Citation: Jones, Corey, Kylan Schatz, and Dominic J. Williamson. "Quantum cellular automata and categorical duality of spin chains." arXiv preprint arXiv:2410.08884 (2024).
Web: https://arxiv.org/abs/2410.08884
Tags: Modular-tensor-categories, Tensor-networks, Spin-chains, Operator-algebras, SPT/SETs
In this paper, the authors introduce a lovely obstruction theory for dualities in spin chains. By *duality* the authors are specifically referring to bounded-spread isomorphisms between symmetric subalgebras of spin chains to symmetric subalgebras of other spin chains, like Kramers-Wannier duality. The question they answer is exactly when these maps on symmetric subalgebras can be lifted to the entire spin chain. As an important motivating example, Kramers-Wannier duality cannot be lifted to the entire spin chain - doing so would necessarily violate locality.
Some important aspects of their results are as follows:
This paper fits into the context of the previous works on DHR bimodules for spin chains:
> "DHR bimodules of quasi-local algebras and symmetric quantum cellular automata", Corey Jones, 2023
> "Local topological order and boundary algebras", Corey Jones, Pieter Naaijkens, David Penneys, Daniel Wallik, 2023
> "An index for quantum cellular automata on fusion spin chains", Corey Jones, Junhwi Lim, 2024
The first two papers are mostly concerned with setting up general theory and proving first consequences, such as the fact that the group of ($\mathbb{Z}_2 \times \mathbb{Z}_2$)-symmetric QCA modulo symmetric finite depth circuits is non-abelian even in one dimension. The present paper is like the 2024 paper, in the sense that it is exploring the consequences of the theory.
I would argue that the results in this paper indicate DHR bimodule theory is an exciting and promising approach to spin chains. I imagine that these techniques could be broadly interesting to experts.
The main weakness of this paper seems to be how specific the scope is. I am not sure if many people have thought about obstructions to lifting dualities from symmetric subalgebras to full algebras before. Kramers-Wannier duality is interesting, but it has already been well-understood from different angles. Seeing as the paper does not contain other interesting examples relevant to physics people already care about, perhaps experts could be uninterested in the paper. DHR bimodules incur a lot of heavy category-theoretic baggage which I expect most people will not be willing to carry for the relatively small benefit.
I'll admit that even though I find this approach promising and interesting, I cannot claim to understand the theory well - it's a lot to learn!
[official-review]