VORTEX SOLUTIONS IN THE EXTENDED SKYRME FADDEEV MODEL NOBUYUKI SAWADO Tokyo University of Science, Japan [email protected] In collaboration with Luiz Agostinho Ferreira, Pawel Klimas (IFSC/USP) Masahiro Hayasaka (TUS) Juha Jäykkä (Nordita) Kouichi Toda (TPU) arXiv:0908.3672 , arXiv:1112.1085, arXiv:1209.6452, arXiv:1210.7523 19 December, 2012 At Miami 2012: A topical conference on elementary particles, astrophysics, and cosmology, 13-20 December, Fort Lauderdale, Florida Objects of Yang-Mills theory (i) Gauge + Higgs composite models Abelian vortex (in U(1)) Abrikosov vortex, graphene, cosmic string, Brane world, etc. ‘tHooft Polyakov monopole GUT, Nucleon catalysis (Callan-Rubakov effect), etc. (ii) Pure Yang-Mills theory Instantons In the Cho-Faddeev-Niemi-Shabanov decomposition Monopole loop N.Fukui,et.al.,PRD86(2012)065020,``Magnetic monopole loops generated from two-instanton solutions: Jackiw-Nohl-Rebbi versus 't Hooft instanton” Condensates in a dual superconductivity Confinement The Skyrme-Faddeev Hopfions, vortices Glueball?, Abrikosov vortex?, Branes? ℒ = ( )2 +( × )2 +( )4 Exotic structures of the vortex…… Semi-local strings M.N.Chernodub and A..S. Nedelin, PRD81,125022(2010) ``Pipelike current-carrying vortices in two-component condensates’’ The Ginzburg-Landau equation P.J.Pereira,L.F.Chibotaru, V.V.Moshchalkov, PRB84,144504 (2011) ``Vortex matter in mesoscopic two-gap superconductor square’’ Summary We got the integrable and also the numerical solutions of the vortices in the extended Skyrme Faddeev Model. A special form of potential is introduced in order to stabilize and to obtian the integrable vortex solutions. We begin with the basic formulation. Cho-Faddeev-Niemi-Shabanov (CFNS) decomposition L.D. Faddeev, A.J. Niemi, Phy.Rev.Lett.82 (1999) 1624, ``Partially dual variables in SU(2) Yang-Mills theory” 3×4 ― 6 = 6 Degrees of freedom electric 2 magnetic 2 remaining terms 1 The Gies lagrangian 1 6 ``Magnetic symmetry’’ + × = 0 H. Gies, Phys. Rev. D63, 125023 (2001),``Wilsonian effective action for SU(2) Yang-Mills theory with Cho-Faddeev-Niemi -Shabanov decomposition t = ln k/L ∈ (−∞ , 0] ``renormalization group time’’ The integrability: the analytical vortex solutions Lagrangian (in Minkowski space) Sterographic project Static hamiltonian Positive definite for The equation of the vortex The vortex solution in the integrable sector L.A.Ferreira, JHEP05(2009)001,``Exact vortex solutions in an extended Skyrme-Faddeev model” O.Alvarez,LAF,et.al,PPB529(1998)689,``A new approach to integrable theories in any dimension” The zero curvature condition = 0 One gets the infinite number of conserved quantity Additional constraint 2 = 1 2 = 0 or 1 + 2 The equation becomes = 0 1 − 2 = − 3 + 0 [ 3 − 0 ] = 0 or (1 +2 )(1 − 2 ) = −(3 + 0 )(3 − 0 ) = = = 1 + 1 2 , = 3 − 2 0 [ +( 3+ 0)] 2 Traveling wave vortex = 1 Vortex solutions in 2 ≠ 1 Ansatz , = 1 − () (++) () = 0 (, cos , sin , ) : = 1− (0≤ ≤ 1) The equation The solution has of the form: = = 2 /(2 + (1 − ) ) We have no solutions for ≠ . . Derrick’s scaling argument G.H.Derrick, J.Math.Phys.5,1252 (1964), ``Comments on nonlinear wave equations as models for elementary particles’’ Consider a model of scalar field: Scaling: 2 4 = () =0 = 4 0 4 and 0 = 0 for = 3 for = 2 We need to introduce form of a potential to stabilize the solution. The baby-skyrmion potential 2 = 1 + 3 2 (1 − 3 ) ≥ 0 > 0 Assume the zero curvature condition , , , = Plug into the equation it is written as 