Short-range intervortex forces

An explicit formula for the interaction energy of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>n</a:mi></a:math> vortices in the Abelian Higgs (or Ginzburg-Landau) model is derived, valid in the regime where all vortices are close to one another. An immediate consequence of this formula is that the interaction energy of a vortex pair with separation <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>d</c:mi></c:math> varies as <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msup><e:mi>d</e:mi><e:mn>4</e:mn></e:msup></e:math>, not <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msup><g:mi>d</g:mi><g:mn>2</g:mn></g:msup></g:math>. The formula contains <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>n</i:mi><i:mo>−</i:mo><i:mn>1</i:mn></i:math> real coefficients, which are fixed by certain spectral data of the Jacobi operator of the cocentered <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>n</k:mi></k:math>-vortex. The coefficients are computed numerically for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>n</m:mi><m:mo>=</m:mo><m:mn>2</m:mn></m:math> and <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>n</o:mi><o:mo>=</o:mo><o:mn>3</o:mn></o:math> for couplings <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mn>0.1</q:mn><q:mo>≤</q:mo><q:mi>λ</q:mi><q:mo>≤</q:mo><q:mn>2.5</q:mn></q:math>. The resulting short-range interaction potentials are compared with the results of full field theory simulations for <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>λ</s:mi><s:mo>=</s:mo><s:mn>0.5</s:mn></s:math> and <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>λ</u:mi><u:mo>=</u:mo><u:mn>2</u:mn></u:math>, with excellent agreement at small to moderate vortex separation.