Citation: Shi, Bowen, and Isaac H. Kim. "Entanglement bootstrap approach for gapped domain walls." Physical Review B 103.11 (2021): 115150.
Web: https://arxiv.org/abs/2008.11793
Tags: MTC-reconstruction, Defects/boundaries
This paper extends the entanglement bootstrap program to the case of gapped domain walls. The axioms are simple. At the domain wall, one can consider the A0 axiom exactly as before. The A1 axiom now has three qualitatively difference cases - both cuts above the domain wall, both cuts below the domain wall, or one on each side. The answer is to only enforce those partitions with one cut on each side. The partitions with both cuts on the same side will typically give nonzero answers. These nonzero answers are topological invariants, however.
The amazing observation of this paper is as follows. Not do these partitions with both cuts on the same side give topologically invariant quantities, these quantities can be expressed in closed form as the logarithm of the global quantum dimension of some object! This global quantum dimension is the sum of squares of quantum dimensions of individual sectors. These sectors are known as parton sectors. These sectors are defined as follows. Consider a half circle which is on one side of the domain wall and dips into the other. This is topologically innequavelnt to the disk. Its information convex is a simplex; its extreme points are superselection sectors; the difference in entropy between those extreme points and the entropy of the reference state is defined to be the logarithm of the quantum dimension. These are the quantum dimensions whose sum of squares gives the entropy discussed above.
In general, there's all sorts of wacky regions one could draw around the boundary. All of these regions have information convexes. The is currently no known general procedure for computing the structure of these complexes! The idea is that there should be some way of interpreting them in terms of category theory and then using algebra to do it, but nobody knows how. It is not clear at all what these new quantities discovered mean algebraically.
A shorter version of the paper with slightly different choice of topics is also online:
> Shi, Bowen, and Isaac H. Kim. "Domain wall topological entanglement entropy." Physical Review Letters 126.14 (2021): 141602.
It is very exciting that the entanglement bootstrap program could be used to uncover new physics!
This new formalism is quite powerful, and applicable to proving theorems. It was used to show that all phases which admit gapped boundaries are equivalent to Levin-Wen string nets (!!!!):
> Kim, Isaac H., and Daniel Ranard. "Classifying 2D topological phases: mapping ground states to string-nets." arXiv preprint arXiv:2405.17379 (2024).