Spontaneous breaking of the SO(2N) symmetry in the Gross-Neveu model

The canonical Gross-Neveu model for <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>N</a:mi></a:math> two-component Dirac fermions in <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mn>2</c:mn><c:mo>+</c:mo><c:mn>1</c:mn></c:math> dimensions suffers a continuous phase transition at a critical interaction <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>g</e:mi><e:mrow><e:mi>c</e:mi><e:mn>1</e:mn></e:mrow></e:msub><e:mo>∼</e:mo><e:mn>1</e:mn><e:mo>/</e:mo><e:mi>N</e:mi></e:math> at large <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>N</g:mi></g:math>, at which its continuous symmetry <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>SO</i:mi><i:mo stretchy="false">(</i:mo><i:mn>2</i:mn><i:mi>N</i:mi><i:mo stretchy="false">)</i:mo></i:math> is preserved and a discrete (Ising) symmetry becomes spontaneously broken. A recent mean-field calculation, however, points to an additional transition at a different critical <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msub><m:mi>g</m:mi><m:mrow><m:mi>c</m:mi><m:mn>2</m:mn></m:mrow></m:msub><m:mo>∼</m:mo><m:mo>−</m:mo><m:mi>N</m:mi><m:msub><m:mi>g</m:mi><m:mrow><m:mi>c</m:mi><m:mn>1</m:mn></m:mrow></m:msub></m:math>, at which <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>SO</o:mi><o:mo stretchy="false">(</o:mo><o:mn>2</o:mn><o:mi>N</o:mi><o:mo stretchy="false">)</o:mo><o:mo stretchy="false">→</o:mo><o:mi>SO</o:mi><o:mo stretchy="false">(</o:mo><o:mi>N</o:mi><o:mo stretchy="false">)</o:mo><o:mo>×</o:mo><o:mi>SO</o:mi><o:mo stretchy="false">(</o:mo><o:mi>N</o:mi><o:mo stretchy="false">)</o:mo></o:math>. To study the latter phase transition we rewrite the Gross-Neveu interaction <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:mi>g</x:mi><x:mo stretchy="false">(</x:mo><x:mover accent="true"><x:mi>ψ</x:mi><x:mo stretchy="false">¯</x:mo></x:mover><x:mi>ψ</x:mi><x:msup><x:mo stretchy="false">)</x:mo><x:mn>2</x:mn></x:msup></x:math> in terms of three different quartic terms for the single (<db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"><db:mi>L</db:mi><db:mo>=</db:mo><db:mn>1</db:mn></db:math>) <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"><fb:mn>4</fb:mn><fb:mi>N</fb:mi></fb:math>-component real (Majorana) fermion, and then extend the theory to <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"><hb:mi>L</hb:mi><hb:mo>></hb:mo><hb:mn>1</hb:mn></hb:math>. This allows us to track the evolution of the fixed points of the renormalization group transformation starting from <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" display="inline"><jb:mi>L</jb:mi><jb:mo>≫</jb:mo><jb:mn>1</jb:mn></jb:math>, where one can discern three distinct critical points which correspond to continuous phase transitions into (1) <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" display="inline"><lb:mi>SO</lb:mi><lb:mo stretchy="false">(</lb:mo><lb:mn>2</lb:mn><lb:mi>N</lb:mi><lb:mo stretchy="false">)</lb:mo></lb:math>-singlet mass-order-parameter, (2) <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" display="inline"><pb:mi>SO</pb:mi><pb:mo stretchy="false">(</pb:mo><pb:mn>2</pb:mn><pb:mi>N</pb:mi><pb:mo stretchy="false">)</pb:mo></pb:math>-symmetric-tensor mass-order-parameters, and (3) <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"><tb:mi>SO</tb:mi><tb:mo stretchy="false">(</tb:mo><tb:mn>2</tb:mn><tb:mi>N</tb:mi><tb:mo stretchy="false">)</tb:mo></tb:math>-adjoint nematic-order-parameters, down to <xb:math xmlns:xb="http://www.w3.org/1998/Math/MathML" display="inline"><xb:mi>L</xb:mi><xb:mo>=</xb:mo><xb:mn>1</xb:mn></xb:math> value that is relevant to the standard Gross-Neveu model. Below the critical value of <zb:math xmlns:zb="http://www.w3.org/1998/Math/MathML" display="inline"><zb:msub><zb:mi>L</zb:mi><zb:mi>c</zb:mi></zb:msub><zb:mo stretchy="false">(</zb:mo><zb:mi>N</zb:mi><zb:mo stretchy="false">)</zb:mo><zb:mo>≈</zb:mo><zb:mn>0.35</zb:mn><zb:mi>N</zb:mi></zb:math> for <dc:math xmlns:dc="http://www.w3.org/1998/Math/MathML" display="inline"><dc:mi>N</dc:mi><dc:mo>≫</dc:mo><dc:mn>1</dc:mn></dc:math> only the Gross-Neveu critical point (1) still implies a diverging susceptibility for its corresponding (<fc:math xmlns:fc="http://www.w3.org/1998/Math/MathML" display="inline"><fc:mi>SO</fc:mi><fc:mo stretchy="false">(</fc:mo><fc:mn>2</fc:mn><fc:mi>N</fc:mi><fc:mo stretchy="false">)</fc:mo></fc:math>-singlet) order parameter, whereas the two new critical points that existed at large <jc:math xmlns:jc="http://www.w3.org/1998/Math/MathML" display="inline"><jc:mi>L</jc:mi></jc:math> ultimately become equivalent to the Gaussian fixed point at <lc:math xmlns:lc="http://www.w3.org/1998/Math/MathML" display="inline"><lc:mi>L</lc:mi><lc:mo>=</lc:mo><lc:mn>1</lc:mn></lc:math>. We interpret this metamorphosis of the <nc:math xmlns:nc="http://www.w3.org/1998/Math/MathML" display="inline"><nc:mi>SO</nc:mi><nc:mo stretchy="false">(</nc:mo><nc:mn>2</nc:mn><nc:mi>N</nc:mi><nc:mo stretchy="false">)</nc:mo></nc:math>-symmetric-tensor fixed point from critical to spurious as an indication that the transition at <rc:math xmlns:rc="http://www.w3.org/1998/Math/MathML" display="inline"><rc:msub><rc:mi>g</rc:mi><rc:mrow><rc:mi>c</rc:mi><rc:mn>2</rc:mn></rc:mrow></rc:msub></rc:math> in the original Gross-Neveu model is turned first-order by fluctuations. Published by the American Physical Society 2024