Vortices in the extended Skyrme-Faddeev model

V ortices in the extended Skyrme-F add eev mo del L. A. F erreira a, 1 ; J. J¨ aykk¨ a b, 2 ; Nobuyuki Saw ado c, 3 ; Kouic hi T o da d, 4 ; a Instituto de F ´ ısica d e S˜ ao Carlos; IFSC/USP; Univ ersidade de S˜ ao Pa ulo - USP Caixa Posta l 369, CE P 13560-97 0, S˜ ao Carlos-SP , Brazil b Sc ho ol of Mathematics, Univ ersit y of Leeds LS2 9JT Leeds, United Kingdom c Departmen t of Physic s, T okyo Univ ersit y of Science, No da, Chiba 278-8 510, Japan d Departmen t of Mathematical Ph ysics, T o y ama Prefectural Univ ersit y , Kurok a w a 5180, Im izu, T o y ama, 939-0398, Japan Abstract W e construct analytical and n umerical vorte x solutions for an exte nded Skyrme- F addeev mo del in a (3 + 1) dimensional Mink o wski space-time. The extension is obtained b y addin g to the L agrangian a quartic term, wh ich is the square of the kinetic term, and a p oten tial wh ic h breaks the S O (3) symm etry do wn to S O (2). The constru ction m ak es use of an ansatz, in v arian t und er the join t action of the in ternal S O (2) and thr ee comm uting U (1) su b groups of the P oincar ´ e group, and whic h r ed uces the equations of motion to an ordinary differen tial equation for a profile fu nction dep ending on the distance to the x 3 -axis. The vortic es hav e fin ite energy p er unit length, an d hav e w a v es prop agating along them w ith the s p eed of ligh t. The analytical vortice s are obtained for sp ecial choice of p otent ials, an d the n umerical ones are constructed using the Su ccessive Over Relaxation m etho d for more general p ote ntia ls. Th e sp ectrum of solutions is analyzed in d etail, sp ecially its d ep endence up on sp ecial com binations of coupling constan ts. 1 e-mail: laf @ifsc .usp. br 2 e-mail: juh aj@ik i.fi 3 e-mail: saw ado@p h.nod a.tus.ac.jp 4 e-mail: kou ichi@ yukaw a.kyoto-u.ac.jp 1 In tro duction The so- called Skyrme-F addeev mo del w as in tro duced in the sev en ties [1] as a generali- zation to (3 + 1) dimensions of the O (3) non-linear sigma mo del in (2 + 1) dimensions [2]. The Skyrme term, quartic in deriv ativ es of the field, balances the quadratic kinetic term and according to Derrick ’s theorem, allow s the existence of stable solutions with non-trivial Hopf top ological c harges. Due to the highly non-linear c haracter of the mo del and the lac k o f symmetries, the first soliton solutions w ere only constructed in the late nineties using numeric al metho ds [3, 4, 5 , 6]. Since then the interes t in the mo del has increased considerably and it has found applications in many areas of ph ysics due mainly to the knotted c haracter of the solutions [7 ]. The numeric al effor ts in the construction of the solutions hav e impro v ed our understanding of the prop erties of the mo del [8] and ev en the scattering of knotted solitons has b een inv es tigated [9]. One of the asp ects of the mo del that has attracted considerable a tten tion has b een its connection with gauge theories. F addeev and Niemi hav e conjectured that it migh t describ e the low energy limit of the pure S U (2) Y ang-Mills theory [1 0]. They based their argumen t on a decomp osition of the ph ysical degrees of freedom of t he S U (2) connection, prop osed in the eighties b y Cho [11], a nd inv o lving a triplet of scalar fields ~ n taking v alues on the sphere S 2 ( ~ n 2 = 1 ). The conjecture, which is quite contro v ers ial [1 2], states that the low energy effectiv e action of the S U (2) Y ang -Mills theory is t he Skyrme-F addeev action, and the knotted solitons would describ e glueballs or eve n v a cuum configurations. The fa ct that the Skyrme-F addeev mo del has an O (3) symmetry , and so p o ssesse s Goldstone b oson excitations, is one of the man y difficulties facing the conjecture, and some m o difications of it w ere in fact prop osed [13]. Any c hec k of suc h t ype of conjectures is of course v ery difficult to p erform since it m ust in v olv e non-p erturbativ e calculations in the strong coupling regime of the Y ang-Mills theory . Ho w ev er, Gies [14] h as calculated the W ilsonian one loo p effectiv e action for the pure S U (2) Y ang-Mills t heory assuming Cho’s decomp osition, and found that the Skyrme-F addeev action is indeed part of it, but additional quartic terms in the deriv at iv es o f the triplet ~ n are unav oidable. In fa ct, the first n umerical Hopf solitons w ere first constructed for the Skyrme -F addeev mo del mo dified b y a quartic term [3] w hic h is the square o f t he kinetic term. Ho w ev er, the solito n solutions in [3] w ere constructed for a sector of the theory where the signs of the coupling constan ts disagree with those indicated b y G ies’ calculations. The addition of quartic terms has t he dra wbac k of making the Lag rangian dep enden t on terms whic h are quartic in time deriv at iv es and so the energy is not p ositiv e definite. How ev er, as a quantum field theory the Skyrme-F addeev mo del is not renormalizable b y p ow er coun ting and ha s to b e considered as a low energy effectiv e theory . In addition, under the Wilsonian renormalization