Citation: Schuch, Norbert, David Pérez-García, and Ignacio Cirac. "Classifying quantum phases using matrix product states and projected entangled pair states." Physical Review B Condensed Matter and Materials Physics 84.16 (2011): 165139.
Web: https://arxiv.org/abs/1010.3732
Tags: Tensor-networks
In this paper, the authors demonstrate that all MPS are in the same phase. This gives a rigorous treatment of the classification of phases in 1-dimensions. Namely, the author shows to construct a gapped path of Hamiltonians between any two MPS. This works by starting with an MPS, going to its isometric form, deforming to the isometric form of the target MPS, and then going to the target MPS.
This paper also discusses the case of symmetries, where the same expected classification is obtained. Namely, the classification is in terms of cocylces in the 2nd cohomology of the symmetry group. The authors offer several different notion of equivalence for states with symmetries. What they find is that these different definitions yield different classifications. They thus emphasize the importance of being precise when dealing with the case of symmetries.
The authors also deal a bit with the two-dimensional case. Of course, they are not able to recover the subtle category-theoretic classification. What they are able to do is introduce a notion of "isometric-form" for PEPS, and they prove that all isometric PEPS are in the trivial phase. This isometric form is not to be confused with the different notion of "isometric PEPS", which can capture all string-net topological orders:
> Zaletel, Michael P., and Frank Pollmann. "Isometric tensor network states in two dimensions." Physical review letters 124.3 (2020): 037201.
An alternate definition of equivalence of phase is finite-depth quasi-local circuit. This is discussed a bit more here:
> Malz, Daniel, et al. "Preparation of matrix product states with log-depth quantum circuits." Physical Review Letters 132.4 (2024): 040404.
the upshot is that all MPS can be approximated arbitrarily well by finite-depth quasi-local circuits, and by log-depth exactly local circuits. Adding feed-forward and classical communication makes things even better.