Chern-Simons Dynamics Research Articles

The effect of Skyrme–Chern–Simons dynamics on gauged Skyrmions in 2+1 dimensions

We study the Skyrmion of the SO(2) gauged O(3) sigma model in 2+1 dimensions in the presence of a Skyrme–Chern–Simons (SCS) term, and compare its properties with the corresponding properties of the Skyrmion in the presence of the usual Chern–Simons (CS) term. We find that these properties are qualitatively largely similar in both cases, meaning that the SCS density can be employed as an alternative to the CS term also in higher dimensions, most importantly in even dimensions where no CS term is defined, e.g., in 3+1 dimensions. The SCS density employed here is defined in terms of the pair of SO(2)×SO(2) gauge fields and an auxiliary O(5) Skyrme scalar, which is contracted to an effective O(3) Skyrme scalar. Technically, this study maps the methods to be applied in higher dimensional examples.

On the effects of the Chern-Simons term in an Abelian gauged Skyrme model in d = 4 + 1 dimensions

We study an Abelian gauged O(5) Skyrme model in 4+1 dimensions, featuring the F4 Maxwell and the Chern-Simons terms. Our aim is to expose the mechanism, discovered in the analogous Abelian gauged O(3) Skyrme model in 2+1 dimensions, which leads to the unusual relation of the mass-energy E to the electric charge Qe and angular momentum J, and, to the change in the value of the “baryon number” q due to the influence of the Abelian field on the Skyrmion. Chern-Simons dynamics together with the dynamics of the gauged Skyrme scalar, allows for solutions with varying asymptotic values of the magnetic field, resulting in these unusual properties listed. Numerical work is carried out on an effective one dimensional subsystem resulting from imposition of an enhanced radial symmetry on R4.

Long range vortex configurations in generalized models with Maxwell or Chern-Simons dynamics

In this work we deal with vortices in Maxwell-Higgs or Chern-Simons-Higgs models that engender long range tails. We find first order differential equations that support minimum energy solutions which solve the equations of motion. In the Maxwell scenario, we work with generalised magnetic permeabilities that lead to vortices described by solutions, magnetic field and energy density with power-law tails that extend farther than the standard exponential ones. We also find a manner to obtain a Chern-Simons model with the same scalar and magnetic field profiles of the Maxwell case. By doing so, we also find vortices with the aforementioned long range feature, which is also present in the electric field in the Chern-Simons model. The present results may motivate investigations on nonrelativistic models, in particular in the case involving Rydberg atoms, which are known to present long range interactions and relatively long lifetimes.

SO(2) gauged Skyrmions in 4+1 dimensions

We study the simplest $SO(2)$ gauged $O(5)$ Skyrme models in $4+1$ (flat) dimensions. In the gauge decoupled limit, the model supports topologically stable solitons (Skyrmions) and after gauging, the static energy of the solutions is bounded from below by a "baryon number". The studied model features both Maxwell and Maxwell--Chern-Simons dynamics. The considered configurations are subject to bi-azimuthal symmetry in the ${\mathbb R}^4$ subspace resulting in a two dimensional subsystem, as well as subject to an enhanced symmetry relating the two planes in the ${\mathbb R}^4$ subspace, which results in a one dimensional subsystem. Numerical solutions are constructed in both cases. In the purely magnetic case, fully bi-azimuthal solutions were given, while electrically charged and spinning solutions were constructed only in the radial (enhanced symmetric) case, both in the presence of a Chern-Simons term, and in its absence. We find that, in contrast with the analogous models in $2+1$ dimensions, the presence of the Chern-Simons term in the model under study here results only in quantitative effects.

Vortices in Vacuumless Systems

We investigate the presence of vortex solutions in potentials without vacuum state. The study is conducted considering Maxwell and Chern-Simons dynamics. Also, we use a first-order formalism that helps us to find the solutions and their respective electromagnetic fields and energy densities. As a bonus, we get to calculate the energy without knowing the explicit solutions. Even though the solutions present a large “tail” which goes far away from the origin, the magnetic flux remains a well defined topological invariant.