43 2 0 = − 1 4 1 + 20 2 2 + 1 + 2 = [+(+)] with the potential: we assume = − /, = + / 2 −3 2 −3 −1 2 2+ and 2−4 4−4 = 4 − ( + 1) 2 − 2− 2 ( 2 −1) 0 2 2 4−4 2−4 Analytical solutions for n = 1, 2 = 2 2 +(1−) and 2 2 0 = 1, 2 = 0.0, = 1.0 2 = 4 2 ( 2 − 1) 0 2 2 0 2 2 = 2, = 0.0, = 1.0 2 2 The energy of the static/traveling wave vortex The static energy per unit of length of the vortex with = 1 4 1 2 − 1) = 2 + ( 3 2 The energy per unit length of the traveling wave vortex with ≥ 2 2 1 2 − 1 2 − 1 = 2 + 3 2 2 2 2 + 2 + 2 − 1 3 The infinite number of conserved current ≔ ∗ − ∗ ℎ ≔ 2 And the equation of motion is written as − 2∗ 1 + 2 2 2 2 = − (1 + || ) 4 ∗ The zero curvature condition = 0 = ∗ ∗ = 0, ∗ = ∗ Thus the current is always conserved: ∵ 2 2 ∗ 2 ∗ ∗ = + + ∗ − ∗2 2 2 ∗ ∗ − ∗ − ∗ 2 2 = − (1 + ||2 )2 ∗ − ∗ − 1+ 4 4 2 2 =0 For ≔ −4/(1 + ||2 ) we get Noether current with ↦ ∗ − ∗ ∗ )( ∗ − ∗ ) 2( 8 2 = −42 − ( − 1) 1+ 2 2 2 (1 + ||2 )4 The components: = 0 The transverse spatial structure of the polar component and the longitudinal component are a pipelike structure. The charge per unit length: = 1 2 0 = −82 2 0 1 1 1 1 2 − 1 + Γ(1 + )Γ(1 − ) 6 2 Broken axisymmetry of the solution The baby-skyrmion exhibits a non-axisymmetric solution depending on a choice of potential I.Hen et. al, Nonlinearity 21 (2008) 399 Old baby skyrmion potential 01 2 = (1 − 3 ) 2 New baby skyrmion potential 11 2 = (1 − 3 )(1 + 3 ) 2 The energy density plot of = 3 for old-, and new-baby potentials Symmetric: Nonsymmetric: = 1 (old) ≥ 2 new ≥ 2 old For the potential 2 2− 2 2+ , the holomorphic solutions appear as a ground state! A sequence of the energy density plots of = 3 for the several 2 for the old-potential 2 = 1.01 2 = 1.1 2 = 2.0 2 = 20.0 A repulsive force between the core of the vortices might appear It might be similar with the force between the Abrikosov vortex. Erick J.Weinberg, PRD19,3008 (1979), ``Multivortex solutions of the Ginzburg-Landau equations” The vortex matter/lattice structure is observed. Summary We got the integrable and the numerical solutions of the vortices in the extended Skyrme Faddeev Model. A special form of potential is introduced in order to stabilize and to obtain the integrable vortex solutions. Our integrable solution thus carries an infinite number of conserved quantity. The model (~two gap model) hides a SU(2) structure even if it describes the U(1) like observation such as SC. Outlook What it the origin of the potential? How can I observe our solutions in Physics? For SC, we may introduce an external magnetic field → = + ×. and see the structure change for the field. Geometrical patterns appear? Lago Mar Resort, USA, 17 Dec.,2012 Thank you！ Tanzan Jinja shrine,Japan, 16 Nov.,2012 The Skyrme-Faddeev model L.Faddeev, A.Niemi, Nature (London) 387, 58 (1997), ``Knots and particles’’ Lagrangian Static hamiltonian R.A.Battye, P.M.Sutcliffe, Phys.Rev.Lett.81,4798(1998) Positive definite for Hopfions(closed vortex) Coordinates: L.A.Ferreira, NS, et.al., JHEP11(2009)124, ``Static Hopfions in the extended Skyrme-Faddeev model” Axially symmetric ansatz Non-axisymmetric case: D.Foster, arXiv:1210.0926 Boundary conditions Hopf charge Hopf charge density (m, n) = (1, 1) (1, 2) (m, n) = (1, 3) (m, n) = (1, 4) (2, 1) (3, 1) (2, 2) (4, 1) Dimensionless energy, Integrability corresponds to the zero curvature condition The solution is close to the Integrable sector, but not exact. 2

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