  We show that by examining the global geometric entanglement it is possible to
identify "elusive" or hard to detect quantum phase transitions. We analyze
several one-dimensional quantum spin chains and demonstrate the existence of
non-analyticities in the geometric entanglement, in particular across a
Kosterlitz-Thouless transition and across a transition for a gapped deformed
Affleck-Kennedy-Lieb-Tasaki chain. The observed non-analyticities can be
understood and classified in connection to the nature of the transitions, and
are in sharp contrast to the analytic behavior of all the two-body reduced
density operators and their derived entanglement measures.
