  Ultrathin sheets of transition metal dichalcogenides (MX$ _2$) with charge
density waves (CDWs) is increasingly gaining interest as a promising candidate
for graphene-like devices. Although experimental data including
stripe/quasi-stripe structure and hidden states have been reported, the ground
state of ultrathin MX$ _2$ compounds and, in particular, the origin of
anisotropic (stripe and quasi-stripe) CDW phases is a long-standing problem.
Anisotropic CDW phases have been explained by Coulomb interaction between
domain walls and inter-layer interaction. However, these models assume that
anisotropic domain walls can exist in the first place. Here, we report that
anisotropic CDW domain walls can appear naturally without assuming anisotropic
interactions: We explain the origin of these phases by topological defect
theory (line defects in a two-dimensional plane) and interference between
harmonics of macroscopic CDW wave functions. We revisit the
McMillan-Nakanishi-Shiba model for monolayer 1$T$-TaS$ _2$ and 2$H$-TaSe$ _2$
and show that CDWs with wave vectors that are separated by $120^\circ$ (i.e.
the three-fold rotation symmetry of the underlying lattice) contain a
free-energy landscape with many local minima. Then, we remove this $120^\circ$
constraint and show that free energy local minima corresponding to the stripe
and quasi-stripe phase appear. Our results imply that Coulomb interaction
between domain walls and inter-layer interaction may be secondary factors for
the appearance of these phases. Furthermore, this model can predict new CDW
phases, hence it may become the basis to study CDW further. We anticipate our
results to be a starting point for further study in two-dimensional physics,
such as explanation of "Hidden CDW states", study the interplay between
supersolid symmetry and lattice symmetry, and application to other van der
Waals structures.