group flo w the square of the 1 kinetic term is as una v oidable as the Skyrme quartic term. Therefore, it is quite natural to in v estigate the prop erties of the Skyrme-F a ddeev mo del with suc h mo difications. In this pap er w e consider an extended Skyrme-F addeev mo del defined b y the La- grangian L = M 2 ∂ µ ~ n · ∂ µ ~ n − 1 e 2 ( ∂ µ ~ n ∧ ∂ ν ~ n ) 2 + β 2 ( ∂ µ ~ n · ∂ µ ~ n ) 2 − V ( n 3 ) (1.1) where ~ n is a triplet of real scalar fields taking v alues on the sphere S 2 , M is a coupling constan t with dimension of (length) − 1 , e 2 and β are dimensionless coupling constan ts, and the p otential is a functional of t he t hird comp o nen t n 3 of the triplet ~ n . Note that the p o ten tial breaks the O (3) symmetry of the original Skyrme-F addeev down to O (2), the gro up of rotat ions on the plane n 1 n 2 , a nd so eliminating tw o of the three Goldstone b oson degrees of f reedom. In this pap er the main role of potential is to stabilize the v ortex solutions. The first exact v ortex solutions for t he t heory ( 1.1) w ere constructed in [15] for the case where the p otential v anishes, and b y exploring the integrabilit y prop erties of a sub- mo del of (1 .1). In order to describ e those exact v ortex solutions it is b etter t o p erform the stereographic pro jection of the targ et space S 2 on to the plane pa rameterized b y the complex scalar field u and related to ~ n b y ~ n =  u + u ∗ , − i ( u − u ∗ ) , | u | 2 − 1  /  1+ | u | 2  (1.2) It w as show n in [15] that the field configurat ions of the for m u ≡ u ( z , y ) u ∗ ≡ u ∗ ( z ∗ , y ) for β e 2 = 1 V = 0 (1.3) are exact solutions of (1.1), where z = x 1 + i ε 1 x 2 and y = x 3 − ε 2 x 0 , with ε a = ± 1, a = 1 , 2, and x µ , µ = 0 , 1 , 2 , 3, are the Cartesian co ordinates of the Mink o wski space-time. Despite the fa ct that (1.3) constitutes a v ery large class o f solutions, no finite energy solutions w ere found within it. If the dep endence of the u field up on the v ariable y is in the fo rm o f phases lik e e i k y , t hen one finds solutions with finite energy p er unit of length along the x 3 -axis. The simplest solution is of the form u = z n e i k y , with n inte ger, a nd it corresp onds to a v ortex parallel to the x 3 -axis a nd with w av es tr a v eling along it with the sp eed of lig h t. More general solutio ns of the class (1.3) we re constructed in [16], including m ulti-v ortices separated fro m each other a nd a ll par allel t o the x 3 -axis. The ideas of [15] w ere generalized to an extended Skyrme-F addeev defined on the target space C P N , p ossessing N − 1 complex scalar fields u i , and the class of solutions constructed is like (1.3), where the fields u i ’s are arbitr ary f unctions of z and y [17]. Note that the solutions (1.3) are no t solutions of the original Skyrme-F addeev mo del, since that correspo nds to 2 β = 0 , a nd ( 1.3) requires the condition β e 2 = 1. If one tak es the limit β → 0 and 1 e 2 → 0 with k eeping the pro duct β e 2 constan t and equal to unit y , o ne observ es that (1.1) reduces to the C P 1 mo del (if V = 0 ). Therefore, the configurations (1.3) a re also solutions of the four dimensional C P 1 mo del. The ideas of [17] w ere then used to construct multi v ortex solutions for the the four dimensional C P N mo del [18, 19]. Aproximate v ortex solutions for the pure Skyrme-F addeev mo del, without the p otential and β terms in (1.1), were constructed in [20 ]. The static energy densit y ( H static = −L ) asso ciated to (1.1) is p ositiv e definite if V > 0, M 2 > 0, e 2 > 0 and β < 0. That is the sector explored in [3] and where Hopf soliton solutions w ere first constructed ( for V = 0). In a ddition, that is also the sector explored in [21] but with a dditional terms in v olving second deriv ativ es of the ~ n field, and where Hopf solitons w ere also constructed. The static energy densit y of (1.1) is also p ositiv e definite for V > 0 if M 2 > 0 ; e 2 < 0 ; β < 0 ; β e 2 ≥ 1 (1.4) That is the sector that agrees with the signature of the terms in the one lo op effectiv e action calculated in [14] and that we will consider in this pap er. Static Hopf solitons w ere constructed in [2 2, 23] for the sector (1.4) (with V = 0) and their quan tum excitations, including comparison with glueball sp ectrum, w ere considered in [24]. An in teresting fea- ture o f the Hopf solitons constructed in [22 ] is that t hey shrink in size and t hen disapp ear as β e 2 → 1, whic h is exactly the p oin t where the v ortex solution of the class (1.3) exists. The a im of the presen t pap er is to in v estigate if v ortex solutions for the mo del (1 .1) con tin ue to exist when the condition β e 2 = 1 is relaxed, and so if they co-exist with the Hopf solitons in [22]. W e also aim at the study of their prop erties and stability . The idea is to kee p the solutions a s close to those of the class (1 .3) as p o ssible. In order to do that, w e follow the ideas of [25] and implemen t an ansatz based on the O (2) internal symmetry giv en b y the transformations u → e iα u , together with t hree comm uting transformations of the P oincar ´ e group giv en