Classically isospinning Skyrmion solutions

We investigate how isospin affects the geometrical shape and energy of classical soliton solutions of topological charges $B=1-4,8$ in the Skyrme model. The novel approach in our work is that we study classically isospinning Skyrmions beyond the rigid-body approximation; that is, we explicitly allow the soliton solutions to deform and to break the symmetries of the static configurations. Our fully three-dimensional relaxation calculations reveal that the symmetries of isospinning Skyrme solitons can differ significantly from the ones of the static configurations. In particular, isospinning Skyrmion solutions can break up into lower-charge Skyrmions, can deform into new solution types that do not exist at vanishing angular frequency $\omega$ or energy degeneracy can be removed. These types of deformations have been largely ignored in previous work on modeling nuclei by quantized Skyrmion solutions.


I. INTRODUCTION
In the SU(2) Skyrme model [1], atomic nuclei of nucleon (or baryon) number B can be identified with soliton solutions of conserved topological charge B which are known as Skyrmions. By numerically relaxing initial Skyrme configurations created with the rational map ansatz [2] and its multi-layer versions [3], minimal energy Skyrmion solutions have been constructed with various baryon numbers up to B = 108 [3,4]. In order to make contact with nuclear physics experiments it is necessary to semi-classically quantize these Skyrmion solutions. Traditionally this is done by treating Skyrmions as rigid bodies that can rotate in both space and isospace. This means that one quantizes just the rotational and isorotational zero modes of each Skyrmion solution of a given B and then determines the spin and isospin quantum numbers [5][6][7][8] which are compatible with the symmetries of the static, classical soliton. This approach neglects any deformations and symmetry changes due to centrifugal effects. This rigid body type approximation resulted in qualitative and encouraging quantitative agreement with experimental nuclear physics data: for even B the allowed quantum states for each Skyrmion often match the experimentally observed states of nuclei, and the energy spectra of a number of light nuclei have been reproduced to a quite good degree of accuracy [9,10]. However, nuclei of odd mass number are not well described by this approach.
Many spin and isospin states do not appear in the right order or are not even predicted by the rigid body quantization of the Skyrmion. One promising way to improve the agreement with experimental data is to allow Skyrmion solutions to deform when they spin and isospin. This can change the symmetries of the solutions and might result in different allowed quantum states [11].
Classically isospinning soliton solutions have been studied recently in the Faddeev-Skyrme [12,13] and the baby Skyrme model [14,15] beyond the rigid body approximation. In the case of the fully (3 + 1)-dimensional Skyrme model a systematic, full numerical investigation of isospinning soliton solutions beyond the rigid body approximation has not been performed yet. To our knowledge, numerical calculations of spinning Skyrmion configurations that take into account deformations originating from the kinematical terms have been carried out exclusively for baryon numbers B = 1 [16][17][18][19] and B = 2 [19]. It is worth mentioning that [16,17] consider classically spinning Skyrmions, whereas [18,19] first calculate the quantum Hamiltonian as a functional of the Skyrme fields and then minimize the Hamiltonian with respect to Skyrme fields for a given quantum state, see also [20] for a related approach. It has been found that allowing for axial deformations drastically reduces the rotational energies and that in order to fit the energies of spinning Skyrmions to the nucleon and delta masses, the pion mass parameter m π of the Skyrme model has to be chosen much larger than its experimental value [17][18][19]. However, all these studies impose spherical or axial symmetry on the spinning Skyrme solitons to simplify the numerical computations.
In this article, we perform numerical full field simulations of isospinning Skyrmion solutions with baryon numbers B = 1 − 4 and B = 8, without imposing any spatial symmetries. The main result of the present paper is that the symmetries and energies of Skyrmion solutions at a given angular frequency ω and for a given mass value µ can significantly differ from the ones of the static soliton solutions. Classically isospinning Skyrmion solutions can break the symmetries of the static solutions and even split apart into lower charge solutions as the angular frequency increases further. A detailed study of the extent to which the inclusion of these classical deformations can result in an improved solitonic description of nuclei is beyond the scope of the present article and will form part of a forthcoming publication. This paper is organized as follows. Section II recalls briefly the SU(2) Skyrme model and describes how we can construct isospinning Skyrmion solutions by solving energy minimization problems numerically. In Section III, we review how suitable initial conditions for Skyrme configurations of a given non-trivial topological charge B and of a specific symmetry G can be created for our numerical relaxation simulations. By relaxing the initial Skyrme fields generated with the methods described in Section III we find Skyrme soliton solutions with topological charges B = 1 − 4, 8 for different values of the pion mass in Section IV. Then, in Section V we investigate how the solitons' geometric shapes, energies, mean charge radii and critical frequencies are affected by the addition of classical isospin. Furthermore, we show in Section VI how the addition of classical isospin induces classical spin, consistent with the Finkelstein-Rubinstein constraints.
This gives a better understanding why some of the isospinnning Skyrmion solutions constructed here tend to stay together, while others prefer to break up into lower charge solutions. A brief summary and conclusion of our results is given in Section VII. For completeness, we explicitly list in Appendix A all diagonal and off-diagonal elements of the inertia tensors for the static B = 1 − 4, 8 Skyrmion solutions investigated in this article.

