La nozione di spin `e basilare per comprendere il mondo che ci circonda. Essa emerge in maniera naturale dalle proprietà di simmetria che modellano le teorie fisiche. Tuttavia una certa classe di fenomeni sembra aggirare le loro costrizioni. Solo un'attenta analisi delle condizioni geometriche e della loro realizzazione fisica spiega come cambiare le regole del gioco.

We consider a chain of SU(2) 4 anyons with transitions to a topologically ordered phase state. For half-integer and integer indices of the type of strongly correlated excitations, we find an effective low-energy Hamiltonian that is an analogue of the standard Heisenberg Hamiltonian for quantum magnets. We describe the properties of the Hilbert spaces of the system eigenstates. For the Drinfeld quantum SU(2)k×SU(2)k doubles, we use numerical computations to show that the largest eigenvalues of the adjacency matrix for graphs that are extended Dynkin diagrams coincide with the total quantum dimensions for the levels k = 2, 3, 4, 5. We also formulate a hypothesis about the reason for the universal behavior of the system in the long-wave limit.

Our goal is to clarify the relation between entanglement and correlation energy in a bipartite system with infinite dimensional Hilbert space. To this aim, we consider the completely solvable Moshinsky's model of two linearly coupled harmonic oscillators. Also, for small values of the couplings, the entanglement of the ground state is nonlinearly related to the correlation energy, involving logarithmic or algebraic corrections. Then, looking for witness observables of the entanglement, we show how to give a physical interpretation of the correlation energy. In particular, we have proven that there exists a set of separable states, continuously connected with the Hartree–Fock state, which may have a larger overlap with the exact ground state, but also a larger energy expectation value. In this sense, the correlation energy provides an entanglement gap, i.e. an energy scale, under which measurements performed on the 1-particle harmonic sub-system can discriminate the ground state from any other separated state of the system. However, in order to verify the generality of the procedure, we have compared the energy distribution cumulants for the 1-particle harmonic sub-system of the Moshinsky's model with the case of a coupling with a damping Ohmic bath at 0 temperature.

Resorting to the Lagrange–Souriau 2-form formalism, a wide class of systems are derived in non-commuting and/or non-canonical variables, nor the Darboux theorem can be helpful, because of the gauge character of all phase-space variables. As a paradigmatic example, the motion of a charged particle in a magnetic monopole field in the presence of a momentum space monopole is considered.

We consider an electrically charged particle simultaneously interacting with a magnetic monopole and a dual monopole in the momentum space. It is a prototype of a three-dimensional system involving noncommuting and/or noncanonical variables but having geometric and also gauge symmetries in both the position and momentum spaces. We discuss the main features of the motions and conservation laws and the analogies to the case of planar motion.

Inspired by the geometrical methods allowing the introduction of mechanical systems confined in the plane and endowed with exotic galilean symmetry, we resort to the Lagrange-Souriau 2-form formalism, in order to look for a wide class of 3D systems, involving not commuting and/or not canonical variables, but possessing geometric as well gauge symmetries in position and momenta space too. As a paradigmatic example, a charged particle simultaneously interacting with a magnetic monopole and a dual monopole in momenta space is considered. The main features of the motions, conservation laws and the analogies with the planar case are discussed. Possible physical realizations of the model are proposed.

Some aspects of the "exotic" particle, associated with the two-parameter central extension of the planar Galilei group are reviewed. A fundamental property is that it has non-commuting position coordinates. Other and generalized non-commutative models are also discussed. Minimal as well as anomalous coupling to an external electromagnetic field is presented. Supersymmetric extension is also considered. Exotic Galilean symmetry is also found in Moyal field theory. Similar equations arise for a semiclassical Bloch electron, used to explain the anomalous/spin/optical Hall effects.

Senza grandi eccezioni, l’arcobaleno Ssembra essere fonte di emozioni po- sitive: un’esperienza che rallegra, rassicura e ispira il senso del bello. Ma da dove traggono origine queste conce- zioni? Ed è sempre stato così? E come si sono evolute le interpretazioni scienti- fiche e quanto hanno inciso nel sentimen- to comune e nella cultura? Infine, quanto la spiegazione del fenomeno particolare ha fornito alla scienza strumenti e spunti per altri temi di ricerca?

