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Ultrathin sheets of transition metal dichalcogenides (MX$ _2$) with chargedensity waves (CDWs) is increasingly gaining interest as a promising candidatefor graphene-like devices. Although experimental data includingstripe/quasi-stripe structure and hidden states have been reported, the groundstate of ultrathin MX$ _2$ compounds and, in particular, the origin ofanisotropic (stripe and quasi-stripe) CDW phases is a long-standing problem.Anisotropic CDW phases have been explained by Coulomb interaction betweendomain walls and inter-layer interaction. However, these models assume thatanisotropic domain walls can exist in the first place. Here, we report thatanisotropic CDW domain walls can appear naturally without assuming anisotropicinteractions: We explain the origin of these phases by topological defecttheory (line defects in a two-dimensional plane) and interference betweenharmonics of macroscopic CDW wave functions. We revisit theMcMillan-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 afree-energy landscape with many local minima. Then, we remove this $120^\circ$constraint and show that free energy local minima corresponding to the stripeand quasi-stripe phase appear. Our results imply that Coulomb interactionbetween domain walls and inter-layer interaction may be secondary factors forthe appearance of these phases. Furthermore, this model can predict new CDWphases, hence it may become the basis to study CDW further. We anticipate ourresults to be a starting point for further study in two-dimensional physics,such as explanation of "Hidden CDW states", study the interplay betweensupersolid symmetry and lattice symmetry, and application to other van derWaals structures.

Origin of Stripe and Quasi-Stripe CDW Structures in Monolayer MX$ _2$ compounds: Multivalley Free Energy Landscape