We consider a six dimensional brane world model, where the brane is described by a localized solution to the baby-Skyrme model extending in the extradimensions. The branes have a cosmological constant modeled by inflating four dimensional slices and we further consider a bulk cosmological constant. We construct solutions numerically and present evidences that the solutions cease to exist for large values of the brane cosmological constant in some particular case. Then we study the stability of the model by considering perturbation of the gravitational part (resp. baby skyrmion) with fixed matter fields (resp. gravitational background). Our results indicate that the perturbation equations do not admit localized solutions for certain type of perturbation. The stability analysis can be alternatively seen as leading to a particle spectrum; we give mass estimations for the baby-Skyrme perturbation and for the graviton.
We consider the six dimensional brane world model, where the brane is described by a localized solution to the baby-Skyrme model extending in the extradimensions. The branes have a cosmological constant modeled by inflating four dimensional slices and we further consider a bulk cosmological constant. We focus on the topological number three solutions and discuss the localization mechanism of the fermions on the above 3-branes. We discuss interpretation of the model in term of quark third generation mass and in terms of the inflation history. We argue that it is possible to track the phases of …
We investigate the existence of non axisymmetric solutions in the 6-dimensional baby-Skyrme brane model. The brane is described by a localized solution to the baby-Skyrme model extending in the extra dimensions. Such non symmetric branes have already been constructed in the original 2+1-dimensional baby-Skyrme model in flat space. We generalize this result to the case of gravitating baby-Skyrme and in the context of extradimensions. These non-trivial deformation from the axisymmetric shape appear for higher values of the topological charge, so we consider the cases of $B=3,4$, where $B$ is the…
We investigate the sigma and baby Skyrme models with an RP^2 target space. We compare these models to models with an S^2 target space. We investigate the interactions between solitons and defects in the RP^2 sigma model.
We report our observations that the baby-Skyrme model in (2+1) dimensions possesses non-topological stationary solutions which we call pseudo-breathers. We discuss their properties and present our results on their interaction with the topological skyrmions.
We discuss one of the most interesting phenomena exhibited by baby skyrmions -- breaking of rotational symmetry. The topics we will deal with here include the appearance of rotational symmetry breaking in the static solutions of baby Skyrme models, both in flat as well as in curved spaces, the zero-temperature crystalline structure of baby skyrmions, and finally, the appearance of spontaneous breaking of rotational symmetry in rotating baby skyrmions.
We construct a method to supersymmetrize higher kinetic terms and apply it to the baby Skyrme model. We find that there exist N=1 supersymmetric extensions for baby Skyrme models with arbitrary potential.
We study the baby Skyrme model as a theory that interpolates between two distinct BPS systems. For this a near-BPS approximation can be used which, however, involves a small deviation from each of the two BPS limits. We provide analytical explanation and numerical support for the validity of this approximation. We then study the set of all possible supersymmetric extensions of the baby Skyrme model with ${\cal N}=1$ and the particular ones with extended ${\cal N}=2$ supersymmetries and relate this to the above mentioned almost-BPS approximation.
We study the structure of minimal-energy solutions of the baby Skyrme models for any topological charge n; the baby multi-skyrmions. Unlike in the (3+1)D nuclear Skyrme model, a potential term must be present in the (2+1)D Skyrme model to ensure stability. The form of this potential term has a crucial effect on the existence and structure of baby multi-skyrmions. The simplest holomorphic model has no known stable minimal-energy solution for n greater than one. The other baby Skyrme model studied in the literature possesses non-radially symmetric minimal-energy configurations that look like `sk…
We consider a class of (2+1) dimensional baby Skyrme models with potentials that have more than one vacum. These potentials are generalisation of old and new baby Skyrme models;they involve more complicated dependence on phi_3.We find that when the potential is invariant under phi_3 -> -phi_3 the configuration corresponding to the baby skyrmions lying "on top of each other" are the minima of the energy. However when the potential breaks this symmetry the lowest field configurations correspond to separated baby skyrmions. We compute the energy distributions for skyrmions of degrees between one …
We derive the Bogomolny decompositions (Bogomolny equations) for: full baby Skyrme model and for its restricted version (so called, pure baby Skyrme model), in (2+0) dimensions, by using so called, concept of strong necessary conditions. It turns out that Bogomolny decomposition can be derived for restricted baby Skyrme model for arbitrary form of the potential term, while for full baby Skyrme model, such derivation is possible only for some class of the potentials.
