Finite-size corrections and scaling for the dimer model on the checkerboard lattice

Lattice models are useful for understanding behaviors of interacting complex many-body systems. The lattice dimer model has been proposed to study the adsorption of diatomic molecules on a substrate. Here we analyze the partition function of the dimer model on a $2M\times 2N$ checkerboard lattice wrapped on a torus and derive the exact asymptotic expansion of the logarithm of the partition function. We find that the internal energy at the critical point is equal to zero. We also derive the exact finite-size corrections for the free energy, the internal energy, and the specific heat. Using the exact partition function and finite-size corrections for the dimer model on a finite checkerboard lattice, we obtain finite-size scaling functions for the free energy, the internal energy, and the specific heat of the dimer model. We investigate the properties of the specific heat near the critical point and find that the specific-heat pseudocritical point coincides with the critical point of the thermodynamic limit, which means that the specific-heat shift exponent $\lambda$ is equal to $\infty$. We have also considered the limit $N\to \infty$ for which we obtain the expansion of the free energy for the dimer model on the infinitely long cylinder. From a finite-size analysis we have found that two conformal field theories with the central charges $c=1$ for the height function description and $c=-2$ for the construction using a mapping of spanning trees can be used to describe the dimer model on the checkerboard lattice.

Figure 1

Unit cell for the dimer model on the checkerboard lattice.

Figure 2

Values of the free-energy asymptotic expansion coefficients (a) $f_1$ and (b) $f_2$ as functions of the aspect ratio $\rho$.

Figure 3

Values of the specific-heat asymptotic expansion coefficients (a) $c_b$ and (b) $c_1$ as functions of the aspect ratio $\rho$.

Figure 4

Value of the asymptotic expansion coefficient $g$ in the fourth derivative of free energy $F_c^{\left(4\right)}$ as a function of the aspect ratio $\rho$.

Figure 5

(a) Free energy $F_{2M,2N}\left(t\right)$, (b) internal energy $U_{2M,2N}\left(t\right)$, and (c) specific heat $C_{2M,2N}\left(t\right)$ as functions of $t$. The aspect ratio $\rho =M/N$ has been set to unity.

Figure 6

(a) $F_{\mathrm{c}}$ as a function of $1/S$ and (b) $C_{\mathrm{c}}-(lnS)/2\pi$ as functions of $1/S$. The aspect ratio $\rho =M/N$ has been set to unity.

Figure 7

Scaling functions (a) ${\mathrm{\Delta }}_F(S,\rho ,\tau )$, (b) ${\mathrm{\Delta }}_U(S,\rho ,\tau )$, and (c) ${\mathrm{\Delta }}_C(S,\rho ,\tau )$ as functions of $\tau$ for different system sizes $S$ with the aspect ratio $\rho =1$, 2, and 4.