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.
Previous results on multi-charged Baby Skyrmion solutions have pointed to a modular structure, comprised of charge two rings and single charge one Skyrmions, which combine to form higher charged structures. In this paper we present numerical evidence which shows an alternative finite chain, multi-charged global energy minimum Baby Skymion solution. We then proceed from the infinite plane, to Baby Skyrmions on a cylinder and then a torus, to obtain the solutions of periodic Baby Skyrmions, of which periodic segments will correspond to sections of large charge Baby Skyrmions in the plane.
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.
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 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 stat…
We numerically simulate the formation of $(2+1)$-dimensional baby Skyrmions and $(3+1)$-dimensional $SU(2)$ Skyrmions from domain wall collisions. It has been suggested that Skyrmion, anti-Skyrmion pairs can be produced from the interaction of two domain walls. This is confirmed, however it is also demonstrated that the process can require quite precise conditions. An alternative, more stable, formation process is proposed as the interaction of more than two segments of domain wall. This is simulated, requiring far less constraints on the initial conditions used. Finally domain wall networks a…
We introduce a consistent ansatz for the baby Skyrme model in (2+1)-dimensions which is able to reduce the complete set of field equations to just one equation for the profile function in situations in which the baby baryon charge can be arbitrary. Many analytic solutions both with and without the inclusion of the effects of the minimal coupling with the Maxwell field are constructed. Linear stability and other physical properties are discussed. These analytic gauged baby Skyrmions generate a persistent $U(1)$ current which cannot be turned off continuously as it is tied to the topological cha…
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…
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 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.
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…
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 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…
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 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.
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 discuss how internal rotation with fixed angular frequency can affect the solitons in the baby Skyrme model in which the global O(3) symmetry is broken to the SO(2). Two particular choices of the potential term are considered, the "old" potential and the "new" double vacuum potential, We do not impose any assumptions about the symmetry on the fields. Our results confirm existence of two types of instabilities determined by the relation between the mass parameter of the potential and the angular frequency.
We analyse the effect of the canonical quantization of the rotational mode of the O(3) $σ$-model which includes the Skyrme term. Numerical evidence is presented that the quantum correction to the mass of the rotationally-invariant charge $n=1,2$ configurations may stabilize the solution even in the limit of vanishing potential. The corresponding range of values of the parameters is discussed.
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.
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 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 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 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-…
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.
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 show that the minimally gauged BPS baby Skyrme model remains a BPS theory after coupling with gravity that is the topologically nontrivial configurations called baby Skyrmions carrying magnetic flux are solutions to a zero pressure equation. Following that the proper mass, magnetic flux and the proper geometric volume are linear function of the topological charge, while the total ADM mass and geometric radius get a contribution due to the gravitational interaction which is quadratic in the topological charge. All these quantities are found exactly as target space integrals (averages) of the…
We introduce a system composed of two (2+1)-dimensional baby-skyrmion models (BSMs) set on parallel planes and linearly coupled by tunneling of fields. This system can be realized in a dual-layer ferromagnetic medium. Unlike dual-core models previously studied\ in nonlinear optics and BEC, here the symmetry-breaking bifurcation (SBB) in solitons (baby skyrmions) occurs with the increase of the inter-core coupling ($κ$), rather than with its decrease, due to the fact that even in the uncoupled system neither core may be empty. Prior to the onset of the symmetry breaking between the two componen…
This paper describes a model for baby Skyrme crystal chunks with arbitrary potential by considering energy contributions from the bulk and surface of a crystal chunk. We focus on two potentials which yield distinct Skyrme lattices: the standard potential $V=m^2(1-φ^3)$ and the easy plane potential $V=\frac{1}{2}m^2 (φ^1)^2$. In both models, the static energy functional is minimized over all $2$-dimensional period lattices, yielding the minimal energy crystal structure(s). For the standard potential, the Skyrmions form a hexagonal crystal structure, whereas, for the easy plane potential, the mi…
We study the baby Skyrme model in a pure AdS background without a mass term. The tail decays and scalings of massless radial solutions are demonstrated to take a similar form to those of the massive flat space model, with the AdS curvature playing a similar role to the flat space pion mass. We also numerically find minimal energy solutions for a range of higher topological charges and find that they form concentric ring-like solutions. Popcorn transitions (named in analogy with studies of toy models of holographic QCD) from an n layer to an n+1-layer configuration are observed at topological c…