b y rotations on the plane x 1 x 2 , and t ranslations in the direc tions x 0 and x 3 . W e imp ose the field configurat ions to b e in v aria n t under the diagonal subgroups of the tensor pro duct of the internal O (2 ) g roup with eac h one of t he three commuting one parameter subgroups of the Poincar ´ e gr oup. The r esulting ansatz is giv en by u ≡ f ( ρ ) e i ( n ϕ + λ z + k τ ) (1.5) where n is an integer for u to b e single v alued, and λ , k are real dimensionles s parameters, and where w e hav e used the dimensionless p olar co ordinates ( ρ, ϕ, z , τ ), defined by x 0 = c t = r 0 τ x 1 = r 0 ρ cos ϕ x 2 = r 0 ρ sin ϕ x 3 = r 0 z (1.6) 3 and where w e hav e in tro duced a length scale r 0 giv en b y r 2 0 = − 4 M 2 e 2 (1.7) whic h is p ositiv e since w e are dealing with e 2 < 0 (see (1.4) ). Not e that when λ = ± k and f ∼ ρ ± n , the configura tions (1.5) ar e of the type (1.3). The ansatz (1.5) is in fact a generalization to (3 + 1) dimensions of the ansatz used in the Bab y Skyrme mo dels [26, 27]. The ty p es of p ot en tial w e will consider in this part are o f the form V ( n 3 ) ≡ µ 2 2 (1 + n 3 ) 2 − 2 N (1 − n 3 ) 2+ 2 N (1.8) where N is a non- v anishing integer, µ a real coupling constant. It is in teresting to note that when the in teger n of (1.5) has the same m o dulus as N of (1.8), one obtains analytical solutions of the form u ( ρ, ϕ, z , τ ) =  ρ a  N e i [ ε N ϕ + k ( z + τ )] (1.9) with ε = ± 1, and where a = | N | h ( − e 2 ) ( β e 2 − 1) M 4 4 µ 2 i 1 / 4 . Such exact solutions are v alid for all v alues of the coupling constan ts. In particular, for the case β = 0, ( 1.9) are exact solutions of the theory L = M 2 ∂ µ ~ n · ∂ µ ~ n − 1 e 2 ( ∂ µ ~ n ∧ ∂ ν ~ n ) 2 − µ 2 2 (1 + n 3 ) 2 − 2 N (1 − n 3 ) 2+ 2 N (1.10) whic h is the prop er Skyrme-F addeev mo del in the presenc e of a po ten tial. In this case we ha v e a = | N | h e 2 M 4 4 µ 2 i 1 / 4 , and µ 2 e 2 > 0. In some cases w e ha v e been unable to find a n umerical solution whic h is exp ected in the analytical set, i.e. (1.9). Reasons for this are presen ted b elow. Apart fro m those cases, w e ha v e che c k ed n umerically the existence of the ab o v e solution. The n umerical sim ulations are p erformed using a standard tec hnique for a differen tial equation, the Succe ssiv e Ov er Relaxation (SOR) metho d. In order to further confirm the accuracy and c orrectness of the SOR co de, some of the results were repro duced b y a n indep enden t co de using Newton’s metho d, giving t ypical differences o f the or der of less than 10 − 4 . The pap er is orga nized as follows. In the next section we briefly describ e the extended Skyrme-F a ddeev mo del. The equations o f motion are also in tro duced in Sec. 2. W e discuss the Hamiltonia n densit y of the mo del in Sec. 3. The metho d and the solutions of the integrable sector of the presen t mo del are discussed in Sec. 4. In Sec. 5, w e sho w the n umerical solutions. Sec. 6 is dev oted t o note briefly p oten tial ph ysical applications of our solutions. A brief summary is presen ted in Sec. 7. 4 2 The mo del In terms of the complex scalar field u in tro duced in (1.2) the L agrangian (1.1) b ecomes L = 4 M 2 ∂ µ u ∂ µ u ∗ (1+ | u | 2 ) 2 + 8 e 2 " ( ∂ µ u ) 2 ( ∂ ν u ∗ ) 2 (1+ | u | 2 ) 4 +  β e 2 − 1  ( ∂ µ u ∂ µ u ∗ ) 2 (1+ | u | 2 ) 4 # − V  | u | 2  (2.1) where w e hav e used the fa ct that n 3 is a functional of | u | 2 only , and so is the p o ten tial. The Euler-Lagrang e equations following from (2.1), or (1.1), reads  1+ | u | 2  ∂ µ K µ − 2 u ∗ K µ ∂ µ u = − u 4  1+ | u | 2  3 V ′ (2.2) where V ′ = ∂ V ∂ | u | 2 , and K µ ≡ M 2 ∂ µ u + 4 e 2 [( ∂ ν u∂ ν u ) ∂ µ u ∗ + ( β e 2 − 1) ( ∂ ν u ∂ ν u ∗ ) ∂ µ u ] (1+ | u | 2 ) 2 (2.3) W e p oint out tha t the theory (2.1) p ossesses an in tegrable sector defined by the conditio n ( ∂ µ u ) 2 = 0 (2.4) Suc h condition was first discov ered in the contex t of the C P 1 mo del using the generalized zero curv ature condition for integrable theories in any dimension [28], and then applied to man y mo dels with target space b eing the sphere S 2 , or C P 1 (see [29] for a r eview). It leads to an infinite num b er of lo cal conserv ed curren ts. Indeed, (2.4) together with the equations of motion (2 .2) imply the conserv ation of the infinity of curren ts g iv en by J G µ ≡ K µ δ G δ u − K ∗ µ δ G δ u ∗ (2.5) where G is an y functional o f | u | 2 only . F or the case where t he p ot en tial v a nishes, the set of conserv ed currents is considerably enlarged since G can b e an arbitrary functional of u and u ∗ , but not of their deriv atives . If in a ddition to the condition (2.4) one takes V = 0 and β e 2 = 1, then the equations of motion reduce to ∂ 2 u = 0. It is in t hat in tegrable sector that the solutions (1.3) lie, and were studied in [15]. F or theories defined b y Lag rangians whic h are functionals of the Skyrme term only (pullbac k of the ar ea f orm of the sphere) the curren ts o f t he form (2.5) are No ether curren ts asso ciated to the area preserving diffeomorphisms of S 2 [30]. It is po ssible to define conditions w eak er than (2.4) that lead t o integrable theories asso ciated to Ab elian subgroups o f the g roup of the area preserving diffeomorphisms [31]. 