Quantum loops in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>1</mml:mn> <mml:mi>T</mml:mi> </mml:math> transition metal dichalcogenides

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 $A{M}^{\ensuremath{'}}{Y}_{2}$ $(A=\text{Li}, \mathrm{Na}; {M}^{\ensuremath{'}}=\text{V}, \mathrm{Nb}; \text{and} Y=\text{O}, \mathrm{S}, \mathrm{Se})$. 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 $1{T}^{\ensuremath{'}}$ 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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