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## "Matrix product states and projected entangled pair states: Concepts, symmetries, theorems", Ignacio Cirac et al., 2021

*Reviewed March 31, 2024*

*Citation:* Cirac, J. Ignacio, et al. "Matrix product states and projected entangled pair states: Concepts, symmetries, theorems." Reviews of Modern Physics 93.4 (2021): 045003.

*Web:* https://arxiv.org/abs/2011.12127

*Tags:* Expository, Tensor-networks

This paper gives a nice review of the tensor network approach to (topological) quantum phases of matter.
In this model, the terms in your Hamiltonian are modeled by local tensors. These tensors fit together in interesting
ways. The entire theory of topological order can be recast in this language, giving anyons a nice
new interpretation. The key idea is that all of the essential features of a topological Hamiltonian
should be understandable by its ground states. By the fundamental theorem of tensor networks,
ground states essentially correspond to tensors up to conjugation. This theory is especially good for describing symmetry enriched topological order:

> Williamson, Dominic J., Nick Bultinck, and Frank Verstraete. "Symmetry-enriched topological order in tensor networks: Defects, gauging and anyon condensation." arXiv preprint arXiv:1711.07982 (2017).

One thing to note about the tensor network approach is that, just like with the Levin-Wen model and subfactors,
it only addresses doubled topological order. To quote the authors: "A systematic
study of these MPO symmetries allows to represent all
fields, including the chiral ones, in terms of the so-called
tube algebra, which is an MPO algebra representing the
Drinfeld Center of the input category".

An interesting feature of tensor networks is that they have Euclidean geometry built into them.
The tensors have dimensions and their interfaces have the appropriate codimensions. This
leads to a setting in which area entanglement laws and q-form symmetries appear naturally, as
well as other important but subtle topological phenomena.