A point particle model of lightly bound skyrmions

A point particle model of lightly bound skyrmions


A simple model of the dynamics of lightly bound skyrmions is developed in which skyrmions are replaced by point particles, each carrying an internal orientation. The model accounts well for the static energy minimizers of baryon number obtained by numerical simulation of the full field theory. For , a large number of static solutions of the point particle model are found, all closely resembling size subsets of a face centred cubic lattice, with the particle orientations dictated by a simple colouring rule. Rigid body quantization of these solutions is performed, and the spin and isospin of the corresponding ground states extracted. As part of the quantization scheme, an algorithm to compute the symmetry group of an oriented point cloud, and to determine its corresponding Finkelstein-Rubinstein constraints, is devised.

1 Introduction

The Skyrme model is an effective theory of nuclear physics in which nucleons emerge as topological solitons in a field whose small amplitude travelling waves represent pions. It thus provides a unified treatment of both nucleons and the mesons which, in the Yukawa picture, are responsible for the strong nuclear forces between them. While the Skyrme model has been superceded as a fundamental model of strong interactions by QCD, interest in the model revived once it was recognized to be a possible low energy reduction of QCD in the limit of large (number of colours) [17, 16], and much work has been conducted to extract phenomenological predictions about nuclei from standard versions of the model [11, 12, 10, 3]. Many of these predictions are in good qualitative agreement with experiment, and recent improvements in skyrmion quantization schemes offer hope of significant further improvement to come [6, 7].

One area in which standard versions of the model perform poorly, however, is that of nuclear binding energies: typically, classical skyrmions are much more tightly bound than the nuclei they are meant to represent (by a factor of 15 or so). In recent years, no fewer than three variants of the model have been proposed which seek to remedy this problem. In each case, the model is, by design, a small perturbation of a Skyrme model in which the binding energies vanish exactly. Perhaps the most radical proposal, due to Sutcliffe and motivated by holography, couples the Skyrme field to an infinite tower of vector mesons [15]. Small but nonvanishing binding energies are (conjecturally) introduced by truncating this infinite tower at some high but finite level. This proposal, while elegant, has so far not been amenable to detailed analysis. A second proposal, due to Adam, Sanchez-Guillen and Wereszczynzki, starts with a model which is invariant under volume preserving diffeomorphisms of space, then perturbs it by mixing with a small fraction of the conventional Skyrme energy [1]. Skyrmions in this model have the attractive feature of being somewhat akin to liquid drops. However, the large (in fact, infinite dimensional) symmetry group of the unperturbed model is extremely problematic for numerical simulations, and the shapes and symmetries of classical skyrmions, even for rather low baryon number () are, so far, not known in this model in the regime of realistically small binding energy [5].

In this paper we will study the third (and arguably least radical) proposal, originally due to one of us [8]. This amounts to making a nonstandard choice of potential term in the standard Skyrme lagrangian and, more importantly, radically shifting the weighting of the derivative terms from the quadratic to the quartic. The resulting model is still amenable to numerical simulation, but its classical solutions are quite different from conventional skyrmions: the lowest energy Skyrme field of baryon number now resembles a loosely bound collection of spherically symmetric unit skyrmions, rather than a tightly bound object in which the skyrmions have merged and lost their individual identities. In the terminology of [14], which studied a dimensional analogue of the model, skyrmions in this lightly bound Skyrme model prefer to hold themselves aloof from one another. Numerical analysis reveals [5] that they also prefer to arrange themselves on the vertices of a face centred cubic spatial lattice, with internal orientations dictated by their lattice position. This suggests that, unlike conventional skyrmions, lightly bound skyrmions can be modelled as point particles, each carrying an internal orientation, interacting with one another through some pairwise interaction potential whose minimum encourages them to sit at a fixed separation with their internal orientations correlated. The aim of this paper is to derive such a simple point particle model, compare its predictions with numerical simulations of the full field theory, and use it to extract, via rigid body quantization, phenomenological predictions about nuclei with baryon number . A similar programme (minus quantization) for the dimensional analogue model was completed in [14].

