CChains of interacting solitons
Ya. Shnir
BLTP, JINR, Dubna 141980, Moscow Region, Russia
We present an overview of multisoliton chains arising in various non-integrable field theories, anddiscuss different mechanisms, which may lead to the occurrence of such axially-symmetric classicalsolutions. We explain the pattern of interactions between different solitons, in particular Q-balls,Skyrmions and monopoles and show how chains of interacting non-BPS solitons may form in adynamic equilibrium between repulsive and attractive forces.
I. INTRODUCTION
There has been astonishing progress over the past sixty years in our understanding of nonlinear phenomena. Till1960s nonlinear systems were given little attention mainly because of their complexity, in the most cases the corre-sponding dynamical equations do not possess analytical solutions. The situation changed drastically with the dawningof computational physics, which made it possible to find reasonably accurate solutions for nearly any properly formu-lated physical problem. This development yields many surprising results and discoveries.One of the most interesting properties of various non-linear systems is that they may support solitons, stable,non-dissipative, localized configurations, behaving in many ways like particles (for review, see, e.g., [1, 2, 4]). Solitonsemerge in diverse contexts in various situations in nonlinear optics, condensed matter, nuclear physics, cosmology,and supersymmetric theories. In some situations, their existence is related to topological properties of the model; inother cases they appear due to balance between the effects of nonlinearity and dispersion. However, unlike the usualparticles, solitons are extended objects, they possess a core in which most energy is localized, and an asymptotictale, which is responsible for the long-range interaction between the well-separated solitons. Further, there is a towerof linearized excitations around a soliton, which belong to the perturbative spectrum. As a result, the pattern ofinteraction between the solitons becomes very complicated, in some situations there are both repulsive and attractiveforces with different asymptotic behavior and the process of the collision between the solitons is very different fromthe simple picture of elastic scattering of point-like particles.The study of the interactions between solitons, the processes of their scattering, radiation and annihilation hasattracted a lot of attention in many different contexts. First, almost immediately after discovery of the solitons inpioneering works [5, 6], the mathematical concept of integrability was developed. It turns out that some models whichmay support solitons, are completely solvable, in other words, all solutions can be presented analytically in closedform. Moreover, in integrable theories the collision between the solitons is always completely elastic. Further, thereis a very special class of so called self-dual solitons, whose exactly saturate the topological energy bound. In sucha case the energy of interaction between the solitons is always zero, then various multisoliton configurations can beconstructed via implication of diverse beautiful differential-geometrical mathods, see e.g. [1, 2].The situation is completely different in non-integrable theories. Perhaps one of the simplest examples of such theoryis a family of (1+1)-dimensional models with a polynomial potential possessing two or more degenerated minima, forexample φ modes with a double-well potential [3]. Another example is the Skyrme model [7], it is very well-known as aprototype of a relativistic field theory which supports non-BPS topological solitons, see, for example [1]. Historically,the 3+1 dimensional Skyrme model was proposed as a model of atomic nuclei, in such a framework baryons areconsidered as solitons with identification of the baryon number and the topological charge of the field configuration.A new development is related to the lower dimensional version of the Skyrme model in 2+1 dimensions since solitonsof that type were experimentally observed in planar magnetic structures and liquid crystals (for a review, see e.q. [9]).Notably, a simplest Skyrmion of topological degree one is rotationally invariant, however, solutions of higher degreesmay possess very interesting geometric shapes [1, 10]. The reason is that the interaction between the Skyrmionsis mediated by the long-range dipole forces, there are both repulsive and attractive channels. Thus, the pattern ofinteraction between the solitons become rather involved. Further, a particular choice of the potential of the modeldefines the structure of multi-soliton configurations, in some cases pairs of solitons in equilibrium may appear.Static solutions of the φ theory and the Skyrme model represent class of topological solitons. There are alsonon-topological solitons, for example stationary field configurations, commonly named Q-balls, that may exist insome models with a suitable self-interaction potential [11–13]. When Q-balls are coupled to gravity so-called bosonstars emerge, they represent compact stationary configurations with a harmonic time dependence of the scalar fieldand unbrocken global symmetry [14, 15]. Notably, the character of the interaction between Q-balls depends on theirrelative phase [16, 80]. In general, a pair of Q-balls is not stable in Minkowski spacetime, however the gravitationalattraction may stabilize it. Further, extended linear chains of rotating boson stars can be formed via this mechanism[18]. a r X i v : . [ h e p - t h ] J a n Another famous example of topological solitons are monopoles, they appear as classical solutions of the non-abelianYang-Mills-Higgs theory in 3+1 dimensions [19], for a review see e.g. [20]. The magnetic charge of the non-abelianmonopole is proportional to the topological charge. However, the pattern of interaction between the monopoles isfar from naive picture of Coulomb interaction of two point-like magnetic charges. The ’t Hooft-Polyakov solution isa coupled topologically stable configuration of gauge and Higgs fields which may have different asymptotic behavior.In particular, in the so-called Bogomol’nyi-Prasad-Sommerfield (BPS) limit of vanishing Higgs potential, both thegauge and the scalar fields become massless, then there is an exact balance of two long-range interactions betweenthe BPS monopoles, which are mediated by the massless photon and the massless scalar particle, respectively. In acontrary, attractive scalar interaction between two non-BPS monopoles becomes stronger than magnetic repulsion, thecorresponding charge two configuration with double zero of the Higgs field at the origin possess axial symmetry [23].On the other hand, it it possible to construct monopole-antimonopole pair solution in a static equilibrium [22, 24].This configuration represent a sphaleron, a saddle point solution of the classical field equations. Similar solutions alsoexist in the Skyrme model [25, 26].Pairs of non-selfdual solitons in a static equilibrium can be considered as basic building blocks of chains of non-selfdual solitons. The main purpose of the present short review is to discuss such solution in a few different models.First, we briefly consider the mechanism of interaction between the kinks in one spatial dimension and describe howstatic multisoliton bound states can be formed due to exchange interaction mediated by the localized fermion states(Section II). Then, we review the interactions between the Skyrmions and their dependency on the structure of thepotential. This yields some insight on the existence of chains of Skyrmions in two and three spatial dimensions(Sections III and IV, respectively). In Section V we discuss interactions between the Q-balls and possible mechanismof formation of pairs of Q-balls in equilibrium. Here we also include the gravitational interaction and consider chainsof boson stars. Finally, in Section VI, we revisit construction of monopole-antimonopole pairs and chains. Conclusionsand remarks are formulated in the last Section.
