Spin Chains and Integrable Systems

P. P. Kulish

Naoto Shiraishi

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.

Zengo Tsuboi, Anton Zabrodin, Andrei Vladimirovich Zotov

For integrable inhomogeneous supersymmetric spin chains (generalized graded magnets) constructed employing Y(gl(N|M))-invariant R-matrices in finite-dimensional representations we introduce the master T-operator which is a sort of generating function for the family of commuting quantum transfer matrices. Any eigenvalue of the master T-operator is the tau-function of the classical mKP hierarchy. It is a polynomial in the spectral parameter which is identified with the 0-th time of the hierarchy. This implies a remarkable relation between the quantum supersymmetric spin chains and classical many-body integrable systems of particles of the Ruijsenaars-Schneider type. As an outcome, we obtain a system of algebraic equations for the spectrum of the spin chain Hamiltonians.

Kazuhiro Hikami, P. P. Kulish, Miki Wadati

Rob Klabbers, Jules Lamers

The Inozemtsev chain is an exactly solvable interpolation between the short-range Heisenberg and long-range Haldane–Shastry (HS) chains. In order to unlock its potential to study spin interactions with tunable interaction range using the powerful tools of integrability, the model’s mathematical properties require better understanding. As a major step in this direction, we present a new generalisation of the Inozemtsev chain with spin symmetry reduced to \textit{U}(1) 𝑈 ( 1 ) , interpolating between a Heisenberg XXZ chain and the XXZ-type HS chain, and integrable throughout. Underlying it is a new quantum many-body system that extends the elliptic Ruijsenaars system by including spins, contains the trigonometric spin-Ruijsenaars–Macdonald system as a special case, and yields our spin chain by ‘freezing’. Our models have potential applications from condensed-matter to high-energy theory, and provide a crucial step towards a general theory for long-range integrability.

A. Kundu, B. Basu-Mallick