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I will describe a theoretical framework to systematically classify and construct generic tensor-network wavefunctions (in d=1,2,3 spatial dimensions) respecting both onsite and spatial symmetries, which turns out to be directly related to topological phases and useful from both conceptual and practical points of view. For example, on the conceptual side, this framework allows us to (partially) classify general symmetry protected topological phases involving spatial symmetries, and to prove a generalized Lieb-Schultz-Mattis theorem involving general magnetic translations. We also identify a generic connection between SPT phases and rather conventional symmetry enriched topological (SET) phases via an anyon condensation mechanism, which may serve as a guideline to search for SPT phases in realistic models. On the practical side, the constructed generic tensor-network wavefunctions are useful for variational numerical simulations.

[References: arXiv:1505.03171, 1509.04358, 1610.02024, 1611.07652, 1705.05421]

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