II. SPINNING AND ISOSPINNING SKYRMIONS
The Lagrangian density of the (3 + 1)-dimensional, massive Skyrme model [1] is defined in SU(2) notation by where the Skyrme field U(t, x) is an SU(2)-valued scalar, R α = (∂ α U) U † its associated righthanded chiral current and µ is a rescaled pion mass parameter. In Lagrangian (1) the energy and length units have been scaled away and are given by F π /4e and 2/eF π , respectively. Here e is a dimensionless parameter and F π is the pion decay constant. The dimensionless pion mass µ is proportional to the tree-level pion mass m π , explicitly µ = 2m π /eF π . Traditionally, the Skyrme parameters e and F π are calibrated so that the physical masses of the nucleon and delta resonance are reproduced when modelling them with a rigidly quantized Skyrmion solution, assuming the experimental value m π = 138 MeV for the pion mass [21,22]. This approach yields the standard values F π = 108 MeV, e = 4.84 and µ = 0.526. Expressed in terms of standard "Skyrme units" [21,22], the energy and length units in Lagrangian (1) are given by 5.58 MeV and 0.755 fm, respectively. Throughout this article we consider pion values µ between 0.5 and 2. These parameter choices are motivated by Refs. [9,10,17,23] where it has been argued that a larger rescaled pion mass parameter µ (in particular, µ > 0.526) yields improved results when applying the Skyrme model to nuclear physics.
Skyrmions arise as static solutions of minimal potential energy in the Skyrme model (1). They can be characterized by their conserved, integer-valued topological charge B which is given by the degree of the mapping U : R 3 → SU(2). To ensure fields have finite potential energy and a well-defined integer degree B the Skyrme field U(t, x) has to approach the vacuum configuration U(x) = 1 2 at spatial infinity for all t. Therefore, the domain can be formally compactified to a 3-sphere S 3 space and the Skyrme field U is then given by a mapping S 3 space → SU(2) ∼ S 3 iso labelled by the topological invariant B = π 3 (S 3 ) ∈ Z. The topological degree B of a static Skyrme soliton solution is explicitly given by where the topological charge density is defined by When modelling atomic nuclei by spinning and isospinning Skyrmion solutions, the topological charge (2) can be interpreted as the mass number or baryon number of the configuration.
Throughout this article, the energies M B of minimal-energy solutions in the Skyrme model will be given in units of 12π 2 , so that the Faddeev-Bogomolny lower energy bound for a charge B Skyrmion takes the form M B ≥ |B|. Recently a stronger lower topological energy bound has been derived in [24,25].
Note that the SU(2) field U can be associated to the scalar meson field σ and the pion isotriplet where τ denotes the triplet of standard Pauli matrices, and the unit vector constraint φ · φ = 1 has to be satisfied. (1) is manifestly invariant under translations in R 3 and rotations in space and isospace. Classically spinning and isospinning Skyrmion solutions are obtained within the collective coordinate approach [21,22]: the 6-dimensional space of zero modes -the space of energy-degenerate Skyrmion solutions which only differ in their orientations in space and isospace -is parametrised by collective coordinates which are then taken to be time-dependent. Here, we are mostly interested in static Skyrmion properties so that we can ignore translational degrees of freedom. Hence, the dynamical Ansatz is given by

The Skyrme Lagrangian
where the matrices A, A ∈ SU(2) are the collective coordinates describing isorotations and rotations around a static minimal energy solution U 0 (x). Substituting (5) in (1) yields the effective where M B is the classical Skyrmion mass given by and Ω k = −iTr τ kȦ A † and ω k = −iTr τ k A †Ȧ are the rotational and isorotational angular veloc-ities, respectively. The inertia tensors U i j , V i j , W i j are given explicitly by the integrals Recall that the moments of inertia (8) are given in units of 1/e 3 F π , that is the mass scale multiplied by the square of the length scale. The conjugate body-fixed spin and isospin angular momenta L and K are given by [9,26] In this article, we focus on the construction of isospinning Skyrmion solutions and consequently (6) simplifies to Uniformly isospinning soliton solutions in Skyrme models are obtained by solving one of the following equivalent variational problems [13] for φ: (1) Extremize the pseudo energy functional F ω (φ) = −L for fixed ω , In this paper, we will use a hybrid of approach (1) and (2). We are considering isospinning Skyrmions, in the sense that we seek stationary Skyrme configurations of the form (5) with A (t) constant, i.e. Ω = 0. We fix the isospin K to be constant. Then we consider the energy which implies Setting Ω = 0 in (9) imposes a constraint on L, namely Hence, in our approach, if W i j is non-zero, then the configuration will obtain classical spin. We will discuss this further in section VI.
We could now express the energy (12) as a function of K and then minimise the energy E.
However, it is more convenient to calculate ω using (13) and then minimize the pseudo energy The minus sign in equation (15) is a consequence of the identity where δ is a derivative and A an invertible matrix. Since we only fix K but not L, the value of ω is not conserved during the minimization. Hence for each step, we recalculate ω using (13).
As a numerical minimisation we use the approach described in [4] namely second order dynamics with a friction term. We rewrite the variational equations derived from (15) in terms of the following modified Newtonian flow equations where M is a symmetric matrix, and we included the time-dependence of the Skyrme Lagrangian (1) to evolve the equations of motion. The dissipation in (16) is added to speed up the relaxation process and the Lagrange multiplier λ imposes the unit vector constraint φ · φ = 1. We do not present the full field equations here since they are cumbersome and not particularly enlightening.
As initial field configurations, we take the static solutions at ω = 0 (see next section) and increase the angular momentum |K| stepwise. Relaxed solutions at lower |K| serve as initial conditions for higher |K|. In order to avoid precession effects in our numerical simulations the Skyrmion solutions have to be oriented in isospace so that their principal axes are aligned with the chosen isorotation axes. The initial configuration is then evolved according to the flow equations (16).
Kinetic energy is removed periodically by settingφ = 0 at all grid points every 50th timestep.
Most of our simulations are performed on regular, cubic grids of (200) 3

