Multiple-quantum dynamics of one-dimensional nuclear spin systems in solids

Multiple-quantum spin dynamics is studied using analytic and numerical methods for one-dimensional finite linear chains and rings of nuclear spins 1/2 coupled by dipole-dipole interactions. An approximation of dipole-dipole interaction between nearest neighbors having the same constants is used to obtain exact expressions for the intensities of the multiple-quantum coherences in the spin systems studied, which are initially in thermal equilibrium and whose evolution is described by a two-spin two-quantum Hamiltonian. An approximation of nearest neighbors with arbitrary dipole-dipole interaction constants is used to establish a simple relationship between the multiple-quantum dynamics and the dynamics of spin systems with an XY Hamiltonian. Numerical methods are developed to calculate the intensities of multiple-quantum coherences in multiple-quantum NMR spectroscopy. The integral of motion is obtained to expand the matrix of the two-spin two-quantum Hamiltonian into two independent blocks. Using the nearest-neighbor approximation the Hamiltonian is factorized according to different values of the projection operator of the total spin momentum on the direction of the external magnetic field. Results of calculations of the multiple-quantum dynamics in linear chains of seven and eight nuclear spins and a six-spin ring are presented. It is shown that the evolution of the intensities of the lowest-order multiple-quantum coherences in linear chains is accurately described allowing for dipole-dipole interaction of nearest and next-nearest neighbors only. Numerical calculations are used to compare the contributions of nearest and remote spins to the intensities of the multiple-quantum coherences.     

On the Sutherland Spin Model of <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">N</Emphasis></Subscript> Type and Its Associated Spin Chain

 The B _N hyperbolic Sutherland spin model is expressed in terms of a suitable set of commuting Dunkl operators. This fact is exploited to derive a complete family of commuting integrals of motion of the model, thus establishing its integrability. The Dunkl operators are shown to possess a common flag of invariant finite-dimensional linear spaces of smooth scalar functions. This implies that the Hamiltonian of the model preserves a corresponding flag of smooth spin functions. The discrete spectrum of the restriction of the Hamiltonian to this spin flag is explicitly computed by triangularization. The integrability of the hyperbolic Sutherland spin chain of B _N type associated with the dynamical model is proved using Polychronakos's ``freezing trick'. Received: 14 February 2002 / Accepted: 19 June 2002 Published online: 10 December 2002 RID="*" ID="*" Corresponding author. E-mail: artemio@fis.ucm.es RID="**" ID="**" On leave of absence from Institute of Mathematics, 3 Tereschenkivska St., 01601 Kyiv-4 Ukraine Communicated by L. Takhtajan     

A Remark on the Notion of Independence of Quantum Integrals of Motion in the Thermodynamic Limit

Studies of integrable quantum many-body systems have a long history with an impressive record of success. However, surprisingly enough, an unambiguous definition of quantum integrability remains a matter of an ongoing debate. We contribute to this debate by dwelling upon an important aspect of quantum integrability—the notion of independence of quantum integrals of motion (QIMs). We point out that a widely accepted definition of functional independence of QIMs is flawed, and suggest a new definition. Our study is motivated by the PXP model—a model of N spins 1/2 possessing an extensive number of binary QIMs. The number of QIMs which are independent according to the common definition turns out to be equal to the number of spins, N. A common wisdom would then suggest that the system is completely integrable, which is not the case. We discuss the origin of this conundrum and demonstrate how it is resolved when a new definition of independence of QIMs is employed.

     

Quadratic algebras for three-dimensional superintegrable systems

The three-dimensional superintegrable systems with quadratic integrals of motion have five functionally independent integrals, one among them is the Hamiltonian. Kalnins, Kress, and Miller have proved that in the case of nondegenerate potentials with quadratic integrals of motion there is a sixth quadratic integral, which is linearly independent of the other integrals. The existence of this sixth integral implies that the integrals of motion form a ternary parafermionic-like quadratic Poisson algebra with five generators. In this contribution we investigate the structure of this algebra. We show that in all the nondegenerate cases there is at least one subalgebra of three integrals having a Poisson quadratic algebra structure, which is similar to the two-dimensional case.     

Symmetries and regular behavior of Hamiltonian systems

The behavior of the phase trajectories of the Hamilton equations is commonly classified as regular and chaotic. Regularity is usually related to the condition for complete integrability, i.e., a Hamiltonian system with n degrees of freedom has n independent integrals in involution. If at the same time the simultaneous integral manifolds are compact, the solutions of the Hamilton equations are quasiperiodic. In particular, the entropy of the Hamiltonian phase flow of a completely integrable system is zero. It is found that there is a broader class of Hamiltonian systems that do not show signs of chaotic behavior. These are systems that allow n commuting "Lagrangian" vector fields, i.e., the symplectic 2-form on each pair of such fields is zero. They include, in particular, Hamiltonian systems with multivalued integrals. (c) 1996 American Institute of Physics.     

