Help with the Derrick scaling argument and topological solitons

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SUMMARY

The discussion centers on the application of Derrick's theorem to topological solitons, specifically in the context of the Skyrme-Faddeev energy formulation. The energy bound E > c|N|^(3/4) is established, where E is the energy of the soliton and N is the Hopf invariant. The confusion arises from the rescaling of the soliton, which seemingly allows E to approach zero without affecting N. However, it is clarified that Derrick's theorem does not apply to theories with higher than quadratic derivative terms, as explained in Manton & Sutcliffe, particularly in the context of baby skyrmions.

PREREQUISITES
  • Understanding of topological solitons and their properties.
  • Familiarity with Derrick's theorem and its implications in field theory.
  • Knowledge of the Skyrme-Faddeev model and its energy formulations.
  • Basic proficiency in calculus and differential equations, particularly in the context of field theory.
NEXT STEPS
  • Study the derivation and implications of Derrick's theorem in various field theories.
  • Explore the Skyrme model and its applications in particle physics.
  • Investigate the role of topological invariants in soliton stability.
  • Review Manton & Sutcliffe's work, focusing on pages 84-85 and 152-153 for detailed explanations of the scaling argument.
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The discussion is beneficial for theoretical physicists, mathematicians specializing in topology, and graduate students studying field theory and solitons. It provides insights into the stability of topological solitons and the application of Derrick's theorem in advanced theoretical contexts.

JackHolmes
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Derrick scaling argument and topological soliton energy bound seem to contradict each other?
I have been reading Manton & Sutcliffe for some time now and can't quite wrap my head around something.

If you take the Hopf invariant N of a topological soliton ϕ then its Skyrme-Faddeev energy (which I hope I've gotten right up to some constants)

E=∫∂iϕ⋅∂iϕ+(∂iϕ×∂jϕ)⋅(∂iϕ×∂jϕ) d3x​

satisfies the bound E>c|N|3/4 (as per page 423). This seems like a really nice result. (BTW ϕ here is a vector field in ℝ3)

However, using the idea behind Derrick's theorem, I can rescale x→μx for some μ∈ℝ. Then the energy for the re-scaled soliton is something like E(μ(x))∽μE(x). Re-scaling space doesn't affect the Hopf invariant though (since it's a topological property) so you can make E arbitrarily small without changing N just by re-scaling the topological soliton. So I don't see how the inequality can hold since E can be made arbitrarily small.

What is wrong with my intuition here?
 
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Topological field theory isn’t really in my wheelhouse, but I think your answer comes on page 84-85 of Manton and Sutcliffe, where it explains that Derrick’s theorem doesn’t apply for theories with terms higher than quadratic in the derivatives of the field. Maybe someone with more experience in topology can give you a better answer.

Edit: this is, in fact, the correct explanation. Manton and Sutcliffe apply Derrick’s theorem to baby skyrmions on page 152-153 (in 2 dimensions, but an analogous argument goes through for 3D).
 
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For posterity, in case folks with similar questions don’t have access to the book:
The scaling argument considers a finite scalar field ##\phi(\mathbf{x})## in ##\mathbb{R}^d## which is scaled by a factor of ##\mu>0## to give ##\phi(\mu\mathbf{x})##. The scaled energy of the field is a function of ##\mu## in general. In particular, the energy for a field with a scalar potential will be:
$$\begin{align*}
E(\phi(\mu\mathbf{x})) &=\int{\left[\mathbf{\nabla}\phi(\mu\mathbf{x})\cdot\mathbf{\nabla}\phi(\mu\mathbf{x}) + U(\phi(\mu\mathbf{x}))\right]d^dx} \\
&=\int{\left[\mu^{2-d}\mathbf{\nabla}\phi(\mu\mathbf{x})\cdot\mathbf{\nabla}\phi(\mu\mathbf{x}) + \mu^{-d}U(\phi(\mu\mathbf{x}))\right]d^d(\mu x)}
\end{align*}$$
So for ##d\geq2##, the energy will, in general, decrease monotonically with increasing ##\mu##. Derrick’s theorem states that, since the scaled energy function has no stationary point, the only static solution of the field equation is the trivial (vacuum) solution. Therefore, there does not exist a stable topological soliton.

However, for the baby Skyrme model:
$$
E(\phi)=\int{\left[\mathbf{\nabla}\phi\cdot\mathbf{\nabla}\phi + (\mathbf{\nabla}\phi\times\mathbf{\nabla}\phi)\cdot(\mathbf{\nabla}\phi\times\mathbf{\nabla}\phi)\right]d^dx}
$$
Calling the first term of the integral ##E_2## and the second term ##E_4##, we see that, in ##\mathbb{R}^3##:
$$E=\mu^{-1}E_2+\mu E_4$$
which has a minimum at finite ##\mu##, meaning that topologically non-trivial solutions to the field equation are not specifically ruled out by Derrick’s theorem.