III. INITIAL CONDITIONS
We use the rational map ansatz [2] to create approximate Skyrme fields of non-trivial topological charge B and of given symmetry type G. Relaxing these initial field configurations with a fully three-dimensional numerical relaxation algorithm [4] we obtain static, minimum energy solutions of the Skyrme model with pion mass µ and baryon number B.
The main idea of the rational map ansatz is to approximate charge B Skyrme configurations φ which can be seen as maps from a 3-sphere in space to a 3-sphere in the target SU(2) by rational maps R : S 2 → S 2 of degree B. Within the rational map ansatz the angular dependence of the Skyrme field φ is described by a rational function where p and q are polynomials in the complex Riemann sphere coordinate z. The z-coordinate can be expressed via standard stereographic projection, z = tan (θ/2) e iφ , in terms of the conventional spherical polar coordinates θ and φ. The radial dependence is encoded in the radial profile function f (r) which has to satisfy f (0) = π and f (∞) = 0 to ensure a well-defined behavior at the origin and finite energy.
The rational function R(z) takes values on the target S 2 , and its value is associated via stereographic projection with the cartesian unit vector The rational map approximation for the Skyrme field is given in terms of the isoscalar σ and the pion isotriplet π of the non-linear sigma model notation by Substituting (19) in the Skyrme energy functional (7) results in an angular integral I which depends on the rational map R(z) and a radial part only dependent on the monotonic function f (r). To find low energy Skyrmion solutions of a given topological charge B and pion mass µ one minimizes I over all maps of algebraic degree B and then solves the Euler-Lagrange equation for f (r) with µ, B and the minimized I occuring as parameters. As starting point for our numerical relaxations we choose initial Skyrme fields generated with the rational maps R(z) given in [2,4,9,27]. Note that the rational maps given in these references are the optimal maps for Skyrmions with massless pions. However, since the angular integral I is independent of the mass parameter µ, the same rational maps are also the minimizing maps for non-zero µ and the main effect of the pion mass is to change the shape function f (r).

IV. STATIC SKYRMION SOLUTIONS WITH BARYON NUMBERS
In this section, we compute low energy static Skyrmion solutions with baryon numbers B = 1 − 4, 8 and with the rescaled pion mass set to µ = 1 by solving the full Skyrme field equations with a numerical three-dimensional relaxation algorithm [4]. Suitable initial Skyrme field configurations of given topological charge B were created using the methods described in the previous section.
For more detailed information on our relaxation procedure we refer the interested reader to the literature [4,12] and to Section II. We list in Table I  All the corresponding off-diagonal elements and inertia tensor elements for different mass values are given in tabular form in Appendix A. The baryon density isosurfaces we obtained can be found in Fig. 1. We can make a rough estimate of the numerical errors by computing the off-diagonal elements of the moment of inertia tensors [10], which should be exactly zero for all the Skyrmion solutions investigated here. We find that in each case the off-diagonal entries are small and are of the order of 10 −2 times the diagonal entries or less. In the following, all inertia tensor elements will be rounded to one decimal place.

A. B=1
The minimal-energy B = 1 Skyrmion solution is spherically-symmetric and substituting the rational map R(z) = z in (19) reproduces the standard hedgehog form where r = |x|, r = x/r and the radial profile function f (r) satisfies the boundary conditions f (0) = π and f (∞) = 0.
Here, we used the collocation method [28,29] to determine the profile function f (r) which minimizes M 1 (21). The rational map approximations for the higher charge Skyrmion solutions will be generated with the profile function f (r) calculated in the charge-1 sector. Note that for an O(3)-symmetric Skyrme configuration (20) the inertia tensors (8) take the simple form in agreement with the moment of inertia Λ = 47.623 calculated within the hedgehog approximation in [8].
For comparison, the energy value M 1 = 1.415 calculated in our full 3D simulation differs by 0.07% compared to the greater accuracy hedgehog ansatz (20). Within our numerical accuracy the inertia tensors (8) computed with our 3D relaxation code are all found to be proportional to the unit matrix with Λ = 47.5. This is in reasonable agreement with the value calculated within the hedgehog ansatz (22) and with the moments of inertia computed in [8] using a three-dimensional non-linear conjugate gradient method.
Note that we cannot confirm the energy value M 1 = 1.465 calculated in the recent article [3] for a B = 1 Skyrmion of mass µ = 1. We double-checked our results using two very diffent numerical approaches (a collocation method [28,29] and a one-dimensional gradient flow method) to solve numerically the Skyrme field equation for the hedgehog ansatz (20). In both cases we obtain an energy value M 1 = 1.416 [44].

