  Recent experiments and simulations have shown that two-dimensional systems
can form tetratic phases with four-fold rotational symmetry, even if they are
composed of particles with only two-fold symmetry. To understand this effect,
we propose a model for the statistical mechanics of particles with almost
four-fold symmetry, which is weakly broken down to two-fold. We introduce a
coefficient $\kappa$ to characterize the symmetry breaking, and find that the
tetratic phase can still exist even up to a substantial value of $\kappa$.
Through a Landau expansion of the free energy, we calculate the mean-field
phase diagram, which is similar to the result of a previous hard-particle
excluded-volume model. To verify our mean-field calculation, we develop a Monte
Carlo simulation of spins on a triangular lattice. The results of the
simulation agree very well with the Landau theory.
