Baby Skyrme Model Research Articles

Baby skyrmion crystals stabilized by vector mesons

In this letter we study soliton crystals in the (2+1)-dimensional analogue model of the (3+1)-dimensional Adkins–Nappi model of nuclear physics. The baby ω-Skyrme model studied here is an O(3) nonlinear σ model coupled to a massive vector meson, the ω-meson. Using recently developed methods in the (3+1)-dimensional ω-Skyrme model we are able to construct soliton crystals in this (2+1)-dimensional baby ω-Skyrme model. The resulting crystals form a hexagonal lattice structure and are qualitatively and quantitatively similar to crystals observed in the standard baby Skyrme model.

A baby–Skyrme model with anisotropic DM interaction: Compact skyrmions revisited

We consider a baby–Skyrme model with Dzyaloshinskii–Moriya interaction (DMI) and two types of potential terms. The model has a close connection with the vacuum functional of fermions coupled with O(3) nonlinear n-fields and with a constant SU(2) gauge background. The energy functional is derived from the heat-kernel expansion for the fermion determinant. The model possesses normal skyrmions with topological charge Q=1. The restricted version of the model also includes both the weak-compacton case (at the boundary, not continuously differentiable) and genuine-compacton case (continuously differentiable). The model consists of only the Skyrme term, and the DMI provides soliton solutions that are known as skyrmions without any potential. The BPS equation in the supersymmetric soliton models implies that the impurity coupling is closely related to the DMI. Therefore, the effect of an exponentially localized DMI is also studied in the present model.

Strong Necessary Conditions and the Cauchy Problem

Some exact solutions of boundary or initial conditions formulated for Bogomolny equations (derived by using the strong necessary conditions and associated with some ordinary equation and some partial differential equations) have been found. The solution obtained for the restricted baby Skyrme model, as well the density of energy for this solution, are localized. Moreover, it turns out that the densities of the ungauged Hamiltonian and the gauged Hamiltonian are correspondingly, non-zero and zero for the found solution of the Cauchy problem associated with the Bogomolny equation of the restricted baby Skyrme model. Hence, a degeneracy of the Hamiltonian for this model has been established. As such, one can see the breaking of some symmetry.

Rotationally invariant isospinning baby-skyrmions dressed by fermions

We couple Fermions to the isospinning (2+1) baby-skyrme model. We show that consistent rotationally invariant localised solutions can be found but the Fermionic solutions to the equations of motion are not in general eigenstates of the Hamiltonian. These do however become full eigenstates in a particular limit. We also find the corresponding Fermionic eigenstates of the Hamiltonian, for which our Fermion ansatz is also well suited. For localised Fermionic states, a novel constraint on the internal isospin frequency exists. Throughout our studies, we include the backreaction of the Fermions on the baby-skyrmion.

Fractional Skyrmion molecules in a \u2102PN\u22121 model

We study fractional Skyrmions in a ℂP2 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 ℂP2.

Self-dual solitons in a Born-Infeld baby Skyrme model

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 supports three types of self-dual solitons profiles: the first describes compacton solitons, the second follows a Gaussian decay law, and the third portrays a power-law decay. Finally, we perform numerical solutions of the self-dual equations and depicted the soliton profiles for different values of the parameters controlling the nonlinearity of the model.

Near-BPS baby Skyrmions with Gaussian tails

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 upepsilon {m}_1^2 . The pions remain lighter than the soliton for any ϵ and therefore the model is physically acceptable, even in the ϵ → 0 limit. The version of the BPS baby Skyrme model we use has BPS solutions with Gaussian tails. We perform full numerical computations in the ϵ → 0 limit and even reach the strict ϵ = 0 case, finding new nontrivial BPS solutions, for which we do not yet know the analytic form.

Analytic baby skyrmions at finite density

We study the baby Skyrme model in (2+1)-dimensions built on a finite cylinder. To this end, we introduce a consistent ansatz which is able to reduce the complete set of field equations to just one equation for the profile function for arbitrary baryon charge. Many analytic solutions both with and without the inclusion of the effects of the minimal coupling with the Maxwell field are constructed. The baby skyrmions appear as a sequence of rings along the cylinder, leading to a periodic shape in the baryon density. 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 charge of the baby Skyrmions themselves. In the simplest non-trivial case of a gauged baby Skyrmion, a very important role is played by the Mathieu equation with an effective coupling constant which can be computed explicitly. These configurations are a very suitable arena to test resurgence in a non-integrable context.

