Citation: Williamson, Dominic J., Arpit Dua, and Meng Cheng. "Spurious topological entanglement entropy from subsystem symmetries." Physical review letters 122.14 (2019): 140506.
Web: https://arxiv.org/abs/1808.05221
Tags: Information-theory, Generalized-symmetries, No-go, SPT/SETs
This paper gives translation-invariant examples in which the Kitaev-Preskill and Levin-Wen prescriptions for topological entanglement entropy give incorrect values. Namely, for well-chosen cuts on a 2D cluster state the topological entanglement entropy will be non-zero despite the fact that cluster states are nonzero. This counterexample is based on the earlier work of
> Zou, Liujun, and Jeongwan Haah. "Spurious long-range entanglement and replica correlation length." Physical Review B 94.7 (2016): 075151.
but goes further because it is able to achieve its results in a translation invariant fashion. The way this is done is by coarse-graining a square lattice, and then choosing cuts which are vertical and horizontal with respect to the lattice.
The idea is that horizontal and vertical lines of sigma_X commute with the stabilizers of the cluster state - these are 1-dimensional string-like (gauge-like) symmetries. When the cuts are chosen in line with these string like symmetries, they will preserve the symmetry and this symmetry preservation implies an extra correctional term to topological entanglement entropy just as described in the work of Zou and Haah.
The authors also introduce a new quantity they call S_dumb. This quantity is computed by using the Kitaev-Preskill formula on three regions A,B,C which form a dumbell (hence, "dumb"). This quantity is equal to zero whenever the entanglement follows a law of the form \alpha * L + \gamma, and is independent of perturbations to the same degree that TEE is. Hence, a nonzero S_dumb is a surefire indication that something has gone wrong in the computation of TEE, probably due to string-like symmetries. The quantity S_topo - S_dumb can sometimes correctly identity topological entanglement entropy, with the spurious entropy cancelling from both sides. This is discussed in detail in the paper.
The authors then give several other examples of systems which give spurious topological entanglement entropy, the most novel of which is a compactified version of Haah's cubic code. Another example which breaks TEE but is not contained in this paper is Klein spin models, which are topologically trivial phases of matter with a large number of 1-dimensional symmetries:
> Nussinov, Zohar, and Gerardo Ortiz. "A symmetry principle for topological quantum order." Annals of Physics 324.5 (2009): 977-1057.
another example, this time NOT based on string-like symmetries, is
> Kato, Kohtaro, and Fernando GSL Brandão. "Toy model of boundary states with spurious topological entanglement entropy." Physical Review Research 2.3 (2020): 032005.