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.

Paritytime symmetry broken by pointgroup symmetry
View Description Hide DescriptionWe discuss a paritytime (PT) symmetric Hamiltonian with complex eigenvalues. It is based on the dimensionless Schrödinger equation for a particle in a square box with the PTsymmetric potential V(x, y) = iaxy. Perturbation theory clearly shows that some of the eigenvalues are complex for sufficiently small values of a. Pointgroup symmetry proves useful to guess if some of the eigenvalues may already be complex for all values of the coupling constant. We confirm those conclusions by means of an accurate numerical calculation based on the diagonalization method. On the other hand, the Schrödinger equation with the potential V(x, y) = iaxy ^{2} exhibits real eigenvalues for sufficiently small values of a. Point group symmetry suggests that PTsymmetry may be broken in the former case and unbroken in the latter one.

Semiclassical properties of Berezin–Toeplitz operators with symbol
View Description Hide DescriptionWe obtain the semiclassical expansion of the kernels and traces of Toeplitz operators with symbol on a symplectic manifold. We also give a semiclassical estimate of the distance of a Toeplitz operator to the space of selfadjoint and multiplication operators.

The SU(1, 1) Perelomov number coherent states and the nondegenerate parametric amplifier
View Description Hide DescriptionWe construct the Perelomov number coherent states for an arbitrary su (1, 1) group operation and study some of their properties. We introduce three operators which act on Perelomov number coherent states and close the su (1, 1) Lie algebra. By using the tilting transformation we apply our results to obtain the energy spectrum and eigenfunctions of the nondegenerate parametric amplifier. We show that these eigenfunctions are the Perelomov number coherent states of the twodimensional harmonic oscillator.

Geometric uncertainty relation for mixed quantum states
View Description Hide DescriptionIn this paper we use symplectic reduction in an Uhlmann bundle to construct a principal fiber bundle over a general space of unitarily equivalent mixed quantum states. The bundle, which generalizes the Hopf bundle for pure states, gives in a canonical way rise to a Riemannian metric and a symplectic structure on the base space. With these we derive a geometric uncertainty relation for observables acting on quantum systems in mixed states. We also give a geometric proof of the classical RobertsonSchrödinger uncertainty relation, and we compare the two. They turn out not to be equivalent, because of the multiple dimensions of the gauge group for general mixed states. We give examples of observables for which the geometric relation provides a stronger estimate than that of Robertson and Schrödinger, and vice versa.

Quantum Information and Computation

Remainder terms for some quantum entropy inequalities
View Description Hide DescriptionWe consider three von Neumann entropy inequalities: subadditivity; Pinsker's inequality for relative entropy; and the monotonicity of relative entropy. For these we state conditions for equality, and we prove some new error bounds away from equality, including an improved version of Pinsker's inequality.

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.
