Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Mizuki Yamaguchi, Naoto Shiraishi

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.

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

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.
Paper Structure (40 sections, 16 theorems, 182 equations, 2 tables)

This paper contains 40 sections, 16 theorems, 182 equations, 2 tables.

Table of Contents

  1. introduction
  2. setup and main result
  3. Partially integrable counterexample I: Type N$^+$
  4. Partially integrable counterexample II: Type C$^-$
  5. New completely integrable systems
  6. Even--odd sensitivity from staggered charges
  7. The Bose-Hubbard model (standard nonintegrable example)
  8. Proof part 1: General treatment
  9. Main idea of the proof
  10. Symbols and useful relations
  11. Analysis of Step 1
  12. Symmetry argument
  13. Analysis of Step 2
  14. Proof part 2: Complete analysis of 3-local charges
  15. Criterion for the existence of 3-local charges
  16. ...and 25 more sections

Key Result

Lemma 1

Let $k$ be an integer with $2 \leq k \leq N/2$, and let $Q_k$ be a $k$-local quantity. Then, the following two conditions are equivalent.

Theorems & Definitions (36)

  • Definition : $k$-local charge
  • Lemma 1: Step 1 analysis
  • Lemma 2: Step 2 analysis
  • Lemma 3: Complete list of models with 3-local charges
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 26 more