We provide analytical and numerical evidence of the existence of classically stable, string-like configurations in a 2+1 dimensional analog of the Skyrme model. The model contains a conserved topological charge usually called the baryon number. Our strings are non-topological solitons which have a constant baryon number per unit length. The energy per length containing one baryon is, however, less than the energy of an isolated baryon (radially symmetric ``baby Skyrmion") in a region of the parameter space, which suggests a degree of stability for our configurations. In a limiting case, our co…
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…
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…
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 …
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 localised 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 …
In the present paper we investigate the causal structure of the baby Skyrme model using appropriate geometrical tools. We discuss several features of excitations propagating on top of background solutions and show that the evolution of high frequency waves is governed by a curved effective geometry. Examples are given for which the effective metric describes the interaction between waves and solitonic solutions such as kinks, antikinks and Hedgehogs. In particular, it is shown how violent processes involving the collisions of solitons and antisolitons may induce metrics which are not globaly h…
The baby Skyrme model is studied with a novel choice of potential, $V=1/2 ϕ_3^2$. This "easy plane" potential vanishes at the equator of the target two-sphere. Hence, in contrast to previously studied cases, the boundary value of the field breaks the residual SO(2) internal symmetry of the model. Consequently, even the unit charge skyrmion has only discrete symmetry and consists of a bound state of two half lumps. A model of long-range inter-skyrmion forces is developed wherein a unit skyrmion is pictured as a single scalar dipole inducing a massless scalar field tangential to the vacuum manif…
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 study a model for two-dimensional skyrmions on a sphere of radius L. Such model simulates a skyrmion lattice of density W/(2 πL^2), where W is the skyrmion winding number. We show that, to a very good approximation, physical results depend only on the product αL^4, where αis the strength of potential term. In the range αL^4 approx. or less than 3 the order parameter vanishes, there is a uniform distribution of the density over the whole surface and the energy of the W=2 sector lies above twice the energy of the W=1 sector. If αL^4 approx. or greater than 6 the order parameter approaches uni…
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 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 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 revisit the quantum Hall system with no Zeeman splitting energy using the noncommutative field theory. We analyze the BPS condition for the delta-function interaction near the filling factor $ν=1$. Multi-skyrmions are shown to saturate the BPS bounds. The dimension of the moduli space of $k$ skyrmions is $4k+2$. Advantage of the noncommutative field description is demonstrated through the derivation of the effective nonlinear $σ$ model Lagrangian.
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 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.
It has been recently shown that solitons are fundamental classical solutions of non-commutative field theories. We reconsider this issue from the standpoint of the Hall effect and identify some solutions with known solutions in the integer Hall effect with no Zeeman coupling.
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 study fractional Skyrmions in a $\mathbb{C}P^2$ baby Skyrme model with a generalization of the easy-plane potential. By numerical methods, we find stable, metastable, and unstable solutions taking the shapes of molecules. Various solutions possess discrete symmetries, and the origin of those symmetries are traced back to congruencies of the fields in homogeneous coordinates on $\mathbb{C}P^2$.
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…
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…
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 matt…
Motivated by describing time-evolutions of noncommutative worlds, I discuss symmetry-preserving evolutions of noncommutative worlds of finite dimensional representation spaces. An interesting issue in such evolutions is that there can be transitions of representation spaces of symmetry, and therefore the evolutions are generally non-unitary or symmetry-violating. The central idea of this paper is that a main world evolves by emitting baby worlds which compensate the violations of the symmetry of the main world. Tracing out the states of the baby worlds, the symmetry-preserving evolutions of th…
We analyze topological solitons in the noncommutative plane by taking a concrete instance of the quantum Hall system with the SU(N) symmetry, where a soliton is identified with a skyrmion. It is shown that a topological soliton induces an excitation of the electron number density from the ground-state value around it. When a judicious choice of the topological charge density $J_{0}(\mathbf{x})$ is made, it acquires a physical reality as the electron density excitation $Δρ^{\text{cl}}(\mathbf{x})$ around a topological soliton, $Δρ^{\text{cl}}(\mathbf{x})=-J_{0}(% \mathbf{x})$. Hence a noncommut…
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…
