Dimer models are well known as prototypes for locally constrained physics. They describe systems in which every site on a lattice must be attached to one dimer. Loop models extend this idea, with the constraint that two dimers must touch at each site. Here, we present a further generalization where every site must have three dimers attached—a trivalent network model. As concrete physical realizations, we discuss $d^3$ transition metal dichalcogenides in the 1T structure—materials with the structural formula ${MX}_2$ $(M=\mathrm{Tc}, \mathrm{Re})$ or ${AM}^{\prime }{X}_2$ $(A=\mathrm{Li} \mathrm{or} \mathrm{Na};$ $M^{\prime }=\mathrm{Mo}, \mathrm{W})$, where X is a chalcogen atom. These materials have a triangular layer of transition metal atoms, each with three valence electrons in $t_{2g}$ orbitals. Each atom forms valence bonds with three of its nearest neighbors. The geometry of the 1T structure imbues each bond with sharp orbital character. We argue that this imposes a “bending constraint” so that two dimers attached to the same site cannot be parallel. This leads to a highly structured space of configurations, with alternating bonds along each line of the underlying triangular lattice. There is no dynamics, as constraints forbid local rearrangements of dimers. We construct a phase diagram, identifying configurations that minimize potential energy. We find a rhombus-stripe phase that explains a distortion pattern seen across several materials. Remarkably, the local constraints in this model lead to a simple example of holography. The bonding configuration in the bulk is completely determined by the configuration at the boundary. We recast the model in terms of three Ising chains that are defined on the boundaries of a triangular cluster. As a testable prediction, we propose that a single impurity will generate long-ranged domain walls.

Close

PrevNext

View figure in article

• Domain walls
• Impurities
• Phase diagrams
• Stripes
• Structural phase transition
• Valence
• Transition metal dichalcogenides
• Triangular lattice
• Exact diagonalization