulk properties of quantum many-body systems
is an intriguing subject of study. One can define a boundary effect function,
which quantifies the change in the ground state as a function of the distance
from the boundary. This function serves as an upper bound for the correlation
functions and the entanglement entropies in the thermodynamic limit. Here, we
perform numerical analyses of the boundary effect function for one-dimensional
free-fermion models. We find that the upper bound established by the boundary
effect fuction is tight for the examined systems, providing a deep insight into
how correlations and entanglement are developed in the ground state as the
system size grows. As a by-product, we derive a general fidelity formula for
fermionic Gaussian states in a self-contained manner, rendering the formula
easier to apprehend.