5 Substituting the a nsatz (1.5) in to the equations of motion (2.2) w e get a n ordinary differen tial equation for the profile function f a s 1 ρ ∂ ρ  ρ f ′ f R  − (1 − f 2 ) (1 + f 2 ) S " Λ −  f ′ f  2 # = r 2 0 4 M 2  1 + f 2  2 ∂ V ∂ | u | 2 (2.6) where the primes denote deriv a tiv es w.r.t. ρ , and where Λ = λ 2 − k 2 + n 2 ρ 2 S = 1 + β e 2 f 2 (1 + f 2 ) 2 " Λ +  f ′ f  2 # (2.7) R = 1 + f 2 (1 + f 2 ) 2 "  β e 2 − 2  Λ + β e 2  f ′ f  2 # With the ch oice of p otential giv en in (1.8) we get that r 2 0 4 M 2  1 + f 2  2 ∂ V ∂ | u | 2 = 2 r 2 0 µ 2 M 2    2 − 2 N  ( f 2 ) ( 1 − 2 N ) −  2 + 2 N  ( f 2 ) ( 2 − 2 N ) (1 + f 2 ) 3   (2.8) W e lo ok for solutions satisfying the f ollo wing b oundary conditio ns ~ n →    (0 , 0 , − 1) for ρ → 0 (0 , 0 , +1) fo r ρ → ∞ (2.9) whic h imply that the profile function should satisfy f → 0 for ρ → 0 and f → ∞ for ρ → ∞ (2.10) Let us then assume the following behavior of the profile function f =    α 0 ρ s 1  1 + α 1 ρ + α 2 ρ 2 . . .  for ρ → 0 β 0 ρ s 2  1 + β 1 1 ρ + β 2 1 ρ 2 . . .  for ρ → ∞ (2.11) where s i > 0 , i = 1 , 2. Substituting that in to the equation (2.6) one gets the b eha vior for small ρ implies tha t s 2 1 = n 2 (2.12) where n is the integer in the ansatz (1.5). The b eha vior of (2 .6) for lar ge ρ implies that the relation b et w een n 2 and s 2 2 dep ends up o n the for m of the p otential, and λ 2 − k 2 =          8 r 2 0 µ 2 M 2 for N = − 1 − 2 r 2 0 µ 2 M 2 for N = − 2 0 for all other N (2.13) 6 where N is the integer app earing in the p otential (1.8), and λ , k are the parameters o f the ansatz ( 1.5). Therefore, except for the cases N = − 1 and N = − 2, the w a v es along the vortex ha v e to tra v el with the sp eed of ligh t since the dep endency up on x 3 and x 0 has to b e of the form x 3 ± c t . F or the dimensionfull constants L := r 0 λ , a nd K := r 0 k , the v elo cit y is defined as K c L =      K c √ K 2 +8 µ 2 / M 2 < c for N = − 1 K c √ K 2 − 2 µ 2 / M 2 > c for N = − 2 (2.14) Therefore, the mo de is tac h y onic for N = − 2. N = − 1 is no t tach yonic, but the energy div erges from the b oundary b ehavior of the p oten tial. In the f ollo wing analysis, w e will concen trate on the a nalysis for N ≧ 1 (th us λ 2 = k 2 ). 3 The Ener gy The Hamilto nian densit y asso ciated to ( 2.1) is not p o sitiv e definite due to the quartic terms in time deriv ativ es. W e shall arrange the Legendre transform of eac h t erm in (2.1) to mak e explicit suc h non p ositiv e contributions, and write the Hamiltonian densit y as (see [32] for details) H = 4 M 2 h | ˙ u | 2 + ~ ∇ u · ~ ∇ u ∗ i (1+ | u | 2 ) 2 − 24 e 2  ~ ∇ u  2  ~ ∇ u ∗  2 (1+ | u | 2 ) 4 "  2 3  2 − F 2 # − 24 ( β e 2 − 1) e 2 h | ˙ u | 2 + 1 3 ~ ∇ u · ~ ∇ u ∗ i h ~ ∇ u · ~ ∇ u ∗ − | ˙ u | 2 i (1+ | u | 2 ) 4 + V  | u | 2  (3.1) where ˙ u denotes the x 0 -deriv at iv e of u , and ~ ∇ u its spatial gradien t, and where we hav e denoted ˙ u 2  ~ ∇ u  2 ≡ 1 3 + F e i Φ (3.2) with F > 0 and 0 ≤ Φ ≤ 2 π , b eing functions of the space-time co o rdinates. Note that H giv en in (3.1) is p ositiv e definite for stat ic configurat ions and for the range of parameters giv en in (1.4). Using the ansatz (1.5 ) a nd the co ordinates (1.6) w e g et ˙ u = i k u /r 0 . The metric on the spatial sub-manifold is giv en b y ds 2 = r 2 0 ( dρ 2 + ρ 2 dϕ 2 + dz 2 ), and so  ~ ∇ u  2 = u 2 r 2 0  Ω − − λ 2  ~ ∇ u · ~ ∇ u ∗ = f 2 r 2 0  Ω + + λ 2  (3.3) 7 where Ω ± =  f ′ f  2 ± n 2 ρ 2 (3.4) In addition one gets that ˙ u 2 ( ~ ∇ u ) 2 = 1 3 + F e i Φ = − k 2 (Ω − λ 2 ) , and since it is r eal it f ollo ws that Φ = 0 or π . Therefore,  2 3  2 − F 2 = (Ω − − λ 2 − 3 k 2 ) (Ω − + k 2 − λ 2 ) / 3 (Ω − − λ 2 ) 2 . So, the Hamiltonian density (3.1) b ecomes H = 4 r 4 0  M 2 r 2 0 f 2 (1 + f 2 ) 2  Ω + + λ 2 + k 2  + 2 f 4 (1 + f 2 ) 4  − 1 e 2  Ω − − λ 2 − 3 k 2   Ω − + k 2 − λ 2  − ( β e 2 − 1) e 2  Ω + + λ 2 + 3 k 2   Ω + + λ 2 − k 2  + µ 2 r 4 0  f 2  − 2 N  (3.5) 4 The in tegrabl e sector It is interes ting to no te that (2.6 )-(2.9) with a sp ecial choice of the p otential (1.8) hav e an analytical solution for eac h top olo gical c harge n . In fa ct, solutions of the integrable equation (2.4) also b ecome the solutions of the presen t mo del. F or λ 2 = k 2 , t he solutions can b e written of the form u ( ρ, ϕ, z , τ ) =  ρ a  n e i [ ε nϕ + k ( z + τ )] (4.1) where ε = ± 1, and a is a dimensionless constant to b e fixed by the equations of motion. Substituting this in to the equation (2.6)-(2 .9), w e get ( β e 2 − 1) 4 n 3 a 4 n 1 +  ρ a  2 n o − 3 h ( n − 1)  ρ a  2 n − 4 − ( n + 1)  ρ a  4 n − 4 i = 2 r 2 0 µ 2 M 2 n 1 +  ρ a  2 N o − 3 h (2 − 2 N )  ρ a  2 N − 4 − (2 + 2 N )  ρ a  4 N − 4 i (4.2) The constant a determines the scale of the v ortex and the equation is satisfied if n = N and a = | n |  M 2 ( β e 2 − 1) r 2 0 µ 2  1 / 4 = | n |  ( − e 2 ) ( β e 2 − 1) M 4 4 µ 2  1 / 4 (4.3) Th us, for all p ossible v alues of β e 2 w e ha v e analytical solutions. All those solutions satisfy the condition (2.4). Clearly the class of solutions con