Transcript Duality and Confinement in the Dual Ginzburg

Duality and Confinement in the
Dual Ginzburg-Landau
Superconductor
Physics Beyond Standard Model – 1st Meeting Rio-Saclay 2006
Leonardo de Sousa Grigorio - Advisor: Clovis Wotzasek
Universidade Federal do Rio de Janeiro
This seminar is organized as follows:
• Duality in usual Maxwell Eletrodynamics;
• The Ginzburg-Landau model of a Dual
Superconductor;
• Confinement between static electric
charges;
• Julia-Toulouse Mechanism;
• Duality between GLDS(Higgs) and JT.
Let us review the symmetry between charges and
electromagnetic fields. Let the Maxwell’s equations:
Dirac introduced magnetic charges exploring the
symmetry of Maxwell’s equations. After that
The symmetry presented by these equations is
In a covariant form these equations read
where
And, as usual
However if we want to describe the fields in terms of
potencials we get a problem. So, if we didn’t have
magnetic monopoles
which is the Bianchi identity. This one can be solved
In order to introduce monopoles we have to violate
Bianchi identity by rewrinting the field strenght as
where the last term is a source, defined by
Let the current created by one electric charge
where
is the world line
of the particle. While associated with
there
is a world sheet.
Applying a divergence and setting
to infinity
The Dirac string defines a region in space where the
gauge potencial becomes ill defined. Summarizing the
duality described above could be seen at the level of
Maxwell’s equations.
How do we describe duality in a most fundamental
way?
Minimizing the action we obtain the equation of
motion,
And the other Maxwell equation comes from the
definition of
So, what is the dual of that Lagrangean? The answer
is
This one can be obtained as follows:
We may get a picture of the couplings
So, lowering the order by a Legendre transformation
Eliminating the vector field we get
By inserting this into the Lagrangean we obtain
This is the dual of the original one.
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• GLDS
Let the dual Abelian Higgs model
where we have a covariant derivative
coupling minimally the vector and matter field and
coupling non-minimally the vector field
to electric charges.
. We may write the Lagrangean in the
Let
following manner
and with an adequate choice of the gauge
All work as if the vector potential absorbed one
of the degrees of freedom of the complex scalar field
and became massive. Let us freeze the remaining degree
of freedom and define
After solving for
we obtain
Going back to the Lagrangean we find
The confinement properties are present in this
form.
• Confinement between static electric
charges
It can be shown that the previous Lagrangean provides
confinement between opposite charges.
Substituting it in the Lagrangean this becomes
In order to find the energy we look at the Hamiltonean
We perform a Fourier transformation
and arrive at
After performing this integral the energy reads
where a cutoff was introduced. It’s physical meaning
is related to a length scale: the size of the vortex core.
• Julia-Toulouse Mechanism
Let us start with this situation
The corresponding Lagrangean is
Field that describes the condensate.
As it turns a field, it must arise three modifications:
-A kinetic term for the condensate;
-The vector field is absorbed by the condensate;
-An interaction that couples the new field to the charges.
Let us work theese ideas through the following
symmetry, that is already present.
A kinetic term wich respects this symmetry is
We can choose
such that
desapears, or in
other words, is absorbed by the condensate. In order to
preserve the symmetry we should have
eats
and gets massive
So
Redefining
It yelds
If we solve, not surprisingly
• Duality between GLDS(Higgs) and JT
We start with the GLDS Lagrangean
By the same methods above we have
Eliminating
and reescaling, we obtain