Columnar Order and Ashkin-Teller Criticality in Mixtures of Hard Squares and Dimers

We show that critical exponents of the transition to columnar order in a mixture of $2\times 1$ dimers and $2\times 2$ hard squares on the square lattice depends on the composition of the mixture in exactly the manner predicted by the theory of Ashkin-Teller criticality, including in the hard-square limit. This result settles the question regarding the nature of the transition in the hard-square lattice gas. It also provides the first example of a polydisperse system whose critical properties depend on composition. Our ideas also lead to some interesting predictions for a class of frustrated quantum magnets that exhibit columnar ordering of the bond energies at low temperature.

Figure 1

(a) Part of a low-density configuration of $2\times 1$ tiles (dimers) and $2\times 2$ tiles (hard squares) on the square lattice, also showing values of the columnar order parameter field $\psi (\overrightarrow{r})$ [see Eq. (2)]. (b) Schematic phase diagram of $Z_{dsv}$. ${\rho }_s,{\rho }_d$, and ${\rho }_v$ are the densities of squares, dimers, and vacancies, respectively, with ${\rho }_s+{\rho }_d+{\rho }_v=1$. Monte Carlo results along the cuts I, II, and III are discussed in the text. (c) Columnar ordered high-density configuration, with stripes running in the vertical direction.

Figure 2

$\mathcal{C}(L)/L^2\sim L^{-\eta (w)}$ with variable exponent $\eta (w)$ in the power-law ordered phase at full packing. Insets: $\mathcal{R}(L)/L^2$ and $\mathcal{I}(L)/L^2$ both scale as $L^{-4\eta (w)}$ with the same $\eta (w)$.

Figure 3

Scaling collapse of $\mathcal{C}(L)/L^{7/4}$ for various $L$ at the hard-square transition yields the estimate ${\nu }^{*}=0.92(3)$ and $v_c^{*}=0.31799(30)$. Inset: $\mathcal{N}(L)/L^{2-{\eta }_2^{*}}$ is a constant for $v=v_c^{*}$ with ${\eta }_2^{*}\approx 0.46(3)$.

Figure 4

Scaling collapse of $\mathcal{C}(L)/L^{7/4}$ for various $L$ along a cut that crosses the AT boundary at an intermediate point $w_c=0.1600(1)$, $v_c=0.0623(1)$ yields the estimate $\nu =1.70(5)$. Inset: Scaling collapse of $\mathcal{R}(L)/L^{2-{\eta }_2}$ yields the estimate ${\eta }_2\approx 0.70(5)$.