Classification of dynamical Lie algebras generated by spin interactions on undirected graphs

TL;DR

This paper classifies the dynamical Lie algebras generated by 2-local spin interactions on undirected graphs, revealing that their structure depends mainly on whether the graph is bipartite, with the one-dimensional case being unique.

Contribution

It extends previous classifications from spin chains to general undirected graphs, highlighting the role of bipartiteness in the algebra's structure.

Findings

Dynamical Lie algebra depends on graph bipartiteness.

One-dimensional case is unique in polynomial size.

Non-one-dimensional cases are restricted in algebra size.

Abstract

We provide a classification of all dynamical Lie algebras generated by 2-local spin interactions on undirected graphs. Building on our previous work where we provided such a classification for spin chains, here we consider the more general case of undirected graphs. As it turns out, the one-dimensional case is special; for any other graph, the dynamical Lie algebra solely depends on whether the graph is bipartite or not. An important consequence of this result is that the cases where the dynamical Lie algebra is polynomial in size are special and restricted to one dimension.