Two-dimensional Potts antiferromagnets with a phase transition at arbitrarily large $q$

We exhibit infinite families of two-dimensional lattices (some of which are triangulations or quadrangulations of the plane) on which the $q$-state Potts antiferromagnet has a finite-temperature phase transition at arbitrarily large values of $q$. This unexpected result is proven rigorously by using a Peierls argument to measure the entropic advantage of sublattice long-range order. Additional numerical data are obtained using transfer matrices, Monte Carlo simulation, and a high-precision graph-theoretic method.

Figure 1

Unit cells of the lattices (a) $G_n$ and (b) $H_n$ (b) for $n=3$.

Figure 2

Unit cells of the lattices (a) $G_n^{\prime }$, (b) $G_n^{\prime \prime }$, and (c) $H_n^{\prime \prime \prime }$ for $n=3$.

Figure 3

The union-jack (UJ) lattice.

Figure 4

Estimated phase boundaries for the $q=4,8,16,32$ Potts models on the UJ lattice, from the Jacobsen-Scullard method (blue curve and numerical values of $v_{\square ,\mathrm{c}}$ at $v_{\times }=-1$) and Monte Carlo simulations (black points).

Figure 5

Estimates of $q_c\left(n\right)$ for the lattices $G_n,G_n^{\prime },H_n,H_n^{\prime }$ divided by their large-$n$ asymptote $2n/W\left(2n\right)$.

Figure 6

The lattice $H_2^{\prime \prime \prime }$ (rotated ${45}^{\circ }$ from Fig. 2) with $L=2$; a unit cell is shown in red, and the transfer direction is indicated with an arrow.

Figure 7

Monte Carlo estimates of $v_c$ for the lattices $H_n^{\prime \prime \prime }$ (black points) and their quadratic fit (blue curves), together with the extrapolated values $q_c$ (red squares).