II. CHAINS OF KINKS
The simplest example of solitons are the kinks, they are classical solutions of the relativistic, nonlinear scalar fieldtheory in 1 + 1 dimensions with Lagrangian density L = 12 ∂ µ ∂ µ φ − V ( φ ) (1)with a smooth non-negative potential possessing some set of minima V ( φ ) = 0. The static kink solution of thismodel interpolates between different vacua φ as space coordinate x runs from −∞ to ∞ . For example, in the φ model the quartic potential V ( φ ) = 12 (cid:0) − φ (cid:1) (2)possesses two vacua, φ = {± } . Then the field equation of the model (1) ∂ µ ∂ µ φ + ∂V∂φ = 0 (3)yields the well-known non-trivial kink solution φ K ( x ) = tanh x interpolating between φ ( −∞ ) = − φ ( ∞ ) = 1.In a contrary, the anti-kink solution φ ¯ K ( x ) = − tanh x connects the vacua φ = 1 and φ = − V ( φ ) = 1 − cos φ supports static kink solution φ K ( x ) = 4 arctan e x which connectstwo neighboring vacua φ = 0 , π . Similarly, the φ model with triple degenerated vacuum V ( φ ) = φ (cid:0) − φ (cid:1) , φ = { , ± } , supports two different kinks φ (0 , = (cid:114) x , φ ( − , = − (cid:114) − tanh x ∂φ∂x = ± ∂W∂φ (5)where a superpotential W ( φ ) is defined as 12 (cid:18) ∂W∂φ (cid:19) = V ( φ ) . The equation (5), often referred to as BPS equation, is a simple realization of the self-duality of the model (1). Inother words, kinks always saturate the topological energy bound, their mass is proportional to the topological chargeof the soliton.Since our discussion focuses on chains of solitons, we are now looking for a possibility to construct such staticmultisoltion solutions in one spatial dimensions. However, the energy of interaction between the kinks is not zero,there is always a force acting between the solitons. This force can be evaluated when we consider an initial configurationof two widely separated kinks, for example in the sine-Gordon model φ ( x ) = φ K ( x + d ) + φ K ( x − d ) − π , (6)where φ K ( x ) is the kink solution mentioned above, and d is the separation parameter. Then we can expand thecorresponding energy of the configuration in powers of 1 /d and subtract the mass of two infinitely separated kinks.This yields the interaction energy E int = 32 e − d . (7)Evidently, it is the Yukawa-type interaction, it is repulsive for the kinks and it is attractive in the case of the kink–anti-kink pair. Further, there is no multisoliton solution in the φ model with double degenerated vacuum, the onlypossible structure of chains of solitons in such a case could be static linear configuration consisting of kinks andanti-kinks in alternating order. However, the energy of interaction between the solitons is not zero again, by analogywith (7) one can find that in the φ model E int = − e − d , so there is an attractive force in the kink–anti-kink pair,again. In other words, there is no static multisoliton solutions in the model (1).However, the situation can be different if we extend the model (1), for example we can consider coupled two-component system with one of the scalar components having the kink structure and the second component being anontopological soliton [30, 31], or modify the model (1), in such a way, that it still supports the kinks but possessa biharmonic spatial derivative term [32]. In the latter case it is possible to construct a static kink-antikink pair.Similarly, such configuration exist as a static solution of the (1+1) dimensional scalar field theory coupled to animpurity [33] which anchors the kinks.Another way to construct a bounded kink-anti-kink pair is to include interaction between the kinks and fermions[34–38]. Such an extended model is defined by the following Lagrangian L = 12 ∂ µ φ∂ µ φ + ¯ ψ [ iγ µ ∂ µ − m − gφ ] ψ − U ( φ ) , (8)where the self-interacting real scalar field φ is coupled with a two-component Dirac spinor ψ and m, g are the baremass of the fermions and the Yukawa coupling constant, respectively. The matrices γ µ are γ = σ , γ = iσ where σ i are the Pauli matrices, and ¯ ψ = ψ † γ .Let us show that the coupling with fermions may provide an additional force stabilizing the kink–anti-kink pair.We make use of the usual parametrization for a two-component spinor ψ = e − i(cid:15)t (cid:18) u ( x ) v ( x ) (cid:19) , it results in the following coupled system of dynamical equations φ xx + 2 guv − U (cid:48) = 0 ; u x + ( m + gφ ) u = (cid:15)v ; − v x + ( m + gφ ) v = (cid:15)u . (9)This system is supplemented by the normalization condition ∞ (cid:82) −∞ dx ( u + v ) = 1 which we impose as a constraint onthe system (9). Clearly, in the decoupled limit g = 0, the model (8) is reduced to the scalar model (1) which supportsthe kinks. We can easily see that for all such solutions, the system (9) possesses a fermionic zero mode (cid:15) = 0 which isexponentially localized on the kink. This mode exist for any value of the Yukawa coupling g , there is no level crossing -5 -4 -3 -2 -1 0 1 2 3 4 5-0.4-0.20.00.20.40.60.81.0 * x FIG. 1: φ kink-antikink pair bounded by fermions. Profiles of the scalar field and fermion density distribution ofthe collective mode at g = 1 (left plot) and scalar field of the configuration bounded to this mode vs Yukawacoupling g (right plot). Reprinted (without modification) from [38], with permission of APS. -3 -2 -1 0 1 2 3-1.0-0.50.00.51.01.5 * x -5 -4 -3 -2 -1 0 1 2 3 4 5-1.00-0.75-0.50-0.250.000.250.500.751.00 * x FIG. 2: φ multikink configurations bounded by fermions. Profiles of the scalar field and fermion densitydistribution of the collective fermionic mode(left plot) and the chain of the kinks ( − ,
1) + (1 , −
1) + ( − ,