B. B=2
The B = 2 Skyrmion has toroidal symmetry D ∞h , and it can be approximated by choosing the rational map R(z) = z 2 in (19). Relaxing this rational map generates an initial Skyrme field configuration. We verify that all inertia tensors are diagonal, with U 11 = U 22 = 97.0, V 11 = V 22 = 153.8 and W 11 = W 22 = 0. Our numerically calculated charge-2 configuration satisfies the relation [18,26] -a manifestation of the axial symmetry. The soliton's energy is M 2 = 2.720, which is reasonably close to the energy value M 2 = 2.77 given in [3]. Note that the moments of inertia U 33 = 68.9, V 33 = 275.4 and W 33 = 137.7 are found to be in close agreement with the corresponding values U 33 = 68.67, V 33 = 274.59 and W 33 = 137.31 stated in [8].
Finally, we can check the results of our fully three-dimensional numerical relaxation with those obtained by minimizing the two-dimensional, total energy functional of an axially symmetric Skyrme configuration. An axially-symmetric ansatz [30] is given by where ψ(ρ, z) = (ψ 1 , ψ 2 , ψ 3 ) is a unit vector that is dependent on the cylindrical coordinates ρ and z. Here, the non-zero winding number n ∈ Z counts the windings of the Skyrme fields in the (x 1 , x 2 )-plane. Substituting (23) in (7) results in the classical soliton mass and the baryon number B of an axially-symmetric configuration ψ is given by substituting (23) in To ensure a configuration of finite energy M B the unit vector ψ has to satisfy the boundary condition ψ → (0, 0, 1) as ρ 2 + z 2 → ∞ together with ψ 1 = 0 and ∂ ρ ψ 2 = ∂ ρ ψ 3 = 0 at ρ = 0. A suitable start configuration with baryon number B = n is given in [30] by where r = ρ 2 + z 2 and f (r) denotes, as usual, a monotonically decreasing profile function sat- Our 2D gradient flow simulation gives U 11 = U 22 = 103.1 and U 33 = 71.5 for an axially symmetric charge 2 Skyrmion solution of mass µ = 1.

C. B=3
The minimal-energy B = 3 Skyrmion has T d symmetry and can be created with the rational Relaxing the tetrahedrally-symmetric B = 3 Skyrme configuration (28), we verify that the inertia We find for the total energy M 3 = 3.969 which is slightly lower than the numerical value M 3 = 4.02 given in [3].

D. B=4
The minimal-energy Skyrmion solution with B = 4 has octahedral symmetry O h and can be approximated by the rational map [9] The inertia tensors U i j , V i j and W i j for the cubically-symmetric, numerically relaxed charge-4 configuration (29) are determined to be diagonal, satisfying U 11 = U 22 = 148.2 and V i j = vδ i j with v = 667.6 and with the cross-term W i j vanishing within the limits of our numerical accuracy. We obtain for the total energy M 4 = 5.177 which agrees with the value M 4 = 5.18 stated in [3].