We develop a one-parameter family of static baby Skyrme models that do not require a potential term to admit topological solitons. This is a novel property as the standard baby Skyrme model must contain a potential term in order to have stable soliton solutions, though the Skyrme model does not require this. Our new models satisfy an energy bound that is linear in terms of the topological charge and can be saturated in an extreme limit. They also satisfy a virial theorem that is shared by the Skyrme model. We calculate the solitons of our new models numerically and observe that their form depe…
We examine non-linear sigma (plus Skyrme term) model in flat space, in particular with a twist, which comprises a twisted baby Skyrmion vortex solution with an added dependence on a twist term $mkz$, where $z$ is the vertical coordinate. We find that the solutions should have asymptotic form. We also find that mass per unit length of string, $μ$, increases roughly linearly with $ζ$, where $ζ=4λ^2K_s(mk)^2$, $λ$ is scaling constant, $K_s$ is Skyrme term, $m$ is integer and $k$ is wave number.
We continue the investigation of supersymmetric extensions of baby Skyrme models in d=2+1 dimensions. In a first step, we show that the CP(1) form of the baby Skyrme model allows for the same N=1 SUSY extension as its O(3) formulation. Then we construct the N=1 SUSY extension of the gauged baby Skyrme model, i.e., the baby Skyrme model coupled to Maxwell electrodynamics. In a next step, we investigate the issue of N=2 SUSY extensions of baby Skyrme models. We find that all gauged and ungauged submodels of the baby Skyrme model which support BPS soliton solutions allow for an N=2 extension such…
In this work we investigate some features of the fermionic sector of the supersymmetric version of the baby Skyrme model. We find that, in the background of BPS compact Skyrmions, fermionic zero modes are confined to the defect core. Further, we show that, while three SUSY generators are broken in the defect core, SUSY is completely restored outside. We study also the effect of a D-term deformation of the model. Such a deformation allows for the existence of fermionic zero modes and broken SUSY outside the compact defect.
In present paper 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 this 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" which describes anisotropic physical systems in terms of multiskyrmion fields with axial symmetry. Some universal characteristics of configurations of this kind are demonstrated, which do not …
We investigate the relation between the BPS baby Skyrme model and its vector meson formulation, where the baby Skyrme term is replaced by a coupling between the topological current $B_μ$ and the vector meson field $ω_μ$. The vector model still possesses infinitely many symmetries leading to infinitely many conserved currents which stand behind its solvability. It turns out that the similarities and differences of the two models depend strongly on the specific form of the potential. We find, for instance, that compactons (which exist in the BPS baby Skyrme model) disappear from the spectrum of …
This is one of the `New Talents' seminars at `Erice International School of Subnuclear Physics 1999' and looks at numerical studies of (2+1)D topological Skyrme-like solitons; the baby skyrmions. We explain the concept of integrable and topological solitons. Then we introduce the non-linear sigma model in 2D in order to discuss the 3D nuclear Skyrme model. We explain that the baby Skyrme models can be viewed as a (2+1)D version of the Skyrme model. We describe the numerical methods needed to study the baby skyrmions. Finally, we present some results on skyrmion-skyrmion scattering and our work…