1) boundedto the higher fermionic mode (right plot). Reprinted (without modification) from [38], with permission of APS.spectral flow in one spatial dimension [34]. Notably, for large values of the Yukawa coupling, other localized fermionicstates with non-zero energy eigenvalues | ε | < | g − m | may appear in the spectrum [34, 35, 37, 39].Consideration of the fermion modes bounded to a kink usually is related with an assumption that the back-reactionof the localized fermions is negligible [34, 35]. However, coupling to the higher localized modes may significantlydistort the φ kink [37]. Further, since such exponentially localized fermion modes may occur in multisoliton systems,localized fermions could mediate the exchange interaction between the solitons.Indeed, numerical solution of the full system of dynamical equations (9) shows that, as the Yukawa coupling increasesslightly above zero, a non-topological soliton emerge in the scalar sector, this lump is linked to a localized fermionicmode extracted from the positive continuum [38]. As g increases further, the lump becomes larger, it representstightly bounded kink–anti-kink pair, as seen in Fig. 1.Further, we found collective fermions localized on various multi-kink configurations. For example, a tower oflocalized fermion modes exist on a coupled pair of sG kinks in the sector of topological degree two [38], similarly,there are bounded pairs of φ kinks and anti-kinks, see Fig. 2.Further, even more complicated bounded multisoliton configuration, which represent multicomponent kink-antikinkchains with localized fermion modes may exist in the extended model (8). As a particular example, in the right plotof Fig. 2, we represent the chain of the φ kinks ( − ,
1) + (1 , −
1) + ( − ,
1) bounded by the higher fermion mode. Notethat similar phenomena are observed in other related models, the mechanism of the fermionic exchange interaction inmulti-soliton congurations is universal [38].FIG. 3: Contour plots of the energy density distributions of the solutions of the planar Skyrme model with thepotential (12) in the sectors of degrees Q = 3 − Q = 10. III. CHAINS OF BABY SKYRMIONS
Clearly, in one spatial dimension any bounded multisoliton configuration represents a chain of solitons. However,chains of solitons exist in many other higher dimensional models. In all cases their existence is warranted due to abalance of repulsive and attractive interactions between the solitons. In this section, as a simple example of suchconfiguration we will consider 2+1 dimensional planar Skyrme model [40–42]. The model is defined by the Lagrangian L = 12 ( ∂ µ φ a ) − (cid:0) ε abc φ a ∂ µ φ b ∂ ν φ c (cid:1) − U ( | φ | ) . (10)Here the triplet of real scalar fields φ a , a = 1 , , φ a · φ a = 1. In otherwords, this is a topological map φ : S → S which is classified by the homotopy group π ( S ) = Z . The planarSkyrme model supports soliton solutions, which are classified in terms of the topological invariant: Q = 18 π (cid:90) d x ε abc ε ij φ a ∂ i φ b ∂ j φ c . (11)The explicit choice of the potential term of the model (10) is important because it defines the asymptotic form ofthe field of the localized soliton. The most common choice is the O (3) symmetry breaking potential U = µ (1 − φ ) , (12)where µ is the rescaled mass parameter. Indeed, the soliton of topological degree one can be constructed using therotationally invariant ansatz φ = cos θ sin f ( r ); φ = sin θ sin f ( r ); φ = cos f ( r ) , (13)where f ( r ) is some monotonically decreasing profile function. Since the field must approach the vacuum on thespacial asymptotic, it satisfies the boundary condition cos f ( r ) → r → ∞ , i.e., f ( ∞ ) →
0. The system of fieldequations of the baby Skyrme model (10) then is reduced to a single ordinary differential equation on the function f ( r ): (cid:18) r + sin fr (cid:19) f (cid:48)(cid:48) + (cid:18) − sin fr + f (cid:48) sin f cos fr (cid:19) f (cid:48) − sin f cos fr − rµ sin f = 0 . (14)Linearizing this equation on the spatial infinity yields the asymptotic tail of the profile function f ( r ) ∼ e − µr / √ r .Evidently, this corresponds to the Yukawa-type decay with the mass of scalar excitation µ . On the other hand,the parameter µ defines the characteristic size of a Skyrmion, for the potential (12) the usual choice µ = 0 . r ∼
1. More precisely, the asymptoticequation on the scalar field of the baby Skyrmion has the form(∆ − µ ) φ a = ppp a · ∇∇∇ δ ( r ) , (15)Hence, the asymptotic field φ a may be thought of as generated by a doublet of orthogonal dipoles, with the strength p , one for each of the massive components φ and φ . The component φ remains massless [43].Let us now consider two widely separated unit charge Skyrmions. The leading term in the energy of interaction ofthe solitons, evaluated by analogy with (7), is E int ∼ µ p π cos χ e − µd √ d , (16)where the separation d between the solitons is supposed to be much larger than the size of the core of the Skyrmion, d (cid:29) /µ , and χ is the relative angle of orientation of two pairs of orthogonal dipoles of equal strength p .Thus, we conclude that the interaction of two separated Skyrmions is most attractive when the two solitons areexactly out of phase, χ = π , and it is most repulsive when the relative phase χ = 0. However, nonlinear effects mayseriously affect this result even at intermediate separation of the solitons, one can expect some deformations of thecore of the solitons may induce a repulsive quasi-elastic force which may balance the long-range attraction.Indeed, numerical simulations confirm, the model with the standard potential (12) supports existence of multi-soliton configurations [43–45]. First, as µ = 0 .
1, the attractive force between two Skyrmions of unit charge isstronger than repulsive quasi-elastic force, which is induced by deformations of the overlapping cores of the solitons.Thus, the global minimum of the sector of degree two is also rotationally invariant, further, the dipole moment of the Q = 2 baby Skyrmion is zero.Situation becomes different for baby Skyrmions of higher topological charges Q ≥ µ = 0 .