E. B=8
For baryon number B = 8, we calculate two very different Skyrmion solutions -one with D 4h and the other with D 6d symmetry. We confirm the results in [27]. To generate suitable initial conditions for our numerical relaxation simulations we approximate an initial field configuration with D 6d symmetry by the rational map [9] where the free parameter is set to a = 0.14. The relaxed Skyrme field resembles a hollow polyhedron, namely a ring of twelve pentagons with a hexagon at the top and at the bottom (see baryon density isosurface plot in Fig. 1). The inertia tensors of the relaxed D 6d -symmetric Skyrme con- Recall that in the massive pion model there exists not only a D 6d symmetric, polyhedral Skyrmion solution but also a bound configuration of two B = 4 cubes [27]. This double cube Skyrmion solution is obtained by relaxing a perturbed, D 4h -symmetric starting configuration approximated with the rational map [9] shape is independent of the angular frequency ω. Analogous calculations of isospinning solitons in the Skyrme-Faddeev model [31,32] that go beyond the rigid body type approximation have been performed in [12,13].
We construct stationary isospinning soliton solutions by numerically solving the energy minimization problem formulated in Section II. Note that spinning Skyrmions with zero pion mass radiate away their energy, see [41] for a detailed discussion. For pion mass µ > 0 stationary solutions exist up to an angular frequency ω crit = µ. At ω crit the values of the energy and angular momentum are finite, and therefore, the corresponding angular momenta K crit (and L crit ) is also finite, see [17]. The situation is different for baby Skyrmions where energy and moment of inertia diverge at ω crit , [14,15,42] for µ < 1. From the point of view of numerics, this behavior is challenging. For ω < ω crit the problem is well-posed, whereas for ω > ω crit the solutions become oscillatory which is difficult to detect in a finite box. Physically, this corresponds to pion radiation, and the fact that stationary solutions do not exist. Numerically, we can find energy minimizers for ω > ω crit but this is an artefact of the finite box approximation. By convention, throughout this paper, when displaying inertia tensors and energies as a function of isospin, we will cut our graphs at the critical isospin value K crit .
Recall that the different orientations of Skyrme solitons in isospace can be visualized using Manton & Sutcliffe's field colouring scheme described in detail in [33]. We illustrate the colouring for a B = 1 Skyrmion solution in Fig. 2: The points where the normalised pion isotriplet π takes the values π 1 = π 2 = 0 and π 3 = +1 are shown in white and those where π 1 = π 2 = 0 and Here, we are purely interested in classically isospinning Skyrmion solutions, hence we will use the classical energy scale F π /4e to calculate the isosrotational energy contributions in Section VII. To estimate the value of [34] for different Skyrme  In this section, we present our numerical results on isospinning soliton solutions of topological charges B = 1 − 4. Our numerical simulations are performed for a range of mass values µ. We verify that for µ = 1 the E tot (ω) graph (calculated with our modified 3D Newtonian flow) shows the same behavior as predicted by an axially-symmetric spinning, charge-1 Skyrme configuration (23). As discussed in section II, in our 3D simulations the soliton's energy is given by E tot (ω) = M 1 + ω 2 U 33 /2 and the energy curve terminates at ω crit = µ = 1. Stable, internally spinnning solutions cease to exist beyond this critical value, but energy and moments of inertia remain finite at ω crit = 1.
For comparison, axially-symmetric deforming, isospinning configurations are constructed by minimizing the total energy E tot = M 1 + K 2 / (2U 33 ) for fixed isospin K with a 2D gradient flow algorithm, where the classical soliton mass M 1 is given by (24) and the relevant moment of inertia U 33 can be found in (27a). Both energy curves agree within the limits of our numerical accuracy (see Fig. 3 (a)). In Fig. 3 (b), we display the mass-isospin relationship E tot (K) for an isospinning B = 1 Skyrmion solution calculated without imposing any symmetry constraints on the configuration. Imposing axial symmetry, we reproduce the same mass-isospin relation. As shown in Fig. 3 (b), the rigid body formula proves to be a good approximation for small isospins (K ≤ 3 × 4π), whereas for higher isospin values E tot (K) deviates from the quadratic behavior. At the critical angular frequency ω crit = 1 (K crit = 6.5 × 4π) the rigid body approximation gives an approximate 7% larger energy value for the isospinning soliton solution. The associated isospin inertia tensor U i j (8a) is diagonal and its diagonal elements as a function of isospin K are shown in Fig. 3 ( where the classical soliton mass M 1 is given by (21) and the associated moment of inertia Λ can be found in (22). The underlying two point boundary value problem -f (0) = π and f (∞) = 0 -is solved with the collocation method [28] for fixed angular frequency ω.
Taking the asymptotic limit (r → ∞, f → 0) of the nonlinear equation for f (r) reveals that a stable, isospinning soliton solution can only exist if the isorotation frequency satisfies ω ≤ √ 3/2 µ.
Stationary solutions cease to exist at ω crit , but energy and moments of inertia remain finite at the This is displayed in Fig. 7(a) for various pion masses. Note that for a spherically-symmetric, hedgehog Skyrme configuration a rotation in physical space is equivalent to a rotation in isospace.
Thus, we expect the same energy curves for a B = 1 Skyrmion classically rotating about the z axis.