As we shall see, the point particle model accounts almost flawlessly for static skyrmions with , where comparison with simulations of the full field theory is available. For , it predicts a rapid proliferation of nearly degenerate skyrmions as grows, all rather close to size subsets of the face centred cubic lattice. In comparison with conventional skyrmions, these typically have rather little symmetry, and anisotropic mass distribution. Determining the symmetries of these configurations is an interesting and important task, nonetheless, as they determine the Finkelstein-Rubinstein constraints on quantization. Usually, symmetries of skyrmions are determined by ad hoc means: one looks at suitable pictures of the skyrmion, predicts a symmetry by eye, then checks it by operating on the numerical data. By contrast, we will develop an algorithm which automatically computes the symmetry group of any point particle configuration. This allows us to completely automate the rigid body quantization scheme. The result is, as a phenomenological model of nuclei, moderately successful: rigid body ground states plausibly account for the lightest nucleus of baryon number for 12 of the 23 values considered. Presumably this can be improved by replacing rigid body quantization by something more sophisticated.

The rest of the paper is structured as follows. In section 2 we review the lightly bound Skyrme model, focussing on its spin-isospin symmetry and associated inertia tensors. In section 3 we introduce the point particle model, then in section 4 we describe a numerical scheme to find its energy minimizers, and present the results of this scheme. In section 5 we formulate the rigid body quantization of our classical energy minimizers, focussing particularly on the Finkelstein-Rubinstein constraints. Some concluding remarks and possible future directions of development are presented in section 6.

2 The lightly bound Skyrme model

The field theory of interest is defined as follows. There is a single Skyrme field , required to satisfy the boundary condition as for all . Such a field, if smooth, has at each , a well-defined integer valued topological charge


the topological degree of the map . Since the field is smooth, is smooth and integer valued, hence automatically conserved. Physically it is interpreted as the baryon number of the field . The right invariant current associated with is , in terms of which the lagrangian density is


Here is the pion decay constant, the pion mass, and , are dimensionless parameters. In [5] the following values were chosen for these parameters so that classical binding energies in the model are comparable with experimentally-measured nuclear binding energies:1


There is certainly room for improvement in this calibration: for example, obtaining the correct pion mass was not a priority in [5], and we expect that a more thorough analysis could result in a parameter set for which is closer to its experimental value of 137MeV. However, the aim in the present paper is not to fine-tune the parameters, but rather to study qualitative properties of static solutions, which we expect to be insensitive to details of the calibration.

It will be convenient to use as a unit of energy and as a unit of length; in these units the lagrangian takes the form , where


and . In the parameter set given above, . Note that when , is the lagrangian of the conventional Skyrme model with pion mass, while for this is a completely unbound model [8]: there is a topological energy bound of the form , but this is attained only when .

The first approximation to a nucleus containing nucleons is a static Skyrme field of degree which minimizes the potential energy . Thus it is important to identify static classical energy minimizers. These are referred to as skyrmions. A better approximation to a nucleus is obtained by allowing solitons to carry spin and isospin. The lagrangian is invariant under a left action of the group , defined by


where we have identified physical space with the Lie algebra via , being the Pauli matrices, to define the action of on . Equivalently,


where is the matrix with entries


The conserved quantities associated with these symmetries are isospin and spin. We refer to transformations as isorotations, in analogy with rotations .

Every defines a one-parameter subgroup of isomorphic to , whose action on a static skyrmion generates a rigidly isorotating and rotating skyrmion, , of constant kinetic energy . The mapping is a quadratic form on , and hence defines a unique symmetric bilinear form called the inertia tensor of the skyrmion . By its definition, vanishes on the subspace of tangent to the isotropy group of (that is, the subgroup which leaves unchanged). If is discrete, as is the case for all the skyrmions studied in this paper except when , then is a positive bilinear form, and thus defines a left invariant Riemannian metric on . In order to identify spin and isospin quantum numbers of skyrmions corresponding to those of nuclei, isorotations and rotations needed to be treated quantum mechanically rather classically. The inertia tensor plays an important role in the simplest quantization scheme, known as rigid body quantization, which will be reviewed in section 5, and amounts to quantizing geodesic motion on , subject to certain symmetry constraints required to give skyrmions fermionic exchange statistics. Clearly, by choosing a basis for , we obtain a basis for which can be used to represent as a real symmetric matrix. We shall consistently represent inertia tensors in this way, having chosen the basis for .