Near-BPS baby Skyrmions

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 → 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 topological sectors greater than one. We develop a perturbative scheme in ϵ with which we can calculate the corrections to the BPS mass. The leading order ( mathcal{O}left({upepsilon}^1right) ) corrections show that the baby Skyrmion with topological charge two is energetically preferred. The binding energy requires us to go to the third order in ϵ to capture the relevant terms in perturbation theory, however, the binding energy contributes to the total energy at order ϵ2. We find that the baby Skyrmions — in the near-BPS regime — are compactons of topological charge two, that touch each other on their periphery at a single point and with orientations in the attractive channel.

Quantum soliton scattering manifolds

We consider the quantum multisoliton scattering problem. For BPS theories one truncates the full field theory to the moduli space, a finite dimensional manifold of energy minimising field configurations, and studies the quantum mechanical problem on this. Non-BPS theories — the generic case — have no such obvious truncation. We define a quantum soliton scattering manifold as a configuration space which satisfies asymptotic completeness and respects the underlying classical dynamics of slow moving solitons. Having done this, we present a new method to construct such manifolds. In the BPS case the dimension of the n-soliton moduli space ℳn is n multiplied by the dimension of ℳ1. We show that this scaling is not necessarily valid for scattering manifolds in non-BPS theories, and argue that it is false for the Skyrme and baby-Skyrme models. In these models, we show that a relative phase difference can generate a relative size difference during a soliton collision. Asymptotically, these are zero and non-zero modes respectively and this new mechanism softens the dichotomy between such modes. Using this discovery, we then show that all previous truncations of the 2-Skyrmion configuration space are unsuitable for the quantum scattering problem as they have the wrong dimension. This gives credence to recent numerical work which suggests that the low-energy configuration space is 14- dimensional (rather than 12-dimensional, as previously thought). We suggest some ways to construct a suitable manifold for the 2-Skyrmion problem, and discuss applications of our new definition and construction for general soliton theories.

Existence criteria for baby skyrmions for a wide range of potentials

Topologically stable field configurations are an important subject of study in various areas such as magnetism, condensed matter physics, nuclear physics, liquid crystals; in the baby Skyrme models (the two dimensional versions of the Skyrme model of nuclear physics) such structures are called baby skyrmions, and they have been widely studied by means of numerical methods. The aim of this paper is to give a simple criterion that guarantees the existence of skyrmions with unit topological charge for most baby Skyrme models considered in the literature. The proof of this criterion is purely analytic, and it is based on the concentration of the energy of minimizing sequences.

Self-dual solitons in a Maxwell-Chern-Simons baby Skyrme model

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 first is described by a solution which decays following an exponential-law ($e^{-\alpha r^2}$, $\alpha>0$); the second is portrayed by a solution having a power-law decay ($r^{-\beta}$, $\beta>0$). On other hand, in both cases the asymptotic behavior of the gauge field is similar to the one presented in the context of the Abelian Higgs models describing Abrikosov-Nielsen-Olesen charged vortices. Other interesting feature we highlight is the localized magnetic flux inversion, a property does not observed in others gauged baby Skyrme models already studied in literature. Numerical results are presented for rotationally symmetrical field configurations by remarking some of its essential features.

Statistical Nature of Skyrme-Faddeev Models in 2+1 Dimensions and Normalizable Fermions

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 $\Pi_3(\mathbb{C}P^1)=\mathbb{Z}$, the term becomes an integer for $N=1$. On the other hand, for $N>1$, it becomes perturbative because $\Pi_3(\mathbb{C}P^N)$ is trivial. The prefactor $\Theta$ 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 ($\mathbb{C}P^N$ Skyrme-Faddeev model). We discuss whether the statistical nature of solitons can be explained using their constituents, i.e., the quarks.