We consider multisolitons with charges 1 =< B =< 5 in the baby Skyrme model for the one-parametric family of potentials U=μ^2 (1-ϕ_3)^s with 0<s =< 4. This class of potentials is a generalization of the `old' (s=1) and `holomorphic' (s=4) baby Skyrme models. We find that for charge one, stable solutions exist for every value of s and they are rotationally-symmetric. For higher charges, stable solutions exist only below s \approx 2. In the charge-two sector the stable solutions are always rotationally-symmetric and ring-like. For charge three and above, rotational symmetry is exhibited only in …
We construct the $\mathcal{N}=2$ supersymmetric extensions of different models containing a Chern-Simons term and study their BPS structure. We find that the coupling of the Chern-Simons term to the general gauged nonlinear $σ$-model allows for general potentials depending on the Kähler metric and preserving two supersymmetries. We construct from supersymmetry the potentials that allow for BPS structures in different $\mathbb{C}P^{N}$ models coupled to Chern-Simons and Maxwell terms. We also couple the Chern-Simons term to a higher-derivative field theory (the baby Skyrme model) and study its …
A restriction of the baby Skyrme model consisting of the quartic and potential terms only is investigated in detail for a wide range of potentials. Further, its properties are compared with those of the corresponding full baby Skyrme models. We find that topological (charge) as well as geometrical (nucleus/shell shape) features of baby skyrmions are captured already by the soliton solutions of the restricted model. Further, we find a coincidence between the compact or non-compact nature of solitons in the restricted model, on the one hand, and the existence or non-existence of multi-skyrmions …
We demonstrate that the gauged BPS baby Skyrme model with a double vacuum potential allows for phase transitions from a non-solitonic to a solitonic phase, where the latter corresponds to a ferromagnetic liquid. Such a transition can be generated by increasing the external pressure $P$ or by turning on an external magnetic field $H$. As a consequence, the topological phase where gauged BPS baby skyrmions exist, is a higher density phase. For smaller densities, obtained for smaller values of $P$ and $H$, a phase without solitons is reached. We find the critical line in the $P,H$ parameter space…
The Bogomolny decompositions (Bogomolny equations) for the gauged baby Skyrme models: restricted and full one, in (2+0)-dimensions, are derived, for some general classes of the potentials. The conditions, which must be satisfied by the potentials, for each of these mentioned models, are also derived.
Using the concept of strong necessary conditions (CSNC), we derive Bogomolny equations and BPS bounds for two modifications of the gauged baby BPS Skyrme model: the nonminimal coupling to the gauge field and k-deformed model. In particular, we study, how the Bogomolny equations and the equation for the potential, reflect these two modifications. In both examples, the CSNC method shows to be a very useful tool.
In this paper we investigate the Q-ball Ansatz in the baby Skyrme model. First, the appearance of peakons, i.e. solutions with extremely large absolute values of the second derivative at maxima, is analyzed. It is argued that such solutions are intrinsic to the baby Skyrme model and do not depend on the detailed form of a potential used in calculations. Next, we concentrate on compact non spinning Q-balls. We show the failure of a small parameter expansion in this case. Finally, we explore the existence and parameter dependence of Q-ball solutions.
