VORTEX SOLUTIONS IN THE EXTENDED
SKYRME FADDEEV MODEL
NOBUYUKI SAWADO
Tokyo University of Science, Japan
sawado@ph.noda.tus.ac.jp
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 = 𝛼(𝜕𝜇 𝑛)2 +𝛽(𝜕𝜇 𝑛 × 𝜕𝜈 𝑛)2 +𝛾(𝜕𝜇 𝑛)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’’
𝜕𝜇 𝑛 + 𝐴𝜇 × 𝑛 = 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
𝜕𝜇 𝑢 𝜕𝜇 𝑢 = 0
One gets the infinite number of conserved quantity
Additional constraint
𝛽𝑒 2 = 1
𝜕 2 𝑢 = 0 or
𝜕1 + 𝑖𝜕2 𝑢
The equation becomes 𝜕𝜇 𝜕𝜇 𝑢 = 0
𝜕1 − 𝑖𝜕2 𝑢 = − 𝜕3 + 𝜕0 𝑢 [ 𝜕3 − 𝜕0 𝑢]
𝜕𝜇 𝑢𝜕𝜇 𝑢 = 0 or (𝜕1 +𝑖𝜕2 )(𝜕1 − 𝑖𝜕2 )𝑢 = −(𝜕3 + 𝜕0 )(𝜕3 − 𝜕0 )𝑢
𝑢 = 𝑣 𝑧 𝑤 𝑦 = 𝑧 𝑛 𝑒 𝑖𝑦 𝑧 = 𝑥 1 + 𝑖𝜖1 𝑥 2 ,
𝑦 = 𝑥 3 − 𝜖2 𝑥 0
𝜌 𝑛 𝑖[𝜖 𝑛𝜑+𝑘(𝑥 3+𝜖 𝑥0)]
2
Traveling wave vortex
=
𝑒 1
𝑎
Vortex solutions in 𝛽𝑒 2 ≠ 1
Ansatz
𝑢 𝒓, 𝑡 =
1 − 𝑔(𝑦) 𝑖(𝑛𝜑+𝜆𝑧+𝑘𝜏)
𝑒
𝑔(𝑦)
𝑥 𝜇 = 𝑟0 (𝜏, 𝜌 cos 𝜑, 𝜌 sin 𝜑 , 𝑧)
𝜌: =
1−𝑦
𝑦
(0≤ 𝑦 ≤ 1)
The equation
The solution has of the form:
𝑓 𝜌 =
𝜌 𝑛
𝑎
𝑔 𝑦 = 𝑎2𝑛 𝑦 𝑛 /(𝑎2𝑛 𝑦 𝑛 + (1 − 𝑦)𝑛 )
We have no solutions for 𝜷𝒆𝟐 ≠ 𝟏. 𝟎.
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:
𝐸2
𝐸4
𝜇=
𝑑𝑒(𝜇)
𝑑𝜇
=0
𝜇=
4
𝐸0
𝐸4
and 𝐸0 = 0
for 𝐷 = 3
for 𝐷 = 2
We need to introduce form of a potential to stabilize the solution.
The baby-skyrmion potential
𝑉𝛼
𝛽
𝜇2
=
1 + 𝑛3
2
𝛼
(1 − 𝑛3 )𝛾 𝛼 ≥ 0 𝛾 > 0
Assume the zero curvature condition
𝜌
𝑢 𝜌, 𝜑, 𝑧, 𝑡 =
𝑎
Plug into the equation
it is written as
4𝑛3
𝜌
2
0 = 𝛽𝑒 − 1 4 1 +
𝑎
𝑎
2𝑟0 2 𝜇2
𝜌
+
1
+
𝑀2
𝑎
𝑁=𝑛
𝑛
𝑒 𝑖[𝜖𝑛𝜑+𝑘(𝑧+𝜏)]
with the potential:
we assume 𝜶 = 𝟐 − 𝟐/𝑵, 𝜸 = 𝟐 + 𝟐/𝑵
2𝑛 −3
2𝑁 −3
𝑛−1
2
2+
𝑁
and
𝜌
𝑎
𝜌
𝑎
2𝑛−4
4𝑁−4
𝑎= 𝑛
4
𝜌
− (𝑛 + 1)
𝑎
2
− 2−
𝑁
𝜌
𝑎
𝑀2 (𝛽𝑒 2 −1)
𝑟0 2 𝜇2
4𝑛−4
2𝑁−4
Analytical solutions for n = 1, 2
𝑔 𝑦 =
𝑎2𝑛 𝑦 𝑛
𝑎2𝑛 𝑦 𝑛 +(1−𝑦)𝑛
and
2 2
𝑟
𝜇
0
𝑛 = 1, 𝑘 2 = 0.0,
= 1.0
𝑀2
𝑎= 𝑛
4
𝑀2 (𝛽𝑒 2 − 1)
𝑟0 2 𝜇2
𝑟0 2 𝜇 2
𝑛 = 2, 𝑘 = 0.0,
= 1.0
𝑀2
2
The energy of the static/traveling wave vortex