tain the sp ecial solution at β e 2 = 1 found previously in [15 ] if we tak e a prop er limit of v anishing p o ten tial, i.e. β e 2 → 1 8 and µ 2 → 0, with β e 2 − 1 µ 2 = constan t. Also, apparently w e hav e no solution at β e 2 6 = 1 without a n y p oten tial b ecause the scale (4 .3) go es t o infinity . Note that the case β = 0 is particularly interesting since it corresp onds to the p rop er Skyrme-F addeev model (without the extra quartic t erm) in the presence of a p oten tial. Therefore, the configurations (4 .1) are exact solutions of the theory (1.10) (for n = N ), with a = | N | h e 2 M 4 4 µ 2 i 1 / 4 , and µ 2 e 2 > 0. As w e men tioned in Sec.2 , for the sector satisfying (2 .4), the mo del p ossesses the infinite set of conserv ed curren ts (2.5). In particular, c ho osing a form of G = − 4 i (1 + | u | 2 ) − 1 , one gets of the No ether curren t for the symmetry of an arbitrary angle α , i.e., u → e iα u J µ = − 4 iM 2 u∂ µ u ∗ − u ∗ ∂ µ u (1 + | u | 2 ) 2 − i 8 e 2 ( β e 2 − 1) 2( ∂ ν u∂ ν u ∗ )( ∂ µ u ∗ u − u ∗ ∂ µ u ) (1 + | u | 2 ) 4 (4.4) F or the solution (4.1), we can ev aluate the c harge p er unit length for the solution Q = Z dx 1 dx 2 J 0 = − 8 π M 2 k a 2 r 0 h I ( n ) + n 6 1 a 2 ( β e 2 − 1) i (4.5) where I ( n ) = 1 n Γ(1 + 1 n )Γ(1 − 1 n ), with Γ b eing the Euler’s Gamma function. Here w e used an in tegral form ula [33 ] Z ∞ 0 x µ − 1 dx ( p + q x ν ) m +1 = 1 ν p m +1  p q  µ ν Γ( µ ν )Γ(1 + m − µ ν ) Γ(1 + m ) (4.6) F or the Hamiltonian (3.5), w e p erform the similar computations. As a r esult, w e get the energy p er unit length b y the integration on the x 1 x 2 plane. F or n = 1, the energy of the static v ortex is (in units of 4 M 2 ) E static = 2 π + 4 π 3 1 a 2 ( β e 2 − 1) (4.7) and for n ≥ 2 t hey are E static = 2 π n + 2 π 3 1 a 2 ( β e 2 − 1)( n 2 − 1) I ( n ) (4.8) Note tha t the first term is prop ortional to the top ological c harge. The energy p er unit of length of the time-dep enden t v ortex diverges for n = 1, and for n ≧ 2 w e obtain E w ave = 2 π n + 2 π 3 1 a 2 ( β e 2 − 1)( n 2 − 1) I ( n ) + k 2 h 2 π a 2 I ( n ) + 2 π 3 ( β e 2 − 1) n i (4.9) F or the limit o f β e 2 → 1 , µ 2 → 0 with k eeping a 2 = n 2 p M 2 ( β e 2 − 1) /r 2 0 µ 2 finite, w e obtain the energy p er unit of length fo und previously in [15]. The energy mono tonically gro ws as k 2 increases. Inte restingly , the static vortex has a minim um a t β e 2 = 1 . 0 and/or µ 2 = 0 . 0 but fo r the time dep endent v ortex there is a minim um of the energy for fixed β e 2 and k 2 and finite µ 2 . The solutions are confirmed n umerically in the subsequen t section. 9 5 The n umerical analys is Although the ansatz (1.5) is given in terms of the p ola r co ordinates, for the n umerical analysis it is more con v enien t to use a new radial co ordinate y , defined b y ρ = q 1 − y y . Accordingly , w e a dopt a function g called the profile function, instead of using f , i.e., f ( ρ ) = q 1 − g ( y ) g ( y ) . The equation (2.6 ) can b e promptly rewritten a s d dy  y (1 − y ) g (1 − g ) g ′ R  +  g − 1 2  S y (1 − y )  Ω −  y (1 − y ) g (1 − g ) g ′  2  = − 1 y 2 r 2 0 µ 2 M 2 (1 − g ) 1 − 2 N g 1+ 2 N n 4 g − 2  1 + 1 N o (5.1) where the primes at this time indicate deriv a tiv es w.r.t. y and where Ω = ( λ 2 − k 2 ) 1 − y y + n 2 (5.2) S = 1 + β e 2 g (1 − g ) y 1 − y  Ω +  y (1 − y ) g (1 − g ) g ′  2  (5.3) R = 1 + g (1 − g ) y 1 − y  ( β e 2 − 2)Ω + β e 2  y (1 − y ) g (1 − g ) g ′  2  (5.4) The energy in the unit of 4 M 2 p er unit length for the time-dep enden t v ortex can b e estimated in terms of f ollo wing four part s of in tegrals of the dimensionless Hamiltonian H := H / 4 M 2 E = 2 π Z ∞ 0 ρdρH ( ρ ) = E 2 + E (1) 4 + ( β e 2 − 1) E (2) 4 + r 2 0 µ 2 M 2 E 0 (5.5) 10 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 g ( y ) y 50.0 10.0 5.0 e 2 = 2.0 1.1 1.01 0 1 2 3 4 0 4 8 12 16 20 24 1.01 50.0 10.0 5.0 1.1 e 2 = 2.0 H ( ) Figure 1 : The n = 1 profile g ( y ) and the corresp onding Hamiltonian densit y of the real space H ( ρ ) of k 2 = 0 . 0 fo r the constan t r 2 0 µ 2 / M 2 = 1 . 0. in whic h the comp onen ts are defined a s E 2 = π Z 1 0 dy y (1 − y )  ( k 2 + λ 2 ) 1 − y y + n 2 +  y (1 − y ) g (1 − g ) g ′  2  g (1 − g ) (5.6) E (1) 4 = π Z 1 0 dy 2(1 − y ) 2  (3 k 2 + λ 2 ) 1 − y y + n 2 −  y (1 − y ) g (1 − g ) g ′  2  ×  ( k 2 − λ 2 ) 1 − y y + n 2 −  y (1 − y ) g (1 − g ) g ′  2   g (1 − g )  2 (5.7) E (2) 4 = π Z 1 0 dy 2(1 − y ) 2  (3 k 2 + λ 2 ) 1 − y y + n 2 +  y (1 − y ) g (1 − g ) g ′  2  ×  ( k 2 − λ 2 ) 1 − y y + n 2 +  y (1 − y ) g (1 − g ) g ′  2   g (1 − g )  2 (5.8) E 0 = 2 π Z 1 0 dy y 2 g 2+ 2 N (1 − g ) 2 − 2 N (5.9) F or the in tegrable sector, w e should choose N = n . Generally sp eaking, v ortex is an ob ject in three spatial dimensions, th us w e hav e explored solutions in t hree spatial dimensions of (1.1). In the three spatial dimensions, the 4th o rder terms in the Lagrangian (including the Skyrme term) successfully av oid the non- existence theorem of static and finite energy solutions by Derric k. How ev er, the equation (2.6 ) of the a nsatz (1.5) is the same as a n equation of corresp onding static t w o spatial dimensions . This means z comp onen t has no essen t ial contribution to the s tability . In fa ct, the Derrick’s theorem for t w o spatial dimensions implies that the con tribution to 11 1 2 3 4 5 6 7 8 9 1 0 6 8 1 0 1 2 1 4 1 6 1 8 2 0 e 2 E s t a t i c @ @ 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 e 2 E 0 0 1 2 3 4 5 6 7 ( e 2 - 1 ) E 4 ( 2 ) 1 2 3 4 5 6 7 8 9 10 0. 