Transfer-matrix study of a hard-square lattice gas with two kinds of particles and density anomaly

Using transfer matrix and finite-size scaling methods, we study the thermodynamic behavior of a lattice gas with two kinds of particles on the square lattice. Only excluded volume interactions are considered, so that the model is athermal. Large particles exclude the site they occupy and its four first neighbors, while small particles exclude only their site. Two thermodynamic phases are found: a disordered phase where large particles occupy both sublattices with the same probability and an ordered phase where one of the two sublattices is preferentially occupied by them. The transition between these phases is continuous at small concentrations of the small particles and discontinuous at larger concentrations, both transitions are separated by a tricritical point. Estimates of the central charge suggest that the critical line is in the Ising universality class, while the tricritical point has tricritical Ising (Blume-Emery-Griffiths) exponents. The isobaric curves of the total density as functions of the fugacity of small or large particles display a minimum in the disordered phase.

Figure 1

Strip of width $L=4$, with periodic (dashed red) and helical (dotted blue) boundary conditions. The lower and upper rows of sites determine the states of the transfer matrix. They are $(0,2,0,1)$ and $(0,0,0,1)$, respectively. The corresponding element is $z_2^{1/2}{z}_1$. The partially overlapping dot-dashed boxes are the states [$(2,0,1,0)$ and $(0,1,0,0)$] for helical boundary conditions, and the corresponding matrix element is 1, since a single empty site is added.

Figure 2

Finite-size and extrapolated transition lines, calculated from the fixed point of the phenomenological renormalization transformation, considering the pair of strip sizes $L,L^{\prime }=L+2$. In the main plot, the differences between the estimates are almost not visible. In the inset the region close to the minimum is amplified, so that the finite-size effects are seen. For a given value of $z_1$, as the width of the strip increases the estimates for the critical value of $z_2$ decrease. The estimated tricritical point, discussed below, is represented by the black circle.

Figure 3

Exponents of finite-size corrections [Eq. (8)] for the tricritical point activities estimated from phenomenological renormalization. In the main plot $y_{1,1}$ and $y_{2,1}$ are extrapolated with $\mathrm{\Delta }=3.38$ and $\mathrm{\Delta }=2.98$, respectively. The behaviors of $y_{1,2}$ and $y_{2,2}$ are shown in the inset.

Figure 4

Central charge $c$ as a function of $z_1$, calculated from $f\left(L^{\prime }\right)-f\left(L\right)$ along the finite-size critical line for the pairs $L,L^{\prime }$ indicated. The dashed horizontal lines indicate the Ising critical (bottom) and tricritical (top) values.

Figure 5

(a) Transition (critical and coexistence) lines calculated for strips of size $L$ in the densities' space, for the fluid (full) and solid (dashed lines) phases. (b) Extrapolated critical (full) and coexistence (dashed) lines considering three-point fits for the sizes $(L-2,L,L+2)$ and the exponents indicated in the text. The square is placed at the localization of the tricritical point estimated by Poland [10].

Figure 6

Total density of particles $\rho ={\rho }_1+2{\rho }_2$, calculated on a strip of size $L=18$, against $z_1$ for several values of pressure $P$. From the left to the right, curves correspond to increased pressures. The minima in the densities curves define the mD line (dashed).

Figure 7

(a) Phase diagram in the variables $z_i/(1+z_i)$ (main plot) and $z_i$ (inset), with $i=1,2$. Critical, coexistence and mD lines are indicated by continuous (red), dashed (blue), and dotted (black) lines, respectively. (b) Phase diagram in the densities ${\rho }_1$ and ${\rho }_2$ space. The densities of fluid (blue) and solid (green) phases at the same point of the coexistence line are connected by tie lines.