Baby Skyrme Research Articles

Scattering of a two skyrmion configuration on potential holes or barriers in a model Landau–Lifshitz equation

The dynamics of a baby-skyrmion configuration, in a model Landau–Lifshitz equation, was studied in the presence of various potential obstructions. The baby-skyrmion configuration was constructed from two Q = 1 hedgehog solutions to the baby-skyrme model in (2+1) dimensions. The potential obstructions were created by introducing a new term into the Lagrangian which resulted in a localized inhomogeneity in the potential terms' coefficient. In the barrier system, the normal circular path was deformed as the skyrmions traversed the barrier. During the same period, it was seen that the skyrmions sped up as they went over the barrier. For critical values of the barrier height and width, the skyrmions were no longer bound and were free to separate. In the case of a potential hole, the baby skyrmions no longer formed a bound state and moved asymptotically along the axis of the hole. It is shown how to modify the definition of the angular momentum to include the effects of the obstructions, so that it is conserved.

Localization of massive fermions on the baby-Skyrmion branes in 6 dimensions

We construct brane solutions in 6 dimensional Einstein-Skyrme systems. A class of baby skyrmion solutions realizes warped compactification of the extra dimensions and gravity localization on the brane for negative bulk cosmological constant. Coupling of the fermions with the brane skyrmions lead to the brane localized fermions. In terms of the level crossing picture, emergence of the massive localized modes as well as the zero mode are observed. Nonlinear nature of the skyrmions brings richer information for the fermions level structure. The level comprises doubly degenerate lowest plus single excited modes. The three generation of the fundamental fermions is based on this structure. The quark/lepton mass hierarchy is successfully obtained in terms of a slightly deformed baby-skyrmions with topological charge three.

The modified London equation, Abrikosov-like vortices and knot solitons in two-gap superconductors

We derive the exact modified London equation for the two-gap superconductor, and compare it with its single-gap counterpart. We show that the vortices in the two-gap superconductor are soft (or continuous) core vortices. In particular, we discuss the topological structure of the finite-energy vortices (Abrikosov-like vortices), and find that they can be viewed as the incarnation of the baby skyrmion stretched in the third direction. Besides, we point out that the knot soliton in the two-gap superconductor is the twisted Abrikosov-like vortex with its two periodic ends connected smoothly. The relation between the magnetic monopoles and the Abrikosov-like vortices is also discussed briefly.

Baby Skyrmions on the two-sphere

We find the static multisoliton solutions of the baby Skyrme model on the two-sphere for topological charges 1< or =B< or =14. Numerical full-field results show that the charge-one Skyrmion is spherical, the charge-two Skyrmion is toroidal, and Skyrmions with higher charge all have point symmetries which are subgroups of O(3). We find that a rational map ansatz yields very good approximations to the full-field solutions. We point out a strong connection between the discrete symmetries of our solutions and those of corresponding solutions of the three-dimensional Skyrme model.

Hexagonal structure of baby Skyrmion lattices

We study the zero-temperature crystalline structure of baby Skyrmions by applying a full-field numerical minimization algorithm to baby Skyrmions placed inside different parallelogramic unit cells and imposing periodic boundary conditions. We find that within this setup, the minimal energy is obtained for the hexagonal lattice, and that in the resulting configuration the Skyrmion splits into quarter Skyrmions. In particular, we find that the energy in the hexagonal case is lower than the one obtained on the well-studied rectangular lattice, in which splitting into half Skyrmions is observed.

WARPED COMPACTIFICATION ON SIGMA MODEL, SKYRMIONS IN SIX DIMENSIONS

We construct two distinct brane solutions in six dimensional effective field theory models. The CP 1 sigma model and the baby skyrmion realize warped compactification of the extra dimensions for negative bulk cosmological constant. Higher winding number solutions of the baby skyrmion are also presented.

TOPOLOGICAL OBJECTS IN THE O(3) NONLINEAR SIGMA MODEL

We study the topological defects in the nonlinear O(3) sigma model in terms of the decomposition of U(1) gauge potential. Time-dependent baby skyrmions are discussed in the (2+1)-dimensional spacetime with the CP1 field. Furthermore, we show that there are three kinds of topological defects–vortex lines, point defects and knot exist in the (3+1)-dimensional model, and their topological charges, locations and motions are determined by the ϕ-mapping topological current theory.

Skyrmion-like states in two- and three-dimensional dynamical lattices

We construct, in discrete two-component systems with cubic nonlinearity, stable states emulating Skyrmions of the classical field theory. In the two-dimensional case, an analog of the baby Skyrmion is built on the square lattice as a discrete vortex soliton of a complex field [whose vorticity plays the role of the Skyrmion's winding number (WN)], coupled to a radial "bubble" in a real lattice field. The most compact quasi-Skyrmion on the cubic lattice is composed of a nearly planar complex-field discrete vortex and a three-dimensional real-field bubble; unlike its continuum counterpart which must have WN=2, this stable discrete state exists with WN=1. Analogs of Skyrmions in the one-dimensional lattice are also constructed. Stability regions for all these states are found in an analytical approximation and verified numerically. The dynamics of unstable discrete Skyrmions (which leads to the onset of lattice turbulence) and their partial stabilization by external potentials are explored too.

