Linked vortices as baryons in the miscible BEC-Skyrme model

We introduce a variation of the Bose-Einstein condensate(BEC)-Skyrme model, with an altered potential for miscible BECs that gives rise to two physical vortex strings. In the ground state of each topological sector, the vortices are linked exactly $B$ times, due to a recently formulated theorem, with $B$ being the baryon number of the solution. The model also possesses metastable states, where the vortices are degenerate and do not lend the interpretation of the baryon number as the linking number of the vortices.


I. INTRODUCTION
More than a century ago, Lord Kelvin imagined that atoms were made of knotted vortices [1], but this idea has not been successful so far. However, recently we have made a connection between not knots, but links of vortices and Skyrmions in the Skyrme model [2]. Skyrmions are solitons of the texture type in 3-dimensional space and possess a topological degree B as they are mapped to a 3-sphere, being the target space or isospin space [3,4]. In the large-N limit of QCD, the baryon is identified with the Skyrmion [5,6], thus providing a solitonic approach to nuclear physics, see e.g. Refs. [7][8][9]. The Skyrmion may also be realized in two-component Bose-Einstein condensates (BECs) [10][11][12][13][14][15][16][17][18][19][20], see Ref. [21] for a review. In Refs. [22][23][24] we introduced a potential inspired by two-component BECs, V ∼ M 2 |φ 1 | 2 |φ 2 | 2 , which deforms the Skyrmions into a twisted vortex ring (or vorton) by explicitly breaking the SU(2) isospin symmetry normally possessed by Skyrmion solutions. Due to the nonlinear sigma model constraint, |φ 1 | 2 + |φ 2 | 2 = 1, the center of the vortex ring in one component confines the other component; it is thus a global analog of Witten's superconducting cosmic string [25]. Similar vortex rings can also be obtained in a different asymmetric potential, V ∼ M 2 |φ 1 | 2 [26]. In addition to a vortex ring, the BEC-inspired potential also allows for a domain wall [22][23][24] into which the vortex ring can be absorbed, creating a vortex handleor rather a link of a handle and a dual handle from both sides of the domain wall [27].
The connection between links of vortices and Skyrmions [2] is made by means of a theorem stating that there exists a projection of the Skyrmion field onto a 2-sphere (as opposed to the 3-sphere that is the target space), which has the properties that two distinct regular points are linked B times in real physical 3-space, with B being the baryon number or topological degree of the Skyrmion. Taking a natural Ansatz for the Skyrmion, like the one giving rise to a B-twisted vortex ring [22][23][24], produces a "physical" vortex with winding number one and a "vacuum" vortex with winding number B. The "vacuum" vortex is not physical in the sense that it strays off to infinity in such a diluted form that the total energy is finite.
As mentioned, the catch is that the points under the projection have to be regular points, which is not always the case. In Ref. [2] we have circumvented the issue by introducing a rotation of the 2-sphere, hence making it possible to find points which are regular and hence giving rise to nondegenerate vortices that thus provide the linking number Q = B. One may consider such a rotation a bit arbitrary and thus wonder about the physical implication thereof.
In this paper, we modify the BEC-inspired potential studied in Refs. [22][23][24]27] by flipping the sign of the potential (which is allowed in a nonlinear sigma model) and this is suitable for the class of BECs called miscible two-component BECs, for which both the components φ 1 and φ 2 develop vacuum expectation values (VEVs) in the vacua. The implications thereof is a rather crucial change in the vacuum structure and indeed the model only possesses a unique vacuum state after the altercation, which we dub the miscible BEC-Skyrme model. This case admits two kinds of vortices having windings in the φ 1 and φ 2 components as in the case of the miscible BEC [21,28,29]. As a consequence, the abovementioned "vacuum" vortex is transformed into a second "physical" vortex, thus realizing, in a physical way, the two linked vortices proposed in Ref. [2].
Moreover there is the issue of regular points leading to nondegenerate links of vortices versus singular points leading to degenerate vortices in the framework of Ref. [2]. This notion takes a very physical form in the presence of the miscible BEC-inspired potential, as the degenerate vortex links generally give rise to higher-energy (metastable) states and the nondegenerate vortex links yield stable (ground) states. This paper is organized as follows. In Sec. II we introduce the miscible BEC-Skyrme model and discuss the vacuum structure of the model. In Sec. III we define the numerical observables, explain the numerical method, define the initial conditions used throughout the paper and discuss the numerical results. Finally, we conclude the paper with a discussion in Sec. IV.

