Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2509.04838 · Physical review. B./Physical review. B

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

Wen-Ming Fan, Kun Hao, Yang-Yang Chen, Kun Zhang, Xiao-Hui Wang, V. E. Korepin

Published Nov 19, 2025• 0 citations
arXivHTMLPDF

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.

DOI
10.1103/2qhs-fz3g

Related papers

The structure of conserved charges in open spin chains

Marek Grabowski, Pierre Mathieu

We study the local conserved charges in integrable spin chains of the XYZ type with non-trivial boundary conditions. The general structure of these charges consists of a bulk part, whose density is identical to that of a periodic chain, and a boundary part. In contrast with the periodic case, only charges corresponding to interactions of even number of spins exist for the open chain. Hence, there are half as many charges in the open case as in the closed case. For the open spin- XY chain, we derive the explicit expressions of all the charges. For the open spin- XXX chain, several lowest-order charges are presented and a general method of obtaining the boundary terms is indicated. In contrast with the closed case, the XXX charges cannot be described in terms of a Catalan tree pattern.

arXiv:hep-th/9604079•Journal of Physics A Mathematical and General•Dec 7, 1996•21 citations

Proof of absence of local conserved quantities in the mixed-field Ising chain

Yuuya Chiba

Absence of local conserved quantities is often required, such as for thermalization or for the validity of response theory. Although many studies have discussed whether thermalization occurs in the Ising chain with longitudinal and transverse fields, rigorous results on local conserved quantities of this model have still been lacking. Here, we rigorously prove that, if all coupling constants are nonzero, this model has no conserved quantity spanned by local operators with support size up to half of the system size other than a trivial one, i.e., a linear combination of the Hamiltonian and the identity. The proof is given not only for the periodic boundary condition but also for the open boundary condition. We also discuss relation to the integrability of the model where the longitudinal field is set to zero. Our results provide the second example of spin models whose nonintegrability is rigorously proved.

arXiv:2307.16703•Physical review. B./Physical review. B•Jan 12, 2024•24 citations

Minimal nonintegrable models with three-site interactions

Wen-Ming Fan, Kun Hao, Yang-Yang Chen +3 more

We study integrability breaking in translationally invariant spin-$1/2$ chains with genuine three-site interactions. Using a two-qubit composite representation, we prove that the deformed Fredkin spin chain is nonintegrable for any nonzero deformation parameter, although its Hamiltonian decomposes into coupled integrable building blocks. We then extract the minimal injective models, defined as the simplest models needed for a rigorous nonintegrability test, from the Hamiltonian density of the deformed Fredkin spin chain. Among the sixteen such models, six satisfy the Reshetikhin condition and are integrable, while two satisfy Hokkyo's criterion and are nonintegrable away from special coefficient values. Returning to one-site translation-invariant spin-$1/2$ chains leaves four genuine three-site models: two integrable models and two generically nonintegrable models. These results give a compact classification of minimal three-site models obtained from the deformed Fredkin spin chain and identify a local mechanism by which coupling individually integrable structures can destroy integrability beyond nearest-neighbor interactions.

quant-pharXiv:2602.07867•Feb 10, 2026•0 citations

Integrability from a single conservation law in quantum spin chains

Akihiro Hokkyo

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.

arXiv:2508.20713•Physical review. B./Physical review. B•Jun 8, 2026•2 citations

Complete classification of integrability and nonintegrability for spin- 1 2 chain with symmetric nearest-neighbor interaction

Mizuki Sanatani, Yuuya Chiba, Naoto Shiraishi

General spin- 1 2 chains with symmetric nearest-neighbor interaction are studied. We rigorously prove that all spin models in this class, except for known integrable systems, are nonintegrable in the sense that they possess no nontrivial local conserved quantities. This result confirms that there are no missing integrable systems; i.e., integrable systems in this class are exactly those that are already known. In addition, this result excludes the possibility of intermediate systems that have a finite number of nontrivial local conserved quantities. Our findings support the expectation that integrable systems are exceptional in quantum many-body systems and most systems are nonintegrable.