B = 2
For the toroidal B = 2 Skyrmion solution we choose two different isospin axes [33]. One is the axis of symmetry -the z axis -with the torus spinning in the xy plane and the other is an axis orthogonal [26,34,35] to it -the y axis -so that the symmetry axis rotates in the xz plane. It was argued in [33] that these two spatial orientations are the relevant ones for describing the rotational states of the deuteron.
With the z axis chosen as our isorotation axis and the mass parameter µ set to 1, we obtain the energy and moment of inertia curves presented in Fig. 8. The corresponding energy density isosurface plots are displayed in Fig. 9. We verify in Fig. 8 (a),(b) that the total energy E tot as a function of ω and K follows in good approximation the behavior expected from an axiallysymmetric charge-2 Skyrme soliton for angular frequencies ω ω crit (K K crit ) with a small deviation near ω = ω crit . An isospinning, axially-symmetric Skyrme configuration can be computed by minimizing the two-dimensional energy functional E tot = M 2 + K 2 / (2U 33 ) for fixed isospin K, where the classical soliton mass M 2 is given by (24) with winding number n = 2 and the relevant moment of inertia U 33 can be found using (27a). Again, we observe that stable isospinning soliton solutions cease to exist at ω crit = µ, but energy and moments of inertia remain finite at ω crit . The rigid-body approximation is shown in Fig. 8  rigid-body formula. Finally we show in Fig. 8 [18,26]. As seen in the inset in Fig. 8 (d) U 11 U 22 close to K crit which is consistent with the energy density contour plots in Fig. 9 where a slight symmetry breaking occurs for K = 7 × 4π.
When repeating our relaxation calculations for higher masses µ, we find (analogous to observations in the conventional baby Skyrme model [14,15]) that the isospinning charge-2 Skyrmion solution breaks axial symmetry at some critical value ω SB and starts to split into its charge-1 constituents that move apart from each other and are orientated in the attractive channel. In Fig. 9 we display baryon density isosurface and contour plots for a range of masses µ. Note that the breakup of isospinning B = 2 Skyrmion solutions into their charge-1 constituents is not as pronounced as the one observed for isospinning solutions in the conventional baby Skyrme model. For pion mass values µ = 1.5 and µ = 2 we find that the breaking of axial symmetry occurs at K SB = 5.7 × 4π (ω SB = 1.1) and K SB = 5.5×4π (ω SB = 1.2), respectively (see the isospin diagonal elements shown as a function of K in Fig. 10).
In Fig. 7 the deviations from the rigid-body approximation are plotted as a function of the angular momentum K for various mass values µ for charge-1 and charge-2 solutions isospinning around their z axis. As the mass µ (or the topological charge B) increases, the rigid body approximation provides more accurate results for the isospinning solutions of the model.
As a further check of our numerics, we calculate numerically the total energy E tot = M 2 + K 2 /(2U 33 ) within the product ansatz approximation [1]. In analogy to [36], we generate a B = 2 Skyrme configuration by superposing two B = 1 solitons centered around x 1 and x 2 : where U H (x) is the hedgehog solution (20) and A(α) = exp (iτ · α/2) with α parametrizing the relative isospin orientation. The two individual Skyrmions are initially arranged so that the attraction is maximal. The Skyrmion-Skyrmion interaction [36] is maximal when one Skyrmion product ansatz approximation is found to provide more accurate results as the pion mass parameter µ increases.
In the following we choose the y axis as our isorotation axes and set µ = 1. In Fig. 12 (a),(c) we display the total energy and the moment of inertia U 22 as a function of ω up to ω crit = 1.
The rigid-body approximation proves to be a valid simplification for isospin values K ≤ 8 × 4π (ω = 0.8). However, at the critical frequency ω crit = 1 (K crit = 11.5 × 4π) the energy values turn out to be ≈ 4% lower than that predicted by the rigid-body approximation. The baryon density isosurfaces of isospinning charge-2 Skyrmions for a range of mass values are displayed in Fig. 13.
Note that we do not observe a breakup into charge-1 constituents for this mass range, instead a "square-like" configuration is formed. The associated moments of inertia as a function of isospin  Fig. 15 (a)). Indeed, we verified that when perturbed slightly, charge two Skyrmion solutions isospinning about (0, 0, 1) can evolve into the lower energy D 4 symmetric solutions with isospin axis K = (0, 1, 0).

B = 3
For B = 3, we isorotate the minimal-energy tetrahedron about its K = (0, 0, 1) axis. Performing a damped field evolution for the mass value µ = 1 we obtain the energy and inertia dependencies on ω and K displayed in Fig. 16. The corresponding baryon density isosurfaces for a range of mass values µ can be found in Fig. 17.
As µ increases, the soliton's deformations due to centrifugal effects become more apparent. For µ = 2 the isospinning charge-3 Skyrmion solution breaks into a toroidal B = 2 Skyrmion solution and a B = 1 Skyrmion before reaching its upper frequency limit ω crit = µ. Note that with increasing angular velocity ω the isospinning B = 3 Skyrmion seems to pass through a distorted "pretzel " configuration -a state that has previously been found to be meta-stable [37,38] for vanishing isospin K. For mass value µ = 1.5 the tetrahedral charge-3 Skyrmion does not break into lower-charge Skyrmions when increasing the angular frequency ω. As ω increases, the charge-3 tetrahedron slowly deforms into the "pretzel "-like configuration which appears to be of lower energy than an isospinning solution with tetrahedral symmetry. Even for µ = 1 the tetrahedral symmetry is broken as ω increases (see inertia-spin relationship shown in Fig. 16 (d) and baryon density isosurfaces in Fig. 17). In fact, the isospinning B = 3 Skyrme soliton (with µ = 1) starts to violate tetrahedral symmetry for angular frequencies ω > ω SB = 0.06 (K SB = 1.1 × 4π). The breaking of tetrahedral symmetry occurs at K SB = 0.8 × 4π (ω SB = 0.1) and K SB = 0.5 × 4π (ω SB = 0.12) for B = 3 Skyrmions with mass µ = 1.5 and µ = 2, respectively. Furthermore, note that the U 11 and U 22 curves shown in Fig. 16 (d) and Fig. 18 lie on top of each other.   As shown in Fig. 24, for pion mass values up to 2 the energy values predicted by the rigid-body formula can be up to 15% higher than those obtained without imposing any spatial symmetries on the isospinning Skyrme configurations.
Similar to the B = 2 case, we find that the Skyrmion configuration of lowest energy for given imate D 4h symmetry we find that when isospinning about their (0, 1, 0) axis there exists a breakup frequency at which the isospinning solution splits into four B = 2 tori (illustrated by the baryon density isosurfaces presented in Fig. 26). Similarly, we observe that the D 6d configuration when isospinning about its (0, 1, 0) axis breaks apart into four B = 2 tori, but aligned in a different way (see baryon density isosurfaces presented in Fig. 28). When choosing K = (0, 0, 1) as isorotation axis, D 4h and D 6d symmetric Skyrmion solutions -are found to break up into two well-separated B = 4 clusters (see Fig. 26 and Fig. 28, respectively). Finally, for B = 8 Skyrmions with approximate D 4h symmetry another possible isorotation axis choice is given by K = (1, 0, 0). Again, as the isospin K increases, we observe a breakup into two B = 4 cubic Skyrme solutions (see Fig. 26).
The associated diagonal elements of the isospin inertia tensor U i j can be found as a function of K in Fig. 27 and Fig. 29, respectively. In Fig. 25 we compare the energy curves for D 4h and D 6d symmetric Skyrmions isospinning about K = (0, 1, 0) and K = (0, 0, 1), respectively. For K = (0, 1, 0) D 4h and D 6d symmetric solutions remain energy degenerate within the limits of our numerical accuracy as the angular velocity increases. However, when choosing the isorotation axis K = (0, 0, 1) the D 4h solution appears to be of lower energy for fixed, non-zero isospin value K.
For D 6d symmetric B = 8 solutions, we cannot decide within the limits of our numerical accuracy which isospin axis results in the lowest energy configuration for fixed isospin (compare energy curves in Fig. 15(c)). For D 4h symmetric B = 8 solutions, the two cube soliton solution with K = (0, 0, 1) has the lowest total energy (see Fig. 15(d))). If slightly perturbed, the solution composed of four aligned B = 2 tori with K = (0, 1, 0) can evolve into two cubes isospinning

