Skyrme model

The Skyrme model is a nonlinear sigma model for an $SU(2)$-valued field supplemented by a four-derivative term that evades Derrick collapse. Its topological degree is baryon number, and finite-energy solitons are Skyrmions.

For a spatially dilated Skyrme model field $U_L(x)=U(x/L)$, the quadratic and quartic derivative energies scale as shown. A static stationary field must satisfy $(d-2)E_2+(d-4)E_4=0$. In three spatial dimensions this permits $E_2=E_4$ and a positive second variation along scale. In two dimensions it requires $E_4=0$, and in four or more it forces a constant finite-energy smooth field. In one dimension the commutator term vanishes identically. Derrick scaling tests a necessary variational condition, not existence or stability under all deformations.

For a smooth finite-energy static planar Skyrme model field, Derrick scaling gives zero quartic energy, hence $[R_1,R_2]=0$ pointwise. The quartic first variation then vanishes, leaving the sigma-model equation ${\partial }_i{R}_i=0$. The right-current Maurer-Cartan equation gives ${\partial }_1{R}_2-{\partial }_2{R}_1=[R_1,R_2]=0$. Each current component is therefore harmonic on the plane and square-integrable. The mean-value estimate on expanding disks makes each component zero, so $U$ is constant. This excludes smooth finite-energy planar static solutions of this potential-free SU(2) model, without misapplying the same conclusion to a pure curved-target sigma model.

The particular quartic-derivative interaction in the Skyrme model, with $L_{\mu }=U^{†}{\partial }_{\mu }U$. Its static energy is positive and scales inversely with soliton size, opposing collapse under Derrick scaling. It is one choice of four-derivative mesonic interaction, not the whole general fourth-order chiral perturbation theory Lagrangian.

The massless derivative Skyrme model has chiral symmetry $U\mapsto LU{R}^{†}$. For the fixed finite-energy boundary condition $U(\infty )=1$, a transformation preserves the same vacuum exactly when $L{R}^{†}=1$, or $L=R$. This leaves conjugation $U\mapsto AU{A}^{†}$, the isospin action. Since $A$ and $-A$ act identically, its effective group is $SO(3)$. Together with spatial translations and rotations it generates the localized zero-mode orbit of a Skyrmion. Independent axial rotations change the massless vacuum and are not additional normalizable rigid orientation modes in that fixed-vacuum sector. A usual positive pion-mass term preserves only conjugation even before the boundary condition is imposed.

A Skyrmion is a finite-energy topological soliton of the Skyrme model. Its baryon number is the degree of the compactified spatial map $S^3\to SU(2)\simeq S^3$.

A charge-$A$ multi-Skyrmion provides an intrinsic field configuration for an atomic nucleus with mass number $A$. Its spatial and isospin collective coordinates must be quantized, with Finkelstein-Rubinstein constraints, before assigning physical nuclear states. A classical intrinsic shape is distinct from rotationally invariant observables in a spin-zero state.

For a centered static Skyrmion $U_0$, the effective rotation-isorotation group is $G=SO(3{)}_{\mathrm{space}}\times SO(3{)}_{\mathrm{iso}}$. Its stabilizer subgroup $H$ consists of pairs obeying $A{U}_0(R^{-1}x)A^{†}=U_0(x)$. Thus distinct rigid orientations form the homogeneous space $G/H$, by the orbit-stabilizer theorem. The unit hedgehog has diagonal $SO(3)$ stabilizer and three orientation coordinates; the toroidal two-Skyrmion has a one-dimensional continuous stabilizer and five; the tetrahedral and cubic cases have discrete stabilizers and six. Adding three translations gives zero-mode orbit dimensions 6, 8, 9, 9. These are symmetry-generated low-energy coordinates, not a claim that non-Bogomolny multi-Skyrmion solutions possess an exact flat moduli space of arbitrary separations.

