The Interaction Energy of Well-Separated Skyrme Solitons

We prove that the asymptotic field of a Skyrme soliton of any degree has a non-trivial multipole expansion. It follows that every Skyrme soliton has a well-defined leading multipole moment. We derive an expression for the linear interaction energy of well-separated Skyrme solitons in terms of their leading multipole moments. This expression can always be made negative by suitable rotations of one of the Skyrme solitons in space and iso-space. We show that the linear interaction energy dominates for large separation if the orders of the Skyrme solitons multipole moments differ by at most two. In that case there are therefore always attractive forces between the Skyrme solitons.     

Solitonic fullerene structures in light atomic nuclei

The Skyrme model is a classical field theory which has topological soliton solutions. These solitons are candidates for describing nuclei, with an identification between the numbers of solitons and nucleons. We have computed numerically, using two different minimization algorithms, minimum energy configurations for up to 22 solitons. We find, remarkably, that the solutions for seven or more solitons have nucleon density isosurfaces in the form of polyhedra made of hexagons and pentagons. Precisely these structures arise, though at the much larger molecular scale, in the chemistry of carbon shells, where they are known as fullerenes.     

The skyrmion-skyrmion interaction

Using the Skyrme effective lagrangian, baryons emerge as topological solitons. This effective lagrangian is used in adiabatic calculations aimed at providing an essentially parameter-free model of the interaction between such solitons. The resulting skyrmion-skyrmion interaction can be understood as terms simulating the exchange of π-, ρ-, and ω-mesons between the solitons. It is readily transformed (by essentially projective techniques) into a low-energy nucleon-nucleon potential. Comparisons of this potential with the best available semiphenomenological nucleon-nucleon interactions are found to be successful at the 30% level. Similarities between the Skyrme model and the quark chiral-bag model are discussed.     

Gravitating Hopfions

We construct solutions of the 3 + 1 dimensional Faddeev–Skyrme model coupled to Einstein gravity. The solutions are static and asymptotically flat. They are characterized by a topological Hopf number. We investigate the dependence of the ADM masses of gravitating Hopfions on the gravitational coupling. When gravity is coupled to flat space solutions, a branch of gravitating Hopfion solutions arises and merges at a maximal value of the coupling constant with a second branch of solutions. This upper branch has no flat space limit. Instead, in the limit of a vanishing coupling constant, it connects to either the Bartnik–McKinnon or a generalized Bartnik–McKinnon solution. We further find that in the strong-coupling limit, there is no difference between the gravitating solitons of the Skyrme model and the Faddeev–Skyrme model.     

Nonlinear chiral models: Soliton solutions and spatio-temporal chaos

Nonlinear effective Lagrangian models with a chiral symmetry have been used to describe strong interactions at low energy, for a long time. The Skyrme model and the chiral quark-meson model are two such models, which have soliton solutions which can be identified with the baryons. We describe the various kinds of soliton states in these nonlinear models and discuss their physical significance and uses in this review. We also study these models from the view point of classical nonlinar dynamical systems. We consider fluctuations around theB=1 soliton solutions of these models (B, being the baryon number) and solve the spherically symmetric, time-dependent systems. Numerical studies indicate that the phase space around the Skyrme soliton solution exhibits spatio-temporal chaos. It is remarkable that topological solitons signifying stability/order and spatio-temporal chaos coexist in this model. In contrast with this, the soliton of the quark-meson model is stable even for large perturbations.     

Fermionic Quantization of Hopf Solitons

In this paper we show how to quantize Hopf solitons using the Finkelstein-Rubinstein approach. Hopf solitons can be quantized as fermions if their Hopf charge is odd. Symmetries of classical minimal energy configurations induce loops in configuration space which give rise to constraints on the wave function. These constraints depend on whether the given loop is contractible. Our method is to exploit the relationship between the configuration spaces of the Faddeev-Hopf and Skyrme models provided by the Hopf fibration. We then use recent results in the Skyrme model to determine whether loops are contractible. We discuss possible quantum ground states up to Hopf charge Q=7.     

