Weakly first-order transition in an athermal lattice gas

We investigate the phase behavior of a two-dimensional athermal lattice gas in which every hard-core particle can have two or fewer nearest neighboring occupied sites on the square lattice. The ground state and close packing density are determined and it is found that at large chemical potential the model undergoes an ordering phase transition with preferential sublattice occupation. Although near the transition point the particle density and entropy exhibit an apparent discontinuity, we find that the order parameter and fluctuations of thermodynamic quantities do not scale with the system volume. These paradoxical results are reconciled by analyzing the size-dependent flow of the thermal exponent by phenomenological renormalization and the curve-crossing method, which lead to a weakly first-order phase transition scenario.

Figure 1

The two symmetric ground state configurations of the ${\mathrm{BM}}_2$ model on the square lattice, corresponding to the close packing density ${\rho }_{\max }=2/3$. Lattice sites are represented by circles, with filled circles standing for particles and empty circles for vacancies. The central shaded $3\times 3$ square represents the unit cell. For the labeling shown here the density of sublattices ${\mathcal{L}}_0$, ${\mathcal{L}}_1$, and ${\mathcal{L}}_2$, in the ground state configurations is ${\rho }_0=0,{\rho }_1={\rho }_2$ in (a), and ${\rho }_0={\rho }_1={\rho }_2$ in (b).

Figure 2

Thermodynamic properties of the two-dimensional ${\mathrm{BM}}_2$ model on the square lattice. Data points (square symbols), extracted from grand-canonical Monte Carlo simulation of a system of linear size $L=75$, are compared with the noninteracting lattice gas behavior (dashed curve) and the Bethe-Peierls, or replica factorized, approximation (dotted curve). (a) Particle density $\rho$ vs chemical potential $\mu$. The horizontal segment indicates the close packing density ${\rho }_{\max }=2/3$. (b) Entropy $s$ vs chemical potential $\mu$. The phase transition is located in the range $\mu \in [2.84,2.87]$.

Figure 3

Binder ratio $U_L$ vs chemical potential $\mu$ in the two-dimensional ${\mathrm{BM}}_2$ model on the square lattice of side $L$. The crossing point is approximately located at ${\mu }_{\mathrm{c}}=2.8559\left(1\right)$. The peaks location shifts from left to right with increasing $L$. Inset: Binder ratio vs scaled distance to the critical point with $\nu =0.54$ and ${\mu }_{\mathrm{c}}=2.8559$.

Figure 4

Finite-size probability density function $P_L(\mu ,q)$ of the order parameter $q$ evaluated at the chemical potential ${\mu }_{\mathrm{c}}^L$ at which the peaks have equal height ($L$ is the square lattice side; larger values of $L$ correspond to higher peaks). Inset: power-law fit of the finite-size shifted critical point ${\mu }_{\mathrm{c}}-{\mu }_{\mathrm{c}}^L$ vs inverse length $1/L$. The estimated value of the infinite volume limit of the critical point and the correlation length exponent are ${\mu }_{\mathrm{c}}=2.8559\left(1\right)$ and $\nu =0.54\left(1\right)$, respectively.

Figure 5

Finite-size scaling properties of the ${\mathrm{BM}}_2$ model on a square lattice of side $L$ for (a) order parameter $q$, (b) susceptibility $\chi$, and (c) compressibility $\kappa$ vs chemical potential $\mu$, and rescaled data in the inset. In panel (a) larger values of $L$ correspond to steeper curves. In both panels (b) and (c) larger values of $L$ correspond to higher peaks. The values of critical point and critical exponents are ${\mu }_{\mathrm{c}}=2.8559$, $\nu =0.54$, $\beta =0.04$, $\gamma =1.1$, and $\alpha =0.85$. Rescaled data are limited to the four largest sizes, $L>90$.

Figure 6

Size-dependent flow of the inverse correlation-length exponent $1/{\nu }_L$ at length scale $L$ for the ${\mathrm{BM}}_2$ model on the square lattice. Squares represent the data obtained from the crossing point analysis with system-size pairs $(L/2,L)$. The dashed green curve is a three-parameter power-law fit, $1/{\nu }_L=1/{\nu }_{\infty }+a{L}^{-\omega }$, where the effective correction-to-scaling exponent is $\omega \simeq 0.07$ and the extrapolated value $1/{\nu }_{\infty }\simeq 3.96$. The dotted blue curve is a two-parameter exponential-law fit $1/{\nu }_L=2+bexp(-L/\xi )$, where the estimated correlation length is $\xi \simeq 132$. The smallest size was excluded from the fits.