The familiar low-energy $B=4$ Skyrmion has a cubic baryon-density shell with six face-hole directions and full density symmetry $O_h$, the symmetry group of a cube. The proper subgroup is $O\cong S_4$, the rotational symmetry group of a cube; its field invariance includes compensating isorotations. A useful rational map approximation for Skyrmions has $R(z)=(z^4+2i\sqrt{3}{z}^2+1)/(z^4-2i\sqrt{3}{z}^2+1)$. The relation $R(iz)=1/R(z)$ pairs a spatial quarter-turn with a target half-turn. Its branch directions are $z=0,\infty ,\pm 1,\pm i$, corresponding to the six coordinate-axis directions. The map gives an approximate field, not an exact analytic energy-minimizing solution.

The familiar low-energy $B=3$ Skyrmion has tetrahedral baryon-density symmetry $T_d$. Proper rotations form the tetrahedral symmetry group $T\cong A_4$, acting on the field with compensating isorotations. An angular approximation is $R(z)=(i\sqrt{3}{z}^2-1)/(z^3-i\sqrt{3}z)$. Its Wronskian of a rational map is proportional to $z^4+2i\sqrt{3}{z}^2+1$. The four branch directions are the face-hole directions of a tetrahedral shell. A rational map and a variational radial profile approximate the actual field, and baryon-density symmetry must be distinguished from invariance of the pion field under an unaccompanied spatial rotation.

The standard low-energy $B=2$ Skyrmion in the ordinary Skyrme model has a toroidal baryon-density shape, with axial density symmetry $D_{\infty h}$. Its proper combined field symmetries have an $O(2)$-type stabilizer. The angular ansatz $R(z)=z^2$ in the rational map approximation for Skyrmions explains the axial relation $R(e^{i\theta }z)=e^{2i\theta }R(z)$: spatial rotation through $\theta$ is accompanied by isorotation through $2\theta$. The angular density vanishes at the two axial directions, yielding the toroidal hole. This describes the familiar minimal branch and a useful approximate field; it is not an assertion that every degree-two field is toroidal.

For a smooth Skyrme model field $U:{\mathbb{R}}^3\to SU(2)$ with $U(\infty )=1$, compactification gives a map $S^3\to S^3$. With $L_i=U^{†}{\partial }_{i}U$, its degree of a map between oriented manifolds is $B=-(24{\pi }^2{)}^{-1}\int {ϵ}_{ijk}\mathrm{tr}(L_i{L}_j{L}_k)d^{3}x$, with sign chosen so the standard decreasing hedgehog profile has $B=1$. The integrand is the pullback of the normalized volume form on $SU(2)$. Hence it is integer-valued and unchanged by smooth finite-energy homotopies. In the rational map approximation for Skyrmions, a degree-$N$ angular map and profile $f(0)=\pi ,f(\infty )=0$ give $B=-(2N/\pi )\int f^{\prime }{\sin }^{2}fdr=N$. This is a topological conservation law, not the Noether charge of isospin.

Under SU(2) as the three-sphere, a Skyrme model field with $U\to I$ at infinity defines $S^3\to S^3$. Choose $T_i=-i{\tau }_i$ and positive left volume form ${\theta }^1\wedge {\theta }^2\wedge {\theta }^3$. The normalized form is $-(24{\pi }^2{)}^{-1}\mathrm{tr}(U^{-1}dU{)}^3$, so the topological baryon number in the Skyrme model equals $\mathrm{deg}U$. Generator and orientation conventions fix the sign.

The local Skyrme baryon density is $\mathcal{B}(x)=-{ϵ}_{ijk}\mathrm{tr}(L_i{L}_j{L}_k)/(24{\pi }^2)$, with $L_i=U^{†}{\partial }_{i}U$ and $B=\int \mathcal{B}{d}^{3}x$. It is invariant under global isorotations and transforms as a scalar under proper spatial rotations, so its contours reveal the geometric shape of a Skyrmion. For the rational map approximation for Skyrmions, $\mathcal{B}=-f^{\prime }{\sin }^{2}f{J}_R/(2{\pi }^2{r}^2)$, where $J_R$ is the angular Jacobian of a rational map. Integrating $J_R$ over the sphere gives $4\pi \mathrm{deg}R$, recovering the integer total charge. A decreasing profile and holomorphic angular map give a nonnegative density, but a general field may have regions of negative density; positivity is not a general topological theorem. The density is different from the energy density.

