  Spontaneous symmetry breaking is well understood through the classical
"Mexican Hat" picture, which describes many quantum phases of matter. Recently,
several new classes of quantum phases of matter, such as topological orders and
symmetry protected topological (SPT) orders, were discovered. In an attempt to
address the transitions between all those phases of quantum matter under the
same framework, we introduced an analogous yet very simple picture for phase
transitions in the context of tensor-networks. Using a very simple iteration
process, we found that both symmetry breaking and some topological phase
transitions (for topological orders described by gauge theory and 1D SPT
orders) could be marked by a sudden change in the symmetry structure of the
so-called "environment matrix". In this process, the environment matrix serves
as an "order parameter" that captures patterns of entanglement in topological
phases. The symmetry change in the environment matrix is very much like the
symmetry breaking of conventional order parameters. We applied this method to
both the transverse Ising model (1D and 2D honeycomb), spin-1 model (1D), and
the Toric Code model in a magnetic field (2D honeycomb), and explored the
corresponding symmetry structure changes in their environment matrices in
details. With just a few variational parameters and a few minutes' run time on
a laptop, we could get the corresponding phase transition points within a few
percent error compared with the Quantum Monte Carlo results.
