  Recently room temperature superconductor was claimed in
Pb$_{10-x}$Cu$_x$(PO$_4$)$_6$O (also known as LK-99) with $x\in (0.9,1.1)$.
Density functional theory (DFT) calculations suggest that the conduction
electrons are from the doped Cu atoms with valence close to $d^{9}$. Motivated
by this picture, we build a two-orbital Hubbard model on a triangular lattice
formed by the $d_{xz}$ and $d_{yz}$ orbitals with total hole density (summed
over spin and orbital) $n=1-p$. When $p=0$, the system is in a Mott insulator
within this model. When $p>0$, we derive a $t-J$ model and perform a
self-consistent slave boson mean field calculation. Interestingly we find a
s-wave pairing in contrast to the one-orbital t-J model which favors $d+id$
pairing. S wave pairing should be more robust to disorder and may lead to high
Tc superconductor with sufficiently large values of $t$ and $J$. However, the
DFT calculations predict a very small value of $t$ and then the $T_c$ is
expected to be small. If LK99 is really a high Tc superconductor, ingredients
beyond the current model are needed. We conjecture that the doped Cu atoms may
distort the original lattice and form local clusters with smaller Cu -Cu
distance and thus larger values of $t$ and $J$. Within these clusters, we may
locally apply our t-J model calculation and expect high Tc s-wave
superconductor. Then the superconducting islands couple together, which may
eventually become a global superconductor, an insulator or even an anomalous
metal depending on sample details.