The static energy per unit of length of the vortex with 𝑛 = 1
𝐸𝑠𝑡𝑎𝑡𝑖𝑐
4𝜋 1
2 − 1)
= 2𝜋 +
(𝛽𝑒
3 𝑎2
The energy per unit length of the traveling wave vortex with 𝑛 ≥ 2
2𝜋 1
2 − 1 𝑛2 − 1 𝐼 𝑛
𝐸𝑤𝑎𝑣𝑒 = 2𝜋𝑛 +
𝛽𝑒
3 𝑎2
2𝜋
2
2
+𝑘 2𝜋𝑎 𝐼 𝑛 +
𝛽𝑒 2 − 1 𝑛
3
The infinite number of conserved current
𝛿𝐺
𝛿𝐺
𝐽𝜇 ≔ ∗ 𝒦𝜇 −
𝒦𝜇 ∗ 𝑤ℎ𝑒𝑟𝑒 𝐺 ≔ 𝐺 𝑢
𝛿𝑢
𝛿𝑢
2
And the equation of motion is written as
𝜕𝜇 𝒦𝜇 − 2𝑢∗ 1 + 𝑢
2
2
𝜇
𝜕𝑉
𝜇
2
2
𝒦𝜇 𝜕 𝑢 = − (1 + |𝑢| )
4
𝜕𝑢 ∗
The zero curvature condition 𝜕𝜇 𝑢𝜕𝜇 𝑢 = 0
𝒦𝜇 𝜕𝜇 𝑢 = 𝒦 ∗𝜇 𝜕𝜇 𝑢∗ = 0,
𝒦 ∗𝜇 𝜕𝜇 𝑢 = 𝒦𝜇 𝜕𝜇 𝑢∗
Thus the current is always conserved:
∵
𝜕𝜇 𝐽𝜇
𝛿 2𝐺 𝜇
𝛿𝐺 𝜇
𝛿 2𝐺 𝜇 ∗
𝛿 2𝐺 𝜇 ∗ ∗
=
𝜕 𝑢𝒦𝜇 +
𝜕 𝒦𝜇 + ∗ 𝜕 𝑢 𝒦𝜇 − ∗2 𝜕 𝑢 𝒦𝜇
2
𝛿𝑢
𝛿𝑢
𝛿𝑢 𝛿𝑢
𝛿𝑢
2
𝛿𝐺 𝜇 ∗
𝛿 𝐺 𝜇
∗
− ∗ 𝜕 𝒦𝜇 −
𝜕
𝑢𝒦
𝜇
𝛿𝑢
𝛿𝑢𝛿𝑢∗
2
𝛿𝐺
𝜇2
𝜕𝑉
𝛿𝐺
𝜇
=
− (1 + |𝑢|2 )2 ∗ − ∗ −
1+ 𝑢
𝛿𝑢
4
𝜕𝑢
𝛿𝑢
4
2 2
𝜕𝑉
=0
𝜕𝑢
For 𝐺 ≔ −4𝑖/(1 + |𝑢|2 )
we get Noether current with 𝑢 ↦ 𝑒 𝑖𝛼 𝑢
∗ − 𝑢∗ 𝜕 𝑢
𝜈 𝑢 ∗ )(𝜕 𝑢 ∗ 𝑢 − 𝑢 ∗ 𝜕 𝑢)
𝑢𝜕
𝑢
2(𝜕
𝑢𝜕
8
𝜇
𝜇
𝜈
𝜇
𝜇
2
𝐽𝜇 = −4𝑖𝑀2
−
𝑖
(𝛽𝑒
−
1)
1+ 𝑢 2 2
𝑒2
(1 + |𝑢|2 )4
The components:
𝐽𝜌 = 0
The transverse spatial structure
of the polar component 𝐽𝜑 and
the longitudinal component 𝐽𝑧
are a pipelike structure.
The charge per unit length:
𝑄=
𝑑𝑥1 𝑑𝑥2 𝐽0 = −8𝜋𝑀2 𝑘𝑎2 𝑟0
𝑛 1
1
1
1
2 − 1 + Γ(1 + )Γ(1 − )
𝛽𝑒
6 𝑎2
𝑛
𝑛
𝑛
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
𝑉 01
𝜇2
= (1 − 𝑛3 )
2
New baby skyrmion
potential
𝑉11
𝜇2
= (1 − 𝑛3 )(1 + 𝑛3 )
2
The energy density plot of 𝑛 = 3 for old-, and new-baby potentials
Symmetric:
Nonsymmetric:
𝑛 = 1 (old)
𝑛 ≥ 2 new
𝑛 ≥ 2 old
For the potential 𝑉
2
2−𝑛
2
2+𝑛
,
the holomorphic solutions
appear as a ground state!
A sequence of the energy density plots of 𝑛 = 3 for the several 𝛽𝑒 2 for
the old-potential
𝛽𝑒 2 = 1.01
𝛽𝑒 2 = 1.1
𝛽𝑒 2 = 2.0
𝛽𝑒 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
𝜕𝜇 → 𝐷𝜇 = 𝜕𝜇 + 𝐴𝜇 𝑛 ×.
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.
𝛽𝑒 2