3 The point particle model

Extensive numerical simulations reported in [5] showed that skyrmions in the lightly bound Skyrme model with invariably resemble collections of particles. Encouraged by this observation, we have developed a point particle model in which a Skyrme field with baryon number is replaced by oriented point particles in .

To explain how the model is derived, we begin by recalling the structure of the simplest skyrmion, which has , and is of “hedgehog” form


with a real function satisfying , as , and . The profile function is determined by solving (numerically) the Euler-Lagrange equation for restricted to fields of hedgehog form, a certain nonlinear second order ODE for . One finds that has total energy , and its energy density is monotonically decreasing with and concentrated around the origin. The 1-skyrmion has a high degree of symmetry: if then

In other words, is the diagonal subgroup of .

This basic skyrmion can be moved and rotated using symmetries of the model. A 1-skyrmion with position and orientation is given by


The energy-minimizers with resemble superpositions of fields of this type [5]. More precisely, their energy densities are concentrated at well-separated points , and near each such point the field is approximately of the above form for some . These positions and orientations are the basic degrees of freedom in our point particle model, and will be allowed to depend on time . The lagrangian for this point particle model takes the form


where and


is an interaction potential.

The terms involving time derivatives of and represent the kinetic energy of a moving skyrmion. Their coefficients could be deduced from the Skyrme model. It is known that the 1-skyrmion has inertia tensor


From this it follows that the kinetic energy of a rigidly rotating skyrmion should take the form , suggesting that in the lagrangian (3.3). Similarly, the kinetic energy of a 1-skyrmion moving with velocity is , where is the potential energy of a static 1-skyrmion. This suggests choosing in the lagrangian. However, we have chosen to fix the coefficients by an alternative phenomenological method that will be explained in the next section.

3.1 Symmetries of the interaction potential

The point particle model inherits an action of from the Skyrme model. The action of on the field defined in equation (3.2) is

Therefore the action of on a point particle configuration is

The point particle lagrangian should be invariant under these transformations, and under translations for . It should be invariant under changes of the signs of any of the , because . It should also be invariant under permutations of the particles, because configurations of particles that are the same up to a re-ordering describe the same Skyrme field. Finally, the Skyrme model is invariant under the inversion

which is equivalent, for a field of the form (3.2), to . Hence, our point particle lagrangian should be invariant under


The kinetic terms in (3.3) obviously have these symmetries. Demanding that the potential (3.4) is also invariant imposes constraints on the function which we now describe.

Translation symmetry implies that depends on the positions of the skyrmions only through their relative position . Isorotation symmetry implies that it depends on only through the isorotation-invariant combination . Thus

for some function on . Invariance under implies


while rotational symmetry demands that


A permutation changes the sign of and inverts , so permutation invariance implies that


Finally, symmetry under inversion (3.5), implies


To proceed further, it is helpful to think of as a one-parameter family of real functions on , parametrized by . We may expand each such function in a convenient basis for , for example, the basis of eigenfunctions of the Laplacian. A natural truncation to finite dimensions is obtained by keeping only eigenfunctions up to a fixed finite eigenvalue. The effect of this truncation is to exclude from terms with fast orientation dependence. This motivates the following definition: for each in the spectrum of , let denote the corresponding eigenspace, and for any ,


Let denote the space of smooth functions on such that for all .

Proposition 1.

Let and be a function in invariant under the symmetries (3.6)-(3.9). Then there exist functions , , such that


Recall that the eigenvalues of the Laplacian on are , , and the corresponding eigenspaces, , are spanned by (the restrictions to of) harmonic homogeneous polynomials in of degree [2]. It follows that the eigenvalues of are with eigenspaces . By (3.6), (3.9), is invariant under both and , so we may restrict and to only even values (homogeneous polynomials of odd degree are parity odd). Further, since with , each restriction lies in


Now acts on both (by rotations of ) and (by conjugation on ), and, by(3.7), each is invariant under the combined action. In fact and carries the irreducible spin representation of , while where, for , carries the irreducible spin representation of . In particular, , on which acts trivially, and decomposes into irreducible representations as


