  We study a monomer-dimer mixture defined on a honeycomb lattice as a toy
model for the spin ice system in a magnetic field. In a low-doping region of
monomers, the effective description of this system is given by the dual
sine-Gordon model. In intermediate- and strong-doping regions, the Potts
lattice gas theory can be employed. Synthesizing these results, we construct a
renormalization-group flow diagram, which includes the stable and unstable
fixed points corresponding to ${\cal M}_5$ and ${\cal M}_6$ in the minimal
models of the conformal field theory. We perform numerical transfer-matrix
calculations to determine a global phase diagram and also to proffer evidence
to check our prediction.