We show the existence of self-dual (topological) solitons in a gauged version of the baby Skyrme model in which the Born-Infeld term governs the gauge field dynamics. The successful implementation of the Bogomol'nyi-Prasad-Sommerfield formalism provides a lower bound for the energy and the respective self-dual equations whose solutions are the solitons saturating such a limit. The energy lower bound (Bogomol'nyi bound) is proportional to the topological charge of the Skyrme field and therefore quantized. In contrast, the total magnetic flux is a nonquantized quantity. Furthermore, the model su…
A new type of gauged BPS baby Skyrme model is presented, where the derivative term is just the Schroers current (i.e., gauge invariant and conserved version of the topological current) squared. This class of models has a topological bound saturated for solutions of the pertinent Bogomolnyi equations supplemented by a so-called superpotential equation. In contrast to the gauged BPS baby Skyrme models considered previously, the superpotential equation is linear and, hence, completely solvable. Furthermore, the magnetic flux is quantized in units of $2π$, which allows, in principle, to define thi…
We have studied the existence de self-dual solitons in a gauged version of the baby Skyrme model in which the gauge field dynamics is governed by the Maxwell-Chern-Simons action. For such a purpose, we have developed a detailed implementation of the Bogomol'nyi-Prasad-Sommerfield formalism providing the self-dual equations whose solutions saturate the energy lower bound. Such a bound related to the topological charge of the Skyrme field becomes quantized whereas both the total magnetic flux and the total electrical charge are not. We have found two types of self-dual Skyrme field profiles: the…
Recent results suggest that multi-Skyrmions stabilized by omega mesons have very similar properties to those stabilized by the Skyrme term. In this paper we present the results of a detailed numerical investigation of a (2+1)-dimensional analogue of this situation. Namely, we compute solitons in an O(3) sigma-model coupled to a massive vector meson and compare the results to baby Skyrmions, which are solitons in an O(3) sigma-model including a Skyrme term. We find that multi-solitons in the vector meson model are surprisingly similar to those in the baby Skyrme model, and we explain this corre…
The baby Skyrme model is a well-known nonlinear field theory supporting topological solitons in two space dimensions. Its action functional consists of a potential term, a kinetic term quadratic in derivatives (the "nonlinear sigma model term") and the Skyrme term quartic in first derivatives. The limiting case of vanishing sigma model term (the so-called BPS baby Skyrme model) is known to support exact soliton solutions saturating a BPS bound which exists for this model. Further, the BPS model has infinitely many symmetries and conservation laws. Recently it was found that the gauged version …
The aloof baby Skyrme model is a (2+1)-dimensional theory with solitons that are lightly bound. It is a low-dimensional analogue of a similar Skyrme model in (3+1)-dimensions, where the lightly bound solitons have binding energies comparable to nuclei. A previous study of static solitons in the aloof baby Skyrme model revealed that multi-soliton bound states have a cluster structure, with constituents that preserve their individual identities due to the short-range repulsion and long-range attraction between solitons. Furthermore, there are many different local energy minima that are all well-…
We consider the baby Skyrme model in a physically motivated limit of reaching the restricted or BPS baby Skyrme model, which is a model that enjoys area-preserving diffeomorphism invariance. The perturbation consists of the kinetic Dirichlet term with a small coefficient $ε$ as well as the standard pion mass term, with coefficient $εm_1^2$. The pions remain lighter than the soliton for any $ε$ and therefore the model is physically acceptable, even in the $ε\to 0$ limit. The version of the BPS baby Skyrme model we use has BPS solutions with Gaussian tails. We perform full numerical computations…
The baby Skyrme model is a well-known nonlinear field theory supporting topological solitons in two space dimensions. In the limit where the term quadratic in derivatives (the "sigma model term") vanishes some additional structure emerges. The resulting ("extreme" or "restricted" or "BPS") baby Skyrme model has exact soliton solutions saturating a BPS bound which exists for this restricted model. Further, the restricted model has infinitely many symmetries and infinitely many conservation laws. Here we consider the gauged version of the restricted baby Skyrme model with gauge group U(1) and th…
The baby Skyrme model is a (2+1)-dimensional analogue of the Skyrme model, in which baryons are described by topological solitons. In this paper we introduce a version of the baby Skyrme model in which the global O(3) symmetry is broken to the dihedral group D_N. It is found that the single soliton in this theory is composed of N partons, that are topologically confined. The case N=3 is studied in some detail and multi-soliton solutions are computed. These soliton solutions are related to polyiamonds, which are plane figures composed of equilateral triangles joined by common edges. It is shown…
