Theoretical Study of Quantum Spin Chains : Integrable Inverse-Square Systems and Non-Integrable Systems

Abstract We study S=1/2 quantum spin chains with shift-invariant and inversion-symmetric next-nearest-neighbor interaction, also known as zigzag spin chains. We completely classify the integrability and non-integrability of the above class of spin systems. We prove that in this class there are only two integrable models, a classical model and a model solvable by the Bethe ansatz, and all the remaining systems are non-integrable. Our classification theorem confirms that within this class of spin chains, there is no missing integrable model. This theorem also implies the absence of intermediate models with a finite number of local conserved quantities.Abstract We study S=1/2 quantum spin chains with shift-invariant and inversion-symmetric next-nearest-neighbor interaction, also known as zigzag spin chains. We completely classify the integrability and non-integrability of the above class of spin systems. We prove that in this class there are only two integrable models, a classical model and a model solvable by the Bethe ansatz, and all the remaining systems are non-integrable. Our classification theorem confirms that within this class of spin chains, there is no missing integrable model. This theorem also implies the absence of intermediate models with a finite number of local conserved quantities.

arXiv:2501.15506Journal of Statistical PhysicsNov 25, 20256 citations

Integrable Quantum Spin Chains and Some Problems Related to Integrable Systems

A. Kundu, B. Basu-Mallick

Research reports in physicsJan 1, 19900 citations

INTEGRABLE SPIN CHAINS AND REPRESENTATION THEORY

P. P. Kulish

WORLD SCIENTIFIC eBooksAug 14, 20138 citations

Integrable Heisenberg-van Vleck chains with variable range exchange

V. I. Inozemtsev

The review of recent results in the s=1/2 quantum spin chains with $1/\sinh^2(κr$ exchange is presented. Related problems in the theory of classical and quantum Calogero-Sutherland-Moser systems with inverse square hyperbolic and elliptic potentials are discussed. The attention is paid to finding the explicit form of corresponding Bethe-Ansatz equations and to connection with generalized Hubbard chains in one dimension.The review of recent results in the s=1/2 quantum spin chains with $1/\sinh^2(κr$ exchange is presented. Related problems in the theory of classical and quantum Calogero-Sutherland-Moser systems with inverse square hyperbolic and elliptic potentials are discussed. The attention is paid to finding the explicit form of corresponding Bethe-Ansatz equations and to connection with generalized Hubbard chains in one dimension.

arXiv:hep-th/0201001Jan 1, 200232 citations

Spin Chains and Integrable Systems

Horaƫiu Năstase

Cambridge University Press eBooksSep 21, 20170 citations

The review of recent results in the s=1/2 quantum spin chains with

1/\sinh^2(κr

exchange is presented. Related problems in the theory of classical and quantum Calogero-Sutherland-Moser systems with inverse square hyperbolic and elliptic potentials are discussed. The attention is paid to finding the explicit form of corresponding Bethe-Ansatz equations and to connection with generalized Hubbard chains in one dimension.