1, themultisoliton solutions represent a stable chain of aligned, charge two baby skyrmions, see Fig. 3. Another linearconfiguration of baby Skyrmions was constructed by Foster [46], it represents a chain on charge one Skyrmion withboth ends capped by two Q = 2 solitons, see Fig. 4. In such a chain each soliton is rotated by π with respect to itsneighbor around the axis of symmetry. The physical picture here is somewhat similar to the mechanism of formationof chain of dipoles in classical electrodynamics, the energy of the chain is minimal with respect to all other possibleconfigurations. Notably, the linear configuration is formed dynamically. The energy of such a chain, which yields aglobal minimum of energy in a given topological sector is slightly lower than the energy of the chain of aligned Q = 2Skyrmions, for example the Q = 10 chain of five aligned Skyrmions of degree two displayed in Fig. 3 has energy about1% higher than the Foster’s cupped chain [46].Another way to construct linear chains of baby Skyrmions is to consider the model (10) on a cylinder R × S imposing anti-periodic boundary conditions [46, 47]( φ ( x, y + β ) , φ ( x, y + β ) , φ ( x, y + β )) = ( − φ ( x, y ) , − φ ( x, y ) , φ ( x, y )) , (17)where β is the period of the chain. This parametrization fixes a relative phase χ = π between the neighboringSkyrmions. Then the energy of an infinitely charged chain becomes a function of the periodicity β , it can be minimizedto find lowest energy configuration. Numerical evaluations suggest that for µ = 1 it corresponds to the period β min ≈ . π [47].Some comments are in order here. First, linear periodic chains of planar Skyrmions exist because of the balance of ashort-range repulsion and a long-range attraction between two single solitons. The long-range attraction is mediatedby the dipole forces, however, the asymptotic form of the scalar field of the baby skyrmion and the character ofinteraction between them, strongly depends on the particular choice of the potential term V ( φ ). For example,the model (10) with the double-vacuum potential V ( φ ) = µ (1 − ( φ ) ) [45] always support rotationally invariantmulti-soliton solutions. In a contrary, the choice of the holomorphic potential V ( φ ) = µ (1 − φ ) [42] invariablyyields repulsive interaction, whatever the separation and relative orientation of the Skyrmions [49] and there are nomultisoliton solutions in such a model. More complicated form of the potential may induce weak attraction, it allowsfor existence of various multi-soliton configuration, including Skyrmions chains [50].Secondly, the existence of chains of baby Skyrmions is an intrinsic property of the model (10), it is not necessarilyto modify it by analogy with 1+1 dimensional scalar model (1). On the other hand, coupling to other fields maysignificantly affect the character of interaction between the solitons, in particular, presence of fermionic modes localizedby the baby Skyrmion yields two additional pairs of asymptotic dipoles [51], then the pattern of interaction betweenFIG. 4: Q = 11 chain of baby Skyrmions in the model (10) with the potential (12): Contour plots of the energydensity distribution and the components of the field φ , φ and φ , respectively, from top to bottom.the solitons becomes more involved. Similarly, the asymptotic forces in the gauged baby Skyrme model includecontribution from the magnetic flux associated with the soliton [52]. Even more complicated the pattern of interactionsbetween planar Skyrmions becomes in the U (1) gauged baby Skyrme model with Chern-Simons term [53]. However,for some set of parameters of the model, there are solutions, which represent linear chains of electrically chargedsolitions with associated magnetic fluxes [53].Note that Skyrmion configurations in 2+1 dimensions have recently been subject of considerable interest sincesolitons of that type were experimentally observed in magnetic structures [9]. A magnetic Skyrmion is a stablevortex-like configuration that exist in a thin film of chiral magnets, or in nematic crystals. However, the usual Skyrmeterm in the model (10) is replaced by the Dzyaloshinskii-Moriya chiral interaction term [55, 56]. Similarly, suchsolutions exist in various condensed matter systems, in particular, in chiral nematic liquid crystals [57]. Further, thesetopological excitations minimize the Oseen-Frank free energy functional, which also includes surface terms, see e.q.[58]. FIG. 5: Chains of planar Skyrmions in a nematic crystal. (Courtesy of Ivan Smalyukh).Notably, there in no multisoliton solutions in the baby Skyrme model with Dzyaloshinskii-Moriya term supplementedby the Zeeman interaction term. The situation here is similar to the case of the usual baby Skyrme model withholomorphic potential [42], the interaction between chiral Skyrmions can be only repulsive. However, modificationof the boundary conditions, or extension of the free energy functional allows for existence of bounded multisolitonsolutions [48, 59–61]. In particular, a strong boundary electric field may generate chains of dynamical planar Skyrmionsin a chiral nematic crystal [60], see Fig. 5. IV. CHAINS OF SKYRMIONS
The above-mentioned scheme of construction of baby Skyrmions can be extended to the original Skyrme model in3+1 dimensions. The field of the model is the unitary, unimodular matrix U ( r , t ) ∈ SU (2), U U † = I , which can bewritten as an expansion in quartet of scalar fields ( σ, π a ) restricted to the surface of the sphere S : U = σ + iπ a · τ a −−−→ r →∞ I . (18)Here τ a are the three usual Pauli matrices. Introducing the quartet of scalar fields φ a = ( σ, π , π , π ), restricted as φ a · φ a = 1, we can write the Lagrangian of the model in the form L = ∂ µ φ a ∂ µ φ a −
12 ( ∂ µ φ a ∂ µ φ a ) + 12 ( ∂ µ φ a ∂ ν φ a )( ∂ µ φ b ∂ ν φ b ) − µ (1 − φ a φ a ∞ ) , (19)where φ a ∞ = (1 , , , σ remains massless,while the triplet of pion fields φ k has a mass µ . The topological charge of the Skyrmion is the winding number Q = − π (cid:90) d x ε abcd ε ijk φ a ∂ i φ b ∂ j φ c ∂ k φ d . (20)Skyrmion solution of degree Q = 1 is spherically symmetric, it can be constructed on the hedgehog ansatz U ( r ) = e if ( r )ˆ r a · τ a = cos f ( r ) + i sin f ( r )ˆ r a · τ a , (21)where f ( r ) is a real monotonically decreasing function of the radial variable with the boundary conditions f (0) = π and f ( ∞ ) = 0. Setting the boundary conditions f (0) = − π and f ( ∞ ) = 0 yields the winding number Q = −
