We study a planar model of a non-relativistic electron in periodic magnetic and electric fields that produce a 1D crystal for two spin components separated by a half-period spacing. We fit the fields to create a self-isospectral pair of finite-gap associated Lamé equations shifted for a half-period, and show that the system obtained is characterized by a new type of supersymmetry. It is a special nonlinear supersymmetry generated by three commuting integrals of motion, related to the parityodd operator of the associated Lax pair, that coherently reflects the band structure and all its peculiarities. In the infinite period limit it provides an unusual picture of supersymmetry breaking.
The dynamical symmetry algebra of the two-dimensional Dirac Hamiltonian with equal scalar and vector Smorodinsky-Winternitz potentials is constructed. It is the Higgs algebra, a cubic polynomial generalization of SU(2). With the help of the Casimir operators, the energy levels are derived algebraically.The concept of dynamical symmetry (DS) is essential and prevalent both in classical and quantum mechanics [1]. Hydrogen atom and isotropic harmonic oscillator are two relatively simple model with DS, whose classical orbits of motion are closed [2]. For simplicity, we only concern with the bound sates of the Coulomb problem in this report. In addition to the orbit angular momentum corresponding to the rotational symmetry, there exist more constants of motion in these systems. They have been proved to be the Rung-Lenz vector [3,4] and the second order tensors [5], in the hydrogen atom and harmonic oscillator respectively. The algebraic relations of these conserved quantities reveal the SO(4) symmetry in the hydrogen atom, and SU(3) in harmonic oscillator. They are called DSs, because the nature of them are not geometrical but the symmetries in the phase space. These symmetries lead to an algebraic approach to determine the energy levels. Generally, the N-dimensional (ND) hydrogen atom has the SO(N+1) and the oscillator has the SU(N) symmetry.Starting from the feature of the classical orbits, Higgs [6] introduced a generalization of the hydrogen atom and harmonic oscillator in a spherical space. The conserved quantities of them construct a cubic polynomial generalization of SU(2), which is called the Higgs algebra now. Its increasing applications have been the focus of very active research in recent year [7,8,9]. Especially, Floreanini and his colleagues [10] find the DS of the twobody Calogero model can be described by the Higgs algebra. Under an orthogonal transformation (Eq. (11) in [10]), their Hamiltonian of the two-body Calogero model is equivalent to the 2D Smorodinsky-Winternitz (SW) system [11,12,13]. This indicates that the DSs, especially which are described by the polynomial Lie algebra, exist not only in the quantum mechanics systems with rotation symmetry but also in some non-central superintegrable potentials [14]. Moreover, the two-body Calogero model is shown related to the concept of hidden nonlinear supersymmetry [15,16]. This nonlinear generalization of supersymmetry is investigated in many systems in recent years [17,18,19,20,21,22,23].In the relativistic quantum mechanics, the motion of spin-1/2 particle satisfies the Dirac equation, which pre-dicts the intrinsic magnet moment naturally. The spinorbit coupling leads to the breaking of DSs in the Dirac hydrogen atom [24] and the Dirac oscillator [25].Recently, in his illuminating work [26,27], Ginocchio has found the U(3) and pseudo-U(3) symmetry in the Dirac equation with scalar and vector harmonic oscillator potentials of equal magnitude. The Dirac Hamiltonian, with scalar and vector potentials of equal magnitude (SVPEM), is said to have the spi...
We show that some simple well studied quantum mechanical systems without fermion (spin) degrees of freedom display, surprisingly, a hidden supersymmetry. The list includes the bound state Aharonov-Bohm, the Dirac delta and the Poschl-Teller potential problems, in which the unbroken and broken N=2 supersymmetry of linear and nonlinear (polynomial) forms is revealed.Comment: 4 pages; refs and comments added, to appear in Ann. Phy
We explain the origin and the nature of a special nonlinear supersymmetry of a reflectionless Pöschl-Teller system by the Aharonov-Bohm effect for a nonrelativistic particle on the AdS 2 . A key role in the supersymmetric structure appearing after reduction by a compact generator of the AdS 2 isometry is shown to be played by the discrete symmetries related to the space and time reflections in the ambient Minkowski space. We also observe that a correspondence between the two quantum non-relativistic systems is somewhat of the AdS/CFT holography nature.
In spite of all this, until very recently no exact analytic solutions of the Skyrme model with non-trivial topological charges were known. One of the reasons is that the Skyrme-BPS bound on the energy cannot be saturated for non-trivial spherically symmetric configurations [14]. Nevertheless, many rigorous results about Skyrmions dynamics have been derived, see for instance [15][16][17].The action of the SU (2) Skyrme system in four dimensional spacetime iswhereHere t i are the SU (2) generators and we set the units = c = 1. The coupling constants K > 0 and λ > 0 are fixed by comparison with experimental data [6]. The presence of the first term of the Skyrme action (1), is mandatory to describe pions while the second is the only covariant term leading to second order field equations in time which supports the existence of Skyrmions in four dimensions.In the present paper, exact spherically symmetric solutions of the Skyrme model with both a non-trivial winding number and a finite soliton mass (topological charge) are presented. Using the formalism introduced in [18][19][20], it is shown that although the BPS bound in terms of the winding cannot be saturated, a new topological charge exists that can be saturated corresponding to a different BPS bound. The baryon number is the homotopy of the space into the group. The simplest choice would be to consider the curved background S 3 as physical space, as already considered in the pioneering papers [21,22]. The second natural choice of special sections with integer homotopy into SU (2) is S 1 × S 2 (or R × S 2 ). This can be represented by a metric of the formIn simple words, this geometry describes tridimensional cylinders whose sections are S 2 spheres of area 4πR 2 0 . The physical meaning of R 0 is that it takes into account finite volume effects. One could put the Skyrme action into, say, a cube. However, this way of proceeding often breaks symmetries. On the other hand, a spherical box of finite radius would lead to difficulties in requiring the Skyrmions approach the identity at the boundary. Therefore, it is much more convenient to choose a metric which at the same time takes into account finite volume effects and keeps the spherical symmetry. We are able construct exact Skyrmions in a finite volume but, instead of putting by hand a cut-off on the coordinates, we leave this task to the geometry. Besides, this geometry is such that the group of the isometries of (2) contains SO(3) as a subgroup and so it includes the spherical symmetry of the Skyrmion in flat space. This fact allows examining how far is the BPS bound from being saturated and to construct an energy bound which can in fact be saturated. This could be of interest both in high energy and solid state physics whose features, after the papers [21,22], have not been thoroughly investigated from the analytical viewpoint.In order to construct the exact solution of the Skyrme model, the following standard parametrization of the
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