Colorings of odd or even chirality on hexagonal lattices

We define two classes of colorings that have odd or even chirality on hexagonal lattices. This parity is an invariant in the dynamics of all loops, and explains why standard Monte Carlo algorithms are nonergodic. We argue that adding the motion of “stranded” loops allows for parity changes. By implementing this algorithm, we show that the even and odd classes have the same entropy. In general, they do not have the same number of states, except for the special geometry of long strips, where a $Z_2$ symmetry between even and odd states occurs in the thermodynamic limit.

Figure 1

Hexagonal lattice with rhombus shape (R) and periodic boundary conditions of the torus geometry. A dimer and color configurations are shown. Here $L=M=4$ and $N=3\times L\times M=48$ edges. The honeycomb lattice is bipartite and is divided into $L\times M$ black and $L\times M$ white sites. The solid and dashed lines show two distinct nonlocal cuts that define topological conserved numbers.

Figure 2

Hexagonal lattice with hexagonal shape (H), here $L=2$ and $N=9L^2=36$ edges ($N_v=6L^2$). A special dimer configuration is shown, sometimes called the “empty room” state in the context of tiling a hexagon with lozenges: the gray colors of the lozenges correspond to the dimer orientations and can be seen as the three walls of an $L^3$ “empty room.” The dashed line shows one nonlocal cut across the boundaries.

Figure 3

Definition of the chirality at a given (white) vertex (the definition is the same for black vertices). The sum of the chirality of all vertices (black and white) can be even or odd and its parity is conserved.

Figure 4

Eigenvalues of the transfer matrix $T(-1)$ (given here in the complex plane for $L=5$) are degenerate and form the multiplets of SU(3), $\left[3\right],\left[6\right],\left[\overline{15}\right],\left[21\right],\left[24\right]$. The colors indicate different charge sectors (the degeneracy due to permutation-related sectors is not shown).

Figure 5

Fraction of odd states as a function of $M$ for a strip of size $N=3\times L\times M$ with rhombus shape and periodic boundary conditions, from transfer matrix. When $M\to \infty$, the fraction goes to 0 for $L=3n$ and 1/2 for $L=3n\pm 1$, thus exhibiting a $Z_2$ symmetry between even and odd states.

Figure 6

First eigenvalues of $T\left(1\right)$ and $T(-1)$ and the angle $\kappa$ defined by ${\mathrm{\Lambda }}_1^{-}=\left|{\mathrm{\Lambda }}_1^{-}\right|e^{\pm ı\kappa }$. The lines are finite-size corrections to Baxter's thermodynamic limit, $s_{\infty }=0.126375$, predicted from conformal invariance.

Figure 7

Triple-stranded loops make the algorithm ergodic by allowing parity changes. Black lozenges are the unit cell; the loops are repeated outside the unit cell for clarity (top, $N=108$; bottom, $N=81$).

Figure 8

A way to implement numerically the flip of triple-stranded loops, by using defects (a) from a perfect three-coloring ($\varnothing$); the three colors of the vertex $i$ marked by a point are exchanged, here clockwise $+$. (b) This generates three defects $q_A$ (BBR), $q_B$ (GGB), $q_C$ (RRG). $q_B$ propagates on a G-R loop, $q_C$ on an R-B loop [double arrows in (b)] until they annihilate $q_B+q_C\to {\overline{q}}_A$ (c). The last two defects propagate on a G-B loop and recombine in a perfect three-coloring $q_A+{\overline{q}}_A\to \varnothing$, resulting in the overall flip of a triple-stranded loop (d).

Figure 9

Fraction of odd states as a function of $1/L$ at fixed aspect ratio $r=1$, depending on $\mathrm{mod}(L,3$), obtained by Monte Carlo algorithm (Sec. 6a), exact enumeration up to $L=8$ (Sec. 3a), and transfer matrix up to $L=12$ (Sec. 5).

Figure 10

Fraction of odd states extrapolated in the thermodynamic limit (squares and triangles), as a function of the aspect ratio $r=L/M$, from Monte Carlo. It depends on whether $L$ or $M$ is a multiple of three. The results from the transfer matrix at fixed $L=10$ (circles) are shown for comparison.

Figure 11

Order-parameter scaling of the $\sqrt{3}\times \sqrt{3}$ order, for both the standard loop Monte Carlo algorithm (squares) and the present ergodic algorithm (circles) with periodic boundary conditions and $L=M=3n$. The systematic error when using the standard algorithm is 4.4%, which reflects the fraction of odd states given in Fig. 9.

Figure 12

Hexagonal lattice with periodic boundary conditions of Klein bottle geometry; see the arrows on the sides. A dimer and color configurations are shown. Here a rhombus (R) shape is chosen with $L=4$ and $N=3L^2=48$ edges. The solid and dashed lines show two distinct nonlocal cuts that define topological conserved numbers.

Figure 13

Fraction of odd states for finite-size clusters of size $L=M$, with the “Klein bottle” geometry. There is a $\mathrm{mod}(L,6$) effect, partially resolved in the thermodynamic limit.