Inhomogeneous condensation in the Gross-Neveu model in noninteger spatial dimensions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>d</mml:mi><mml:mo><</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math>. II. Nonzero temperature and chemical potential
Adrian Koenigstein, Laurin Pannullo
Abstract
We continue previous investigations of the (inhomogeneous) phase structure of the Gross-Neveu model in a noninteger number of spatial dimensions (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mn>1</a:mn><a:mo>≤</a:mo><a:mi>d</a:mi><a:mo><</a:mo><a:mn>3</a:mn></a:math>) in the limit of an infinite number of fermion species (<c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>N</c:mi><c:mo stretchy="false">→</c:mo><c:mi>∞</c:mi></c:math>) at (non)zero chemical potential <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mi>μ</f:mi></f:math> [L. Pannullo, Inhomogeneous condensation in the Gross-Neveu model in noninteger spatial dimensions <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mrow><h:mn>1</h:mn><h:mo>≤</h:mo><h:mi>d</h:mi><h:mo><</h:mo><h:mn>3</h:mn></h:mrow></h:math>, ]. In this work, we extend the analysis from zero to nonzero temperature <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi>T</j:mi></j:math>. The phase diagram of the Gross-Neveu model in <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mn>1</l:mn><l:mo>≤</l:mo><l:mi>d</l:mi><l:mo><</l:mo><l:mn>3</l:mn></l:math> spatial dimensions is well-known under the assumption of spatially homogeneous condensation with both a symmetry broken and a symmetric phase present for all spatial dimensions. In <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mi>d</n:mi><n:mo>=</n:mo><n:mn>1</n:mn></n:math> one additionally finds an inhomogeneous phase, where the order parameter, the condensate, is varying in space. Similarly, phases of spatially varying condensates are also found in the Gross-Neveu model in <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi>d</p:mi><p:mo>=</p:mo><p:mn>2</p:mn></p:math> and <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mi>d</r:mi><r:mo>=</r:mo><r:mn>3</r:mn></r:math>, as long as the theory is not fully renormalized, i.e., in the presence of a regulator. For <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:mi>d</t:mi><t:mo>=</t:mo><t:mn>2</t:mn></t:math>, one observes that the inhomogeneous phase vanishes, when the regulator is properly removed (which is not possible for <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:mi>d</v:mi><v:mo>=</v:mo><v:mn>3</v:mn></v:math> without introducing additional parameters). In the present work, we use the stability analysis of the symmetric phase to study the presence (for <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:mn>1</x:mn><x:mo>≤</x:mo><x:mi>d</x:mi><x:mo><</x:mo><x:mn>2</x:mn></x:math>) and absence (for <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:mn>2</z:mn><z:mo>≤</z:mo><z:mi>d</z:mi><z:mo><</z:mo><z:mn>3</z:mn></z:math>) of these inhomogeneous phases and the related moat regimes in the fully renormalized Gross-Neveu model in the <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"><bb:mrow><bb:mi>μ</bb:mi></bb:mrow></bb:math>, <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"><db:mrow><db:mi>T</db:mi></db:mrow></db:math>-plane. We also discuss the relation between “the number of spatial dimensions” and “studying the model with a finite regulator” as well as the possible consequences for the limit <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"><fb:mi>d</fb:mi><fb:mo stretchy="false">→</fb:mo><fb:mn>3</fb:mn></fb:math>. Published by the American Physical Society 2024