Absence of local conserved charges of the Fredkin spin chain and its truncated versions

TL;DR

The paper addresses whether the three-site Fredkin spin chain is quantum integrable by searching for local conserved charges. Using a Shiraishi style column-expression approach, it proves that no nontrivial $k$-local conserved charges exist for $4\le k\le N/2$ under periodic and open boundary conditions. It extends the analysis to truncated Fredkin chains obtained by removing some three-site terms, showing nonintegrability persists unless all three-site terms are removed. The results clarify the integrability landscape for constrained and medium-range spin models and point to future work on quasi-local charges and deformed variants.

Abstract

Conservation laws serve as the hallmark of integrability. The absence of conserved charges typically implies that the model is nonintegrable. The recently proposed Fredkin spin chain exhibits rich structures, and its ground state is analytically known. However, whether the Fredkin spin chain is integrable remains an open question. In this work, through rigorous analytical calculations, we demonstrate that the Fredkin spin chain, under both periodic and open boundary conditions, lacks local conserved charges, thereby confirming its nonintegrable nature. Furthermore, we find that when one or a portion of the Hamiltonian terms are removed (referred to as the truncated Fredkin spin chain), local conserved charges are still absent. Our findings suggest that in models involving three-site interactions, integrable models are generally rare.

Theorems & Definitions (9)

• Theorem 1
• Theorem 2
• Lemma 1
• proof
• Lemma 2
• Theorem 3
• proof
• Theorem 4
• proof