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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