Classifying Quantum Phases Using Tensor-Network State Ching-Yu Huang ^{1*}, Tzu-Chieh Wei^{2,3}, Román Orús^{4}^{1}Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan^{2}C. N. Yang Institute for Theoretical Physics, Stony Brook, New York, USA^{3}Department of Physics and Astronomy, State University of New York at Stony Brook, Stony, Stony Brook, New York, USA^{4}Institute of Physics, Johannes Gutenberg University, Mainz, Germany* Presenter:Ching-Yu Huang, email:ayajor827@gmail.com Symmetry-protected topological (SPT) phases exhibit nontrivial order if symmetry is respected but are adiabatically connected to the trivial product phase if symmetry is not respected. However, unlike the symmetry breaking phase, there is no local order parameter for SPT phases. Here we employ a tensor-network method to compute the topological invariants characterized by the simulated modular S and T matrices to study transitions in a few families of two-dimensional (2D) wave functions which are Z
_{N} (N = 2 and 3) symmetric. We find that in addition to the topologically ordered phases, the modular matrices can be used to identify nontrivial SPT phases and detect transitions between different SPT phases as well as between symmetric and symmetry-breaking phases. Therefore modular matrices can be used to characterize various types of gapped phases in a unifying way. On the other hand, we show also how corner properties can be used to pinpoint quantum phase transitions, topological or not, without the need for observables. Moreover, for a chiral topological PEPS we show by examples that corner tensors can be used to extract the entanglement spectrum of half a system, with the expected symmetries of the SU(2)_{k} Wess-Zumino-Witten model describing its gapless edge for k = 1,2.Keywords: tensor-network state, topological order, symmetry protected topological order |