II. THE MISCIBLE BEC-SKYRME MODEL
We consider the generalized Skyrme model which contains the kinetic term, the Skyrme term [3,4] and the BPS-Skyrme term [30][31][32] as well as a potential where σ a are the Pauli matrices, φ ≡ (φ 1 (x), φ 2 (x)) T is a complex two-vector of scalar fields which obey the nonlinear sigma model constraint φ † φ = |φ 1 | 2 + |φ 2 | 2 , the spacetime indices µ, ν, ρ, σ run over 0 through 3 and the flat Minkowski metric is taken to be of the mostly positive signature. The relation between the complex 2-vector of scalar fields, φ, and the chiral Lagrangian field usually used in Skyrme-type models is given by which translates the constraint φ † φ = det U = 1 into the usual one in terms of U .
With the potential turned off, the target space manifold is given by O(4)/O(3) SU(2) S 3 which is a unit 3-sphere. This is due to the model without a potential possessing O (4) symmetry, which however would spontaneously break to O(3) by the requirement of finite energy. The maps with finite energy necessarily have vanishing derivatives at spatial infinity, which effectively point compactifies R 3 to R 3 ∪ {∞} S 3 . Hence, topological solitons are supported with topological degree B ∈ π 3 (S 3 ) = Z, where B is given by where i, j, k = 1, 2, 3 are spatial indices.
In this paper we will augment the model with the miscible BEC-inspired potential which is the same as the BEC-inspired potential used in Refs. [22][23][24] Turning on the potential (7) explicitly breaks the symmetry from O(4) to where the group actions defined by the above symmetries are given by with a 2 + b 2 = 1. The (Z 2 ) 1,2 exchanges φ 1 and φ 2 as with e iγ = b+ia. U(1) 3 is acting on (Z 2 ) 1,2 transforming γ such that they define a semidirect product which we have denoted by .
In stark contrast to the immiscible BEC-inspired potential, the miscible version accom-modates only a connected vacuum state spontaneously breaking the symmetry G into a Z 2 subgroup (12) with which we denote by (Z 2 ) α−β . The target space is thus given by which does not allow for domain walls because of but accommodates two types of vortices supported by as the case of miscible two-component BECs [21,28,29].
The absence of the domain wall is in contrast to the case of immiscible BEC-Skyrme model with the spontaneously breaking of (Z 2 ) 1,2 admitting a domain wall. The immiscible BEC-Skyrme model was shown to contain vortex rings in Ref. [27] and when pushed toward a domain wall -present for the immiscible BEC-inspired potential -the vortex ring turned into a handle (in say φ 1 ) on the domain wall, albeit creating a second dual handle (in φ 2 ). The two seemingly different vortices were always present at the same time and hence topologically the immiscible BEC-inspired potential of Ref. [27] gives rise to a symmetry breaking supporting only one topological number for the vortices. In principle, in this model we can have two independent numbers of vortices.
The parameter space of the model with c 4 , c 6 , and M is rather large and for this reason we will choose only two model points 2 + 4 model : 2 + 6 model : which disentangles the effects of the Skyrme term (c 4 = 0) and the BPS-Skyrme term (c 6 = 0).

A. Local observables
In order to observe the linked vortices in Skyrmions we need some observables to identify the vortices. We will consider the baryon charge density B, the static energy density E = −L, the potential energy (density) V as well as a vorticity density. In order to define the latter, we will construct a term inspired by BPS vortices in the Abelian-Higgs model at critical coupling. Starting with the BPS equation and aiming at constructing an expression for the topological vortex charge we can solve for the gauge field from Eq. (21) obtaining from which we can readily construct F 12 : which is an expression for the topological vortex charge density for a vortex pointing in the x 3 direction. A simple extension of the charge density to cover a vortex pointing in any direction in R 3 can thus be made as where we have added a label for the two complex fields in our model.
For definiteness, we will define the total energy by the integral expression where the subscript B labels the topological sector at hand.

B. Numerical method
In order to explore the model, we need to find numerical solutions to the equations of motion, which we will solve by using the arrested Newton flow method [33] for a cubic 100 3 lattice, where the spatial derivatives are discretized using a standard finite difference scheme with a fourth-order 5-point stencil. The arrested Newton flow is updated in the "time direction" using a fourth-order Runge-Kutta method.

