Critical dynamics within the real-time FRG approach

The Schwinger-Keldysh functional renormalization group developed by Y.-y. Tan [Real-time dynamics of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>O</a:mi><a:mo stretchy="false">(</a:mo><a:mn>4</a:mn><a:mo stretchy="false">)</a:mo></a:math> scalar theory within the fRG approach, ] is employed to investigate critical dynamics related to a second-order phase transition. The effective action of model A is expanded to the order of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>O</e:mi><e:mo stretchy="false">(</e:mo><e:msup><e:mo>∂</e:mo><e:mn>2</e:mn></e:msup><e:mo stretchy="false">)</e:mo></e:math> in the derivative expansion for the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>O</i:mi><i:mo stretchy="false">(</i:mo><i:mi>N</i:mi><i:mo stretchy="false">)</i:mo></i:math> symmetry. By solving the fixed-point equations of effective potential and wave function, we obtain static and dynamic critical exponents for different values of the spatial dimension <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>d</m:mi></m:math> and the field component number <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>N</o:mi></o:math>. It is found that one has <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>z</q:mi><q:mo>≥</q:mo><q:mn>2</q:mn></q:math> in the whole range of <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mn>2</s:mn><s:mo>≤</s:mo><s:mi>d</s:mi><s:mo>≤</s:mo><s:mn>4</s:mn></s:math> for the case of <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>N</u:mi><u:mo>=</u:mo><u:mn>1</u:mn></u:math>, while in the case of <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>N</w:mi><w:mo>=</w:mo><w:mn>4</w:mn></w:math>, the dynamic critical exponent turns to <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>z</y:mi><y:mo><</y:mo><y:mn>2</y:mn></y:math> when the dimension approach towards <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi>d</ab:mi><ab:mo>=</ab:mo><ab:mn>2</ab:mn></ab:math>.