Condensates, Crystals, and Renormalons in the Gross-Neveu Model at Finite Density

We study the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>O</a:mi><a:mo stretchy="false">(</a:mo><a:mn>2</a:mn><a:mi>N</a:mi><a:mo stretchy="false">)</a:mo></a:mrow></a:math> symmetric Gross-Neveu model at finite density in the presence of a <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>U</e:mi><e:mo stretchy="false">(</e:mo><e:mn>1</e:mn><e:mo stretchy="false">)</e:mo></e:math> chemical potential <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>h</i:mi></i:math> for a generic number <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mrow><k:mi>a</k:mi><k:mo>≤</k:mo><k:mi>N</k:mi><k:mo>−</k:mo><k:mn>2</k:mn></k:mrow></k:math> of fermion fields. By combining perturbative quantum field theory, semiclassical large <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>N</m:mi></m:math>, and Bethe ansatz techniques, we show that at finite <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>N</o:mi></o:math> two new dynamically generated scales <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msub><q:mi mathvariant="normal">Λ</q:mi><q:mi mathvariant="normal">n</q:mi></q:msub></q:math> and <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msub><u:mi mathvariant="normal">Λ</u:mi><u:mi mathvariant="normal">c</u:mi></u:msub></u:math> appear in the theory, governing the mass gap of neutral and charged fermions, respectively. Above a certain threshold value for <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>h</y:mi></y:math>, <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi>a</ab:mi></ab:math>-fermion bound states condense and form an inhomogeneous configuration, which at infinite <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mi>N</cb:mi></cb:math> is a crystal spontaneously breaking translations. At large <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:mi>h</eb:mi></eb:math>, this crystal has mean <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:msub><gb:mi mathvariant="normal">Λ</gb:mi><gb:mi mathvariant="normal">n</gb:mi></gb:msub></gb:math> and spatial oscillations of amplitude <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:msub><kb:mi mathvariant="normal">Λ</kb:mi><kb:mi mathvariant="normal">c</kb:mi></kb:msub></kb:math>. The two scales also control the nonperturbative corrections to the free energy, resolving a puzzle concerning fractional-power renormalons and predicting new ones.