0 5. 0x 10 - 1 2 1. 0x 10 - 1 1 1. 5x 10 - 1 1 e 2 E 4 ( 1 ) 6. 275 6. 280 6. 285 6. 290 6. 295 6. 300 6. 305 6. 310 6. 315 E 2 Figure 2: The static energy a nd its comp onen ts corresp onding to the solutions of Fig.1. 12 the energy p er unit length from quartic terms and the p oten tial m ust b e equal, namely E (1) 4 + ( β e 2 − 1) E (2) 4 = r 2 0 µ 2 M 2 E 0 (5.10) Since the solutions of the integrable sector satisfy E (1) 4 = 0, putting to gether with β e 2 = 1 and µ 2 = 0, one can confirm that t he solution without the p oten tial found in [15] satisfies the ab o v e condition automat ically . This fact indicates that the Derrick’s argumen t t o the energy p er unit length also works w ell outside of the in tegrable sector. The definition of the scale parameter a giv en in (4.3) indicates the existence of the analytical solution for the same sign of β e 2 − 1 and µ 2 . In our previous study o f Hopfions on the extended Skyrme-F addeev mo del, we confirmed n umerically the solutions exist only for β e 2 > 1 [22 ], so we b egin our analysis with the case of β e 2 > 1. W e shall give commen ts for the p ossibility of finding solution of β e 2 < 1 in the next subsection. 5.1 The solutions of the in tegrable sector The analytic profiles ( 4.1) can b e written in the co ordinate y as g ( y ) =        a 2 y a 2 y + 1 − y for n = 1 a 4 y 2 a 4 y 2 + (1 − y ) 2 for n = 2 (5.11) where a is determined via (4.3). Apparently ( 5.11) are solutions of ( 5.1). Next, w e will see whether the solutions app ear o r no t when w e n umerically solve (5.1) without an y constrain t. Also, for the obtained solutio ns w e will che c k the zero curv ature condition (2.4). Since (5.1) is an ordinary second order differen tial equation, of course there are sev eral metho ds to in v estigate. Ho w ev er, it is easily noticed that the equation (5.1) may exhibit singular-lik e b ehav ior at the b oundary b ecause of t he term g (1 − g ) of the denominator. Once the computation contains a small nume rical error, the equation quic kly dive rges. The n umerical metho d whic h can safely solve suc h a difficult y is we ll-know n, the SOR metho d. Essen tially we ha v e solv ed the follow ing diffusion equation for a field ˜ g ( y , t ) ∂ ˜ g ∂ t = ω A h ˜ g , ∂ ˜ g ∂ y , ∂ 2 ˜ g ∂ y 2 i (5.12) in whic h we employ (5.1) as A . Here ω is called as a relaxation factor whic h is usually c hosen ω = 1 . 0 ∼ 2 . 0. After a h uge n um b er of iteration steps, the field is relaxed to the static one, i.e, ˜ g ( y , t ) → g ( y ), whic h we are finding. 13 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 2.0 g ( y ) y 50.0 10.0 5.0 e 2 = 1.1 1.01 0 2 4 6 8 0 2 4 6 8 1.01 50.0 10.0 5.0 2.0 e 2 = 1.1 H ( ) Figure 3: The n = 2 profile g ( y ) and the corresp onding Hamiltonian densit y of the real space H ( ρ ) of k 2 = 0 . 0 fo r the constan t r 2 0 µ 2 / M 2 = 1 . 0. The first case is n = 1 . F rom (1.8), the explicit form of the p otential is V n =1 = µ 2 2 (1 − n 3 ) 4 (5.13) In Fig.1 w e presen t the numerical solution g ( y ) and the corresponding Hamiltonian densit y H ( ρ ) for n = 1 . Fig .2 is the energy per unit length and its comp onen ts for sev eral v alues of β e 2 with fixed µ . The function g ( y ) in Fig.1 p erfectly agr ees with the analytical solution (5.11). W e shall giv e a few commen ts for the compo nen ts of the energy . F or the in tegrable solution, the top olo gical con tribution of the energy , i.e., E 2 should b e a constant. Also, E (1) 4 is exactly zero for the integrable solution. The v alue o f the comp onent E (1) 4 in the n umerical solution is not exactly zero, but compatible with zero within the n umerical precision. Note that the plot seems to blow up for the vicinit y of β e 2 = 1 . 0, but the v alue is still up to order ∼ 10 − 8 , so it is still negligible. This clearly means that our nume rical solutions satisfy the zero curv at ure condition and th us b elong to the in tegrable sector. These nu merical errors are pro bably orig inated in the finite n um b er of the mesh p oints. In a usual case, we used the n um b er N mesh = 1000. When w e employ a la rger n um b er, the v alue o f E 2 should b e conv erg e to the constant, i.e., 2 π . F or the n = 2, form of the p oten tial is V n =2 = µ 2 2 (1 + n 3 )(1 − n 3 ) 3 (5.14) th us the p oten tial is zero at b o th the origin and the infinity . Fig.3 is the profile function and the Hamiltonian densit y for n = 2. Aga in the n umerical profile and the a nalytical one (5.11) coalesce. Con trary to the case of n = 1, the density has ann ular shap e . Fig.4 14 1 2 3 4 5 6 7 8 9 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 e 2 E s t a t i c @ @ 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 e 2 E 0 0 1 2 3 4 5 6 7 8 9 ( e 2 - 1 ) E 4 ( 2 ) 1 2 3 4 5 6 7 8 9 10 0. 