We investigate the cluster structure of ${}^{13}\mathrm{N}$ using a microscopic $\alpha +\alpha +\alpha +p$ four-body cluster model. The calculated spectra reasonably agree with the observed spectra in the low-lying states. We calculate the reduced width amplitudes and spectroscopic factors to investigate the Hoyle-analog state in ${}^{13}\mathrm{N}$. Our calculations show that the $3/2_3^{-}$ state at $E_x=10.8$ MeV is primarily composed of ${}^{12}\mathrm{C}\left(0_2^{+}\right)+\mathit{p}$ and ${}^9\mathrm{B}(3/2^{-})+\alpha$ components. This finding is generally consistent with the newly observed $3/2^{-}$ state at $E_x=11.3$ MeV via the $3\alpha +\mathit{p}$ decay channel. Moreover, considering the calculated root-mean-square radius and isoscalar monopole transition strength, the $3/2_3^{-}$ state emerges as a candidate for the Hoyle-analog state with the ${}^{12}\mathrm{C}\left(0_2^{+}\right)+\mathit{p}$ cluster structure.

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• Cluster models
• Nuclear structure & decays
• 6 ≤ A ≤ 19