C. Initial conditions
In this paper, the linked vortices will naturally be physical due to the potential (7).
Nevertheless, it is not a priori guaranteed that the linked vortices are not degenerate, obscuring the interpretation that the linking number is equal to the topological degree of the Skyrmions, as proved by the theorem given in Ref. [2]. One could expect that increasing the coupling of the potential M 2 , would eventually force the linked vortices to be nondegenerate.
This will be investigated numerically in the next subsection.
As initial conditions we use both the rational map approximations of Ref. [34], whose linked vortices were interpreted and studied in Ref. [2], and composite Skyrmions constructed from lower-charge solutions using the symmetric product Ansatz As already mentioned in the introduction, the linked vortices in Ref. [2] naturally contain a vacuum vortex, which is not particularly physical, because it extends to infinity in R 3 although its flux is diluted such that the energy is finite. In Ref. [2] we introduced a rotation of the 2-sphere after applying the Hopf map to a Skyrmion map, which effectively eliminated the degeneracy of the linked vortices, allowing for the interpretation of the linking number as the topological charge or baryon number of the Skyrmions.
In this paper, we do not need to perform such a rotation of the 2-sphere, but it is necessary to rotate the standard frame of a Skyrmion such that the vacuum is compatible with the potential (7) used here. That is, a normal Skyrmion is a map φ : R 3 → S 3 with the vacuum chosen to be which translates to the vacuum in φ: which is obviously not the vacuum (13) in this model, i.e. with the potential (7), but is indeed compatible with a normal pion mass term tr (1 2 − U ) or any generalization thereof.
Therefore, for all initial conditions, which are constructed to match the standard frame, we perform the following rotatioñ which is an SO(4) transformation transforming the standard frame vacuum (29) intõ The initial conditions are now ready to be studied using the numerical method described in the previous subsection and the results will be given next.

D. Numerical results
We are now ready to present the numerical results in the 2+4 and 2+6 models for various values of M , with M 2 the coefficient of the miscible BEC-inspired potential. The color scheme utilized in the first 3 columns is the one used for standard Skyrmions, where the three real components φ 1 , φ 2 and φ 2 are normalized where n 1 + in 2 = exp(iH) is mapped to the hue H and n 3 determines the lightness, such that n 3 = 1 is white, n 3 = −1 is black and n 3 = 0 is a color determined by H: H = 0 is red, H = 2π/3 is green and H = 4π/3 is blue. The intent is simply to show how the surfaces are mapped to the target space S 3 . The last column in each panel of Fig. 1 shows the vorticities given in Eq. (25) such that the vorticity density in φ 1 (φ 2 ) is given by Q 1 (Q 2 ) and shown with red (blue) surfaces. All the isosurfaces are shown at half-maximum of the corresponding observable, except for the vorticities, which are shown at a quarter of the maximum vorticity. themselves in the topological baryon charge density or the total energy density for the 2+4 model ( Fig. 1(a)). All that happens is that the baryon charge density and energy density isosurfaces are slightly deformed, which is expected due to the broken SU(2) symmetry. The standard frame would make one of the vortices a "vacuum" vortex and the other a "physical" vortex, see Ref. [2]. The transformation (30) is thus equivalent to one of the rotations of the 2-sphere performed in Ref. [2]. with the 2+4 model, the total energy density isosurface (at half-maximum density) is quickly transformed from a sphere into the shape given by two linked vortices, see the 2nd column of Fig. 1  One could now expect that the degenerate soliton becomes unstable for a sufficiently large M and decays into the nondegenerate state. This begs the question, what is the phase diagram of these two states as function of M . In particular, which is the stable state for smaller values of M . In order to investigate this, we will use the B = 2 Skyrmion with broken degeneracy (see the last lines of Figs. 2(a,b)) as the initial guess and decrease M to zero -and while doing so, we calculate the total energy. The result is shown in Fig. 3 and for both models, the linked vortices are nondegenerate all the way as M tends to zero, thus allowing one to count the topological degree as the linking number of the two vortex species, see Ref. [2]. In order to conclude which of the two solutions is the stable one, we scattering of two B = 1 Skyrmions in the attractive channel. The initial state was prepared by means of the asymmetric product Ansatz of two 1-Skyrmions, which was then transformed into the frame compatible with this model's vacuum using Eq. (30). Surprisingly, the result for M = 1 was the metastable "degenerate" state instead of the stable ground state.
We will now consider the B = 3 topological sector, for which the results are shown in symmetric (which is the dual symmetry of the cube). First we obtain solutions for various values of M using the initial data which are transformed rational map solutions for the 2+4 model in Fig. 6 and for the 2+6 model in Fig. 7. In both the 2+4 and the 2+6 model, the solution obtained for small values of M is degenerate according to definition 1, see the 3rd metastable. Beyond that value of the mass parameter, we do not trust the accuracy of the code for the lattices used in these simulations.  Fig. 9 shows the stable nondegenerate soliton solution, which is nondegenerate in both the red (φ 1 ) and blue (φ 2 ) vortices. In order back up our claim, we calculate the energies for the different metastable and the stable solutions as functions of M and the result is shown in Fig. 10. It is clear both from the energies in Fig. 10 and from inspecting the baryon charge (1st column) and energy (2nd column) isosurfaces of Figs. 6-9, that the M = 0 solution is energetically and physically the same solution. Nevertheless, the vorticities plots are clearly different in Figs. 6-7 versus Fig. 8 versus Fig. 9. This is because, for vanishing potential, the difference is merely a rotation of the 2-sphere in the language of Ref. [2], which is a subgroup of O(4)  and hence does not change the energy or the physics of the Skyrmion with massless pions.
Once, the potential (7) is turned on, this rotation is no longer a symmetry and the potential picks out the nondegenerate vortex links as the ground state. Perhaps surprisingly, there are metastable states with residual degeneracy or even partial residual degeneracy left. This was not expected.
Although the fully degenerate state shown in Figs. 6-7 is clearly heavier on the energy graph 10(a) than the other two solutions, the difference between the lower-in-energy metastable state of Fig. 8 and the stable nondegenerate solution of Fig. 9 is actually so small, that we have plotted the difference between the energies of the degenerate and the nondegenerate states in Fig. 10(b). Clearly the partially degenerate solution with only the red vortices being degenerate, has almost the same energy as the nondegenerate stable solution.
This was not expected in the model with the potential (7) turned on.
We will now summarize the numerical results so far. The B = 1 case does not possess degeneracy in any frame, see Ref. [2], but the second vortex is not necessarily physical, as it can go off to infinity as a "vacuum" vortex. For B = 2 the standard frame after the transformation (30) turns out to give a degenerate solution. Cranking up the potential parameter turns this metastable degenerate state into an unstable state and the true nondegenerate "ground state" is found. For B = 3 the found solution does not possess degeneracy. For B = 4 the standard frame after rotation yields a metastable state with degeneracy which is higher in energy than another metastable state with partial degeneracy. The stable solution, however, in all cases is the nondegenerate one.
Since we have shown the first four Skyrmion solutions in great detail, we will only depict the stable ones of the next four Skyrmions (B = 5 through B = 8) and only for M = 0, 5: For the 2+4 (2+6) model the results are shown in Fig. 11 (Fig. 12). The story for the first seven baryon numbers (one through seven) was that the solutions with degenerate vortices had higher energies than solutions with nondegenerate vortices.
For the B = 8 Skyrmion it turns out to be more complicated. First of all, there exists both a fullerene-like solution, which is predicted by the rational map approximation [34] as well as a solution that is made up of two cubes attached to each other [7]. In the Skyrme model with a standard pion mass there are in fact two solutions, which are made of two cubes M . This collapse of the twisted double cube B = 8 Skyrmion has been observed also in other Skyrme-like models, see Ref. [33]. The existence of another fullerene-like solution has also been observed in Ref. [33]. Of course, such existence or nonexistence of the different solutions and which one is the global minimizer of the energy functional, is very dependent on the model and in particular on the potential.

