Gross-Neveu-Yukawa theory of $\text{SO}(2N)\rightarrow \text{SO}(N) \times \text{SO}(N)$ spontaneous symmetry breaking

Abstract

We construct and study the relativistic Gross-Neveu-Yukawa field theory for the $\text{SO}(2N)$ real symmetric second-rank tensor order parameter coupled to $N_f$ flavors of $4N$-component Majorana fermions in 2+1 dimensions. Such a tensor order parameter unifies all Lorentz-invariant mass-gap orders for $N$ two-component Dirac fermions in two dimensions except for the $\text{SO}(2N)$-singlet anomalous quantum Hall state. The value $N_f=1$ corresponds to the canonical Gross-Neveu model. Within the leading-order $\epsilon$-expansion around the upper critical dimension of $3+1$ the field theory exhibits a critical fixed point in its renormalization group flow which describes spontaneous symmetry breaking to $\text{SO}(N)\times \text{SO}(N)$ for the number of flavors of Majorana fermions higher than a critical value $N_{f,c2}\approx 2N$. For $N_{f, c1}< N_f < N_{f,c2}$ , with $N_{f,c1} \approx N$ the critical fixed point resides in the unstable region of the theory where the effective potential is unbounded from below, whereas for $N_f < N_{f,c1}$ there is no real critical fixed point, and the flow runs away. In either case, for $N_f < N_{f,c2}$ the transition should become fluctuation-induced first-order, and we discuss the dependence of its size on the parameters $N$ and $N_{f}$ in the theory. One-loop critical exponents for the universality class at $N_{f, c2}< N_f $ are computed and the flow diagram in various regimes is discussed.