Spontaneous symmetry breaking of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>SO</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> <

It was recently established that the paradigmatic Gross-Neveu model with <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>N</a:mi> </a:math> copies of two-dimensional Dirac fermions features an <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>SO</c:mi> <c:mo stretchy="false">(</c:mo> <c:mn>2</c:mn> <c:mi>N</c:mi> <c:mo stretchy="false">)</c:mo> </c:math> symmetry if certain interactions are suppressed. This becomes evident when the theory is rewritten in terms of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mn>2</g:mn> <g:mi>N</g:mi> </g:math> copies of two-dimensional Majorana fermions. Mean-field theory for the <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> model predicts, besides the chiral Ising transition at <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>1</m:mn> </m:mrow> </m:msub> </m:math> , a second critical point <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:msub> <o:mi>g</o:mi> <o:mrow> <o:mi>c</o:mi> <o:mn>2</o:mn> </o:mrow> </o:msub> </o:math> where <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mi>SO</q:mi> <q:mo stretchy="false">(</q:mo> <q:mn>2</q:mn> <q:mi>N</q:mi> <q:mo stretchy="false">)</q:mo> </q:math> is broken down to <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi>SO</u:mi> <u:mo stretchy="false">(</u:mo> <u:mi>N</u:mi> <u:mo stretchy="false">)</u:mo> <u:mo>×</u:mo> <u:mi>SO</u:mi> <u:mo stretchy="false">(</u:mo> <u:mi>N</u:mi> <u:mo stretchy="false">)</u:mo> </u:math> . A subsequent Wilsonian renormalization group analysis directly in <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:mi>d</ab:mi> <ab:mo>=</ab:mo> <ab:mn>3</ab:mn> </ab:math> supports its existence in a generalized theory, where <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:msub> <cb:mi>N</cb:mi> <cb:mi>f</cb:mi> </cb:msub> </cb:math> copies of the <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:mn>4</eb:mn> <eb:mi>N</eb:mi> </eb:math> -component Majorana fermions are introduced. This allows us to track the evolution of a (i) quantum anomalous Hall Gross-Neveu-Ising, (ii) symmetric-tensor, and (iii) adjoint-nematic fixed point separately. However, it turns out that (ii) and (iii) lose their criticality when approaching <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"> <gb:msub> <gb:mi>N</gb:mi> <gb:mi>f</gb:mi> </gb:msub> <gb:mo>=</gb:mo> <gb:mn>1</gb:mn> </gb:math> , suggesting that the transition is first order. In this work, we approach the problem from the lower-critical dimension of two. We construct a Fierz-complete renormalizable Lagrangian, compute the leading order <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"> <ib:mi>β</ib:mi> </ib:math> functions, fermion anomalous dimension, as well as the order parameter anomalous dimensions, and resolve the three universality classes corresponding to (i)–(iii). Before becoming equal to the Gaussian fixed point at <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"> <kb:msub> <kb:mi>N</kb:mi> <kb:mi>f</kb:mi> </kb:msub> <kb:mo>=</kb:mo> <kb:mn>1</kb:mn> </kb:math> , (ii) remains critical for all values of <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"> <mb:mrow> <mb:msub> <mb:mrow> <mb:mi>N</mb:mi> </mb:mrow> <mb:mrow> <mb:mi>f</mb:mi> </mb:mrow> </mb:msub> <mb:mo>&gt;</mb:mo> <mb:msubsup> <mb:mrow> <mb:mi>N</mb:mi> </mb:mrow> <mb:mrow> <mb:mi>f</mb:mi> <mb:mo>,</mb:mo> <mb:mi>c</mb:mi> </mb:mrow> <mb:mrow> <mb:mi>ST</mb:mi> </mb:mrow> </mb:msubsup> <mb:mo stretchy="false">(</mb:mo> <mb:mi>N</mb:mi> <mb:mo stretchy="false">)</mb:mo> <mb:mo>≈</mb:mo> <mb:mn>0.56</mb:mn> <mb:mo>+</mb:mo> <mb:mn>1.48</mb:mn> <mb:mi>N</mb:mi> <mb:mo>+</mb:mo> <mb:mi mathvariant="script">O</mb:mi> <mb:mo stretchy="false">(</mb:mo> <mb:mi>ε</mb:mi> <mb:mo stretchy="false">)</mb:mo> </mb:mrow> </mb:math> , which compares well with the estimate of previous studies. We further find that (iii) becomes equal to (i) when approaching <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"> <tb:msub> <tb:mi>N</tb:mi> <tb:mi>f</tb:mi> </tb:msub> <tb:mo>=</tb:mo> <tb:mn>1</tb:mn> </tb:math> . An instability is, however, only present in the susceptibility corresponding to the Gross-Neveu-Ising order parameter.