  The characterization of an infinite-order quantum phase transition (QPT) by
entanglement measures is analyzed. To this aim, we consider two closely related
solvable spin-1/2 chains, namely, the Ashkin-Teller and the staggered XXZ
models. These systems display a distinct pattern of eigenstates but exhibit the
same thermodynamics, i.e. the same energy spectrum. By performing exact
diagonalization, we investigate the behavior of pairwise and block entanglement
in the ground state of both models. In contrast with the XXZ chain, we show
that pairwise entanglement fails in the characterization of the infinite-order
QPT in the Ashkin-Teller model, although it can be achieved by analyzing the
distance of the pair state from the separability boundary. Concerning block
entanglement, we show that both XXZ and Ashkin-Teller models exhibit identical
von Neumann entropies as long as a suitable choice of blocks is performed.
Entanglement entropy is then shown to be able to identify the quantum phase
diagram, even though its local extremes (either maximum or minimum) may also
appear in the absence of any infinite-order QPT.
