We develop the Gutzwiller approximation method to obtain the renormalized Hamiltonian of the SU(4) $t$-$J$ model, with the corresponding renormalization factors. Subsequently, a mean-field theory is employed on the renormalized Hamiltonian of the mod
el on the honeycomb lattice under the scenario of a cooperative condensation of carriers moving in the resonating valence bond state of flavors. In particular, we find the extended $s$-wave superconductivity is much more favorable than the $d+id$ superconductivity in the doping range close to quarter filling. The pairing states of the SU(4) case reveal the property that the spin-singlet pairing and the spin-triplet pairing can exist simultaneously. Our results might provide new insights into the twisted bilayer graphene system.
A theoretical formalism of Andreev reflection is proposed to provide a theoretical support for distinguishing the singlet pairing and the triplet pairing by the point contact Andreev reflection (PCAR) experiments, in contrast to the previous models d
esigned only for the singlet pairing case. We utilize our theoretical curves to fit the data of the PCAR experiment on the unconventional superconductivity in the $Bi/Ni$ bilayer [arXiv:1810.10403], and find the Anderson-Brinkman-Morel state satisfies the main characteristics of the experimental data. Moreover, the Andreev reflection spectra of the Balian-Werthamer state and the chiral $p$-wave are also presented.
