  Motivated by the experiments on the triangular lattice bilayer colbaltate
K$_2$Co$_2$(SeO$_3$)$_3$, we formulate a theory to explore the underlying
physics from a couple observation. The model is composed of interacting
Co$^{2+}$ dimers on the triangular lattice, where the Co$^{2+}$ ion provides an
effective spin-1/2 local moment via the spin-orbit coupling and the crystal
field effect. The intra-dimer interaction is dominant and would simply favor
the local spin singlet, and the inter-dimer interactions compete with the
inter-dimer interaction, leading to rich behaviors. With the easy-axis
anisotropy, it is shown that, in the ground state manifold of the intra-dimer
Ising interaction, the system realizes an effective transverse field Ising
model, where the ground state is either a three-sublattice order or Ising
disordered. The finite temperature regime naturally realizes the
Berezinskii-Kosterlitz-Thouless physics. To explore the full excitations, we
incorporate the excited state manifold of the intra-dimer Ising interaction and
establish the emergent Kugel-Khomskii physics. Thus, the triangular lattice
bilayer colbaltate is an excellent platform to explore the interplay between
geometrical frustration and anisotropic interactions as well as the emergent
effective models and the resulting physics.
