  The TiSiCO-family monolayers have recently been attracting significant
attention due to their unique valley-layer coupling (VLC). In this work, we
present a minimal, four-band tight-binding (TB) model to capture the low-energy
physics of the TiSiCO-family monolayers $X_{2}Y$CO$_{2}$ ($X=$ Ti, Zr, Hf; $Y=$
Si, Ge) with strong VLC. These monolayers comprise two $X$ atom layers
separated by approximately $4$ \AA ~in the out-of-plane direction. Around each
valley ($X$ or $X'$), the conduction and valence bands are mainly dominated by
the $A_{1}\{d_{z^{2}(x^{2}-y^{2})}\}$ and $B_{2}\{d_{yz}\}$ orbitals of the top
$X$ atoms,and the $A_{1}\{d_{z^{2}(x^{2}-y^{2})}\}$ and $B_{1}\{d_{xz}\}$
orbitals of the bottom $X$ atoms. Using these four states as a basis, we
construct a symmetry-allowed TB model. Through parameter fitting from
first-principles calculations, the four-band TB model not only reproduces the
electronic band structure, but also captures the strong VLC, high-order
topology, and valley-contrasting linear dichroism of the monolayers.
Furthermore, the TB model reveals that these monolayers may exhibit various
intriguing topological phases under electric fields and biaxial strains. Hence,
the TB model established here can serve as the starting point for future
research exploring the physics related to VLC and the $X_{2}Y$CO$_{2}$
monolayers.
