In a quantum fluid, we know that it is difficult to add or destroy a

[stephenhky]

In a quantum fluid, we know that it is difficult to add or destroy a vortex because of the conservation of angular momentum.

Suppose now we have a skyrmion replacing a vortex. What the conservation law is skyrmion associated with?

 

[tom.stoer]

In nucleon physics (the original Skyrme model) skyrmions are collective solitonic modes of a fundamental pion field; these collective modes can be seen a nucleons (with the correct quantum numbers) emerging from pions.

The conservation law is not based on Noether's theorem but is due to a so-called topological charge. This charge is something like a winding number due to a non-trivial map from the SU(2) manifold of the pion field into the target manifold S³ (which is compactified space with R³ + the point at infinity).

The winding number of the skyrmion is related to the third homotpy group of SU(2) (third because the map is into S³). The third homotopy group of SU(2) characterizing the mapping is Z (the group of integer numbers).

Unwinding the field configuration to trivial topology (= vacuum) would require an infinite amount of energy.

 

[stephenhky]

I understand that unwinding the field configurations to trivial topology requires an infinite amount of energy, or in other words we can say a continuous deformation of the field from one topology to another is impossible. But is there a physical reason that leads to this consequence? I do not know about nucleon physics, but I am curious why people are interested in Skyrme model. So what is the probability of unwinding a field configuration to a trivial topology despite the infinite energy barrier?

 

[tom.stoer]

Unwinding the field configuration is forbidden classically due to topology. The simplest example is a one-dim. Sine-Gordon model where you compactify the one-dim. space dimension from R to S¹. The winding number 1 cannot be contiuously deformed to 0 due to a conserved charge.

In quantum mechanics or quantum field theory one has to care about quantum fluctuations as well.This is where the energy barrier comes in. Due to the infirnite barrier even quantum fluctuations cannot destroy the soliton. This can be seen by calculating it explicitly for the Sine-Gordon model. Note that the "path" from winding numer 1 to zero is parameterized by a t-dependend function which is not a solution of the field equation. But this is not necessary as the quantum fluctuations need not be classical solutions.

The idea of the Skyrme model is based on chiral-effective theories taking into account low-energy degress of freedoms like pions only; or - in more general approaches - vector mesons in addition to pseudo-scalar mesons. One can even think about replacing SU(2)_Flavor by SU(3)_Flavor. These models a rather successful in describing pion-pion and pion-photon coupling, scattering cross section, form factors and things like that. But they do not contain spin 1/2 baryons and one is therefore interested in enlarging the models such that they can describe pion-nucleon coupling as well. Instead of introducing explicit nucleon spinor fields one tries to stay with the pions and allow them to form stable nucleon states. Now the problem is that one would need a kind of potential term for the pion fields, but one knows that pion-pion coupling is weak - which rules out a normal potential term. So instead of using a potential which binds pions to nucleons one tries to stabalize the nucleons with a topological effect.

It was clear from the very beginning that the so-called non-linear sigma-models with SU(2) pion fields contain a topological sector due to the winding number, but unfortunately such a soliton would collaps to zero size when one starts to minimize its energy. In the Skyrme model (or in vector-meson models) the nucleon is stabilized due to additional self-interactions of the meson fields (this self interaction does not change the soft two-pion interaction at low energies). The Skyrme model is an effective model where other vector-meson contributions have been integarted out.