Citation: Siva, Karthik, et al. "Universal tripartite entanglement signature of ungappable edge states." Physical Review B 106.4 (2022): L041107.
Web: https://arxiv.org/abs/2110.11965
Tags: MTC-reconstruction
In this paper the authors introduce a new entanglement measure for ground states of topological order, based on the notion of a "Markov gap". The measure works as follows. First, you consider two adjacent regions A and B, and you call C the complement of AB. The way the invariant is defined is by first minimizing over all possible unitaries which can be applied at the two trijunction points, and then taking the Markov gap between regions A and B. The authors argue that this invariant is equal to log(2)/3 times the minimal total central charge of any CFT edge theory.
This invariant is nice, because it is a bulk invariant which diagnoses chiral topological order. The inconvenient part is that it involves this minimization over unitaries around the trijunction points. This minimization is in a real sense choosing a boundary theory. The fact that disentangling the regions around the trijunction points is tantamount to picking a boundary theory is used in the argument for why the invariant should be related to CFT. So, in a real sense this is not quite a bulk invariant.
A point to keep in mind here is that if we are too liberal, it is easy to define a bulk invariant which determines whether or not a theory admints a gapped boundary. Namely, look at that theory on a disk and ask if there is any unitary on the boundary that can be applied such that if you remove the edge of the boundary then the boundary becomes gapped. By asking about all the different ways of creating boundary theories, you can easily extract the minimum total central charge, possible chiral central charges etc...
So, when defining bulk invariants, it is important to make sure you aren't accidentally defining a boundary invariant. This paper is definitely toeing the line.
When discussing why the trijunction points need to be treated differently, the authors say that "The trisections can contribute a lattice-scale non-universal contribution to h(A : B). Intuitively, UV physics can dress the trisection with an entangled tripartite state, which can contribute a finite h". Perhaps this is the sort of language which would help some people understand spurious phenomena.