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.
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