Volume 55, Issue 4, April 2014
Index of content:

The timedependent GinzburgLandau formalism for (d + s)wave superconductors and their representation using auxiliary fields is investigated. By using the link variable method, we then develop suitable discretization of these equations. Numerical simulations are carried out for a mesoscopic superconductor in a homogeneous perpendicular magnetic field which revealed peculiar vortex states.
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Partial Differential Equations

Numerical solution of the time dependent GinzburgLandau equations for mixed (d + s)wave superconductors
View Description Hide DescriptionThe timedependent GinzburgLandau formalism for (d + s)wave superconductors and their representation using auxiliary fields is investigated. By using the link variable method, we then develop suitable discretization of these equations. Numerical simulations are carried out for a mesoscopic superconductor in a homogeneous perpendicular magnetic field which revealed peculiar vortex states.

Representation Theory and Algebraic Methods

Filiform Lie algebras of order 3
View Description Hide DescriptionThe aim of this work is to generalize a very important type of Lie algebras and superalgebras, i.e., filiform Lie (super)algebras, into the theory of Lie algebras of order F. Thus, the concept of filiform Lie algebras of order F is obtained. In particular, for F = 3 it has been proved that by using infinitesimal deformations of the associated model elementary Lie algebra it can be obtained families of filiform elementary lie algebras of order 3, analogously as that occurs into the theory of Lie algebras [M. Vergne, “Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie nilpotentes,” Bull. Soc. Math. France98, 81–116 (1970)]. Also we give the dimension, using an adaptation of the module Method, and a basis of such infinitesimal deformations in some generic cases.

A braided monoidal category for free superbosons
View Description Hide DescriptionThe chiral conformal field theory of free superbosons is generated by weight one currents whose mode algebra is the affinisation of an abelian Lie superalgebra with nondegenerate supersymmetric pairing. The mode algebras of a single free boson and of a single pair of symplectic fermions arise for evenodd dimension 10 and 02 of , respectively. In this paper, the representations of the untwisted mode algebra of free superbosons are equipped with a tensor product, a braiding, and an associator. In the symplectic fermion case, i.e., if is purely odd, the braided monoidal structure is extended to representations of the twisted mode algebra. The tensor product is obtained by computing spaces of vertex operators. The braiding and associator are determined by explicit calculations from three and fourpoint conformal blocks.

Kitaev models based on unitary quantum groupoids
View Description Hide DescriptionWe establish a generalization of Kitaev models based on unitary quantum groupoids. In particular, when inputting a KitaevKong quantum groupoid , we show that the ground state manifold of the generalized model is canonically isomorphic to that of the LevinWen model based on a unitary fusion category . Therefore, the generalized Kitaev models provide realizations of the target space of the TuraevViro topological quantum field theory based on .

On infinitedimensional 3Lie algebras
View Description Hide DescriptionIn this paper, we study some properties of w ∞ 3Lie algebra and SDiff(T ^{2}) 3Lie algebra and prove that they do not have nontrivial central extensions.

Curvature and geometric modules of noncommutative spheres and tori
View Description Hide DescriptionWhen considered as submanifolds of Euclidean space, the Riemannian geometry of the round sphere and the Clifford torus may be formulated in terms of Poisson algebraic expressions involving the embedding coordinates, and a central object is the projection operator, projecting tangent vectors in the ambient space onto the tangent space of the submanifold. In this note, we point out that there exist noncommutative analogues of these projection operators, which implies a very natural definition of noncommutative tangent spaces as particular projective modules. These modules carry an induced connection from Euclidean space, and we compute its scalar curvature.

Quantum Mechanics

Stokes' theorem, gauge symmetry and the timedependent AharonovBohm effect
View Description Hide DescriptionStokes' theorem is investigated in the context of the timedependent AharonovBohm effect—the twoslit quantum interference experiment with a time varying solenoid between the slits. The time varying solenoid produces an electric field which leads to an additional phase shift which is found to exactly cancel the timedependent part of the usual magnetic AharonovBohm phase shift. This electric field arises from a combination of a nonsingle valued scalar potential and/or a 3vector potential. The gauge transformation which leads to the scalar and 3vector potentials for the electric field is nonsingle valued. This feature is connected with the nonsimply connected topology of the AharonovBohm setup. The nonsingle valued nature of the gauge transformation function has interesting consequences for the 4dimensional Stokes' theorem for the timedependent AharonovBohm effect. An experimental test of these conclusions is proposed.

Hermite polynomials and quasiclassical asymptotics
View Description Hide DescriptionWe study an unorthodox variant of the BerezinToeplitz type of quantization scheme, on a reproducing kernel Hilbert space generated by the real Hermite polynomials and work out the associated quasiclassical asymptotics.

Galilei invariant technique for quantum system description
View Description Hide DescriptionProblems with quantum systems models, violating Galilei invariance are examined. The method for arbitrary nonrelativistic quantum system Galilei invariant wave function construction, applying a modified basis where centerofmass excitations have been removed before Hamiltonian matrix diagonalization, is developed. For identical fermion system, the Galilei invariant wave function can be obtained while applying conventional antisymmetrization methods of wave functions, dependent on single particle spatial variables.

A regular version of Smilansky model
View Description Hide DescriptionWe discuss a modification of Smilansky model in which a singular potential “channel” is replaced by a regular, below unbounded potential which shrinks as it becomes deeper. We demonstrate that, similarly to the original model, such a system exhibits a spectral transition with respect to the coupling constant, and determine the critical value above which a new spectral branch opens. The result is generalized to situations with multiple potential “channels.”

