Quantum three-coloring dimer model and the disruptive effect of quantum glassiness on its line of critical points

We construct a quantum extension of the (classical) three-coloring model introduced by Baxter [J. Math. Phys. 11, 784 (1970)] for which the ground state can be computed exactly along a continuous line of Rokhsar-Kivelson solvable points. The quantum model, which admits a local spin representation, displays at least three different phases; an antiferromagnetic phase, a line of quantum critical points, and a ferromagnetic phase. We argue that, in the ferromagnetic phase, the system cannot reach dynamically the quantum ground state when coupled to a bath through local interactions, and thus lingers in a state of quantum glassiness.

Figure 1

(Color online) Two configurations, ${\mathcal{C}}_{\mathit{AF}}$ in panel (a) and ${\mathcal{C}}_F$ in panel (b), from the three-coloring model.

Figure 2

(Color online) Two examples of decorated loops of the types $BCBC\dots$ and $CBCB\dots$ in the three-coloring model are shown. They are related by the color-swap defined in the text. Notice the information encoded in both loops, (1) the backbone structure (circled lattice sites) that is specified by all the sites visited by the sequence of bonds of color $B$ or $C$ emanating from a seed, (2) the two-color pattern ($BCBC\dots$ and $CBCB\dots$) along each loop, and (3) the colors and locations of all the dangling bonds (thick lines). The connecting bonds that link the backbone of a decorated loop to its dangling bonds are represented by wavy lines for clarity.

Figure 3

(Color online) Conjectured phase diagram of the classical partition function (2b) along the reduced temperature axis $\beta J$. The right end point of the AF phase is a hexagon. The left and right end points of the critical phase are a circle and a square, respectively. The circle corresponds to $\beta J=0$ and realizes the $\mathrm{SU}{\left(3\right)}_1$ WZW critical theory.

Figure 4

(Color online) Two configurations ${\mathcal{C}}_{\mathit{AF}}$ in panel (a) and ${\mathcal{C}}_F$ in panel (b) in the brick wall representation. In panel (a) $N_A=N_B=N_C$. In panel (b) $N_A$ is maximum with $N_B=N_C=0$ for ${\mathcal{C}}_F$.