We perform full two-dimensional (2D) numerical relaxations of isospinning soliton solutions in the baby Skyrme model in which the global $O(3)$ symmetry is broken by the 2D analogue of the pion mass term in the Skyrme model. In our calculations we explicitely allow the isospinning solitons to deform and to break the symmetries of the static configurations. We find that stable isospinning baby Skyrme solutions can be constructed numerically for all angular frequencies $ω\le \text{min}(μ,1)$, where $μ$ is the mass parameter of the model. Stable, rotationally-symmetric baby Skyrmion solutions for…
We construct a molecule of fractional vortices with fractional topological lump charges as a baby Skyrmion with the unit topological lump charge in the anti-ferromagnetic (or XY) baby Skyrme model, that is, an O(3) sigma model with a four derivative term and an anti-ferromagnetic or XY-type potential term quadratic in fields. We further construct configurations with topological lump charges Q <= 7 and find that bound states of vortex molecules constitute regular polygons with 2Q vertices as vortices, where the rotational symmetry SO(2) in real space is spontaneously broken into a discrete subg…
We show that a suitable choice for the potential term in the two-dimensional baby Skyrme model yields solitons that have a short-range repulsion and a long-range attraction. The solitons are therefore aloof, in the sense that static multi-soliton bound states have constituents that preserve their individual identities and are sufficiently far apart that tail interactions yield small binding energies. The static multi-soliton solutions are found to have a cluster structure that is reproduced by a simple binary species particle model. In the standard three-dimensional Skyrme model of nuclei, sol…
We have shown the existence of self-dual solitons in a type of generalized Chern-Simons baby Skyrme model where the generalized function (depending only in the Skyrme field) is coupled to the sigma-model term. The consistent implementation of the Bogomol'nyi-Prasad-Sommerfield (BPS) formalism requires the generalizing function becomes the superpotential defining properly the self-dual potential. Thus, we have obtained a topological energy lower-bound (Bogomol'nyi bound) and the self-dual equations satisfied by the fields saturating such a bound. The Bogomol'nyi bound being proportional to the …
The static baby Skyrme model is investigated in the extreme limit where the energy functional contains only the potential and Skyrme terms, but not the Dirichlet energy term. It is shown that the model with potential $V=\frac12(1+ϕ_3)^2$ possesses solutions with extremely unusual localization properties, which we call semi-compactons. These minimize energy in the degree 1 homotopy class, have support contained in a semi-infinite rectangular strip, and decay along the length of the strip as $x^{-\log x}$. By gluing together several semi-compactons, it is shown that every homotopy class has line…
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 Skyrmion, 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.
We consider a restricted baby Skyrme-Maxwell scenario enlarged via the inclusion of a nontrivial magnetic permeability. We then proceed with the minimization of its total energy by means of the Bogomol'nyi-Prasad-Sommerfield (BPS) prescription, from which we get that the self-dual potential now depends on the magnetic permeability itself. As a result, we obtain not only the lower bound for the energy, but also the self-dual equations whose solutions saturate that bound. In such a context, we focus our attention on those time-independent gauged skyrmions with radial symmetry and no electric cha…
We subject the baby Skyrme model to a Moyal deformation, for unitary or Grassmannian target spaces and without a potential term. In the abelian case, the radial BPS configurations of the ordinary noncommutative sigma model also solve the baby Skyrme equation of motion. This gives a class of exact analytic noncommutative baby Skyrmions, which have a singular commutative limit but are stable against scaling due to the noncommutativity. We compute their energies, investigate their stability and determine the asymptotic two-Skyrmion interaction.
We present a study of $U(1)$ gauged modification of the 2+1 dimensional planar Skyrme model with a particular choice of the symmetry breaking potential term which combines a short-range repulsion and a long-range attraction. In the absence of the gauge interaction the multi-solitons of the model are aloof, they consist of the individual constituents which are well separated. Peculiar feature of the model is that there are usually several different stable static multi-soliton solutions of rather similar energy in a topological sector of given degree. We investigated the pattern of the solutions…
We find the static multi-soliton solutions of the baby Skyrme model on the two-sphere for topological charges 1 =< B =< 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 3D Skyrme model.
The noncommutative baby Skyrme model is a Moyal deformation of the two-dimensional sigma model plus a Skyrme term, with a group-valued or Grassmannian target. Exact abelian solitonic solutions have been identified analytically in this model, with a singular commutative limit. Inside any given Grassmannian, we establish a BPS bound for the energy functional, which is saturated by these baby Skyrmions. This asserts their stability for unit charge, as we also test in second-order perturbation theory.