Spectra of random operators with absolutely continuous integrated density of states
View Description Hide DescriptionThe structure of the spectrum of random operators is studied. It is shown that if the density of states measure of some subsets of the spectrum is zero, then these subsets are empty. In particular follows that absolute continuity of the integrated density of states implies singular spectra of ergodic operators is either empty or of positive measure. Our results apply to Anderson and alloy type models, perturbed Landau Hamiltonians, almost periodic potentials, and models which are not ergodic.

The Berry phase and the phase of the determinant
View Description Hide DescriptionWe show that under very general assumptions the adiabatic approximation of the phase of the zetaregularized determinant of the imaginarytime Schrödinger operator with periodic Hamiltonian is equal to the Berry phase.

General Relativity and Gravitation

An instability of hyperbolic space under the YangMills flow
View Description Hide DescriptionWe consider the YangMills flow on hyperbolic 3space. The gauge connection is constructed from the framefield and (not necessarily compatible) spin connection components. The fixed points of this flow include zero YangMills curvature configurations, for which the spin connection has zero torsion and the associated Riemannian geometry is one of constant curvature. We analytically solve the linearized flow equations for a large class of perturbations to the fixed point corresponding to hyperbolic 3space. These can be expressed as a linear superposition of distinct modes, some of which are exponentially growing along the flow. The growing modes imply the divergence of the (gauge invariant) perturbative torsion for a wide class of initial data, indicating an instability of the background geometry that we confirm with numeric simulations in the partially compactified case. There are stable modes with zero torsion, but all the unstable modes are torsionfull. This leads us to speculate that the instability is induced by the torsion degrees of freedom present in the YangMills flow.

Dynamical Systems

Topological horseshoes in travelling waves of discretized nonlinear wave equations
View Description Hide DescriptionApplying the concept of antiintegrable limit to coupled map lattices originated from spacetime discretized nonlinear wave equations, we show that there exist topological horseshoes in the phase space formed by the initial states of travelling wave solutions. In particular, the coupled map lattices display spatiotemporal chaos on the horseshoes.

Classical Mechanics and Classical Fields

Electromagnetic momentum and the energy–momentum tensor in a linear medium with magnetic and dielectric properties
View Description Hide DescriptionIn a continuum setting, the energy–momentum tensor embodies the relations between conservation of energy, conservation of linear momentum, and conservation of angular momentum. The welldefined total energy and the welldefined total momentum in a thermodynamically closed system with complete equations of motion are used to construct the total energy–momentum tensor for a stationary simple linear material with both magnetic and dielectric properties illuminated by a quasimonochromatic pulse of light through a gradientindex antireflection coating. The perplexing issues surrounding the Abraham and Minkowski momentums are bypassed by working entirely with conservation principles, the total energy, and the total momentum. We derive electromagnetic continuity equations and equations of motion for the macroscopic fields based on the material fourdivergence of the traceless, symmetric total energy–momentum tensor. We identify contradictions between the macroscopic Maxwell equations and the continuum form of the conservation principles. We resolve the contradictions, which are the actual fundamental issues underlying the Abraham–Minkowski controversy, by constructing a unified version of continuum electrodynamics that is based on establishing consistency between the threedimensional Maxwell equations for macroscopic fields, the electromagnetic continuity equations, the fourdivergence of the total energy–momentum tensor, and a fourdimensional tensor formulation of electrodynamics for macroscopic fields in a simple linear medium.

Topologically massive YangMills: A HamiltonJacobi constraint analysis
View Description Hide DescriptionWe analyse the constraint structure of the topologically massive YangMills theory in instantform and nullplane dynamics via the HamiltonJacobi formalism. The complete set of hamiltonians that generates the dynamics of the system is obtained from the Frobenius’ integrability conditions, as well as its characteristic equations. As generators of canonical transformations, the hamiltonians are naturally linked to the generator of Lagrangian gauge transformations.

Nonlinear periodic waves solutions of the nonlinear selfdual network equations
View Description Hide DescriptionThe new classes of periodic solutions of nonlinear selfdual network equations describing the breather and soliton lattices, expressed in terms of the Jacobi elliptic functions have been obtained. The dependences of the frequencies on energy have been found. Numerical simulations of soliton lattice demonstrate their stability in the ideal lattice and the breather lattice instability in the dissipative lattice. However, the lifetime of such structures in the dissipative lattice can be extended through the application of ac driving terms.

Methods of Mathematical Physics

Derivatives of the Pochhammer and reciprocal Pochhammer symbols and their use in epsilonexpansions of Appell and Kampé de Fériet functions
View Description Hide DescriptionUseful expressions of the derivatives, to any order, of Pochhammer and reciprocal Pochhammer symbols with respect to their arguments are presented. They are building blocks of a procedure, recently suggested, for obtaining the ɛexpansion of functions of the hypergeometric class related to Feynman integrals. The procedure is applied to some examples of such kind of functions taken from the literature.

General displaced SU(1, 1) number states: Revisited
View Description Hide DescriptionThe most general displaced number states, based on the bosonic and an irreducible representation of the Lie algebra symmetry of su (1, 1) and associated with the CalogeroSutherland model are introduced. Here, we utilize the BarutGirardello displacement operator instead of the KlauderPerelomov counterpart, to construct new kind of the displaced number states which can be classified in nonlinear coherent states regime, too, with special nonlinearity functions. They depend on two parameters, and can be converted into the wellknown BarutGirardello coherent and number states, respectively, depending on which of the parameters equal to zero. A discussion of the statistical properties of these states is included. Significant are their squeezing properties and antibunching effects which can be raised by increasing the energy quantum number. Depending on the particular choice of the parameters of the above scenario, we are able to determine the status of compliance with flexible statistics. Major parts of the issue is spent on something that these states, in fact, should be considered as new kind of photonadded coherent states, too. Which can be reproduced through an iterated action of a creation operator on new nonlinear BarutGirardello coherent states . Where the latter carry, also, outstanding statistical features.
