vortex solutions in the extended skyrme faddeev model in collaboration with luiz agostinho ferreira,...
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VORTEX SOLUTIONS IN THE EXTENDED SKYRME FADDEEV MODEL
In collaboration with γγ Luiz Agostinho Ferreira, Pawel Klimas (IFSC/USP) Masahiro Hayasaka (TUS) γγγ Juha JΓ€ykkΓ€ (Nordita) Kouichi Toda (TPU)
NOBUYUKI SAWADOTokyo University of Science, Japan
γγγ arXiv:0908.3672 , arXiv:1112.1085, arXiv:1209.6452, arXiv:1210.7523
At Miami 2012: A topical conference on elementary particles, astrophysics, and cosmology,
13-20 December, Fort Lauderdale, Florida
19 December, 2012
Objects of Yang-Mills theory
β=πΌ (ππβοΏ½βοΏ½)2+π½ (ππ
βοΏ½βοΏ½ΓππβοΏ½βοΏ½)2+πΎ(ππ
βοΏ½βοΏ½)4
(i) Gauge + Higgs composite models
Abelian vortex (in U(1))
γγγ Abrikosov vortex, graphene, cosmic string, Brane world,
etc.
βtHooft Polyakov monopole
γγγγγγ GUT, Nucleon catalysis (Callan-Rubakov effect),
etc.
The Skyrme-Faddeev Hopfions, vortices Glueball?, Abrikosov vortex?, Branes?
(ii) Pure Yang-Mills theory
Instantons
In the Cho-Faddeev-Niemi-Shabanov decomposition
Monopole loop
Condensates in a dual superconductivity γγγγ Confinement
N.Fukui,et.al.,PRD86(2012)065020,``Magnetic monopole loops generated from two-instanton solutions: Jackiw-Nohl-Rebbi versus 't Hooft instantonβ
Exotic structures of the vortexβ¦β¦
M.N.Chernodub and A..S. Nedelin, PRD81,125022(2010)``Pipelike current-carrying vortices in two-component condensatesββ
P.J.Pereira,L.F.Chibotaru, V.V.Moshchalkov, PRB84,144504 (2011)``Vortex matter in mesoscopic two-gap superconductor squareββ
Semi-local strings The Ginzburg-Landau equation
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
electric magnetic remaining terms
22 1 1
3Γ4 β 6 = 6Degrees of freedom
6
L.D. Faddeev, A.J. Niemi, Phy.Rev.Lett.82 (1999) 1624,``Partially dual variables in SU(2) Yang-Mills theoryβ
t = ln k/ L ``renormalization group timeββ
H. Gies, Phys. Rev. D63, 125023 (2001),``Wilsonian effective action for SU(2) Yang-Mills theory with Cho-Faddeev-Niemi-Shabanov decomposition
The Gies lagrangian
``Magnetic symmetryββ
Lagrangian (in Minkowski space)
Sterographic project
Static hamiltonian
Positive definite for
The integrability: the analytical vortex solutions
The equation of the vortex
The zero curvature condition πππ’πππ’=0
π½π2=1 πππππ’=0The equation becomes
or
or )(
π’=π£ (π§ )π€ (π¦ )=π§ππππ¦π§=π₯1+ ππ1π₯2 , π¦=π₯3βπ2π₯
0
ΒΏ (ππ )π
ππ [π1ππ+π (π₯3+π2 π₯0)]
Traveling wave vortex
The vortex solution in the integrable sectorL.A.Ferreira, JHEP05(2009)001,``Exact vortex solutions in an extended Skyrme-Faddeev modelβ O.Alvarez,LAF,et.al,PPB529(1998)689,``A new approach to integrable theories in any dimensionβ
One gets the infinite number of conserved quantity
Additional constraint
(0)
The equation
The solution has of the form:
We have no solutions for
π’ (π , π‘ )=β 1βπ (π¦ )π (π¦ )ππ(ππ+ππ§+ππ )
π₯π=π 0(π ,π cosπ ,π sinπ , π§)
Vortex solutions in
Ansatz
= 0
and for
for
Derrickβs scaling argument G.H.Derrick, J.Math.Phys.5,1252 (1964),``Comments on nonlinear wave equations as models for elementary particlesββ
Scaling:
Consider a model of scalar field:
We need to introduce form of a potential to stabilize the solution.
