  A systematic analysis is performed for quantum phase transitions in a
two-dimensional anisotropic spin 1/2 anti-ferromagnetic XYX model in an
external magnetic field. With the help of an innovative tensor network
algorithm, we compute the fidelity per lattice site to demonstrate that the
field-induced quantum phase transition is unambiguously characterized by a
pinch point on the fidelity surface, marking a continuous phase transition. We
also compute an entanglement estimator, defined as a ratio between the
one-tangle and the sum of squared concurrences, to identify both the
factorizing field and the critical point, resulting in a quantitative agreement
with quantum Monte Carlo simulation. In addition, the local order parameter is
"derived" from the tensor network representation of the system's ground state
wave functions.