Now the tensor product contains no trivial subrepresentation if , and exactly one if . Hence, of the summands in (3.12), , and each contain a one-dimensional subspace on which acts trivially (while does not) and, by (3.7), lies in the three-dimensional space spanned by these. Clearly which is spanned by the constant function . Consider the functions


These are manifestly invariant and extend to homogeneous polynomials on of bidegree and respectively. Furthermore, one may readily check that these polynomials are harmonic (separately with respect to and ). Hence, they span and respectively. Noting that on , the claim follows. ∎

From now on, we assume that lies in the truncated function space , so that it has the structure prescribed by Proposition 1.

Recall that, in the standard Skyrme model, the interaction potential for well separated skyrmion pairs can be modelled using the dipole formalism [13]: far from its centre, a unit skyrmion looks like the field induced in the linearization of the Skyrme model about the vacuum, , by an orthogonal triplet of scalar dipoles placed at the skyrmion’s centre. The interaction potential for a skyrmion pair with relative position and orientation can then be approximated by the interaction energy of a pair of triplets of dipoles held at relative displacement and orientation , interacting via the linear theory. This approximation introduces another useful constant associated with the unit skyrmion, namely the strength of the (necessarily equal) dipoles. In practice this is determined numerically by reading off a coefficient in the large asymptotics of the skyrmion profile function. This formalism is readily adapted to the lightly bound Skyrme model, producing an interaction potential of the form (3.11) with


The dipole strength (for and ) is found numerically to be . These formulae reproduce the usual prediction of attractive and repulsive channels for well-separated skyrmions. That is, is maximally attractive (increases fastest with ) if the orientations of the skyrmions differ by a rotation by about any direction orthogonal to , is maximally repulsive if the orientations differ by a rotation by about , and is nonmaximally repulsive if their orientations are equal. We refer to these three situations as the attractive, repulsive and product channels respectively.

The existence of these three channels allows us to fix the functions numerically by conducting scattering simulations of skyrmion pairs in the full field theory, in similar fashion to Salmi and Sutcliffe’s work on the dimensional model [14]. We begin with a Skyrme field of the form


where is large and is a unit hedgehog skyrmion defined (numerically) in a ball of radius less than (so for all , and the product above commutes). Such a field represents a pair of skyrmions located at , that is, with separation , in the attractive channel. Here, and henceforth, we define the skyrmion positions of a Skyrme field to be those points where . We now allow to evolve with time according to the dynamics defined by the lagrangian (2.2), using the fourth order spatial discretization employed by the energy minimization scheme of [5], and a fourth order Runge-Kutta scheme with fixed time step for the time evolution. This numerical scheme conserved total energy to extremely high accuracy,


for all the dynamical processes presented here. As the dipole model predicts, the skyrmions with these initial data slowly move towards one another, attain a minimum separation, then recede again. By recording their separation and potential energy at each time step, we recover a numerical approximation to the attractive channel interaction potential which, according to (3.11) is related to by


We then repeat the process with intial data


which are in the repulsive and product channels respectively. To make the skyrmions approach one another and interact, we now Galilean boost them towards one another at low speed (v=0.1). Note that the reflexion symmetries of the initial data trap these fields in their respective channels for all time. From these numerical solutions we obtain numerical approximations to the repulsive and product channel interaction potentials, which are related to by


It is clear that uniquely determine and hence, within the ansatz (3.11), .

Figure 1: Interaction energies of skyrmions pairs with separation in the attractive (blue), product (red) and repulsive (green) channels. In each case the thick curve represents numerical data extracted from a scattering process, the thin curve is a fit to this, and the dashed curve is the interaction energy predicted by the dipole model.

