Phases of matter are sharply defined in the thermodynamic limit. One major challenge of accurately simulating phase diagrams of interacting quantum systems is due to the fact that numerical simulations usually deal with the energy density, a local observable, while identifying different quantum phases generally rely on long-range physics. In this talk we discuss constructions of generic fully symmetric quantum wavefunctions under certain assumptions, using a type of tensor networks: projected entangled pair states, and provide practical simulation algorithms based on them. We find that different quantum phases can be organized into crude classes distinguished by short-range physics, which is related to the fractionalization of both on-site symmetries and space-group symmetries. Consequently, our simulation algorithms, which are useful to study long-range physics as well, are expected to be able to sharply determine crude classes in interacting quantum systems efficiently. Examples of these crude classes, limitations and generalizations of our results are discussed.
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