The baby-skyrmion potential
Plug into the equation it is written as
and
Assume the zero curvature condition
0=( π½π2β1 ) 4π3
π4 {1+( ππ )2π}
β 3
[ (πβ1 )( ππ )2πβ4
β(π+1)( ππ )4πβ4 ]
+2π02π2
π 2 {1+(ππ )2π}
β3
[(2+ 2π )( ππ )4πβ 4
β(2β 2π )( ππ )
2π β4 ]
with the potential: we assume
π’ (π ,π ,π§ ,π‘ )=(ππ )π
ππ [πππ+π (π§+π)]
π πΌπ½=π2
2(1+π3β)πΌ(1βπ3β)πΎπΌ β₯0πΎ>0
Analytical solutions for n = 1, 2
π=|π|4βπ2(π½π2β1)π 0
2π2 and
π=1 ,π2=0.0 ,π 0
2π2
π 2 =1.0 π=2 ,π2=0.0 ,π02π2
π 2 =1.0
The energy per unit length of the traveling wave vortex with
The static energy per unit of length of the vortex with
The energy of the static/traveling wave vortex
πΈπ π‘ππ‘ππ=2π+4 π31
π2(π½π2β1)
The infinite number of conserved current
π½πβπΏπΊπΏπ’βπ¦πβ
πΏπΊπΏπ’
π¦πβ hπ€ ππππΊβπΊ (|π’|2 )
ΒΏπΏπΊπΏπ’
ΒΏ
Thus the current is always conserved:
And the equation of motion is written as
πππ¦πβ2π’β (1+|π’|2 )π¦ππ
ππ’=β π2
4ΒΏΒΏ
The zero curvature condition π¦ππππ’=π¦β
ππππ’β=0 ,π¦β
ππππ’=π¦ππ
ππ’β
π½π=0The transverse spatial structure of the polar component and the longitudinal component are a pipelike structure.
The charge per unit length:
π=β«ππ₯1ππ₯2 π½ 0=β8π π 2ππ2π0 [π6 1π2 ( π½π2β1 )+ 1πΞ (1+
1π)Ξ (1β
1π)]
For
we get Noether current with
π½π=β4 ππ2 π’πππ’
ββπ’β πππ’
(1+|π’|2 )2β π 8π2
(π½π2β1)2(πππ’π
ππ’β)(πππ’βπ’βπ’βπππ’)
ΒΏΒΏΒΏ
The components:
Broken axisymmetry of the solution
The energy density plot of for old-, and new-baby potentials
Old baby skyrmion potential
New baby skyrmion potential
γγγγγγγγγγγγ (old) γγγγγγγγγγγγ newNonsymmetric: old
For the potential , the holomorphic solutions appear as a ground state!
Symmetric:
The baby-skyrmion exhibits a non-axisymmetric solution depending on a choice of potential I.Hen et. al, Nonlinearity 21 (2008) 399
A sequence of the energy density plots of for the several for
the old-potential
π½ π2=1.01 π½ π2=1.1 π½ π2=2.0 π½ π2=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 equationsβ
The vortex matter/lattice structure is observed.
SummaryWe 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 toobtain the integrable vortex solutions.
OutlookWhat 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?
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.
Thank you οΌTanzan Jinja shrine,Japan, 16 Nov.,2012
Lago Mar Resort, USA, 17 Dec.,2012
The Skyrme-Faddeev modelL.Faddeev, A.Niemi, Nature (London) 387, 58 (1997),``Knots and particlesββ
R.A.Battye, P.M.Sutcliffe, Phys.Rev.Lett.81,4798(1998)
Lagrangian
Static hamiltonian
Positive definite for
Boundary conditions
Coordinates:
Hopfions(closed vortex)
Hopf charge
L.A.Ferreira, NS, et.al., JHEP11(2009)124, ``Static Hopfions in the extended Skyrme-Faddeev modelβ
Axially symmetric ansatz
Non-axisymmetric case:D.Foster, arXiv:1210.0926
(m, n) = (1, 1) (1, 2) (2, 1)
(m, n) = (1, 3)
(m, n) = (1, 4) (2, 2) (4, 1)
Hopf charge density
(3, 1)
corresponds to the zero curvature condition
Dimensionless energy, Integrability
The solution is close to the Integrable sector, but not exact.
π½π2