1, thisis the anti-Skyrmion solution. The profile function f ( r ) satisfies the ordinary differential equation of second order( r + 2 sin f ) f (cid:48)(cid:48) + 2 rf (cid:48) − sin 2 f (cid:18) − f (cid:48) + sin fr (cid:19) + µ sin f = 0 . (22)The solution of this equation can be found numerically.As it was outlined above, the character of long-range interaction between two separated solitons depends on theasymptotic form of the field φ a . As r → ∞ , cos f ( r ) → f ( r ) ∼ f ( r ) →
0. Then the asymptotic form of thesolution of the equation (22) is f ( r ) ∼ d πr + O (cid:18) r (cid:19) , as r → ∞ , (23)where d is some constant. Therefore the corresponding asymptotic triplet of massive pion fields, π i = sin f ( r )ˆ r i ,represents the field of three mutually orthogonal scalar dipoles of equal dipole strength d : π i = dr i πr . (24)Consequently, the pattern of the long-distance interactions of two separated Skyrmions depends on their relativeorientation [54]. If the solitons are aligned, they repel each other, by analogy with above consideration of planarSkyrmions. The strongest repulsive force occurs as one of the Skyrmions is rotated by π about the axis joining theSkyrmions. The attractive channel in interaction of the solitons corresponds to the case when one of the Skyrmionsis rotated by π about an axis perpendicular to axis R . However, for the usual choice of the potential function in(19), the attractive interaction in that channel is stronger than quasi-elastic forces of deformation of the core, theresulting charge two configuration is axially-symmetric [63–66]. Note that, asymptotically, the field of the axiallysymmetric Q = 2 Skyrmion has only one non-vanishing dipole component, associated with the axis of symmetry ofthe configuration. Indeed, composing two Skyrmions into the axially symmetric configuration, we cancel the dipolefields which are orthogonal to the axis of symmetry while the components directed along this axis will add. Thus, thedipole strength of the Q = 2 Skyrmion is approximately twice larger than Q = 1 asymptotic dipole field.In order to construct pair of bounded Skyrmions, one has to consider a nonstandard choice of potential term, whichcombines both repulsive and attractive interactions [67, 68]. However, the dipoles forces of the lightly bounded pair,which are orthogonal to the axis of symmetry, do not support existence of a linear chain of Skyrmions of unit charge.On the other hand, a Skyrme chain can be constructed in a way analogous to the construction of arrays of babySkyrmions outlined above [69]. The field of the chain is supposed to be periodic in z -direction, i.e., U ( x, y, z ) = RU ( x, y, z + β ) where matrix of iso-rotations R ∈ SO (3). Hence, each Skyrmion in the chain is iso-rotated by R withrespect to its neighbors. Effectively, it corresponds to compactification of z -axis onto a circle S . This constructioncan be thought of as a one-dimensional reduction of Skyrme crystals [70–72].Another possibility is to consider Skyrmion–anti-Skyrmion (SAS) pair in a static equilibrium, such configurationrepresents a saddle point solution, a sphaleron [25]. The SAS pair solution corresponds to a middle of a non-contractibleloop on the functional space of the Skyrme model. Physically, as said above, the charge 2 axially-symmetric Skyrmionpossesses only one asymptotic dipole field, the dipole interaction may stabilize the SAS pair.The SAS saddle point solution can be constructed numerically using the axially symmetric parametrization of thefield U ( r ) U ( r ) = cos f ( r, θ ) + iπ a · τ a sin f ( r, θ ) , (25)where the triplet of pion fields is π = sin g ( r, θ ) cos nϕ ; π = sin g ( r, θ ) sin nϕ ; π = cos g ( r, θ ) , (26)and σ = cos f ( r, θ ). In this parametrization the integer n ∈ Z counts the winding of the field in the x − y plane.For the SAS pair the boundary conditions imposed on the functions f ( r, θ ) , g ( r, θ ) are f (0 , θ ) = π ; g ( r,
0) = 0; f ( ∞ , θ ) = 0; g ( r, π ) = 2 π , (27)This yields a configuration with zero net topological charge, as one can see directly from (20). Further generalizationof this construction is possible, if we impose g ( r, π ) = mπ , where m is an integer number, which counts the number0FIG. 6: Skyrmion–anti-Skyrmion chains. Reprinted (without modification) from [26], with STM Permission.of constituents in the resulting SAS chain configuration [26]. Together with the winding number n of each individualSkyrmion appearing in (26), it yields the net topological charge of the axially symmetric chain: Q = n − ( − m ) . (28)Clearly, the case m = 1 corresponds to the (multi)-Skyrmions of topological charge Q = n , while m = 2 gives a pairwith zero net topological charge consisting of a charge Q = n Skyrmion and a charge Q = − n anti-Skyrmion. Moregeneral, for odd values of m the winding number n coincides with the topological charge of the Skyrmion Q whereaseven values of m correspond to the deformations of the topologically trivial sector. Thus we can construct a chain ofcharge n Skyrmions and charge − n anti-Skyrmions placed along the axis of symmetry in alternating order. Note thatthe solitons are dynamically arranged in such a linear configuration, in a contrary, the periodic Skyrmions constructedon the ansatz U ( x, y, z ) = RU ( x, y, z + β ), where R is a rotation matrix, by definition represent a one-dimensionalcluster of equally spaced Skyrmions. Such a chain, may only contract or extend itself as the period β varies [69].In Fig. 6 we represent the energy density isosurfaces of the | Q | = 2 SAS chains for zero pion mass. Notably, thereis no such a saddle point solution for a single | Q | = 1 Skyrmion–anti-Skyrmion pair [25, 26]. However, coupling togravity may provide an additional attractive force, it stabilizes the SAS pair in curved spacetime [73, 98]. V. CHAINS OF Q-BALLS AND BOSON STARS
In the previous sections we considered linear chains of topological solitons in various spatial dimensions. Anothertype of chains can be constructed in models which support non-topological solitons. One of the simplest examplesin flat space is given by Q-balls, stationary spinning configurations of a complex scalar field with a suitable self-interaction potential [11–13]. When Q-balls are coupled to gravity so-called boson stars emerge, which representcompact stationary configurations with a harmonic time dependence of the scalar field [14, 15].Both Q-balls and boson stars carry a Noether charge associated with an unbroken continuous global symmetry.This charge is proportional to the angular frequency of the complex boson field and represents the boson particlenumber of the configurations [12, 13].Localized Q-ball solutions appear in a simple 3+1 dimensional complex scalar model with the Lagrangian [13] L = | ∂ µ φ | − V ( | φ | ) , (29)and appropriate choice of the non-linear potential V ( | φ | ). The fundamental spherically symmetric solution in thiscase represents a stationary spinning configuration φ = f ( r ) e iωt where f ( r ) is the real function of radial variable.This function satisfied the equation of motion d fdr + 2 r dfdr + ω f = 12 dUdf . (30)1with the boundary conditions ∂ r f ( r ) | r =0 = 0 and f ( r ) | r = ∞ = 0. Then the solution of the equation (30) must decayasymptotically as f ∼ r e − √ µ − ω r + O (1 /r ) , (31)where µ = U (cid:48)(cid:48) (0) is the mass of the scalar excitation. In other words, the configuration is exponentially localizedat the origin.Evidently, the properties of the Q-balls depend on the particular choice of the potential and its parameters. It isconvenient to make use of the non-linear sextic potential [75–78] U ( | φ | ) = a | φ | − b | φ | + c | φ | , (32)where the positive parameters are taken as a = 1 . b = 2 and c = 1. Setting b = c = 0 reduces the model to theusual Klein-Gordon system in the flat space, which does not support any localized solutions.As the angular frequency tends to its upper critical value, ω ∼