C. Critical Angular Frequencies
It has been observed [12][13][14][15] that isospinning soliton solutions in models of the Skyrme family suffer from two different types of instabilities: One is related to the deformed metric in the pseudoenergy functional F ω (φ) becoming singular at ω 1 = 1 [13] and the other to the Hamiltonian no longer being bounded from below at ω 2 . The first critical frequency ω 1 is independent of the concrete potential choice, whereas the second critical frequency ω 2 crucially depends on the particular choice of the potential term.
Note that we do not observe the same pattern of critical frequencies in the full Skyrme model When choosing the double vacuum potential V = 2µ 2 (1 − σ 2 ) in (7), we obtain for the isospinning B = 1 Skyrmion solution the energy and isospin curves shown in Fig. 31. Again, we observe that Skyrmions cannot spin with angular frequencies ω > √ 2µ -the meson mass of the modelsince they become unstable to the emission of radiation. for a range of mass values µ. We display the total energy E tot and isospin K as function of angular frequency ω and isospin K at µ = 0.5, 1, 1.5, 2. The isorotation axis is chosen to be K = (0, 0, 1). We perform the 3D relaxation calculations on a (100) 3 grid with grid spacing ∆x = 0.2.

D. Mean Charge Radii
In Fig. 32 Similar to our observations of isospinning soliton solutions in the standard baby Skyrme model [14], we note that the changes in the Skyrmion's shape are reflected by the changes in slopes of the mean charge radius curves in Fig. 32. For example for B = 1 and mass value µ = 1 we find that the radius < r 2 > 1/2 grows with the isospin and that there exists an inflection point. For B = 2 and B = 3 we observe that the inflection point occurs at higher isospin values. These inflection points are in reasonable agreement with the maximal isospin values stated in Section V A up to which the rigid-body approximation is a good simplification. Furthermore, as µ increases the inflection point of the mean charge radius < r 2 > 1/2 occurs at increasingly higher isospin values. This confirms that the rigid body approximation becomes more accurate as B or µ increases.