arXiv:2411.02162•Physical Review Research•Jun 9, 2026•1 citation

Proof of nonintegrability of the spin-1 bilinear-biquadratic chain model

HaRu K. Park, SungBin Lee

Spin-1 chain models have been extensively studied in condensed matter physics, significantly advancing our understanding of quantum magnetism and low-dimensional systems, which exhibit unique properties compared to their spin-$1/2$ counterparts. Despite substantial progress in this area, a rigorous proof of nonintegrability for the bilinear-biquadratic chain model has been lacking. While integrable solutions are known for specific parameter values, a comprehensive understanding of the model's general integrability has remained elusive. In this paper, we present a rigorous proof of nonintegrability for general spin-1 bilinear-biquadratic chain models, by extending the proofs of nonintegrability on spin-$1/2$ models. Our proof not only confirms the nonintegrability of widely studied models but also provides offer deeper insights into several areas. These include the identification of the absence of local conserved quantities in quantum many-body scar systems with perfect fidelity revivals, such as the AKLT model. This paper marks a significant step toward understanding the complex dynamics of spin-1 systems.

arXiv:2410.23286•Physical review. B./Physical review. B•Apr 28, 2025•9 citations
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.
We study the local conserved charges in integrable spin chains of the XYZ type with non-trivial boundary conditions. The general structure of these charges consists of a bulk part, whose density is identical to that of a periodic chain, and a boundary part. In contrast with the periodic case, only charges corresponding to interactions of even number of spins exist for the open chain. Hence, there are half as many charges in the open case as in the closed case. For the open spin- XY chain, we derive the explicit expressions of all the charges. For the open spin- XXX chain, several lowest-order charges are presented and a general method of obtaining the boundary terms is indicated. In contrast with the closed case, the XXX charges cannot be described in terms of a Catalan tree pattern.
Absence of local conserved quantities is often required, such as for thermalization or for the validity of response theory. Although many studies have discussed whether thermalization occurs in the Ising chain with longitudinal and transverse fields, rigorous results on local conserved quantities of this model have still been lacking. Here, we rigorously prove that, if all coupling constants are nonzero, this model has no conserved quantity spanned by local operators with support size up to half of the system size other than a trivial one, i.e., a linear combination of the Hamiltonian and the identity. The proof is given not only for the periodic boundary condition but also for the open boundary condition. We also discuss relation to the integrability of the model where the longitudinal field is set to zero. Our results provide the second example of spin models whose nonintegrability is rigorously proved.
We study integrability breaking in translationally invariant spin-1/21/21/2 chains with genuine three-site interactions. Using a two-qubit composite representation, we prove that the deformed Fredkin spin chain is nonintegrable for any nonzero deformation parameter, although its Hamiltonian decomposes into coupled integrable building blocks. We then extract the minimal injective models, defined as the simplest models needed for a rigorous nonintegrability test, from the Hamiltonian density of the deformed Fredkin spin chain. Among the sixteen such models, six satisfy the Reshetikhin condition and are integrable, while two satisfy Hokkyo's criterion and are nonintegrable away from special coefficient values. Returning to one-site translation-invariant spin-1/21/21/2 chains leaves four genuine three-site models: two integrable models and two generically nonintegrable models. These results give a compact classification of minimal three-site models obtained from the deformed Fredkin spin chain and identify a local mechanism by which coupling individually integrable structures can destroy integrability beyond nearest-neighbor interactions.
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.
General spin- 1 2 chains with symmetric nearest-neighbor interaction are studied. We rigorously prove that all spin models in this class, except for known integrable systems, are nonintegrable in the sense that they possess no nontrivial local conserved quantities. This result confirms that there are no missing integrable systems; i.e., integrable systems in this class are exactly those that are already known. In addition, this result excludes the possibility of intermediate systems that have a finite number of nontrivial local conserved quantities. Our findings support the expectation that integrable systems are exceptional in quantum many-body systems and most systems are nonintegrable.
Spin-1 chain models have been extensively studied in condensed matter physics, significantly advancing our understanding of quantum magnetism and low-dimensional systems, which exhibit unique properties compared to their spin-1/21/21/2 counterparts. Despite substantial progress in this area, a rigorous proof of nonintegrability for the bilinear-biquadratic chain model has been lacking. While integrable solutions are known for specific parameter values, a comprehensive understanding of the model's general integrability has remained elusive. In this paper, we present a rigorous proof of nonintegrability for general spin-1 bilinear-biquadratic chain models, by extending the proofs of nonintegrability on spin-1/21/21/2 models. Our proof not only confirms the nonintegrability of widely studied models but also provides offer deeper insights into several areas. These include the identification of the absence of local conserved quantities in quantum many-body scar systems with perfect fidelity revivals, such as the AKLT model. This paper marks a significant step toward understanding the complex dynamics of spin-1 systems.