Solitons in low-dimensional sigma models

The aim of this thesis is to study topological soliton solutions in classical field theories, called sigma models, on a three-dimensional space. In chapter 1 we review the general field-theoretical framework of classical soliton solutions and exemplify it on the main features of the 0(3) σ-model and the Abehan Higgs model in (2+1) dimensions. In chapter 2 a U(l)-gauged 0(3) σ-model is discussed, where the behaviour of the gauge field is determined by a Chern-Simons term in the action. We find numerical solutions for radially symmetric fields and discuss those of degree one and two. They carry a non-vanishing angular momentum and can be interpreted as classical anyons. A similar model is studied in chapter 3. Here the potential is of Higgs-type and chosen to produce a Bogomol'nyi model where the energy is bounded from below by a linear combination of the topological degree of the matter fields and the local U(l)-charge. Depending on internal parameters, the solutions are solitons or vortices. We study them numerically and prove for a certain range of the matter field's vacuum value that there cannot be a 1-soliton.In chapter 4 we discuss a modified 0(3) σ-model in (3+0) dimensions. The topological stability of the solitons is here imphed by the degree of the map S(^3) → S(^2), which provides a lower boundon the potential energy of the configuration. Numerical solutions are obtained for configurations of azimuthal symmetry and the spectrum of slowly rotating solitons is approximated. Chapter 5 deals with a theory where the fields are maps IR(^2+1) → CP(^2). The Lagrangian includes a potential and a fourth-order term in the field-gradient. We find a family of static analytic solutions of degree one and study the 2-soIiton configuration numerically by using a gradient-flow equation on the moduli space of solutions. We conclude this thesis with a brief summary and give an outlook to open questions

Faddeev-Hopf knots: Dynamics of linked un-knots

We have studied numerically Faddeev-Hopf knots, which are defined as those unit-vector fields in $R^3$ that have a nontrivial Hopf charge and minimize Faddeev's Lagrangian. A given initial configuration was allowed to relax into a (local) minimum using the first order dissipative dynamics corresponding to the steepest descent method. A linked combination of two un-knots was seen to relax into different minimum energy configurations depending on their charges and their relative handedness and direction. In order to visualize the results we plot certain gauge-invariant iso-surfaces.Comment: 11 pages with 6 figures (3 in color

Bogomol'nyi solitons in a gauged O(3)O(3) sigma model

The scale invariance of the $O(3)$ sigma model can be broken by gauging a $U(1)$ subgroup of the $O(3)$ symmetry and including a Maxwell term for the gauge field in the Lagrangian. Adding also a suitable potential one obtains a field theory of Bogomol'nyi type with topological solitons. These solitons are stable against rescaling and carry magnetic flux which can take arbitrary values in some finite interval. The soliton mass is independent of the flux, but the soliton size depends on it. However, dynamically changing the flux requires infinite energy, so the flux, and hence the soliton size, remains constant during time evolution.Comment: 10 pages, Latex, 2 postscript figure

Interaction energy of Chern-Simons vortices in the gauged O(3) sigma model

The purpose of this Letter is to present a computation of the interaction energy of gauged O(3) Chern-Simons vortices which are infinitely separated. The results will show the behaviour of the interaction energy as a function of the constant coupling the potential, which measures the relative strength of the matter self-coupling and the electromagnetic coupling. We find that vortices attract each other for $\lambda > 1$ and repel when $\lambda < 1$. When $\lambda =1$ there is a topological lower bound on the energy. It is possible to saturate the bound if the fields satisfy a set of first order partial differential equations.Comment: 12 pages, LateX, 7 figures available on request from author. [email protected]

Knot Solitons

The existence of ring-like and knotted solitons in O(3) non-linear sigma model is analysed. The role of isotopy of knots/links in classifying such solitons is pointed out. Appearance of torus knot solitons is seen.Comment: Latex 9 pages + 2 eps figure

Comment on ``Reduction of static field equation of Faddeev model to first order PDE'', arXiv:0707.2207

The authors of the article Phys. Lett. B 652 (2007) 384, (arXiv:0707.2207), propose an interesting method to solve the Faddeev model by reducing it to a set of first order PDEs. They first construct a vectorial quantity $\bm \alpha$, depending on the original field and its first derivatives, in terms of which the field equations reduce to a linear first order equation. Then they find vectors $\bm \alpha_1$ and $\bm \alpha_2$ which identically obey this linear first order equation. The last step consists in the identification of the $\bm \alpha_i$ with the original $\bm \alpha$ as a function of the original field. Unfortunately, the derivation of this last step in the paper cited above contains an error which invalidates most of its results

Maxwell--Chern-Simons gauged non-relativistic O(3) model with self-dual vortices

A non-relativistic version of the 2+1 dimensional gauged Chern-Simons O(3) sigma model, augmented by a Maxwell term, is presented and shown to support topologically stable static self-dual vortices. Exactly like their counterparts of the ungauged model, these vortices are shown to exhibit Hall behaviour in their dynamics.Comment: 12 pages, LateX, to appear in Mod. Phys. Lett. 199

Exact Self-dual Soliton Solutions in a Gauged O(3) Sigma Model with Anomalous Magnetic Moment Interaction

It is shown that a gauged nonlinear $O(3)$ sigma model with anomalous magnetic moment interaction in $2+1$ dimensions is exactly integrable for static, self-dual field configurations. The matter fields are exactly equivalent to those of the usual ungauged nonlinear $O(3)$ sigma model. These static soliton solutions can be mapped into an Abelian purely magnetic vortex solutions through a suitable reduction of the non-Abelian gauge group. A relativistic Abelian model in $2+1$ dimensions is also presented where these purely magnetic vortices can be realized.Comment: A discussion on $CP^N$ case has been made. New references have been added. To appear in Physics Letters B. RevTeX, 13 pages, no figur

Estimation of the Lin-Yang bound of the least static energy of the Faddeev model

Lin and Yang's upper bound E_Q <= cQ^(3/4) of the least static energy E_Q of the Faddeev model in a sector with a fixed Hopf index Q is investigated. By constructing an explicit trial configuration for the Faddeev field n, a possible value of the coefficient c is obtained numerically, which is much smaller than the value obtained quite recently by analytic discussions.Comment: 11 pages, 2 figure