VI. SPIN INDUCED FROM ISOSPIN
If W i j is non-zero, then Skyrme configurations will obtain classical spin when isospin is added.
However, if we choose K = (0, 0, 1) as isospin axis, the B = 2 Skyrmion gains spin L = −W 33 K as K increases, see Fig. 33 (b). For µ = 1 the axial symmetry remains unbroken and hence L depends linearly on K. We confirm that the numerically calculated slope −1.97 agrees well with the expected one L/K = −W 33 /U 33 = −2 as W 33 = 2U 33 for an axially symmetric charge two configuration. For larger mass values µ isospinning around the K = (0, 0, 1) leads to the break up into two B = 1 Skyrmions orientated in the attractive channel. The axial symmetry is broken at K SB , and the B = 2 Skyrmion solution starts to split into two B = 1 Skyrmions. As the isospin K increases further L is approximately −K. In this regime the isospinning configuration is well described by two separated, axially symmetric deformed B = 1 Skyrmions. This is consistent with head-on scattering of two spinning B = 1 Skyrmions in the attractive channel [39] where the configuration of closest approach is not the torus but a configuration of two separated Skyrmions.
The attractive channel has also been discussed in [34] when quantising the Deuteron. This degree of freedom was essential for comparing the spatial probability distribution of the deuteron with experimental values [26,33,35].
In summary, for B = 2 we observe that Skyrme configurations with non-zero W and hence non-zero spin L show centrifugal effects and separate out whereas states with W = 0 tend to stay more compact. Isospinning the charge 2 Skyrmion about its (0, 0, 1)-axis results in the breakup into two well separated charge one Skyrmions, whereas isospinning about (0, 1, 0) yields compact D 4 -symmetric B = 2 configurations of lower energy.
For B = 3 we display in Fig. 33 (c) the L(K) graphs for pion masses µ = 1, 1.5, 2. We observe that as long as the tetrahedral symmetry remains unbroken the spin L inreases linearly with K.
Breaking of the tetrahedral symmetry results in a lower W and hence a lower increase in L for higher K values.
For B = 4 there are two different isospin axes: K = (0, 0, 1) and K = (0, 1, 0). For K = (0, 0, 1) we find that the octahedral symmetry remains unbroken (see baryon density isosurfaces shown in Fig. 22). The mixed inertia tensor W and hence L vanish for all classically allowed isospin values K. When isospinning the cubic B = 4 Skyrmion solution about K = (0, 1, 0) we observe breakup into two D 4 -symmetric charge 2 Skyrmions (see Fig. 22). Again, the mixed inertia tensor W and hence spin L is found to vanish for all classically allowed K values.
For B = 8 it is interesting to observe that all configurations are breaking up into constituents, either into two B = 4 parts or into four B = 2 parts. Note that physically B = 8 may describe beryllium 8 Be which is unstable to splitting up into two α particles, see e.g. [40]. For D 6dsymmetric Skyrmion solutions we find that when isospinning about K = (0, 0, 1) there exist a critical isospin value at which the soliton solution splits up into two B = 4 cubes. This breakup process is reflected in the L(K) graph shown in Fig. 33 (d). For K ≤ 17 × 4π (L ≥ −6.6 × 4π) the spin |L| grows linearly with K and the isospinning Skyrme configuration preserves its dihedral symmetry. For higher isospin values K the dihedral symmetry is broken and the isospinning solution starts to break apart into two B = 4 cubes of zero total spin L.  (iv) Spin generated from Isospin: If W i j is non-zero, then Skyrme configurations will obtain classical spin when isospin is added. For example for B = 1 this gives states with spin and isospin opposite, as required by the Finkelstein-Rubinstein constraints [6]. For B = 2 we observe that Skyrme configurations with non-zero mixed inertia tensor W i j show centrifugal effects and separate out whereas states with W i j = 0 tend to stay more compact. Isospinning around K = (0, 0, 1) leads to the breakup into two B = 1 Skyrmions orientated in the attractive channel and with L crit given by approximately −K crit . Isospinning around K = (0, 1, 0) leads to a novel configuration with D 4 symmetry of vanishing total spin for all classically allowed isospin values K.
Finally, we also investigated numerically the critical behavior of isospinning Skyrmion solutions. Contrary to previous numerical studies on isospinning soliton configurations in the baby Skyrme [14,15] and Skyrme-Faddeev model [12,13] we found numerically only one type of instability being present. Skyrmion solutions can isospin up to a critical angular frequency ω crit that is given by the meson mass of the model. For ω > ω crit Skyrmion solutions become unstable to pion radiation. Recall that at ω crit the values of the energy and angular momenta are finite, and therefore, the corresponding angular momenta K crit (and L crit ) are also finite, see [17]. The situation is different for baby Skyrmions where energy and moment of inertia diverge at ω crit [42].
In Table II we give a classical bound on how fast Skyrme configurations of baryon number B = 1 − 4, 8 for given pion mass µ and isorotation axis K are allowed to isospin. For simplicity, we took the following approach: We fixed the Skyrme parameters e and F π as in Refs. [21,22] and calculated the induced pion mass via m π = 2µ/eF π . The critical isospin values K crit up to which stationary solutions exist are given in units, where = 46.8 for the standard parameter set F π = 108 MeV and e = 4.84. The associated energy values are obtained by multiplying our numerical energy values by the energy scale F π /4e = 5.58 MeV. Note that our findings are in qualitative agreement with earlier literature [17]. In particular, we find that the nucleon (E N tot = 939 MeV) and delta (E ∆ tot = 1232 MeV) cannot be reproduced if the pion mass mass value is set to its experimental value m π = 138 MeV (µ = 0.526) and the standard values for the Skyrme parameters e and F π are used. However, if the pion mass is taken to be larger than its experimental value, then we can reproduce nucleon and delta masses. Note that the calibration of the Skyrme model is still an open problem [17,43].
The types of deformations observed in this article have been largely ignored in previous work [9,10] on modelling nuclei by quantized Skyrmion solutions and are exactly the ones we would like to take into account when quantizing the Skyrme model. Spin and isospin quantum numbers of ground states and excited states have so far almost exclusively been calculated within the rigid body approach [5][6][7][8], that is by neglecting any deformations and symmetry changes due to centrifugal effects. Our numerical full field simulations clearly demonstrate the limitations of this simplification. The symmetries of isospinning soliton solutions can change drastically and the solitons are found to be of substantially lower energies than predicted by the rigid body approach.
This work offers interesting new insights into the classical behavior of Skyrmions and gives an indication of which effects have to be taken into account when quantising Skyrmions.
1. µ = 1 TABLE III: We list the off-diagonal elements of the isospin (U i j ), spin (V i j ) and mixed (W i j ) inertia tensors, the mean charge radii < r 2 > 1/2 and the symmetries G of the Skyrme solitons. The mass parameter µ is chosen to be 1.