Integrability from a single conservation law in quantum spin chains

Abstract

We prove that, for translationally invariant quantum spin chains with finite-range interactions, the existence of a specific conservation law implies the presence of infinitely many local conserved quantities, i.e., integrability. This shows that the standard hierarchy of local conserved quantities arising for nearest-neighbor Hamiltonians obtained via the canonical Yang-Baxter construction is already encoded in the lowest nontrivial conservation law, known as the Reshetikhin condition. Combined with recent rigorous results on nonintegrability, our theorem strongly restricts the possibility of partially integrable systems that admit only a finite but large number of local conserved quantities. Our work establishes a rigorous foundation for the systematic identification of new integrable models and deepens the algebraic understanding of conservation-law structures in quantum spin chains.

15 pages

References in corpus (8)

• Thermalization and its mechanism for generic isolated quantum systems
• Experimental Observation of a Generalized Gibbs Ensemble
• Open XXZ spin chain: Nonequilibrium steady state and strict bound on ballistic transport
• Current operators in integrable spin chains: lessons from long range deformations
• Classifying nearest-neighbour interactions and deformations of AdS
• Complete classification of integrability and non-integrability of S=1/2 spin chains with symmetric next-nearest-neighbor interaction
• Bootstrapping the $R$-matrix
• Dichotomy theorem separating complete integrability and non-integrability of isotropic spin chains