Semiflexible Fully Packed Loop Model and Interacting Rhombus Tilings

Motivated by a recent adsorption experiment [M. O. Blunt et al., Science 322, 1077 (2008)], we study tilings of the plane with three different types of rhombi. An interaction disfavors pairs of adjacent rhombi of the same type. This is shown to be a special case of a model of fully packed loops with interactions between monomers at distance two along a loop. We solve this model using Coulomb gas techniques and show that its critical exponents vary continuously with the interaction strength. At low temperature it undergoes a Kosterlitz-Thouless transition to an ordered phase, which is predicted from numerics to occur at a temperature $T\sim 110\mathrm{K}$ in the experiments.

Figure 1

Dimer covering of the hexagonal lattice (thick white lines), the complementary fully packed loops (thick black lines), and corresponding rhombus tiling of the plane (a). Height mapping for the rhombus tilings (b) and for the fully packed loop model (c).

Figure 2

The upper group of three configurations for a vertex pair has weights $\chi \equiv e^{i\pi e_0/3}$ which are invariant under the cyclic permutation of labels ($v_0$, $v_{+}$, $v_{-}$), whereas the weights of the lower group differ by factors of $w$.

Figure 3

Each column shows a pair of configurations with identical link positions, but different loop orientations. The middle column comes with a weight $w$.

Figure 4

Numerical simulations of the IRT model. As functions of $w$: (a) Central charge $c$, obtained from three-point fits of the free energy on cylinders of circumference $L$. (b) Coupling constant ratio $\gamma$, found from two-point fits for the critical exponent $x_1$ (TM data), and from winding number fluctuations on a $768\times 256$ sample (MC data). The long-dashed vertical line denotes $w_c=0.635$.