A new finite-dimensional classical integrable system and a new quantum integrable system are generated from a spectral problem for the MKdV hierarchy through the nonlinearization technique of Lax systems. Our classical integrable system is an example of Gaudin magnet with boundary and relates to the finite band solutions of the MKdV hierarchy. Its Lax representation and r -matrix is given, and its separation of variables is performed. Based on a direct link between r -matrix formulas for classical systems and quantum problems, a quantum integrable system with separated variables is presented.A new finite-dimensional classical integrable system and a new quantum integrable system are generated from a spectral problem for the MKdV hierarchy through the nonlinearization technique of Lax systems. Our classical integrable system is an example of Gaudin magnet with boundary and relates to the finite band solutions of the MKdV hierarchy. Its Lax representation and r -matrix is given, and its separation of variables is performed. Based on a direct link between r -matrix formulas for classical systems and quantum problems, a quantum integrable system with separated variables is presented.

Journal of the Physical Society of JapanDec 15, 199815 citations

Quantum integrable systems in one dimension

Michael Fowler

arXiv:cond-mat/9401051Physica D Nonlinear PhenomenaSep 1, 19952 citations

QUANTUM GROUP AND QUANTUM INTEGRABLE SYSTEMS

Mo‐Lin Ge

Jul 1, 199245 citations

Integrability and nonintegrability of quantum systems: Quantum integrability and dynamical symmetry

Wei-Min Zhang, D.H. Feng, Jian‐Min Yuan +1 more

In this paper we discuss the concepts of quantum integrability and nonintegrability. Based on the concept of a complete set of commuting observables and the Hilbert-space structure of a quantum system, the definitions are given for the quantum-dynamical degrees of freedom and quantum phase space from which the quantum integrability is defined. A criterion for quantum integrability then emerges; the system is integrable if it possesses dynamical symmetry. Breaking of dynamical symmetry is connected with the nonintegrability of systems and thus is the inherent mechanism of chaotic motion. A number of examples are discussed.In this paper we discuss the concepts of quantum integrability and nonintegrability. Based on the concept of a complete set of commuting observables and the Hilbert-space structure of a quantum system, the definitions are given for the quantum-dynamical degrees of freedom and quantum phase space from which the quantum integrability is defined. A criterion for quantum integrability then emerges; the system is integrable if it possesses dynamical symmetry. Breaking of dynamical symmetry is connected with the nonintegrability of systems and thus is the inherent mechanism of chaotic motion. A number of examples are discussed.

Physical Review AJul 1, 198961 citations

Quantum theory of non-integrable systems

Tomio Yamakoshi Petrosky, I. Prigogine, S. Tasaki

Physica A Statistical Mechanics and its ApplicationsApr 1, 1991206 citations

An extensive number of independent local integrals of motion in a nonintegrable quantum many-body system

Oleg Valentinovich Lychkovskiy

Studies of integrable quantum many-body systems have a long history with an impressive record of success. However, quite surprisingly, an unambiguous definition of quantum integrability remains a matter of an ongoing debate. We contribute to this debate by presenting several observation motivated by a particular quantum model of $N$ spins $1/2$ . Remarkably, although this system is nonintegrable, it possesses a large number of mutually commuting local binary integrals of motion. We address the question of the number of independent integrals of motion. Strikingly, we find that, according to a common definition of independence of quantum integrals of motion (QIMs), this number equals to the number of spins, $N$ . A common wisdom would then suggest that the system is completely integrable, which is not the case. To resolve this conundrum, we dwell upon the notion of independence of QIMs. We argue that the common definition should be amended in order to match the intuitive notion of independence. We suggest a new definition, which, in particular, implies that some of the $N$ QIMs of the considered model are dependent. Still, even according to this strict definition, $N/2$ integrals of motion remain independent. This challenges the intuition on the integrable quantum many-body systems as well as some of the most elaborated definitions of quantum integrability.Studies of integrable quantum many-body systems have a long history with an impressive record of success. However, quite surprisingly, an unambiguous definition of quantum integrability remains a matter of an ongoing debate. We contribute to this debate by presenting several observation motivated by a particular quantum model of $N$ spins $1/2$. Remarkably, although this system is nonintegrable, it possesses a large number of mutually commuting local binary integrals of motion. We address the question of the number of independent integrals of motion. Strikingly, we find that, according to a common definition of independence of quantum integrals of motion (QIMs), this number equals to the number of spins, $N$. A common wisdom would then suggest that the system is completely integrable, which is not the case. To resolve this conundrum, we dwell upon the notion of independence of QIMs. We argue that the common definition should be amended in order to match the intuitive notion of independence. We suggest a new definition, which, in particular, implies that some of the $N$ QIMs of the considered model are dependent. Still, even according to this strict definition, $N/2$ integrals of motion remain independent. This challenges the intuition on the integrable quantum many-body systems as well as some of the most elaborated definitions of quantum integrability.

Dec 11, 20180 citations

N

spins

1/2

. Remarkably, although this system is nonintegrable, it possesses a large number of mutually commuting local binary integrals of motion. We address the question of the number of independent integrals of motion. Strikingly, we find that, according to a common definition of independence of quantum integrals of motion (QIMs), this number equals to the number of spins,

N

. A common wisdom would then suggest that the system is completely integrable, which is not the case. To resolve this conundrum, we dwell upon the notion of independence of QIMs. We argue that the common definition should be amended in order to match the intuitive notion of independence. We suggest a new definition, which, in particular, implies that some of the

N

QIMs of the considered model are dependent. Still, even according to this strict definition,

N/2

integrals of motion remain independent. This challenges the intuition on the integrable quantum many-body systems as well as some of the most elaborated definitions of quantum integrability.