Vortices of SO(2) gauged Skyrmions in 2+1 dimensions

Vortices the $SO(2)$ gauged planar Skyrme model, with a) only Maxwell, b) only Chern-Simons, and c) both Maxwell and Chern-Simons dynamics are studied systematically. In cases a) and b), where both models feature a single parameter $\lambda$ (the coupling of the potential term), the dependence of the energy on $\lambda$ is analysed. It is shown that the plots of the energy $vs.$ $\lambda$ feature discontinuities and branches. In case c), the emphasis is on the evolution of the topological charge, taking non-integer values. Throughout, the properties studied are contrasted with those of the corresponding Abelian Higgs models.

Planar ringlike vortices

We investigate the presence of vortex structures in generalized Maxwell–Higgs and Chern–Simons–Higgs models in the three-dimensional spacetime. Despite the important difference between the Maxwell and Chern–Simons dynamics, we have been able to introduce first order differential equations that solve the equations of motion for static and rotationally symmetric field configurations. In both cases, solutions of the first order equations engender minimum energy, and we have found vortex configurations whose internal structure unveils interesting and unusual ringlike profile.

Effect of Chern-Simons dynamics on the energy of electrically charged and spinning vortices

We study the effect of a Chern-Simons term on the electrically charged and spinning solitons of several $U(1)$ gauged models in $2+1$ dimensions. These are vortices of complex scalar field theories, both with and without symmetry breaking dynamics, and the $O(3)$ Skyrme model. In all cases the gauge decoupling limits are also considered. It is well known that the effect of the Chern-Simons dynamics is to endow vortices with electric charge $Q_e$ and spin $J$, but our main aim here is to reveal a new feature: that the mass-energy $E$ of the electrically charged vortex can be lower than that of the electrically neutral one, in contrast to the usual monotonic increase of $E$ with $Q_e$. These effects of Chern-Simons dynamics were observed previously in $3+1$ dimensional systems, and the present results can be viewed as corroborating the latter. Moreover, the usual energy-spin relationship is likewise altered. We carry out a detailed quantitative analysis of azimuthally symmetric vortices and describe their qualitative features by constructing the solutions numerically.

General algorithm for nonrelativistic diffeomorphism invariance

An algorithmic approach towards the formulation of non-relativistic diffeomorphism invariance has been developed which involves both matter and gauge fields. A step by step procedure has been provided which can accommodate all types of (abelian) gauge interaction. The algorithm is applied to the problem of a two dimensional electron moving under an external field and also under the Chern-Simons dynamics.

Chern–Simons vortices in the Gudnason model

We present a series of existence theorems for multiple vortex solutions in the Gudnason model of the N=2 supersymmetric field theory where non-Abelian gauge fields are governed by the pure Chern–Simons dynamics at dual levels and realized as the solutions of a system of elliptic equations with exponential nonlinearity over two-dimensional domains. In the full plane situation, our method utilizes a minimization approach, and in the doubly periodic situation, we employ an inequality-constrained minimization approach. In the latter case, we also obtain sufficient conditions under which we show that there exist at least two gauge-distinct solutions for any prescribed distribution of vortices. In other words, there are distinct solutions with identical vortex distribution, energy, and electric and magnetic charges.

Adiabatic Limit and the Slow Motion of Vortices in a Chern-Simons-Schrödinger System

We study a nonlinear system of partial differential equations in which a complex field (the Higgs field) evolves according to a nonlinear Schroedinger equation, coupled to an electromagnetic field whose time evolution is determined by a Chern-Simons term in the action. In two space dimensions, the Chern-Simons dynamics is a Galileo invariant evolution for A, which is an interesting alternative to the Lorentz invariant Maxwell evolution, and is finding increasing numbers of applications in two dimensional condensed matter field theory. The system we study, introduced by Manton, is a special case (for constant external magnetic field, and a point interaction) of the effective field theory of Zhang, Hansson and Kivelson arising in studies of the fractional quantum Hall effect. From the mathematical perspective the system is a natural gauge invariant generalization of the nonlinear Schroedinger equation, which is also Galileo invariant and admits a self-dual structure with a resulting large space of topological solitons (the moduli space of self-dual Ginzburg-Landau vortices). We prove a theorem describing the adiabatic approximation of this system by a Hamiltonian system on the moduli space. The approximation holds for values of the Higgs self-coupling constant close to the self-dual (Bogomolny) value of 1. The viability of the approximation scheme depends upon the fact that self-dual vortices form a symplectic submanifold of the phase space (modulo gauge invariance). The theorem provides a rigorous description of slow vortex dynamics in the near self-dual limit.