Graphs of , determined numerically as described above, are presented in figure 1. These curves also show the potentials predicted by the dipole model (with dipole strength ). Clearly, the dipole formulae (3.15) do not provide an accurate quantitative picture of skyrmion interactions in the lightly bound model at any separation where the interactions are not negligible. This is, perhaps, not surprising, since the dipole formalism replaces the full field theory by terms originating only in the quadratic and pion mass potential terms of the lagrangian, and these are precisely the terms which are given very low weighting, , in the lightly bound regime. The qualitative predictions of the dipole picture are reliable however: the interaction potentials appear to decay exponentially fast, and the three channels identified have the behaviour predicted (attractive, repulsive, more weakly repulsive). For later use, it is convenient to have explicit functions which approximate the numerical data for . For our purposes, it is important that these functions decay exponentially with and accurately fit the numerical data for , where is somewhat smaller then the equilibrium separation defined by (that is, the separation at which is minimal). The behaviour for is not so important, provided the formulae introduce a repulsive core interaction, and is, in any case, inaccessible to our numerical scheme (since close approach of lightly bound skyrmions is forbidden in low energy scattering processes). Figure 1 also depicts the following fit functions


Of these, the most elaborate is , a Padé approximant on spliced to an exponentially decaying tail, the splice being chosen so that is continuously differentiable. Unlike and , is well defined at , where it is chosen to equal the static energy of the axially symmetric solution (a saddle point of the Skyrme energy), obtained numerically by a different scheme, a choice made mainly for aesthetic reasons.

From now on, we choose to be the function defined by (3.11), where

and are the functions defined in (3.23). It is straightforward to show that this function is bounded below as, on physical grounds, it should be.

3.2 The FCC lattice

We have seen that the interaction potential prefers particles to be in the attractive channel, i.e. such that their relative orientation corresponds to a rotation about an axis perpendicular to their line of separation through angle . It is therefore desirable to find a way to pack them together such that all neighbouring pairs of particles are in the attractive channel. The face-centred-cubic (FCC) lattice provides a solution to this problem.

The face-centred cubic lattice may be defined to be

with defining a lattice scale. The underlying cubic lattice is given by points for which are all even. Those points for which some of the coordinates are odd lie on faces of the underlying cubic cells.

We assign orientations to these points as follows: those points on the vertices have orientation , those on faces perpendicular to the -axis have orientation , those on faces perpendicular to the -axis have orientation , and those perpendicular to the -axis have orientation . Here we have implicitly identified elements with unit quaternions , such that , , and is the identity matrix. Put differently, the orientation of a particle at lattice site is such that

The reader may verify that any pair of nearest neighbours, separated by a distance , is in the attractive channel.

One might expect that minimizers of the potential energy derived from (3.3) resemble subsets of the FCC lattice. This was certainly true of all global minima of the Skyrme energy identified in [5], and all but one of the local minima.

3.3 Inertia tensors

The point particle model (3.3) makes simple predictions for the inertia tensors of lightly bound skyrmions. These are obtained by calculating the kinetic energy of a rotating and isorotating oriented point cloud.

Let be a minimizer of the potential energy derived from (3.3). Choose any pair of angular velocities . It is useful to identify each with a vector by choosing , , as a basis for (so ). Consider the following configuration, which is isorotating and rotating at constant angular velocity :

We find


Therefore the kinetic energy is


where the inertia tensor is


The point particle model predicts that this is a good approximation to the inertia tensor of a lightly bound degree skyrmion. We will test this prediction in the next section.

4 Energy minimizers in the point particle model

4.1 Light nuclei

Having introduced the point particle model for lightly bound skyrmions, in this section we present our results for energy-minimizing configurations of point particles. We begin by discussing our results for eight particles or fewer, where comparison can be made with energy minima in the lightly bound Skyrme model found in [5].

We have developed an iterative zero-temperature annealing algorithm to minimize the energy of a configuration of particles. We applied this algorithm both to randomly-chosen initial ensembles of particles and to initial ensembles that are subsets of the FCC lattice. We ran a large number of simulations for each value of , typically obtaining several local energy minima, and record here only the lowest local minimum and up to two closest competitors. Energies of these local minima with are presented in table 1. The particle ensembles themselves are depicted in figure 2. The corresponding binding energies in the lightly bound Skyrme model are also recorded in the table. These are defined to be the energy of the -skyrmion minus times the energy of the 1-skyrmion.

