How are vortices the finite energy time independent solutions for 2+1 dimensions abelian Higgs model? Doesn't it violate Derrick's theorem that there are no finite energy time independent solutions in dimensions higher than 1+1?
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2 Answers
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No, it's not a violation. The conclusion of Derrick's theorem is modified in the presence of gauge fields, cf. e.g. this Phys.SE post.
answered Nov 23, 2017 at 21:07
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No, it does not violate. Derrick's theorem is actually very restricted. It says that there is no finite energy time independent solution for $$L=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-V(\phi),$$ in dimensions larger than $1+1$. The vortice in the Abelian-Higgs model appears for the lagrangian $$L=-\frac 14F_{\mu\nu}F^{\mu\nu}+(D_\mu\phi)^*D^\mu\phi-\frac{\lambda}{4}(|\phi|^2-v^2)^2,$$ in $2+1$ or $3+1$. This is a gauge theory and the gauge field provides "more structure" to the theory, allowing for finite time independent solutions.
The intuition is that the for the first theory, there is a term $\int rdr\partial_i\phi\partial_i\phi$ in the energy density of the possible vortex but this diverges logarithmically. The corresponding term for the second theory is $\int rdr (D_i\phi)^*D_i\phi$ and the presence of the gauge field $A$ turns it possible that $D_i\phi=\partial_i\phi+ieA_i\phi$ decays sufficiently fast.
answered Nov 23, 2017 at 21:10