  Geometric entanglement(GE), as a measure of multipartite entanglement, has
been investigated as a universal tool to detect phase transitions in quantum
many-body lattice models. We outline a systematic method to compute GE for
two-dimensional (2D) quantum many-body lattice models based on the
translational invariant structure of infinite projected entangled pair state
(iPEPS) representations. By employing this method, the $q$-state quantum Potts
model on the square lattice with $q \in \{2, 3 ,4, 5\}$ is investigated as a
prototypical example. Further, we have explored three 2D Heisenberg models,
such as the spin-$\frac{1}{2}$ XXX and antiferromagnetic anisotropic XYX models
in an external magnetic field, and the spin-1 antiferromagnetic XXZ model. We
find that continuous GE does not guarantee a continuous phase transition across
a phase transition point. We observe and thus classify three different types of
continuous GE across a phase transition point: (i) GE is continuous with
maximum value at the transition point and the phase transition is continuous,
(ii) GE is continuous with maximum value at the transition point but the phase
transition is discontinuous, and (iii) GE is continuous with non-maximum value
at the transition point and the phase transition is continuous. For the models
under consideration we find that the second and the third types are related to
a point of dual symmetry and a fully polarized phase, respectively.