Dynamical systems in the theory of solitons in the presence of nonlocal interactions

The behavior of solitons in models which take into account complex dispersion or nonlocal interaction of nonlinear waves is examined. A method is proposed to reduce this problem to one involving special trajectories (homoclinic and heteroclinic) of the dynamic system. This method involves replacing the nonlinear integrodifferential equation with the differential equations which link the original nonlinear field with the auxiliary linear fields. The interaction of fields in such a model is a local interaction. The number of introduced linear fields is determined by the Laplace transform of the integral operator kernel of the basic integrodifferential equation. The problem involving topological solitons for the nonlocal generalization of the Klein-Gordon equation is considered. Nonlocal interactions are found to lead to a number of singularities (unrestricted increase in the slope of the topological soliton front, break in the solutions, and other singularities).     

The indecomposable representation of SO0(2, 2) on the one-particle space of the massless field in 1 + 1 dimension

In the continuum O(3) sigma model in two spatial dimensions, there are topological solitons whose size can be stabilized by adding Skyrme and potential terms. This Letter describes a lattice version, namely a natural way of modifying the 2D Heisenberg model to achieve topological stability on the lattice.     

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.     

Effect of time-odd fields on odd-even mass differences of semi-magic nuclei

The effect of time-odd fields of Skyrme interaction on neutron odd-even mass differences is studied in the framework of axially deformed Skyrme Hartree-Fock(DSHF)+BCS model. To this end, we take into account both the time-even and time-odd fields to calculate the one-neutron and two-neutron separation energies and pairing gaps of semi-magic Ca, Ni, and Sn isotopic chains. In the calculations, a surface-type pairing interaction(IS pairing) and an isospin dependent contact pairing interaction(IS+IV pairing)are adopted on top of Skyrme interactions SLy4, SLy6 and Sk M*, respectively. We find that the time-odd fields have in general small effects on pairing gaps, but achieve better agreement with experimental data using SLy4 and Sly6 interactions, respectively.It is also shown that the calculations with IS+IV pairing reproduce the one-neutron separation energies of Sn isotopes better than those with the IS pairing interaction when the contributions of the time-odd fields are included.     

Entropic information for travelling solitons in Lorentz and CPT breaking systems

In this work we group four research topics apparently disconnected, namely solitons, Lorentz symmetry breaking, supersymmetry, and entropy. Following a recent work (Gleiser and Stamatopoulos, 2012), we show that it is possible to construct in the context of travelling wave solutions a configurational entropy measure in functional space, from the field configurations. Thus, we investigate the existence and properties of travelling solitons in Lorentz and CPT breaking scenarios for a class of models with two interacting scalar fields. Here, we obtain a complete set of exact solutions for the model studied which display both double and single-kink configurations. In fact, such models are very important in applications that include Bloch branes, Skyrmions, Yang–Mills, Q-balls, oscillons and various superstring-motivated theories. We find that the so-called Configurational Entropy (CE) for travelling solitons shows that the best value of parameter responsible to break the Lorentz symmetry is one where the energy density is distributed equally around the origin. In this way, the information-theoretical measure of travelling solitons in Lorentz symmetry violation scenarios opens a new window to probe situations where the parameters responsible for breaking the symmetries are arbitrary. In this case, the CE selects the best value of the parameter in the model.     

