Loop arrangements and their quantum superpositions describe several interesting many-particle states. We propose that they also describe bonding in a class of transition metal dichalcogenides. We present an effective quantum loop model for monolayers with a $1T$ structure and a $d^2$ valence electron configuration: materials of the form $M{X}_2$ $(M=\text{Mo}, \mathrm{W} \text{and} X=\text{S}, \mathrm{Se}, \mathrm{Te})$ and $A{M}^{\prime }{Y}_2$ $(A=\text{Li}, \mathrm{Na}; M^{\prime }=\text{V}, \mathrm{Nb}; \text{and} Y=\text{O}, \mathrm{S}, \mathrm{Se})$. Their $t_{2g}$ orbitals exhibit strongly directional overlaps between neighboring atoms, favoring the formation of valence bonds. A transition metal atom forms two valence bonds, each with one of its neighbors. When connected, these bonds form loops that cover the triangular lattice. We construct a minimal Rokhsar-Kivelson-like model with resonance processes that cut and reconnect loops that run in proximity. The resulting dynamics is more constrained than in traditional quantum dimer models, with a “bending” constraint that arises from orbital structure. In the resulting phase diagram, we find phases that resemble distorted phases seen in materials, viz., the $1T^{\prime }$ and trimerized phases. As a testable prediction, we propose that a single $d^1$ or $d^3$ impurity will terminate a loop and give rise to a long-ranged texture. For example, a Ti/Cr defect in ${\mathrm{LiVO}}_2$ will produce one or more domain walls that propagate outward from the impurity. We discuss the possibility of a loop-liquid phase that can emerge in these materials.

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• Domain walls
• Impurities
• Phase diagrams
• Structural phase transition
• Monolayer films
• Strongly correlated systems
• Transition metal dichalcogenides
• Triangular lattice