0 1. 0x 10 - 1 0 2. 0x 10 - 1 0 3. 0x 10 - 1 0 4. 0x 10 - 1 0 e 2 E 4 ( 1 ) 12. 5660 12. 5662 12. 5664 12. 5666 12. 5668 12. 5670 E 2 Figure 4: The static energy and its comp o nen ts corresp onding to the solutions of Fig.3. 15 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 2.0 g ( y ) y 50.0 10.0 5.0 e 2 = 1.1 1.01 0 2 4 6 8 10 0 1 2 3 4 5 6 1.01 50.0 10.0 5.0 2.0 e 2 = 1.1 H ( ) Figure 5: The n = 3 profile g ( y ) and the corresp onding Hamiltonian densit y of the real space H ( ρ ) of k 2 = 0 . 0. for the constant r 2 0 µ 2 / M 2 = 1 . 0. is the energy p er unit length a nd its comp onen ts fo r sev eral v alues of β e 2 and fixed µ 2 . Again w e confirmed that the v alue of the comp onent E (4) 1 is regarded as zero within the n umerical uncertaint y . F or n = 3, form of the p ot en tial is V n =3 = µ 2 2 (1 + n 3 ) 4 / 3 (1 − n 3 ) 8 / 3 (5.15) th us aga in the p oten tial is zero at b o th the origin and the infinity . Fig .5 is the profile function and t he Hamiltonian dens ity for n = 3. As is easily seen the radius of the ann ulus is larger than that o f n = 2. Fig .6 is the energy p er unit length a nd its comp onents for sev eral v alues o f β e 2 and fixed µ 2 . In this case, w e face a num erical difficult y . During the computation by the SOR metho d, the solution g tends to oscillate around the true v alue and sometimes it acciden tally go es b elo w zero at the vicinit y o f the orig in, a nd then the computation f ails b ecause o f the term g 1+2 /n in (5.1). In order to a v oid it, w e emplo y a finer mesh, i.e., the n um b er is at least N mesh = 3000 . Un til now, w e hav e examined in the case of β e 2 > 1. The formalism leading to (4.3 ) suggests that the c hoice β e 2 < 1 and µ 2 < 0 might also b e p ossible and the scale is now defined as a = | n |  M 2 (1 − β e 2 ) − r 2 0 µ 2  1 / 4 for β e 2 < 1 , µ 2 < 0 (5.16) The result is plotted in Fig.7. How eve r, existence of such solution seems dubious; the energy turns negative at a critical v alue of β e 2 th us the solution has no energy low er b ound. Also, num erically the change of sign of the p otential in the equation of motion quic kly breaks the computation. 16 1 2 3 4 5 6 7 8 9 1 0 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2 3 4 3 6 3 8 4 0 e 2 E s t a t i c @ @ 1 2 3 4 5 6 7 8 9 1 0 0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 e 2 E 0 0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 ( e 2 - 1 ) E 4 ( 2 ) 1 2 3 4 5 6 7 8 9 1 0 0 . 0 2 . 0 x 1 0 - 9 4 . 0 x 1 0 - 9 6 . 0 x 1 0 - 9 e 2 E 4 ( 1 ) 1 8 . 8 4 9 0 1 8 . 8 4 9 2 1 8 . 8 4 9 4 1 8 . 8 4 9 6 1 8 . 8 4 9 8 1 8 . 8 5 0 0 E 2 Figure 6: The static energy and its comp o nen ts corresp onding to the solutions of Fig.5. 17 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 g ( y ) y - 4 8 . 0 - 8 . 0 - 3 . 0 e 2 = 0 . 0 0 . 9 0 . 9 9 0 1 2 3 4 - 4 0 4 8 1 2 1 6 - 4 8 . 0 - 8 . 0 - 3 . 0 0 . 0 e 2 = 0 . 9 0 . 9 9 H ( ) Figure 7: T he n = 1 profile functions and the energy densit y for the case of β e 2 < 1 , for the constan t r 2 0 µ 2 / M 2 = − 1 . 0. 5.2 The solutions outside of the in tegrable sector Although w e hav e obtained the analytical solutions fo r a sp ecial f orm of the p o ten tial (1.8), w e hav e many options for choice of the p otential. The p oten tials which w e emplo y ed in this pap er essen tially b elong to a class of the generalized Bab y-Skyrmion (BS) p oten tial formally written as V BS = µ 2 2 v α γ , v α γ = (1 + n 3 ) α (1 − n 3 ) γ (5.17) F or α = 0, the p otential is called ( the class of ) the old - BS p oten tial whic h w as in tro duced in [34], while α 6 = 0 is (the class of ) the new - BS p oten tial [35]. Our case (1.8) corr esp onds to α = 2 − 2 /n, γ = 2 + 2 /n . Note that the p ossible choice of the p otential is certainly restricted b y the analys is of the limiting b eha vior (2.11) for s ev eral p oten tials. The results are summarized in T able 1. W e can obtain man y numeric al solutions for the sev eral t yp es of the p oten tials. W e sho w the result of n = 2 for the p oten tials v 0 2 , v 0 4 , v 4 / 3 8 / 3 ; of course these are not of the form of the analytical solution. Fig.8 presen ts the energies and the comp o nen t E (1) 4 for these p oten tials. F or n = 2, the ol d -BS p oten tials give higher tot al energy than the new one. This indicates tha t the same class o f p otentials gives the similar energy and then, for n = 2 the energy of the new t yp e p oten tial v 4 / 3 8 / 3 is closest to the in tegrable sector, whic h is also plotted in Fig. 8 fo r reference. 18 1 2 3 4 5 6 7 8 9 10 10 15 20 25 30 35 40 45 E s t at i c e 2 ( 1 - n 3 ) 4 ( 1 - n 3 ) 2 ( 1 - n 3 ) 3 ( 1 + n 3 ) ( 1 - n 3 ) 8 / 3 ( 1 + n 3 ) 4 / 3 1 2 3 4 5 6 7 8 9 1 0 1 E - 4 1 E - 3 0 . 0 1 0 . 1 1 ( 1 + n 3 ) 4/3 ( 1 - n 3 ) 8/3 ( 1 - n 3 ) 2 ( 1 - n 3 ) 4 E 4 ( 1 ) e 2 Figure 8: The n = 2 static energy and the comp onen t E (1) 4 for sev eral t yp e of p oten tials v i j . k 2 = 0 and r 2 0 µ 2 / M 2 = 1 . 0. 