1, the Q-balls become linked to the perturbativeexcitations of the Klein-Gordon model, φ ∼ √ r J l + ( r ) Y ln ( θ, ϕ )where J l + ( r ) is the Bessel function of the first kind of order l and Y ln ( θ, ϕ ) = (cid:115) l + 14 π ( l − n )!( l + n )! P nl (cos θ ) e inϕ are the usual spherical harmonics with n ∈ [ − l, l ]. Here P nl (cos θ ) are the associated Legendre functions. Thespherically symmetric fundamental Q-ball corresponds to the spherical harmonic Y while the simplest non-sphericalexcitation corresponds to the harmonic Y , and induces a pair of oscillating perturbations with opposite phases. Suchexcitations can be considered as droplets of bosonic condensate, they may exist in various models, in particular in aBose-Einstein condensate with dipole-dipole interaction [79].It was pointed out that, similarly to the case of dipole-dipole interactions between the Skyrmions, the character ofthe interaction between Q-balls in Minkowski spacetime depends on their relative phase [16, 80]. If the Q-balls are inphase, the interaction is attractive, if they are out of phase, there is a repulsive force between them. Thus, an axiallysymmetric excitation of the complex scalar field of the form Y is in general not stable in Minkowski spacetime,however the gravitational interaction may stabilize it. Consequently, a decrease of the angular frequency increases thesize of the configuration producing a binary system of boson stars spinning in opposite phases. Such a pair representsa building block of linear chains of Q-ball in curved space-time [81, 82].Let us now consider a self-interacting complex scalar field φ , which is minimally coupled to Einstein gravity. Thecorresponding action of the system is S = (cid:90) √− g (cid:18) R α − L (cid:19) d x, (33)where R is the Ricci scalar curvature with respect to the metric g µν , g denotes the determinant of the metric, α = 4 πG is the gravitational coupling constant, G is Newton’s constant, and L is the matter field Lagrangian (29). Below weconsider axially-symmetric configurations, which can be parameterized by the ansatz φ = φ ( r, θ ) e iωt and make use ofthe Lewis-Papapetrou metric ds = − f dt + mf (cid:0) dr + r dθ (cid:1) + r sin θ lf dϕ (34)where the metric functions f, m and l are functions of r and θ only.Below we will consider axially symmetric configurations, composed of several constituents, whose centers are lo-calized on the symmetry axis. Such solutions can be obtained numerically [81, 82], the results show that, indeed,there are multi-soliton linear solutions with k nodes on the symmetry axis. For relatively small values of gravitationalcoupling α and the angular frequencies a bit lower than the mass threshold, all solitons assembled into a chain possesssimilar sizes, shapes and distance from their next neighbors. However, this very democratic picture changes as wemove along the set of branches that form as the angular frequency ω is varied (for a given coupling α ). In Fig.7 wedisplayed a few examples of such chains of boson stars at α = 0 .
25 and ω = 0 . Chains of boson stars: Energy density isosurfaces on the fundamental branch for α = 0 .
25 at ω = 0 .
80 (in differentscales).
As said above, pairs and chains of boson stars do not possess flat space limit, repulsive interaction between con-stituents of the chain should be balanced by some attractive force. Such a force can appear if we consider spinning U (1) gauged Q-balls with non-zero angular momentum [83].Notably, there are two families of the spinning Q-balls with positive and negative parity, the corresponding solutionsare symmetric or anti-symmetric with respect to reflections in the equatorial plane [78]. Apart the the above-mentionedfundamental spherically symmetric Q-balls, there are both radially and angularly excited Q-balls [17, 76–78]. Theradially excited solutions are still spherically symmetric, however the scalar field possesses some set of radial nodes.Such radially excited gauged Q-balls also exist in the U (1) gauged model [84]. The angularly excited solutions withsome set of nodes in θ -direction, can be parity-even, or parity-odd.The angularly excited axially symmetric Q-balls with non-zero angular momentum possess an additional azimuthalphase factor of the spinning field [75, 76, 78]. In the U (1) gauged theory such configurations induce a toroidal magneticfield [85, 86], these solutions can be viewed as vortons, the finite energy localized spinning loops with non-zero angularmomentum and magnetic flux [75, 87–89].We consider two-component U (1) gauged Friedberg-Lee-Sirlin-Maxwell model [83, 86], which describes a coupledsystem of a real self-interacting scalar field ψ and a complex scalar field φ , minimally interacting with the Abeliangauge field A µ . The corresponding Lagrangian density is L = − F µν F µν + ( ∂ µ ψ ) + | D µ φ | − m ψ | φ | − U ( ψ ) , (35)where D µ = ∂ µ + igA µ denotes the covariant derivative. Here F µν = ∂ µ A ν − ∂ ν A µ is the electromagnetic fieldstrength tensor, g is the gauge coupling constant and m is the scalar coupling constant. The symmetry breakingquartic potential of the real scalar field ψ is U ( ψ ) = µ (1 − ψ ) , Notably, the model (35) can be considered as ageneralization of the Abelian Higgs model, in other words gauged Q-ball behaves like a superconductor [90] with thefield component ψ playing a role of the order parameter.The Lagrangian (35) is invariant under the local U (1) gauge transformations of the fields, the correspondingconserved Noether current is j µ = i ( φD µ φ ∗ − φ ∗ D µ φ ) . (36)This current is a source in the Maxwell equation ∂ µ F µν = gj ν (37)Two other dynamical equations correspond to the variations of the Lagrangian (35) with respect to the