Multisolitons in a two-dimensional Skyrme model

Abstract

The Skyrme model can be generalised to a situation where static fields are maps from one Riemannian manifold to another. Here we study a Skyrme model where physical space is twodimensional euclidean space and the target space is the two-sphere with its standard metric. The model has topological soliton solutions which are exponentially localised. We describe a superposition procedure for solitons in our model and derive an expression for the interaction potential of two solitons which only involves the solitons' asymptotic fields. If the solitons have topological degree 1 or 2 there are simple formulae for their interaction potentials which we use to prove the existence of solitons of higher degree. We explicitly compute the fields and energy distributions for solitons of degrees between one and six and discuss their geometrical shapes and binding energies. A Skyrme Model in Two DimensionsThe Skyrme model is a non-linear theory for SU(2) valued fields in 3 (spatial) dimensions which has soliton solutions. Each soliton has an associated integer topological charge or degree which Skyrme identified with the baryon number [1]. A soliton with topological charge one is called a Skyrmion; suitably quantised it is a model for a physical nucleon.Solitons of higher topological charge, called multisolitons, are classical models for higher nuclei.In this paper we study multisolitons in a two-dimensional version of the Skyrme model. The model was first considered in [2], but the motivation there is somewhat different from the approach taken here. For the present purpose it is important to be clear in what sense our model resembles Skyrme's model. In this section we will therefore briefly review a general framework for the Skyrme model due to Manton [3] and explain how our model fits into that framework. In [3] the usual Skyrme energy functional is interpreted in terms of elasticity theory and a static Skyrme field is a map π : S → Σ (1.1) from physical space S to the target space Σ. Both S and Σ are assumed to be Riemannian manifolds with metrics t and τ respectively. The energy of a configuration π is expressed in terms of its strain tensor D. To calculate the strain tensor one introduces coordinates p i on S and π α on Σ and orthonormal frame fields s m on S and σ µ on Σ (1 ≤ i, m ≤ dimS, 1 ≤ α, µ ≤ dimΣ). The Jacobian of the map π is, in orthonormal coordinates, J mµ = s i m ∂π α ∂p i σ µα (1.2) and the strain tensor D is defined via

Aloof baby Skyrmions

“…The numerical studies in [8,11], for the potential (2.2), selected the value m 2 = 0.1, so to aid comparison with previous studies we take this same value for m.…”

“…The long-range attraction can be understood analytically by calculating the asymptotic interaction energy between two well-separated solitons at the linear level. This calculation has been performed for the standard baby Skyrme model [8] and the result can be applied directly to the aloof model with µ ∈ (0, 1], because the last term in the potential (2.7) does not contribute to the interaction potential at the linear level. For two solitons with a separation R and a relative phase χ the asymptotic result for the large R interaction energy…”

“…The standard baby Skyrme model [8] is a two-dimensional analogue of the standard Skyrme model. Furthermore, a version of the baby Skyrme model with repulsive solitons has been known for a long time [9], and in fact this has the same potential term as in the recently discovered repulsive Skyrme model [6].…”

Some aspects of self-duality and generalised BPS theories

“…And, indeed, field theories based on versions of the above energy functional [7]- [15] are known to support both Bogomolnyi bounds and topological BPS solitons [10]- [15]. Among these BPS theories there are BPS submodels of some well-known and relevant non-linear field theories like, e.g., the Skyrme [16][17][18] and baby Skyrme [19][20][21][22][23] models or the abelian Higgs model, which makes them all the more interesting also from a phenomenological perspective (see, e.g., [24,25]). Given these results, one rather obvious question then arises -as to whether a version of the "deformation procedure" mentioned above generalises to the higher-dimensional theories, and whether this procedure can be used also in these cases to find soliton solutions of many different theories starting from a soliton of a certain "seed" theory.…”

Baby Skyrmions on the two-sphere

“…The baby Skyrme model, first introduced by [4], is a two dimensional version of the original Skyrme model, with R 2 as its domain. As its older brother, it is known to give rise, under certain settings, to structured multi-Skyrmion configurations [4,5,6,7,8].…”