Violating the All-or-Nothing Picture of Local Charges in Non-Hermitian Bosonic Chains

TL;DR

This paper provides counterexamples to the common belief that local charges in non-Hermitian bosonic chains exhibit all-or-nothing behavior, revealing more complex charge structures and limitations of existing integrability tests.

Contribution

It constructs explicit non-Hermitian models with selective local charges and offers a complete classification of conditions for k-local charges in these systems.

Findings

Existence of models with 3-local charges but no other nontrivial local charges.

Models with k-local charges for all k except k=4.

Grabowski--Mathieu integrability test is not universally applicable.

Abstract

We present explicit counterexamples to a widespread empirical expectation that local commuting charges display all-or-nothing behavior. In the class of bosonic chains with symmetric nearest-neighbor hopping and arbitrary on-site terms (including non-Hermitian terms), we exhibit systems that possess k-local charges for some but not all k. Concretely, we construct non-Hermitian models with a 3-local charge but no other nontrivial local charges and models with k-local charges for all k except k = 4. These results show that the Grabowski--Mathieu integrability test based on 3-local charges is not universally applicable. We further give necessary and sufficient conditions for the existence of k-local charges in this class, yielding an exhaustive classification and uncovering additional integrable models.