We extend our previous analysis of the classical integrable models of Calogero in several respects. Firstly we provide the algebraic resaons of their quantum integrability.Secondly we show why these systems allow their initial value problem to be solved in closed form . Furthermore we show that due to their similarity with the above models the classical and quantum Heisenberg magnets with long rang interactions in a magnetic field are also intergrable. Explicit expressions are given for the integrals of motion in involution in the classical case and for the commuting operators in the quantum case.
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.
We review the current (as of Fall 2016) status of the studies on the emergent integrability in many‐body localized models. We start by explaining how the phenomenology of fully many‐body localized systems can be recovered if one assumes the existence of a complete set of (quasi)local operators which commute with the Hamiltonian (local integrals of motions, or LIOMs). We describe the evolution of this idea from the initial conjecture, to the perturbative constructions, to the mathematical proof given for a disordered spin chain. We discuss the proposed numerical algorithms for the construction of LIOMs and the status of the debate on the existence and nature of such operators in systems with a many‐body mobility edge, and in dimensions larger than one. image