Hoyle-analog state in ${}^{13}$N

We investigate the cluster structure of $\rm {}^{13}N$ using a microscopic $\alpha+\alpha+\alpha+p$ four-body cluster model. The calculated spectra agree well 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 $\rm {}^{13}N$. Our calculations show that the $3/2_3^-$ state at $E_x=10.8$ MeV is primarily constituted by $\rm {}^{12}C(0^+_2)+\mathit{p}$ and $\rm {}^{9}B(3/2^-)+\alpha$ 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 strengths, the $3/2_3^-$ state emerges as a candidate for the Hoyle-analog state with the $\rm {}^{12}C(0^+_2)+\mathit{p}$ cluster structure.