Reinterpretation of Faddeev–Niemi knot in Skyrme theory

We identify the Faddeev–Niemi knot in Skyrme theory as a vortex ring made of a helical baby skyrmion (a twisted chromomagnetic vortex which is periodic in z-coordinate) with the periodic ends connected together. This allows us to interpret the knot as two quantized magnetic flux rings linked together, the first one winding the second m times and the second one winding the first n times, whose linking number mn is fixed by the Chern–Simon index of the magnetic potential. This interpretation strongly suggests that the Skyrme theory could also describe a very interesting low energy physics in a completely different environment, which puts the theory in a totally new perspective.

Approximate analytical solutions of the baby Skyrme model

We show that many properties of the baby skyrmions, which have been determined numerically, can be understood in terms of an analytic approximation. In particular, we show that the approximation captures properties of the multiskyrmion solutions (derived numerically) such as their stability towards decay into various channels, and that it is more accurate for the “new baby Skyrme model” describing anisotropic physical systems in terms of multiskyrmion fields with axial symmetry. Some universal characteristics of configurations of this kind are demonstrated that are independent of their topological number.

Finite temperature induced fermion number in the nonlinearσmodel in2+1dimensions

We compute the finite temperature induced fermion number for fermions coupled to a static nonlinear sigma model background in (2+1) dimensions, in the derivative expansion limit. While the zero temperature induced fermion number is well known to be topological (it is the winding number of the background), at finite temperature there is a temperature dependent correction that is nontopological -- this finite T correction is sensitive to the detailed shape of the background. At low temperature we resum the derivative expansion to all orders, and we consider explicit forms of the background as a CP^1 instanton or as a baby skyrmion.

Collision-induced decay of metastable baby Skyrmions

Many extensions of the standard model predict heavy metastable particles which may be modeled as solitons (skyrmions of the Higgs field), relating their particle number to a winding number. Previous work has shown that the electroweak interactions admit processes in which these solitons decay, violating standard model baryon number. We motivate the hypothesis that baryon-number-violating decay is a generic outcome of collisions between these heavy particles. We do so by exploring a 2+1 dimensional theory which also possesses metastable skyrmions. We use relaxation techniques to determine the size, shape and energy of static solitons in their ground state. These solitons could decay by quantum mechanical tunneling. Classically, they are metastable: only a finite excitation energy is required to induce their decay. We attempt to induce soliton decay in a classical simulation by colliding pairs of solitons. We analyze the collision of solitons with varying inherent stabilities and varying incident velocities and orientations. Our results suggest that winding-number violating decay is a generic outcome of collisions. All that is required is sufficient (not necessarily very large) incident velocity; no fine-tuning of initial conditions is required.

Baby Skyrme modelsfor a class of potentials

We consider a class of (2+1)-dimensional baby Skyrme models withpotentials that have more than one vacuum. These potentials aregeneralizations of the old and new baby Skyrme models in that theyinvolvea more complicated dependence on ϕ3 (V(ϕ3)⩾0). Theboundary conditions are such that ϕ3 = 1 corresponds to the vacuumand ϕ3 = -1 at the position of each skyrmion. We find that when thepotential vanishes at ϕ3 = -1 the configurations corresponding tothe baby skyrmions lying `on top of each other' are the minima of theenergy. However, when V(ϕ3 = -1)≠0 the lowest fieldconfigurations correspond to separated baby skyrmions. We determine theenergy distributions for skyrmions of degrees between one and eight anddiscuss their geometrical shapes and binding energies. We also comparethe two-skyrmion states for these potentials. Most of our work has beenperformed numerically with the model being formulated in terms of threereal scalar fields (satisfying one constraint).

Geometrical Aspects of Skyrmions, Reflection Group, and the Internal Symmetry of Hadrons

The topological aspects of skyrmions are studied and it is shown that hadronscan be viewed as composite states of baby skyrmions when the internal symmetrySU(3) is generated from reflection. It is shown that in an anisotropic space aparticle can move with l = 1/2 with a specific lx value, and a bosonic constituentmoving with l = 1/2 will appear as a baby skyrmion and a fermionic constituentwill appear as if a spin carrier is attached to a baby skyrmion. The associatedmagnetic field causes a strong statistical attraction which helps to form the boundstate of such constituents. The doublet of such particles having opposite lx valuesform a conformal spinor when each member behaves as a Cartan semispinor.The conformal reflection then helps us to generate the internal SU(3) symmetry,which splits as SU(3) → SU(2) × U(1), giving rise to the hadronic spectra. Thestrong interaction involves a composite cluster in such a bound system whenrearrangement of the constituents takes place preserving the direction vectors,and an elementary constituent can take part in a weak interaction, causing parityviolation. These features help us to consider elementary constituents as knownparticles like leptons.