Integrability of spinning particle motion in higher-dimensional rotating black hole spacetimes

We study the motion of a classical spinning particle (with spin degrees of freedom described by a vector of Grassmann variables) in higher-dimensional general rotating black hole spacetimes with a cosmological constant. In all dimensions n we exhibit n bosonic functionally independent integrals of spinning particle motion, corresponding to explicit and hidden symmetries generated from the principal conformal Killing-Yano tensor. Moreover, we demonstrate that in 4-, 5-, 6-, and 7-dimensional black hole spacetimes such integrals are in involution, proving the bosonic part of the motion integrable. We conjecture that the same conclusion remains valid in all higher dimensions. Our result generalizes the result of Page et?al. [Phys. Rev. Lett. 98, 061102 (2007)] on complete integrability of geodesic motion in these spacetimes.     

Quantum and classical spin clusters: disappearance of quantum numbers and Hamiltonian chaos

We present a direct link between manifestations of classical Hamiltonian chaos and quantum nonintegrability effects as they occur in quantum invariants. In integrable classical Hamiltonian systems, analytic invariants (integrals of the motion) can be constructed numerically by means of time averages of dynamical variables over phase-space trajectories, whereas in near-integrable models such time averages yield nonanalytic invariants with qualitatively different properties. Translated into quantum mechanics, the invariants obtained from time averages of dynamical variables in energy eigenstates provide a topographical map of the plane of quantized actions (quantum numbers) with properties which again depend sensitively on whether or not the classical integrability condition is satisfied. The most conspicuous indicator of quantum chaos is the disappearance of quantum numbers, a phenomenon directly related to the breakdown of invariant tori in the classical phase flow. All results are for a system consisting of two exchange-coupled spins with biaxial exchange and single-site anisotropy, a system with a nontrivial integrability condition.     

Differential Geometry and Integrability of the Hamiltonian System of a Closed Vortex Filament

The system of a closed vortex filament is an integrable Hamiltonian one, namely, a Hamiltonian system with an infinite sequence of constants of motion in involution. An algebraic framework is given with the aim of describing the differential geometry of this system and a geometrical structure related to the integrability of this system is revealed. It is not a bi-Hamiltonian structure but a similar one. As a related topic, a remark on the inspection of J. Langer and R. Perline (J. Nonlinear Sci. 1 (1991), 71) is given.     

Binary nonlinearization of the super classical-Boussinesq hierarchy

The symmetry constraint and binary nonlinearization of Lax pairs for the super classical-Boussinesq hierarchy is obtained.Under the obtained symmetry constraint,the n-th flow of the super classical-Boussinesq hierarchy is decomposed into two super finite-dimensional integrable Hamiltonian systems,defined over the super-symmetry manifold with the corresponding dynamical variables x and t n.The integrals of motion required for Liouville integrability are explicitly given.     

Singularities motion equations in 2-dimensional ideal hydrodynamics of incompressible fluid

In this Letter, we have obtained motion equations for a wide class of one-dimensional singularities in 2D ideal hydrodynamics. The simplest of them, are well known as point vortices. More complicated singularities correspond to vorticity point dipoles. It has been proved that point multipoles of a higher order (quadrupoles and more) are not the exact solutions of two-dimensional ideal hydrodynamics. The motion equations for a system of interacting point vortices and point dipoles have been obtained. It is shown that these equations are Hamiltonian ones and have three motion integrals in involution. It means the complete integrability of two-particle system, which has a point vortex and a point dipole.     

The Conformal Group SU(2, 2) and Integrable Systems on a Lorentzian Hyperboloid

Eleven different types of “maximally superintegrable” Hamiltonian systems on the real hyperboloid (s⁰)² – (s¹)² + (s²)² – (s³)² = 1 are obtained. All of them correspond to a free Hamiltonian system on the homogeneous space SU(2, 2)/U(2, 1), but to reductions by different maximal abelian subgroups of SU(2, 2). Each of the obtained systems allows 5 functionally independent integrals of motion, from which it is possible to form two or more triplets in involution (each of them includes the hamiltonian). The corresponding classical and quantum equations of motion can be solved by separation of variables on the O(2, 2) space.     

Preparation of quantum states of two spin-12 particles in the form of the Schmidt decomposition

We consider a system of two spins that are coupled via an isotropic Heisenberg Hamiltonian. For the first time, a two-step method for the preparation of an arbitrary quantum state of two qubits in the form of the Schmidt decomposition is proposed. The simplified version of this method is applied to the physical system of an atom with a nuclear spin 1/2 and one valence electron. As an example, the preparation of two-spin quantum states in the ³¹P system is considered.     

