  The recently observed abnormal bifurcation of the double binding energy
differences $\delta V_{pn}$ between the odd-odd and even-even nuclei along the
$N=Z$ line from Ni to Rb has challenged the nuclear theories. To solve this
problem, a shell-model-like approach based on the relativistic density
functional theory is established, by treating simultaneously the
neutron-neutron, proton-neutron, and proton-proton pairing correlations both
microscopically and self-consistently. Without any \textit{ad hoc} parameters,
the calculated results well reproduce the observations, and the mechanism for
this abnormal bifurcation is found to be due to the enhanced proton-neutron
pairing correlations in the odd-odd $N=Z$ nuclei, compared with the even-even
ones. The present results provide an excellent interpretation for the abnormal
$\delta V_{pn}$ bifurcation, and provide a clear signal for the existence of
the proton-neutron pairing correlations for nuclei close to the $N=Z$ line.
