arXiv:2008.06825 · Symmetry Integrability and Geometry Methods and Applications

Perfect Integrability and Gaudin Models

Kang Lu

Published Dec 10, 2020• 4 citations

Abstract

We suggest the notion of perfect integrability for quantum spin chains and conjecture that quantum spin chains are perfectly integrable. We show the perfect integrability for Gaudin models associated to simple Lie algebras of all finite types, with periodic and regular quasi-periodic boundary conditions.

DOI: 10.3842/sigma.2020.132

Related papers

The Bethe-Ansatz for Gaudin Spin Chains

Kowalik, Ilona

We investigate a special case of the quantum integrable Heisenberg spin chain known as Gaudin model. The Gaudin model is an important example of quantum integrable systems. We study the Gaudin model for the Lie algebra s[z(

Jun 9, 20080 citations

INTEGRABILITY OF OPEN SPIN CHAINS WITH QUANTUM ALGEBRA SYMMETRY

Luca Mezincescu, Rafael I. Nepomechie

We construct an open quantum spin chain from the “twisted” [Formula: see text]R matrix in the fundamental representation which has the quantum algebra symmetry Uq[ su (2)]. This anisotropic spin-1 chain is different from the Uq[ su (2)]-invariant chain constructed from the “untwisted” [Formula: see text] spin-1 R matrix (namely, the spin-1 XXZ chain of Fateev-Zamolodchikov with boundary terms) but, nevertheless, is also completely integrable. We discuss the general case of an R matrix of the type g(k), where k∈{1, 2, 3}, and g is any simple Lie algebra.

arXiv:hep-th/9206047International Journal of Modern Physics ADec 10, 1991132 citations

Quantum-classical duality for Gaudin magnets with boundary

M. Vasilyev, Anton Vladimirovich Zabrodin, Andrei Vladimirovich Zotov

We establish a remarkable relationship between the quantum Gaudin models with boundary and the classical many-body integrable systems of Calogero-Moser type associated with the root systems of classical Lie algebras (B, C and D). We show that under identification of spectra of the Gaudin Hamiltonians HjG with particles velocities q˙j of the classical model all integrals of motion of the latter take zero values. This is the generalization of the quantum-classical duality observed earlier for Gaudin models with periodic boundary conditions and Calogero-Moser models associated with the root system of the type A.

arXiv:1911.11792Nuclear Physics BJan 21, 20204 citations

A new class of completely integrable quantum spin chains

Tomaž Prosen

A large (infinitely-dimensional) class of completely integrable (possibly non-autonomous) spin chains is discovered associated to an infinite-dimensional Lie Algebra of infinite rank. The complete set of integrals of motion is constructed explicitly, as well as their eigenstates and spectra. As an example we outline kicked Ising model: Ising chain periodically kicked with transversal magnetic field.

arXiv:solv-int/9711007Journal of Physics A Mathematical and GeneralMay 29, 199829 citations

ADDENDUM: INTEGRABILITY OF OPEN SPIN CHAINS WITH QUANTUM ALGEBRA SYMMETRY

Luca Mezincescu, Rafael I. Nepomechie

In the above reference, we prove that certain quantum spin chains with quantumalgebra symmetry are integrable. Specifically, these are the quantum-algebra-invariant open chains associated with the affine Lie algebras A (1) 1, A(2) 2n, A(2) 2n−1, B(1) n, C (1) n, and D (1)

International Journal of Modern Physics ASep 10, 199270 citations

Lie algebra deformations and the exact solutions of integrable quantum many-body systems

Yuan-Harng Lee

In this body of work, we will examine two classes of Lie algebra deformations and their appli- cation to the exact solutions of various quantum integrable systems. The first class of Lie algebra deformations are polynomial deformations of the A1 Lie algebra. We construct unitary representations of these algebras in terms of bosonic and spin operators as well as obtain their single-variable differential operator realizations. Using these results, we obtain the exact Bethe ansatz solutions of a variety of quantum integrable systems such as the Bose-Hubbard and Tavis-Cummings model. The second class of Lie algebra deformations are the so-called quasi-Gaudin algebra. This algebra was introduced to construct integrable systems which are only quasi-exactly solvable. Using a suitable representation this algebra, we obtain partial solutions for a class of bosonic models which do not preserve U(1) symmetry

Jan 1, 20120 citations