Integrable perturbations of the N-dimensional isotropic oscillator

Two new families of completely integrable perturbations of the N-dimensional isotropic harmonic oscillator Hamiltonian are presented. Such perturbations depend on arbitrary functions and N free parameters and their integrals of motion are explicitly constructed by making use of an underlying h₆-coalgebra symmetry. Several known integrable Hamiltonians in low dimensions are obtained as particular specializations of the general results here presented. An alternative route for the integrability of all these systems is provided by a suitable canonical transformation which, in turn, opens the possibility of adding (N−1) ‘Rosochatius’ terms that preserve the complete integrability of all these models.     

Integrals of the motion,green functions,and coherent states of dynamical systems

The connection between the integrals of the motion of a quantum system and its Green function is established. The Green function is shown to be the eigenfunction of the integrals of the motion which describe initial points of the system trajectory in the phase space of average coordinates and moments. The explicit expressions for the Green functions of theN-dimensional system with the Hamiltonian which is the most general quadratic form of coordinates and momenta with time-dependent coefficients is obtained in coordinate, momentum, and coherent states representations. The Green functions of the nonstationary singular oscillator and of the stationary Schrödinger equation are also obtained.     

Quasi-integrable and superintegrable systems from a ‘deformed-mass’ two-photon algebra

A quantum deformation of the two-photon (or Schrödinger) Lie algebra is introduced in order to construct newn-dimensional classical Hamiltonian systems which have (n?2) functionally independent integrals of motion in involution; we say that such Hamiltonians define quasi-integrable systems. Furthermore, Hopf subalgebras of this quantum two-photon algebra (quantum extended Galilei and harmonic oscillator algebras) provide another set of (n?1) integrals of motion for Hamiltonians defined on these Hopf subalgebras, so that they lead to superintegrable systems.     

Heisenberg-Integrable Spin Systems

We investigate certain classes of integrable classical or quantum spin systems. The first class is characterized by the recursively defined property P saying that the spin system consists of a single spin or can be decomposed into two uniformly coupled or disjoint subsystems with property P. For these systems the time evolution can be explicitly calculated. The second class consists of spin systems where all non-zero coupling constants have the same strength (spin graphs) possessing N − 1 independent, commuting constants of motion of Heisenberg type. These systems are shown to have the above property P and can be characterized as spin graphs not containing chains of length four as vertex-induced sub-graphs. We completely enumerate and characterize all spin graphs up to N = 5 spins. Applications to the construction of symplectic numerical integrators for non-integrable spin systems are briefly discussed.      

Integrable and nonintegrable classical spin clusters

This study investigates the nonlinear dynamics of a pair of exchange-coupled spins with biaxial exchange and single-site anisotropy. It represents a Hamiltonian system with 2 degrees of freedom for which we have already established the (nontrivial) integrability criteria and constructed the integrals of the motion provided they exist. Here we present a comparative study of the phase-space trajectories for two specific models with the same symmetry properties, one of which (the XY model with exchange anisotropy) is integrable, and the other (the XY model with single-site anisotropy) nonintegrable. In the integrable model, the integrals of the motion (analytic invariants) can be reconstructed numerically by means of time averages of dynamical variables over all trajectories. In the nonintegrable model, such time averages over trajectories define nonanalytic invariants, where the nonanalyticities are associated with the presence of chaotic trajectories. A prominent feature in the nonintegrable model is the occurrence of very long time scales caused by the presence of low-flux cantori, which form sticky coats on the boundary between chaotic regions and regular islands or leaky walls between different chaotic regions. These cantori dominate the convergence properties of time averages and presumably determine the long-time asymptotic properties of dynamic correlation functions. Finally, we present a special class of integrable systems containing arbitrarily many spins coupled by general biaxial exchange anisotropy.     

Delocalized 2-magnons eigenstates in long-range correlated random Heisenberg chains

We consider the one-dimensional quantum disordered Heisenberg ferromagnetic chain model with long-range correlated exchange couplings and study the nature of collective two-spin excitations. By using an exact diagonalization of the Hamiltonian in the two-spin flip subspace, we compute the spin wave participation number to characterize the localized or delocalized nature of the two-magnon states. For strongly correlated random exchange couplings, extended two-spin excitations with finite energy appear. Integrating the time-dependent Schroedinger equation, we follow the time-evolution of an initially localized two-spin state. We find that, associated with the emergence of extended spin waves, the wave-packet mean-square displacement develops a ballistic spread. Further, the single-spin wave-packet acquires an asymmetric profile due to the kinematic interaction between the excited spins.     

Generalized Landau-Lifshitz models on the interval

We study the classical generalized gln Landau-Lifshitz (L-L) model with special boundary conditions that preserve integrability. We explicitly derive the first non-trivial local integral of motion, which corresponds to the boundary Hamiltonian for the sl₂ L-L model. Novel expressions of the modified Lax pairs associated to the integrals of motion are also extracted. The relevant equations of motion with the corresponding boundary conditions are determined. Dynamical integrable boundary conditions are also examined within this spirit. Then the generalized isotropic and anisotropic gln Landau-Lifshitz models are considered, and novel expressions of the boundary Hamiltonians and the relevant equations of motion and boundary conditions are derived.