IV. DISCUSSION
In this paper, we have considered the miscible BEC-Skyrme model, which is the gen- The conjecture is verified by the calculations in the topological sectors B = 1 through B = 7 in this paper.
A curious exception seems to occur for the B = 8 topological sector, where the lowestenergy solution we have found -for any value of the potential parameter M -turns out to be partially degenerate. That is, the red vortices (i.e. those of φ 1 ) are degenerate, whereas the blue vortices (i.e. those of φ 2 ) are nondegenerate. We did find a Skyrmion solution in this topological sector with nondegenerate vortices, which is a solution made of two B = 4 cubes joined together. It turned out, however, to have a higher energy than the partially degenerate fullerene-like solution. It would be interesting if this is an exception, or there actually exists a lower-energy solution for the B = 8 sector that we somehow did not find.
As an argument in favor of this hypothesis, we found that the difference in energy between the nondegenerate and the partially degenerate Skyrmion solutions in the B = 4 sector turned out to be extremely small. Therefore, increasing the potential parameter, M , may not be a viable technique in such a situation, as incredibly large values of M could be needed for finding the lowest-energy state (if such a state exists). Other approaches for searching for the lowest-energy state may be needed.
It would be interesting to study the case in which a "Josephson (or Rabi) term" φ * 1 φ 2 +c.c. is added to the potential. In the case of two-component BECs, this term induces a sine-Gordon soliton stretching between two types of vortices, exhibiting vortex confinement [36][37][38][39][40][41][42][43]. Several aspects of vortices in this case have been studied extensively, in particular, in two spatial dimensions, such as a vortex lattice [38], confinement [39,41], dynamics of vortices [40], collisions of vortices [43], as well as the phase structure [42]. In our case, two linked vortex lines will be connected by a minimal-surface soliton sheet, like a soap film.
In Ref. [2] we found that there exists a certain projection with which a baryon contains linked vortices. In this paper, we have shown that with a certain potential, those linked