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Symmetry-protected topological (SPT) phases exhibit nontrivial order ifsymmetry is respected but are adiabatically connected to the trivial productphase if symmetry is not respected. However, unlike the symmetry-breakingphase, there is no local order parameter for SPT phases. Here we employ atensor-network method to compute the topological invariants characterized bythe simulated modular $S$ and $T$ matrices to study transitions in a fewfamilies of two-dimensional (2D) wavefunctions which are $\mathbb{Z}_N$ ($N=2\,\&3$) symmetric. We find that in addition to the topologically ordered phases,the modular matrices can be used to identify nontrivial SPT phases and detecttransitions between different SPT phases as well as between symmetric andsymmetry-breaking phases. Therefore, modular matrices can be used tocharacterize various types of gapped phases in a unifying way.

physical review. b./physical review. b

Detecting and identifying 2D symmetry-protected topological,  symmetry-breaking and intrinsic topological phases with modular matrices via  tensor-network methods