Schrödinger–Chern–Simons vortex dynamics

We study the motion of vortices in the planar Ginzburg–Landau model with Schrödinger–Chern–Simons dynamics. We compare the moduli space approximation with the results of numerical simulations of the full field theory and find that there is agreement if the coupling constant is very close to the critical value separating type I from type II superconductors. However, there are significant qualitative differences even for modest deviations from the critically coupled regime. Radiation effects produce forces which are of the same order of magnitude as the intervortex force and therefore have a significant impact on vortex motion. We conclude that the moduli space approximation does not provide a good description of the dynamics in this regime.

The symmetries of the Manton superconductivity model

The symmetries and conserved quantities of Manton’s modified superconductivity model with non-relativistic Maxwell–Chern–Simons dynamics (also related to the Quantized Hall Effect) are obtained in the “Kaluza–Klein type” framework of Duval et al.

Solitons in gauged Sigma models: 2 dimensions

A prescription for the gauging of 2-dimensional Sigma models is studied. This prescription satisfies the criterion that the U(1) gauged Sigma model supports topologically stable solitons, like its ungauged counterpart. The gauging is carried out separately for both Maxwell and Chern-Simons dynamics of the U(1) field and it is shown that independently of this dynamics and of the inclusion of all possible Skyrme terms, our criterion is fulfilled. In the Maxwell case, the vortices are presented in detail and are seen to have some interesting properties of their own.

THE CHERN-SIMONS SOURCE AS A CONFORMAL FAMILY AND ITS VERTEX OPERATORS

In a previous work, a straighforward canonical approach to the source-free quantum Chern—Simons dynamics was developed. It makes use of neither gauge conditions nor functional integrals and needs only ideas known from QCD and quantum gravity. It gives Witten’s conformal edge states in a simple way when the spatial slice is a disc. Here we extend the formalism by including sources as well. The quantum states of a source with a fixed spatial location are shown to be those of a conformal family, a result also discovered first by Witten. The internal states of a source are not thus associated with just a single ray of a Hilbert space. Vertex operators for both abelian and nonabelian sources are constructed. The regularized abelian Wilson line is proved to be a vertex operator. We also argue in favor of a similar nonabelian result. The spin-statistics theorem is established for Chern-Simons dynamics even though the sources are not described by relativistic quantum fields. The proof employs geometrical methods which we find are strikingly transparent and pleasing. It is based on the research of European physicists about “fields localized on cones.”

Self-Dual Chern-Simons Solitons

These lectures are devoted to the two-year old topic of Chern-Simons solitons. Solitons have interested particle theorists since the mid-1970’s[1]; Chern-Simons dynamics, since the early 1980’s [2]. Now in the decade of the 1990’s, we are combining the two subjects.

Gravitational anyonization.

The statistics of scalar matter changes when it is coupled to a U(1) gauge field with Chern-Simons dynamics, because all particles carry a magnetic flux and therefore give rise to Aharonov-Bohm phases when they move around each other. We argue that also the ``dual'' version of the Aharonov-Bohm effect, the Aharonov-Casher effect, can give rise to Berry phases which transmute ordinary particles into anyons. The Aharonov-Casher effect consists of an extra topological phase in the wave function of a magnetic moment moving around an electric charge. Considering (2+1)-dimensional Dirac fermions at low energies, both effects are present, and the fermions are turned into (interacting) anyons even though there is no Chern-Simons term included in the action. We study in detail the gravitational analogue of this mechanism. The post-Newtonian approximation is applied to the gravitational interaction of (2+1)-dimensional particles with spin, and to stringlike matter distributions with internal angular momentum in 3+1 dimensions. The action for gravity is taken to be the pure Einstein-Hilbert term. In the adiabatic limit one finds scrA\ensuremath{\cdot}v-type interactions where scrA is a long-range vortex field. These interactions give rise to various kinds of Berry phases, in particular to the gravitational analogues of the Aharonov-Bohm and the Aharonov-Casher phases. The former occurs when a mass moves around a particle with spin, and the latter arises when a particle with spin moves in the Newtonian scalar potential of a second (spinless) particle. These Berry phases lead to a ``self-anyonization'' of particles with nonzero spin. The topological term in their effective action has the same structure as the one which obtains when spinless particles are considered, but with a gravitational Chern-Simons term included in the action for the gravitational field.