Quantized Lax Equations and Their Solutions

The O(3) sigma model and abelian Higgs model in two space dimensions admit topological (Bogomol'nyi) lower bounds on their energy. This paper proposes lattice versions of these systems which maintain the Bogomol'nyi bounds. One consequence is that instantons/solitons/vortices on the lattice then have a high degree of stability. Received: 29 February 1996 / Accepted: 5 August 1996     

Hopf solitons and Hopf Q-balls on S3

Field theories with a S²-valued unit vector field living on S³×ℝ space-time are investigated. The corresponding eikonal equation, which is known to provide an integrable sector for various sigma models in different spaces, is solved giving static as well as time-dependent multiply knotted configurations on S³ with arbitrary values of the Hopf index. Using these results, we then find a set of hopfions with topological charge Q_H=m², m∈Z, in the integrable subsector of the pure CP¹ model. In addition, we show that the CP¹ model with a potential term provides time-dependent solitons. In the case of the so-called “new baby Skyrme” potential we find, e.g., exact stationary hopfions, i.e., topological Q-balls. Our results further enable us to construct exact static and stationary Hopf solitons in the Faddeev–Niemi model with or without the new baby Skyrme potential. Generalizations for a large class of models are also discussed.     

Solitons on Tori and Soliton Crystals

Necessary conditions for a soliton on a torus \({M = \mathbb{R}^m/\Lambda}\) to be a soliton crystal, that is, a spatially periodic array of topological solitons in stable equilibrium, are derived. The stress tensor of the soliton must be L ² orthogonal to \({\mathbb{E}}\) , the space of parallel symmetric bilinear forms on TM, and, further, a certain symmetric bilinear form on \({\mathbb{E}}\) , called the hessian, must be positive. It is shown that, for baby Skyrme models, the first condition actually implies the second. It is also shown that, for any choice of period lattice Λ, there is a baby Skyrme model which supports a soliton crystal of periodicity Λ. For the three-dimensional Skyrme model, it is shown that any soliton solution on a cubic lattice which satisfies a virial constraint and is equivariant with respect to (a subgroup of) the lattice symmetries automatically satisfies both tests. This verifies, in particular, that the celebrated Skyrme crystal of Castillejo et  al., and Kugler and Shtrikman, passes both tests.     

The nucleon as a soliton

We review the Skyrme model which treats baryons as chiral solitons.     

The nontrivial states in one-dimensional nonlinear bichromatic superlattices

We study topological properties of one-dimensional nonlinear bichromatic superlattices and unveil the effect of nonlinearity on topological states. We find the existence of nontrivial edge solitions, which distribute on the boundaries of the lattice with their chemical potential located in the linear gap regime and are sensitive to the phase parameter of the superlattice potential. We further demonstrate that the topological property of the nonlinear Bloch bands can be characterized by topological Chern numbers defined in the extended two-dimensional parameter space. In addition, we discuss that the composition relations between the nolinear Bloch waves and gap solitions for the nonlinear superlattices. The stabilities of edge solitons are also studied.     

Possible heavy solitons in the strongly coupled Higgs sector

In a presumed dynamically broken, minimally coupled SU(2) model, a natural Higgs mass of order 1 TeV marks the onset of a strongly interacting Higgs sector probably rich in resonance structure and inaccessible to perturbation theory. In the spirit of the chiral dynamics approach to low-energy hadron physics, the heavy Higgs sector is here assumed to be well described up to one-loop effects by an SO(4) non-linear σ-model of the Skyrme type. Taken as an effective zeroth-order lagrangian, the latter is shown to admit two varieties of finite-energy, three-dimensional localized solitons which may exist in nature. They are given by the S³ → S³ Chern-Pontryagin maps and the S³ → S² twisted toroid Hopf maps, respectively. Upper and lower bounds on the masses of the hedgehog and twisted ring with kink-number one are found to lie in the few TeV range. By a topological theorem of Finkelstein et al., both types of solitons provide classical analogues of superheavy fermion states. The connection between these solitons with other extended objects predicted by Nambu and Huang, and their possible experimental signatures are sketched. Finally, the extension of our results to the more realistic SU(2) × U(1) Weinberg-Salam model is discussed.     

On topological aspects of the birth of the universe

The topological requirements are considered of extending matter fields in a three-dimensional universe to the whole compact four-manifold in quantum cosmology. In some cases, especially in the Skyrme model, the winding number of matter fields φ must be zero for a large class of manifolds.