Scaling dimension of the 4π -flux monopole operator in four-flavor three-dimensional QED using lattice simulation

We numerically address the issue of which monopole operators are relevant under renormalization group flow in three-dimensional parity-invariant noncompact QED with four flavors of massless two-component Dirac fermion. Using lattice simulation and finite-size scaling analysis of the free energy to introduce monopole-antimonopole pairs in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>N</a:mi><a:mo>=</a:mo><a:mn>4</a:mn></a:math> and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>N</c:mi><c:mo>=</c:mo><c:mn>12</c:mn></c:math> flavor noncompact <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:msub><e:mrow><e:mi>QED</e:mi></e:mrow><e:mrow><e:mn>3</e:mn></e:mrow></e:msub></e:mrow></e:math>, we estimate the infrared scaling dimensions of monopole operators that introduce <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mn>2</g:mn><g:mi>π</g:mi></g:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mn>4</i:mn><i:mi>π</i:mi></i:math> fluxes around them. We first show that the estimates for the monopole scaling dimensions are consistent with the large-<k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>N</k:mi></k:math> expectations 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>12</m:mn></m:math> <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mrow><o:msub><o:mrow><o:mi>QED</o:mi></o:mrow><o:mrow><o:mn>3</o:mn></o:mrow></o:msub></o:mrow></o:math>. Applying the same procedure in <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>N</q:mi><q:mo>=</q:mo><q:mn>4</q:mn></q:math> <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mrow><s:msub><s:mrow><s:mi>QED</s:mi></s:mrow><s:mrow><s:mn>3</s:mn></s:mrow></s:msub></s:mrow></s:math>, we estimate the scaling dimension of <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mn>4</u:mn><u:mi>π</u:mi></u:math> flux monopole operator to be 3.7(3), which allows the possibility of the operator being irrelevant. This finding offers support to the scenario in which higher-flux monopoles are irrelevant deformations to the Dirac spin liquid phase that could be realized on certain non-bipartite lattices by forbidding <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mn>2</w:mn><w:mi>π</w:mi></w:math>-flux monopoles.