Self-dual solitons in a generalized Chern-Simons baby Skyrme model

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 topological charge of the Skyrme field is quantized whereas the total magnetic flux is not. Such as expected in a Chern-Simons model the total magnetic flux and the total electrical charge are proportional to each other. Thus, by considering the superpotential a well-behaved function in the whole target space we have shown the existence of three types of self-dual solutions: compacton solitons, soliton solutions whose tail decays following an exponential-law $e^{-\alpha r^{2}}$ ($\alpha>0$), and solitons having a power-law decay $r^{-\beta}$ ($\beta>0$). The profiles of the two last solitons can exhibit a compactonlike behavior. The self-dual equations have been solved numerically and we have depicted the soliton profiles, commenting on the main characteristics exhibited by them.

Gravitating gauged BPS baby Skyrmions

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 so-called superpotential. A complete classification of possible mass-radius curves is provided.As an example we consider the model with the pion like mass potential, for which, an approximated but analytical form of the superpotential is provided.

SUSY Chern–Simons and baby Skyrme models and their BPS structures

We construct the supersymmetric extensions of different nonlinear -models containing a Chern–Simons and/or Maxwell terms. A general classification of the BPS potentials, BPS-types and BPS equations in terms of the auxiliary fields is given. The supersymmetric extension of the Chern–Simons baby Skyrme model is also discussed and its possible BPS structure is analyzed.

Exactly solvable gravitating perfect fluid solitons in (2 + 1) dimensions

The Bogomolnyi-Prasad-Sommerfield (BPS) baby Skyrme model coupled to gravity is considered. We show that in an asymptotically flat space-time the model still possesses the BPS property, i.e., admits a BPS reduction to first order Bogomolnyi equations, which guarantees that the corresponding proper energy is a linear function of the topological charge. We also find the mass-radius relation as well as the maximal mass and radius. All these results are obtained in an analytical manner, which implies the complete solvability of this selfgravitating matter system.If a cosmological constant is added, then the BPS property is lost. In de Sitter (dS ) space-time both extremal and non-extremal solutions are found, where the former correspond to finite positive pressure solutions of the flat space-time model. For the asymptotic anti-de Sitter (AdS ) case, extremal solutions do not exist as there are no negative pressure BPS baby Skyrmions in flat space-time. Non-extremal solutions with AdS asymptotics do exist and may be constructed numerically. The impact of the negative cosmological constant on the mass-radius relation is studied. We also found two potentials for which exact multi-soliton solutions in the external AdS space can be obtained. Finally, we elaborate on the implications of these findings for certain three-dimensional models of holographic QCD.

Broken phase solitons in the baby Skyrme model

The baby Skyrme model is studied in the presence of an isospin chemical potential. The inclusion of the chemical potential leads to an asymmetric vacuum. As a result, the residual $SO(2)$ symmetry of the model is broken into the discrete ${\mathbb{Z}}_{2}$ group. Nontrivial topological field configurations are found to be allowed in the model. We obtained numerical solutions for the solitons on top of the new vacuum by minimizing the energy functional using a simulated annealing algorithm. All the calculations were performed by relaxing the rotational symmetry inherent to the hedgehog approximation. The corresponding soliton solutions exhibit discrete symmetries imposed by the new vacuum.

Gauged BPS baby Skyrmions with quantized magnetic flux

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 $2\pi$, which allows, in principle, to define this theory on a compact manifold without boundary, unlike all gauged baby Skyrme models considered so far.

Gauged baby Skyrme model with a Chern-Simons term

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 configuration remains unbroken. It is shown that coupling of the planar multi-Skyrmions to the electric and magnetic field strongly affects the pattern of interaction between the constituents. We analyze the dependency of the structure of the solutions, the energies, angular momenta, electric and magnetic fields of the configurations on the gauge coupling constant $g$, and the electric potential. It is found that, generically, the coupling to the Chern-Simons term strongly affects the usual pattern of interaction between the skyrmions, in particular the electric repulsion between the solitons may break the multisoliton configuration into partons. We show that as the gauge coupling becomes strong, both the magnetic flux and the electric charge of the solutions become quantized although they are not topological numbers.