19 v α γ n = 1 n = 2 n = 3 old -BS: 1 − n 3 × × × (1 − n 3 ) 2 ×   (1 − n 3 ) 3    (1 − n 3 ) 4    new -BS: (1 + n 3 )(1 − n 3 ) × × × (1 + n 3 )(1 − n 3 ) 3    (1 + n 3 ) 4 3 (1 − n 3 ) 8 3    T a ble 1: The analysis of the limiting b eha vior o f the solutions at b oth y = 0 , 1 fo r sev eral c hoice of the p oten tial, and  ( × ) which indicates that there exist (no) solutions. 6 P oten tial ph ysical applications of the sol utions Since the mo del (1.1) w as prop osed in the con text of Wilsonian renormalizat ion g roup argumen t of the S U (2) Y ang-Mills theory , we e xp ect that the vortex s olutions constructed in this pa p er should describ e some features of strong coupling regimes, suc h as the dual sup erconductor picture [36]. Apart from that, v ortices app ear in sev eral areas of ph ysics. The Nielsen-Olesen (NO) v ortices in the Ab elian Higg s mo del [37] w ere a pplied for type I I sup erconductors (SC) a nd later they ha v e extensiv ely b een studied in the context of cosmology , i.e., the cosmic string [38] and the brane-world scenario [39]. The mo del has a close relationship with the standard electro w eak theory , sp ecially when one considers the case of a globa l S U (2 ) and a lo cal U (1 ) breaking in to a global U (1), where the mo del reduces to an Ab elian Higg s mo del with t w o c harged scalar fields [40]. It is interes ting to note that the v ortices of such mo del carry the so- called longitudinal electromagnetic curren ts [41, 42]. In (4 .4) w e give the No ether curren t asso ciated to a global U (1), i.e., u → e iα u , and so one can straightforw ardly compute the lo ngitudinal curren t in the in tegrable/nonin tegrable sector. In Fig.9, w e plot the t ypical results of t he tra nsv erse spatial structure of the p olar comp onen t of the curren t in the case o f the in tegrable sector. (Using (4.1) and (4.4), one can easily see that the ra dial comp onen t of the current is alw a ys zero.) Note that for higher winding num bers as well for unit winding num ber the solutions exhibit the pip e-lik e structure, whic h w as observ ed in the analysis of [43]. Our mo del enjo ys a symmetry breaking of the t yp e O (3 ) global → O (2) global whic h is similar to S U (2) global ⊗ U (1 ) lo cal → U (1) global . A notable difference b etw een the NO v ortices and ours is t hat the gauge degrees of fr eedom ar e absen t in o ur mo del. If one 20 0 2 4 6 8 10 0.0 0.5 1.0 1.5 2.0 2.5 n = 6 n = 3 n = 2 J ( M 2 ) e 2 = 5.0 = 1.1 n = 1 Figure 9: The tra nsv erse spatial structure of the p olar comp onen t of t he curren ts (in units of − 1 / 8 M 2 ) for the top olog ical ch arge n = 1 , 2 , 3 and 6, with the pa rameters β e 2 = 1 . 1 , 5 . 0. W e set r 2 0 µ 2 / M 2 = 1 . 0. All solutions hav e a pip e-lik e structure. wishes to discuss the existence of the ga uge field in ty p e I I SC, how ev er, the ga uging of the mo del according to [44] should w ork. Confinemen t or squeezing of the magnetic field in to t yp e I I SC should b e realized in terms of the lo calization of the gauge field in to our v ortices. 7 Summary W e ha v e studied v ortex solutions of the extended Skyrme-F addeev mo del esp ecially for the outside of the in tegrable constrain t β e 2 = 1. In order to find the solutions, w e in tro duced p otentials of the extension of the Ba b y-Skyrmion t yp e. W e f ound sev eral analytical solutions of the mo del. W e also confirmed the existence of the solutions in terms of the n umerical analysis. By using the standard SOR metho d, w e obtained the axially symmetric solutions f or charge up t o n = 3 with sev eral form of the p o ten tial for v ario us v a lue of the mo del parameters. In this work, we imp osed the axial symmetry to the solution ansatz. How ever, solu- tions with lo w er symmetry , such as Z 2 -symmetry w ere fo und by a n umerical sim ulation for the Baby -Skyrme mo del [45]. It w ould b e in teresting to inv es tigate whether suc h de- 21 formed solutions app ear in the extended Skyrme-F addeev mo del. F urthermore, full 3D sim ulations of the mo del will certainly clarify the detailed structure of the v ortices. The analysis implemen ting these issues will b e discussed in a forthcoming pap er. Ac kno wledgmen ts W e a re gr ateful to the anony mous referee fo r a careful reading of the man uscript and for v aluable remarks on the ph ysical applications. W e would like to thank W o jtek Zakrzewski and Pa w e l Klimas for man y useful discuss ions. NS would lik e to thank the kind hospitalit y at Instituto de F ´ ıs ica de S˜ ao Carlos, Univ ersidade de S˜ ao Paulo. He a lso ac kno wledges the financial supp or t o f F APESP (Brazil). JJ would also lik e to thank the UK Engineering and Ph ysical Science s Researc h Council for supp o rt. LAF is partially supp orted by CNPq-Brazil. 22 References [1] L. D . F addeev, “Quan tization of solitons,”, Princeton preprin t IAS Prin t-75-QS70 (1975); in 40 Y e ars in Mathem atic a l Physics , (W orld Scien tific, 1 995). [2] A. M. P oly ak o v and A. A. Bela vin, “Metastable Sta tes of Tw o-Dimensional Isotropic F erromagnets,” JETP Lett. 22 , 245 (1975 ) [Pisma Zh. Eksp. T eor. Fiz. 22 , 503 (1975)]. [3] J. Gladiko ws ki and M. 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