fields ψ and φ , respectively: ∂ µ ∂ µ ψ = − m ψ | φ | + 2 µ ψ (cid:0) − ψ (cid:1) ,D µ D µ φ = − m ψ φ , (38)3FIG. 8: Axially-symmetric n = 1 chain of gauged Q-balls: The field components X (upper left) and Y (upper right), theelectric charge density distribution (bottom left), and the magnitude of the magnetic field distribution (bottom right) of the k = 6 chain [83]. Below we consider stationary spinning axially-symmetric solutions of the model (35). The correspondingparametrization of the scalar fields is ψ = X ( r, θ ) , φ = Y ( r, θ ) e i ( ωt + nϕ ) , (39)where ω is the angular frequency of the spinning complex field φ , and n ∈ Z is the azimuthal winding number. Further,in the static gauge the electromagnetic potential is A µ dx µ = A ( r, θ ) dt + A ϕ ( r, θ ) sin θdϕ , (40)hence, the spinning gauged Q-ball with non-zero angular momentum possess both electric charge and magnetic flux.Clearly, both electric and magnetic field of the Q-ball contribute to the energy of interaction between two solitons,together with Yukawa interactions mediated by the scalar fields.In order to construct saddle point solution, which represent a pair of Q-balls in a static equilibrium, we have tobalance the attractive and repulsive forces. Note that the real scalar component ψ may induce only attractive interac-tion, while the spinning complex field φ of two Q-balls in opposite phases generates repulsive force. Further, electriccharge of the components of the pair always yields a repulsive Coulomb interaction, this piece can be compensated bythe solenoidal magnetic field of the pair. Indeed, this pattern is confirmed with numerical simulations [83], we foundlinear chains of spinning gauged Q-balls, located symmetrically with respect to the origin along the symmetry axis.These solutions are classified by the winding number n and the number of constituents k . In Fig. 8 we displayed anexample of such a chain with 6 components. Note that the neighboring complex components of the system are inopposite phases, as expected. Interestingly, there are two branches of solutions of the system (35), the chains existonly in a frequency range, which is restricted from below by some critical value of angular frequency. The electricrepulsion provides a leading contribution to the interaction energy on the lower in energy branch, in a contrary,the magnetic energy rapidly grows along the upper, magnetic branch, it extends forward as the frequency increases.Peculiar feature if this branch is that the strong magnetic field of the vortex destroys the superconductive phase insome region inside the spinning Q-ball [83, 86]. VI. MONOPOLE-ANTI MONOPOLE CHAINS
As we have seen in previous section, some localized configurations may possess a few different types of asymptoticfields. In such a situation all of the corresponding interactions in a pair of separated solitons have to be balanced4FIG. 9:
Monopole-anti-monopole chains: The magnetic charge density distributions (upper plots), the energy densitydistributions (middle plots)) and the magnitude of the Higgs field (bottom plots), of the n = 1 , m = 3 and n = 1 , m = 6chains. to provide a zero net force. An interesting example of such a system with different types of asymptotic fields is themonopole-antimonopole chains in the non-Abelian Yang-Mills-Higgs theory [22, 24, 91–95].The SU (2) Yang-Mills-Higgs theory has the Lagrangian density L = 12 Tr ( F µν F µν ) + 14 Tr ( D µ Φ D µ Φ) + λ Tr (Φ − (41)with gauge potential A µ = A aµ τ a , field strength tensor F µν = ∂ µ A ν − ∂ ν A µ + ie [ A µ , A ν ], and covariant derivative ofthe Higgs field D µ Φ = ∂ µ Φ + ie [ A µ , Φ]. Here e is the gauge coupling and λ is the strength of the scalar self-coupling.The static solutions of the corresponding field equations can be constructed numerically by employing of the axially-symmetric ansatz [91–93] A µ dx µ = (cid:18) K r dr + (1 − K ) dθ (cid:19) τ ( n ) ϕ e − n sin θ (cid:32) K τ ( n,m ) r e + (1 − K ) τ ( n,m ) θ e (cid:33) dϕ , Φ = H τ ( n,m ) r + H τ ( n,m ) θ , (42)where the su (2) matrices τ ( n,m ) r , τ ( n,m ) θ , and τ ( n ) ϕ are defined as a product of these vectors with the usual Pauli5matrices τ a : τ ( n,m ) r = sin( mθ ) τ ( n ) ρ + cos( mθ ) τ z ,τ ( n,m ) θ = cos( mθ ) τ ( n ) ρ − sin( mθ ) τ z ,τ ( n ) ϕ = − sin( nϕ ) τ x + cos( nϕ ) τ y , where τ ( n ) ρ = cos( nϕ ) τ x + sin( nϕ ) τ y and ρ = (cid:112) x + y = r sin θ . Note that the ansatz (42) is axially symmetric, aspatial rotation around the z -axis can be compensated by an Abelian gauge transformation U = exp { iω ( r, θ ) τ ( n ) ϕ / } .Variation of the Lagrangian (41) yields a system of six second-order non-linear partial differential equations in thecoordinates r and θ , these equations can be solved numerically, see [91–93]. The well-known spherically symmetric ‘tHooft–Polyakov ansatz is recovered as we impose the constraints K = K = H = 0 , K = K = K ( r ) , H = H ( r ).The generalized monopoles (42) are characterized by two integers, the winding number m in polar angle θ andthe winding number n in azimuthal angle ϕ . Making use of the usual definition of the topological charge of theconfiguration and taking into account the boundary conditions of the profile functions, we obtain [93] Q = 18 π (cid:90) S Tr ( ˆΦ d ˆΦ ∧ d ˆΦ) = n − ( − m ) , (43)where ˆΦ is the su (2) normalized Higgs field. Hence, the configurations with even values of the winding number m areaxially-symmetric deformations of the topologically trivial sector, while the configurations with odd values of m aredeformations of the fundamental ‘t Hooft–Polyakov solution [93]. Note that here we discuss solutions of the secondorder equations, they do not satisfy the first order BPS monopole equations.Simplest non-trivial solution represent a monopole-anti-monopole pair in a static equilibrium. The reason of exis-tence of such a solution, a magnetic dipole, is related to an exact balance of short-range Yukawa interactions mediatedbe the vector and scalar fields.Indeed, the pattern of interaction between non-abelian monopoles does not correspond to a naive picture of elec-tromagnetic Coulomb interaction between two point-like magnetic charges. First, there is an attractive force betweenwell separated monopoles, it is mediated by the A µ component of the Yang-Mills field. However, this field is masslessonly on the spatial infinity, such interaction is short-ranged. On the other hand, there is a scalar attraction mediatedby a massive Higgs boson, so the monopoles attract each other with double force.The situation is different in the BPS limit, then both the gauge and the scalar field possess long-range Coulombasymptotic. Further, in such a case repulsive gauge interaction between the