Skyrmions in quantum Hall ferromagnets as spin waves bound to unbalanced magnetic-flux quanta

A microscopic description of (baby)skyrmions in quantum Hall ferromagnets is derived from a scattering theory of collective (neutral) spin modes by a bare quasiparticle. We start by mapping the low lying spectrum of spin waves in the uniform ferromagnet onto that of free moving spin excitons, and then we study their scattering by the defect of charge. In the presence of this disturbance, the local spin stiffness varies in space, and we translate it into an inhomogeneus metric in the Hilbert space supporting the excitons. An attractive potencial is then required to preserve the symmetry under global spin rotations, and it traps the excitons around the charged defect. The quasiparticle now carries a spin texture. Textures containing more than one exciton are described within a mean-field theory, the interaction among the excitons being taken into account through a new renormalization of the metric. The number of excitons actually bound depends on the Zeeman coupling, that plays the same role as a chemical potencial. For small Zeeman energies, the defect binds many excitons which condensate. As the bound excitons have a unit of angular momentum, provided by the quantum of magnetic flux left unbalanced by the defect of charge, the resulting texture turns out to be a topological excitation of charge 1. Its energy is that given by the non-linear sigma model for the ground state in this topological sector, i.e. the texture is a skyrmion.

Mesons, baryons and waves in the baby Skyrmion model

We study various classical solutions of the baby-Skyrmion model in $(2+1)$ dimensions. We point out the existence of higher energy states interpret them as resonances of Skyrmions and anti-Skyrmions and study their decays. Most of the discussion involves a highly exited Skyrmion-like state with winding number one which decays into an ordinary Skyrmion and a Skyrmion-anti-Skyrmion pair. We also study wave-like solutions of the model and show that some of such solutions can be constructed from the solutions of the sine-Gordon equation. We also show that the baby-Skyrmion has non-topological stationary solutions. We study their interactions with Skyrmions.

Mesons, baryons and waves in the baby Skyrmion model

We study various classical solutions of the baby-Skyrmion model in (2+1) dimensions. We point out the existence of higher energy states, interpret them as resonances of Skyrmions and anti-Skyrmions and study their decays. Most of the discussion involves a highly exited Skyrmion-like state with winding number one which decays into an ordinary Skyrmion and a Skyrmion-anti-Skyrmion pair. We also study wave-like solutions of the model and show that some of such solutions can be constructed from the solutions of the sine-Gordon equation. We also show that the baby-Skyrmion model has non-topological stationary solutions. We study their interactions with Skyrmions.

Quantum Numbers of Textured Hall Effect Quasiparticles.

We propose a class of variational wave functions with slow variation in spin and charge density and simple vortex structure at infinity, which properly generalize both the Laughlin quasiparticles and baby Skyrmions. We argue that the spin of the corresponding quasiparticle has a fractional part related in a universal fashion to the properties of the bulk state, and propose a direct experimental test of this claim. We show that certain spin-singlet quantum Hall states can be understood as arising from primary polarized states by Skyrmion condensation.

Baby Skyrmion strings

We consider two limiting cases of the 2+1 dimensional analog of the Skyrme model, where we show the existence of string-like solutions to the equations of motion. The model contains a conserved topological charge usually called the baryon number. Our strings are solitons which have a constant baryon number per unit length. In one limiting case, our configuration saturates a Bogomolnyi-type bound and is degenerate in energy per baryon with the baby Skyrmion which is the analog of the spherically symmetric Skyrmion soliton. Hence here the string is energetically stable. In the other limiting case these energies are still degenerate but neither saturates the corresponding Bogomolnyi-type bound. Here we find invariance under area preserving diffeomorphisms. Both cases are solvable analytically. For intermediate ranges of the parameters we provide numerical evidence of the existence of the string-like configurations in a region of the parameter space. We speculate that the string is classically stable here as the energy per length containing one baryon is less than the energy of an isolated baryon (radially symmetric “baby Skyrmion”).

Dynamics of baby Skyrmions

Baby Skyrmions are topological solitons in a (2+1)-dimensional field theory which resembles the Skyrme model in important respects. We apply some of the techniques and approximations commonly used in discussions of the Skyrme model to the dynamics of baby Skyrmions and directly test them against numerical simulations. Specifically we study the effect of spin on the shape of a single baby Skyrmmion, the dependence of the forces between two baby Skyrmions on the baby Skyrmions' relative orientation and the forces between two baby Skyrmions when one of them is spinning.