  Finding suitable indicators for characterizing quantum phase transitions
plays an important role in understanding different phases of matter. It is
especially important for fracton phases where a combination of topology and
fractionalization leads to exotic features not seen in other known quantum
phases. In this paper, we consider the above problem by studying phase
transition in the X-cube model in the presence of a non-linear perturbation.
Using an analysis of the ground state fidelity and identifying a discontinuity
in the global entanglement, we show there is a first order quantum phase
transition from a type I fracton phase with a highly entangled nature to a
magnetized phase. Accordingly, we conclude that the global entanglement, as a
measure of the total quantum correlations in the ground state, can well capture
certain features of fracton phase transitions. Then, we introduce a non-local
order parameter in the form of a foliated operator which can characterize the
above phase transition. We particularly show that such an order parameter has a
geometric nature which captures specific differences of fracton phases with
topological phases. Our study is specifically based on a well-known dual
mapping to the classical plaquette Ising model where it shows the importance of
such dualities in studying different quantum phases of matter.