monopoles is always balanced by thescalar interaction for any separation, any system of BPS monopoles can be static.It was pointed out by Taubes [21] that, for non-BPS monopoles, the massive vector bosons A ± µ also mediate theshort-range Yukawa interactions between the monopoles and contribute to the interaction energy. Furthermore, thesign of this contribution to the net interaction potential depends on the relative orientation of the monopoles. Themonopole-anti-monopole pair is a saddle point configuration where the attractive short-range forces, mediated both bythe A µ vector boson and the Higgs boson, are balanced by the repulsive interaction due to massive vector bosons A ± µ with opposite orientation in the group space [95]. The effective net potential of the interaction between a monopoleand an anti-monopole is attractive for large separation and it is repulsive on a short distance, it resembles as that ofwell known Van der Waals molecular potential. The pair is a sphaleron solution in the topologically trivial sector, itcorresponds to the middle of non-contractible loop on the configuration space of the system. This loop correspondsto the creation of a monopole-antimonopole pair with relative orientation in the internal space − π from the vacuum,separation of the pair, rotation of the monopole by 2 π , and annihilation of the pair back into vacuum [21].Furthermore, each topological sector of the Yang-Mills-Higgs model (41) contains besides the (multi)monopolesolutions further regular, finite mass solutions, which do not satisfy the first order Bogomolnyi equations, but onlythe set of second order field equations, even for vanishing Higgs potential [21]. Such solutions form saddlepoints ofthe energy functional, and possess a mass above the BPS bound.The simplest solution of that type, m = 2, n = 1 monopole-anti-monopole pair, posses two zeros of the Higgs fieldlocated symmetrically on the positive and negative z axis, the peaks of the energy density distribution are associatedwith these zeros. Further generalizations of this solution correspond to the chains of monopoles and anti-monopoles,each carrying charge n = ± m = 3 , n = 1 at λ = 0 .
5. The chains with even number of constituents m are deformations of the topologicallytrivial sector, while the chains with odd values of m represent deformations of the charge n monopole. Positionsof the partons in a chain depend on these integers, for n = 1 , λ the solitons are located on the symmetry axis. Note that the asymptotic field of the monopole-anti-monopole pair6represents a magnetic dipole [91] A µ dx µ ∼ d r sin θ τ dϕ (44)where d is the dipole moment. The emergency of the chains can be explained as formation of the system of aligneddipoles, just in the same way as Skyrmion chains are formed.Note that, as the charge n of the monopole and anti-monopole in the chain increases beyond n = 2, it becomesfavorable for the monopole-anti-monopole system to form a system of vortex rings, in which the Higgs field vanisheson a ring centered around the symmetry axis [92–94]. For larger values of the scalar mass also more complicatedconfigurations can appear, which consist of monopole-anti-monopole pairs or chains as well as vortex rings, thesituation becomes much more complex as gravity is included into consideration [96], and/or an electric charge isadded to the monopoles in the chain [97].Remarkably, that there is certain similarity with the Skyrmion-anti-Skyrmion chains we discussed previously inthe Section 4. In both cases the soliton-anti-soliton chains represent axially symmetric saddle-point sphaleron-typesolutions which are characterized by two integers, one of which yields the number of constituents in the chain, andanother corresponds to the absolute value of the topological charge of a component. Further, the component of theSkyrme field σ shows a clear relation to the corresponding behavior of the magnitude of the Higgs field | Φ | in theYang-Mills-Higgs system [25]. However, the dipole-dipole interaction between the components of the SAS pair isweaker than the short-range Yukawa interactions in the monopole-anti-monopole pair, in the flat space the SAS pairmay exist only if the topological charge of the components is higher than two [25]. However, the effect of the pionmass term [26] or coupling to gravity [73, 98], open a possibility for existence of the Skyrmion-anti-Skyrmion chainswith constituents carrying unit topological charge.Our final remark is related with a possibility to construct Euclidean counterparts of the MAP solutions in fourdimensional Yang-Mills theory [99]. These non-self-dual instanton-antiinstanton saddle point configurations are ob-tained numerically by analogy with the construction of the above-mentioned axially-symmetric MAP solution. Sincethe holonomy of Yang-Mills instantons provides a very good approximations to Skyrmions [100], it is of no surprisethat the holonomy of the chains of interpolating calorons-anticalorons gives a nice approximation to the correspondingSkyrmion-anti-Skyrmion chains [101]. VII. CONCLUSIONS
The main purpose of this short review was to provide a comparative analysis of different types of linear chainsof non-self dual solitons in various models. Such solutions exist because of balancing of repulsive and attractiveinteractions, in most cases they represent sphaleron-like field configurations. Apparently, any multisoliton solutionin one spatial dimension represents a chain, however such static configuration may exist only if there is a dynamicequilibrium between the repulsive and attractive forces. A particular example is a real scalar theory coupled tothe Dirac fermions, here scalar repulsion between the kinks is evened out by fermionic exchange interaction. Otherhigher-dimensional examples include self-gravitating chains of boson stars, where the scalar repulsion is compensatedby the gravitational attraction of the solitons and Skyrmion-anti-Skyrmion chains, which are stabilized by the dipole-dipole interactions. The pattern of interactions between the constituents of linear chains of spinning gauged Q-ballsin two-component Friedberg-Lee-Sirlin model is more complicated, it includes both scalar and electromagnetic forces.Similarly, the monopole-anti-monopole chains exist because of precise balance of the scalar and gauge short-rangeYukawa interactions mediated by the Higgs boson and by the massive vector bosons A ± µ , A µ , respectively. Notably,there is a certain similarity between the monopole-anti-monopole chains in the non-abelian Yang-Mills-Higgs model,and Skyrmion-anti-Skyrmion chains, related with the Atiyah-Manton construction [100]. Acknowledgements
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