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?

And the other Maxwell equation comes from the definition of

So, what is the dual of that Lagrangean? The answeris

Minimizing the action we obtain the equation ofmotion,

We may get a picture of the couplings

So, lowering the order by a Legendre transformation

This one can be obtained as follows:

This is the dual of the original one.

*

By inserting this into the Lagrangean we obtain

Eliminating the vector field we get

• 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.

Let . We may write the Lagrangean in the

following manner

and with an adequate choice of the gauge

All work as if the vector potential absorbed oneof the degrees of freedom of the complex scalar fieldand became massive. Let us freeze the remaining degreeof 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

Substituting it in the Lagrangean this becomes

It can be shown that the previous Lagrangean provides confinement between opposite charges.

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 meaningis 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