Citation: Kim, Isaac H., et al. "Strict area law implies commuting parent Hamiltonian." arXiv preprint arXiv:2404.05867 (2024).
Web: https://arxiv.org/abs/2404.05867
Tags: MTC-reconstruction, Approximate-structure
In this lovely paper, the authors show that any quantum state satisfying the entanglement bootstrap axioms A0+A1 has a commuting parent Hamiltonian. In particular, since a strict area law implies A0+A1, this means that any state with a strict area law has a commuting parent Hamiltonian. An intriguing consequence of this fact is that one expects a commuting parent Hamiltonian to not have any bulk 2-current, and as such its chiral central charge should vanish. So, states satisfying exact area laws only can represent a very small class of states. More general topological states will still have area laws, but they will have error terms which decay with system size.
A very nice feature of this paper is how conceptually clean the proof is. The point is that whenever ABC are adjacent squares in a sequence, A1 states that I(A,C:B)=0. Now, whenever I(A,C:B)=0 then the projector onto the support of AB commutes with the projector onto the support of BC. So, by making a Hamiltonian out of terms which overlap at regions with the right geometry, one can make a commuting parent Hamiltonian out of projectors onto supports of reduced density matrices. By making these regions have overlaps which cover the plane, one can ensure the LTQO condition.
This paper is a big motivation for using approximate axioms instead of exact ones - you're missing out on some real physics by using exact axioms.
One of my favorite features of this paper is how it interacts with the following one:
> Kim, Isaac H., et al. "Chiral central charge from a single bulk wave function." Physical Review Letters 128.17 (2022): 176402.
Namely, for states with A0+A1 there's a known procedure for robustly extracting the chiral central charge. The conjectured conclusion is that for states satisfying A0+A1 (the only states where they can prove any theorems), the chiral central charge they extract is always 0! Proving this is still an open problem, and it is not clear how well-behaved the modular commutator is for states which approximately satisfy A1.