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Critical structure and emergent symmetry of Dirac fermion systems

Emergent symmetry in Dirac system means that the system acquires an enlargement of two basic symmetries at some special critical point. The continuous quantum criticality between the two symmetry broken phases can be described within the framework of Gross-Neveu-Yukawa (GNY) model. Using the first-order $ε$ expansion in $4-ε$ dimensions, we study the critical structure and emergent symmetry of the chiral GNY model with $N_f$ flavors of four-component Dirac fermions coupled strongly to an $O(N)$ scalar field under a small $O(N)$-symmetry breaking perturbation. After determining the stable fixed point, we calculate the inverse correlation length exponent and the anomalous dimensions (bosonic and fermionic) for general $N$ and $N_f$. Further, we discuss the emergent-symmetry and the emergent supersymmetric critical point for $N\geq4$ on the basis of $O(N)$-GNY model. It turns out that the chiral emergent-$O(N)$ universality class is physically meaningful if and only if $N<2N_f+4$. On this premise, the small $O(N)$-symmetry breaking perturbation is always irrelevant in the chiral emergent-$O(N)$ universality class. Our studies show that the emergent symmetry in Dirac systems has an upper boundary $O(2N_f+3)$, depending on the flavor numbers $N_f$. As a result, the emergent-$O(4)$ and $O(5)$ symmetries are possible to be found in the systems with fermion flavor $N_f=1$, and the emergent-$O(4)$, $O(5)$, $O(6)$ and $O(7)$ symmetries are expected to be found in the systems with fermion flavor $N_f=2$. Our result also suggests some rich transitions with emergent-$Z_2\times O(2)\times O(3)$ symmetry and so on. Interestingly, in the emergent-$O(4)$ universality class, there is a supersymmetric critical point which is expected to be found in the systems with fermion flavor $N_f=1$.