  We study a finite spin-$\frac{1}{2}$ Ising chain with a spatially alternating
transverse field of period 2. By means of a Jordan-Wigner transformation for
even and odd sites, we are able to map it into a one-dimensional model of free
fermions. We determine the ground-state energies in the positive- and
negative-parity subspaces (subspaces with an even or odd total number of down
spins, respectively) and compare them in order to establish the ground-state
energy for the entire Hamiltonian. We derive closed-form expressions for this
energy gap between the different parity subspaces and analyze its behavior and
dependence on the system size in the various regimes of the applied field.
Finally, we suggest an expression for the correlation length of such a model
that is consistent with the various values found in the literature for its
behavior in the vicinity of critical points.