The properties of the multisoliton solutions of the (2+1)-dimensional Maxwell-Chern-Simons-Skyrme model are investigated numerically. Coupling to the Chern-Simons term allows for existence of the electrically charge solitons which may also carry magnetic fluxes. Two particular choices of the potential term is considered: (i) the weakly bounded potential and (ii) the double vacuum potential. In the absence of the gauge interaction in the former case the individual constituents of the multisoliton configuration are well separated, while in the latter case the rotational invariance of the configu…
The BPS baby Skyrme models are submodels of baby Skyrme models, where the nonlinear sigma model term is suppressed. They have skyrmion solutions saturating a BPS bound, and the corresponding static energy functional is invariant under area-preserving diffeomorphisms (APDs). Here we show that the solitons in the BPS baby Skyrme model, which carry a nontrivial topological charge $Q_{b} \in π_2(S^2)$ (a winding number), are dual to vortices in a BPS vortex model with a topological charge $Q_{v}\in π_1(S^1)$ (a vortex number), in the sense that there is a map between the BPS solutions of the two m…
We show that the BPS Skyrme model, as well as its (2+1) dimensional baby version (restricted), can be coupled with an impurity in the BPS preserving manner. The corresponding Bogomolny equations are derived.
We consider the baby-Skyrme model in the regime close to the so-called restricted baby-Skyrme model, which is a BPS model with area-preserving diffeomorphism invariance. The perturbation takes the form of the standard kinetic Dirichlet term with a small coefficient $ε$. Classical solutions of this model, to leading order in $ε$, are called restricted harmonic maps. In the BPS limit ($ε\to 0$) of the model with the potential being the standard pion-mass term, the solution with unit topological charge is a compacton. Using analytical and numerical arguments we obtain solutions to the problem for…
For the baby Skyrme model with a specific potential, compacton solutions, i.e., configurations with a compact support and parabolic approach to the vacuum, are derived. Specifically, in the non-topological sector, we find spinning Q-balls and Q-shells, as well as peakons. Moreover, we obtain compact baby skyrmions with non-trivial topological charge. All these solutions may form stable multi-soliton configurations provided they are sufficiently separated.
This paper examines the classical dynamics of false vacuum regions embedded in surrounding regions of true vacuum, in the thin-wall limit. The dynamics of all generally relativistically allowed solutions -- most but not all of which have been previously studied -- are derived, enumerated, and interpreted. We comment on the relation of these solutions to possible mechanisms whereby inflating regions may be spawned from non-inflating ones. We then calculate the dynamics of first order deviations from spherical symmetry, finding that many solutions are unstable to such aspherical perturbations. T…
The magnetothermodynamics of skyrmion type matter described by the gauged BPS baby Skyrme model at zero temperature is investigated. We prove that the BPS property of the model is preserved also for boundary conditions corresponding to an asymptotically constant magnetic field. The BPS bound and the corresponding BPS equations saturating the bound are found. Further, we show that one may introduce pressure in the gauged model by a redefinition of the superpotential. Interestingly, this is related to non-extremal type solutions in the so-called fake supersymmetry method. Finally, we compute the…
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…
The Skyrme-Faddeev model has planar soliton solutions with target space $\mathbb{C}P^N$. An Abelian Chern-Simons term (the Hopf term) in the Lagrangian of the model plays a crucial role for the statistical properties of the solutions. Because $Π_3(\mathbb{C}P^1)=\mathbb{Z}$, the term becomes an integer for $N=1$. On the other hand, for $N>1$, it becomes perturbative because $Π_3(\mathbb{C}P^N)$ is trivial. The prefactor $Θ$ of the Hopf term is not quantized, and its value depends on the physical system. We study the spectral flow of the normalizable fermions coupled with the baby-Skyrme model …
