The fate of non-supersymmetric Gross-Neveu-Yukawa fixed point in two dimensions

We investigate the fate of the non-supersymmetric Gross-Neveu-Yukawa fixed point found by Fei et al in $4-\epsilon$ dimensions with a two-component Majorana fermion continued to two dimensions. Assuming that it is a fermionic minimal model which possesses a chiral $\mathbb{Z}_2$ symmetry (in addition to fermion number parity) and just two relevant singlet operators, we can zero in on four candidates. Assuming further that the least relevant deformation leads to the supersymmetric Gross-Neveu-Yukawa fixed point (i.e. fermionic tricritical Ising model), we can rule out two of them by matching the spin contents of the preserved topological defect lines. The final candidates are the fermionic $(11,4)$ minimal model if it is non-unitary, and the fermionic $(E_{6}, A_{10})$ minimal model if it is unitary. If we further use a constraint from the double braiding relation proposed by one of the authors, the former scenario is preferable.