A system of partial differential equations, describing one-dimensional nematic liquid crystals is studied by Lie group analysis. These equations are the Euler–Lagrange equations associated with a free energy functional that depends on the mass density and the gradient of the mass density. The group analysis is an algorithmic approach that allows us to show all the point symmetries of the system, to determine all possible symmetry reductions and, finally, to obtain invariant solutions in the form of travelling waves. The Hamiltonian formulation of the dynamical equations is also considered and the conservation laws found by exploiting the local symmetries.

The Liouville equation is well known to be linearizable by a point transformation. It has an infinite dimensional Lie point symmetry algebra isomorphic to a direct sum of two Virasoro algebras. We show that it is not possible to discretize the equation keeping the entire symmetry algebra as point symmetries. We do however construct a difference system approximating the Liouville equation that is invariant under the maximal finite subgroup ##IMG## [http://ej.iop.org/images/1751-8121/48/2/025204/jpa504755ieqn1.gif] {$S{{L}_{x}}(2,mathbb{R})otimes S{{L}_{y}}(2,mathbb{R})$} . The invariant scheme is an explicit one and provides a much better approximation of exact solutions than a comparable standard (noninvariant) scheme and also than a scheme invariant under an infinite dimensional group of generalized symmetries.

The Skyrme-Faddeev model admits exact analytical non localized solutions, which describe magnetic domain wall solutions when multivalued singularities appear or, differently, always regular periodic nonlinear waves, which may degenerate into linear spin waves or solitonic structures. Here both classes of solutions are derived and discussed and a general discusssion about the existence of integrable subsectors of the model is addressed.

In order to derive a large set of Hamiltonian dynamical systems, but with only first order Lagrangian, we resort to the formulation in terms of Lagrange-Souriau 2-form formalism. A wide class of systems derived in different phenomenological contexts are covered. The non-commutativity of the particle position coordinates are a natural consequence. Some explicit examples are considered.

This paper is devoted to a study of the connections between three different analytic descriptions for the immersion functions of 2D-surfaces corresponding to the following three types of symmetries: gauge symmetries of the linear spectral problem, conformal transformations in the spectral parameter and generalized symmetries of the associated integrable system. After a brief exposition of the theory of soliton surfaces and of the main tool used to study classical and generalized Lie symmetries, we derive the necessary and sufficient conditions under which the immersion formulas associated with these symmetries are linked by gauge transformations. We illustrate the theoretical results by examples involving the sigma model.

The main purpose of this article is to show how symmetry structures in partial differential equations can be preserved in a discrete world and reflected in difference schemes. Three different structure preserving discretizations of the Liouville equation are presented and then used to solve specific boundary value problems. The results are compared with exact solutions satisfying the same boundary conditions. All three discretizations are on four point lattices. One preserves linearizability of the equation, another the infinite-dimensional symmetry group as higher symmetries, the third one preserves the maximal finite-dimensional subgroup of the symmetry group as point symmetries. A 9-point invariant scheme that gives a better approximation of the equation, but significantly worse numerical results for solutions is presented and discussed.

In the present article we perform the symmetry analysis of the Skyrme-Faddeev system at the level of point symmetries. We provide a new simple approximate expression for the axisymmetric solutions, with an accuracy of 10−4 in the static energy value. Furthermore, we proceed to a reduction with respect to all discrete Platonic rotation subgroups, providing the equation for the radial profile function.

We show that the Skyrme–Faddeev model can be reduced in different ways to completely integrable sectors; the corresponding classes of solutions can be parametrized by specific sets of arbitrary functions. Moreover, using the ansatz of a phase and pseudo-phase reduction, the corresponding ordinary nonlinear wave solutions can be integrated in terms of elliptic functions, leading to periodic solutions. The Whitham averaging method has been exploited in order to describe a slow deformation of periodic wave states, leading to a Hamiltonian system, the integrability of which has been studied.