The unit Skyrmion has a spherically symmetric ansatz with Pauli matrices $\mathbf{\sigma }$, $f(0)=\pi$ and $f(\infty )=0$. The baryon number isThe radial profile is obtained from the Skyrme model variational equation. Spatial rotations and isospin rotations act on the same hedgehog orientation, leaving only three independent rotational collective coordinates.

Write $U=A{U}_0{A}^{-1}$, $A\in SU(2)$, and $A^{-1}\dot{A}=i\mathbf{\omega }\cdot \mathbf{\sigma }/2$. The rotational kinetic energy is $\Lambda |\mathbf{\omega }{|}^2/2$. Since $A$ and $-A$ give the same classical field, the physical orientation space is $SO(3)$, covered by $SU(2)$. For fermionic unit baryons the Finkelstein-Rubinstein constraints require $\Psi (-A)=-\Psi (A)$. The SU(2) representations then give half-integer $j$, with spin angular momentum and isospin both of magnitude $j$. The rigid-rotor energies are $M+j(j+1){\hslash }^2/(2\Lambda )$; the lowest multiplets model the nucleon and Delta baryon. Large rotational energies can excite deformation and radiation, beyond the rigid approximation.

The rational map approximation writes the angular dependence of a Skyrmion using a rational map $R:S^2\to S^2$ and determines a radial profile variationally. The degree of $R$ equals the baryon number, while zeros of its Wronskian mark directions of vanishing angular baryon density.

This degree-four rational map obeys $R(iz)=1/R(z)$ and $R((iz+1)/(1-iz))=e^{2\pi i/3}R(z)$. These pair the generators of the rotational symmetry group of a cube with target rotations, giving combined spatial-isospin symmetry in the Skyrme model. Its spatial half-turn kernel is a Klein four-group. The map gives a cubic four-Skyrmion approximation; preserving only its Wronskian zero set does not guarantee the same rotational symmetry.

For a degree-$N$ rational map, the angular Jacobian of a rational map has average $N$, so the Cauchy-Schwarz inequality gives $\mathcal{I}\ge N^2$. In dimensionless Skyrme model units the radial energy within the rational-map ansatz isFirst minimize $\mathcal{I}$ over admissible maps, then minimize over profiles with $f(0)=\pi$, $f(\infty )=0$. This gives a restricted variational approximation and an upper bound on the unrestricted minimum in that topological sector, not an exact multi-Skyrmion solution. Houghton, Manton and Sutcliffe developed this construction in arxiv.org/abs/hep-th/9705151.

Finkelstein-Rubinstein constraints impose the allowed signs under combined spatial and isospin rotations of a quantized Skyrmion. They encode the fermionic or bosonic exchange behavior of topological solitons.

A rigid Skyrmion orientation is quantized on a cover of its Skyrmion collective-coordinate orbit. A lifted combined rotation in the static field's stabilizer subgroup identifies the same classical field and imposes ${\hat{D}}^J(R){\hat{D}}^I(A)\Psi ={\chi }_{\mathrm{FR}}\Psi$, where the sign is the chosen Finkelstein-Rubinstein constraints character of that configuration-space loop. Choosing the nontrivial character models unit baryons as fermions. A $2\pi$ spatial rotation or isorotation of charge $B$ then has sign $(-1{)}^B$: deforming the field to separated unit lumps adds the unit-lump loop classes in the fundamental group ${\mathbb{Z}}_2$, while the labelled orbital paths can be contracted in three dimensions. The resulting Finkelstein-Rubinstein constraints sign is the product of the unit-lump signs. Hence spin and isospin are half-integer for odd $B$ and integer for even $B$. Further stabilizer constraints restrict which pairs $(J,I)$ and which body-fixed states occur. The energy operator comes from the inertia tensors on the orbit; symmetry selects allowed states but does not by itself determine their energies. A bosonic choice of the trivial topological character is mathematically possible but does not model a fermionic nucleon.