Name bonds particle energy Skyrme interaction energy
2a 2 -0.310 -0.36
3a 3 -0.931 -0.92
4a 6 -1.862 -1.71
5a 8 -2.338 -2.20
5b 8 -2.185 -2.00*
6a 12 -3.229 -2.85
6b 11 -3.117 -2.87
6c 11 -3.046 -2.79*
7a 15 -4.057 -3.58*
7b 14 -3.895 -3.52
8a 18 -4.889 -4.47
8b 18 -4.869 -4.37
8c 18 -4.781 -4.34*
Table 1: Energies and numbers of bonds of the lowest-energy local minima in the point particle model, and energies of the corresponding lightly bound skyrmions (taken from [5], except those marked , which result from new simulations conducted with the same numerical scheme).
3 4 5(a) 5(b)
6(a) 6(b) 6(c) 7(a)
7(b) 8(a) 8(b) 8(c)
Figure 2: Local energies minimizers in the point particle model. Each ball is centred on a point skyrmion position and its colour represents the internal orientation . Each picture also depicts the FCC lattice configuration of size to which the minimizer best fits. Thick grey line segments indicate interskyrmion bonds no more than 10% longer than the FCC bond length, while thin magenta line segments show nearest neighbour bonds in the best fit lattice configuration. In most cases the fit is so good that the thin bonds are not visible. They show quite clearly on 5(a), 5(b), 6(b), 6(c), 7(b) and 8(c) however.

Our results are almost entirely consistent with the results obtained for the lightly bound Skyrme model in [5]. For we obtained the same global minima as in the lightly bound Skyrme model. For multiple local minima were previously obtained in the lightly bound Skyrme model. All of these occured as local minima in the point particle model. For the ordering of energies in the point particle also agreed with the ordering of energies in the lightly bound Skyrme model. The only failure of the point particle model is for 6 particles: here the energies of the two lowest-energy local minima appear in the wrong order.

In addition to reproducing previously-known minimizers from the lightly bound Skyrme model our point particle model also predicted some new local minima. Most interestingly, the global energy-minimizer in the point particle model for , labelled in figure 2, did not correspond to any solution of the lightly bound Skyrme model found in [5]. Based on this discovery, we constructed an approximate Skyrme field with a similar shape to the point particle energy-minimizer, and minimized its energy using the same numerical scheme that was used in [5]. After relaxation this Skyrme field had a lower energy than any of the configurations discovered in [5], as predicted by the point particle model. Thus we have a new candidate global energy minimizer at charge seven. Similarly, new simulations find local energy minimizers in the lightly bound model of similar shape to , and , and these have energies ordered exactly as the point particle model predicts (so , for example).

In every case, the minimizers found look, to the naked eye, like subsets of the FCC lattice.2 It is an interesting problem to measure this property quantitatively. Given an oriented point cloud , we wish to identify the FCC subset of size which best approximates it. To do this, we consider the orbit of under the group of similitudes of ,


For each , we define to be the squared distance from to the FCC lattice, i.e.


Now, given a neighbouring triple of particles in (a particle , its nearest neighbour and next-nearest neighbour ), we construct a similitude which maps to , to and to the plane spanned by , . We then solve the gradient flow equation of , with , to find a local minimum of close to . Repeating over all neighbouring triples, we keep the lowest local minimum of found (note that never has a global minimum since can be made arbitrarily close to by taking sufficiently large). In this way we identify the closest FCC subset to and its root mean square distance from , namely . Having found , the FCC colouring rule predicts the internal orientations the particles should have. These should be compared with , bearing in mind that orientations are defined only up to sign, and that the system is isospin invariant. Thus we minimize


over , again by gradient flow. This gives us a measure of the root mean squared distance of the internal orientations of the configuration from those imposed by the colouring rule applied to its closest FCC approximant, namely . It also allows us to “coarse grain” the internal orientations, that is, map each to the element of to which is closest. We used this method to determine the particle colours and FCC bonds in figure 2. We will present graphs of and in the next section.

In addition to comparing energies we have also compared inertia tensors in the point particle and lightly bound Skyrme models. Under isorotations and rotations inertia tensors transform as

In comparing the inertia tensors of a charge skyrmion, obtained by solving the field theory, and a charge point particle energy minimizer, we must account for the fact that the orientations of these two objects are completely unrelated. We do this by introducing a standard form for inertia tensors which fixes these symmetries. We say that an inertia tensor is in standard form if