Multiple phase transitions in extended hard-core lattice gas models in two dimensions

We study the $k$-NN hard-core lattice gas model in which the first $k$ next-nearest-neighbor sites of a particle are excluded from occupation by other particles on a two-dimensional square lattice. This model is the lattice version of the hard-disk system with increasing $k$ corresponding to decreasing lattice spacing. While the hard-disk system is known to undergo a two-step freezing process with increasing density, the lattice model has been known to show only one transition. Here, based on Monte Carlo simulations and high-density expansions of the free energy and density, we argue that for $k=4,10,11,14,\cdots$, the lattice model undergoes multiple transitions with increasing density. Using Monte Carlo simulations, we confirm the same for $k=4,...,11$. This, in turn, resolves an existing puzzle as to why the 4-NN model has a continuous transition against the expectation of a first-order transition.

Figure 1

The lattices sites that are excluded by a particle (black filled circle). The labels denote the sites that are the $k^{\mathrm{th}}$ next nearest neighbors. For $k$-NN HCLG, all sites with labels less than or equal to $k$ are excluded.

Figure 2

The Monte Carlo algorithm illustrated for 1-NN. (a) A typical valid configuration. The lattice consists of particles (filled circles), excluded sites (empty circles), and empty sites where a particle may be added to obtain a new valid configuration. (b) A row is chosen at random (denoted by arrow) and all particles on that row are removed. The row is now divided into intervals of contiguous empty sites by the excluded sites. (c) The row is reoccupied with a new valid configuration with the correct equilibrium weight.

Figure 3

An example of the algorithm for 3-NN. (a) A valid configuration of two particles (filled circles) and excluded sites (empty circles). (b) Adding particles in a row (denoted by arrow) by deposition. If a particle is added at A, then its exclusion range includes B. Thus, the occupation of the empty intervals are not independent. (c) Adding particles in a diagonal oriented in the $\pi /4$ direction (denoted by arrow). Now the occupation of the empty intervals are independent of each other.

Figure 4

The sublattice labeling for 4-NN. (a) All sites belonging to a diagonal oriented in the $\pi /4$ direction have same label. If the site has coordinates $(x,y)$, then the label is $\left[\right(x-y\left)\mathrm{mod}4\right]$. (b) All sites belonging to a diagonal oriented in the $3\pi /4$ direction have same label. If the site has coordinates $(x,y)$, then the label is $\left[\right(x+y)\mathrm{mod}4+4]$. In addition, all sites with label 0 or 2 (or, equivalently, 4 and 6) will be called sublattice $A$ and all sites with label 1 or 3 (or, equivalently, 5 and 7) will be called sublattice $B$.

Figure 5

Snapshots of typical configurations of the 4-NN model. The particles are colored according to the sublattice to which it belongs (see Fig. 4). In the left panel, the colors are light blue for 0, dark magenta for 1, deep blue for 2, and plum for 3. In the right panel, the colors are dark green for 4, red for 5, light green for 6, and pink for 7. (a) Low-density disordered phase where all four colors are present in both panels. (b) Intermediate density sublattice phase, where even or odd sublattices are preferentially occupied. (c) High-density columnar phase, where one sublattice in one of the panels (left in figure) and two sublattices in the other panel are preferentially occupied.

Figure 6

The variation of the order parameters $Q_{\mathrm{sl}}$ and $Q_{\mathrm{cl}}$ with chemical potential $\mu$ for different system sizes $L$. $Q_{\mathrm{sl}}$ becomes nonzero at a smaller value of $\mu$ than $Q_{\mathrm{cl}}$.

Figure 7

The data for cumulants of $Q_{sl}$ for the 4-NN model near the first transition from the disordered phase to sublattice phase. (a) The Binder cumulants $U_{\mathrm{sl}}$ for different $L$ intersect at ${\mu }_c=4.705$. The data for (b) $U_{\mathrm{sl}}$, (c) $Q_{\mathrm{sl}}$, and (d) ${\chi }_{\mathrm{sl}}^{\left(2\right)}$ for different $L$ collapse onto a single curve when scaled as in Eq. (4) with the Ising exponents $\beta /\nu =1/8,\gamma /\nu =7/4$, and $\nu =1$. (e) The variation of the maximum value ${\chi }_{max}$ of the susceptibility $\chi$ with $L$. The solid line is a best fit to the data with slope $\gamma /\nu =1.72\pm 0.04$. (f) The variation of the order parameter $Q_{\mathrm{sl}}$ at the critical point with $L$. The solid line is a best fit to the data with slope $-\beta /\nu =-0.136\pm 0.017$. (g) Variation of ${\mu }_c$ for different $L$ with $1/L$. The straight line intersects the $\mu$ axis at ${\mu }_c=4.705\pm 0.005$. (h) The variation of the compressibility $\kappa$ at the critical point with $L$. The straight line shows $\kappa \left({\mu }_c\right)\sim lnL$.

Figure 8

The data for cumulants of $Q_{\mathrm{cl}}$ for the 4-NN model near the second transition from the sublattice phase to the columnar phase. (a) The Binder cumulant $U_{\mathrm{cl}}$ for different $L$ intersect at ${\mu }_c=5.07\pm 0.01$. The data for (b) $U_{\mathrm{cl}}$, (c) $Q_{\mathrm{sl}}$ and (d) ${\chi }_{\mathrm{sl}}^{\left(2\right)}$ for different $L$ collapse onto a single curve when scaled as in Eq. (4) with critical exponents $\beta /\nu =1/8,\gamma /\nu =7/4$, and $\nu =0.83\pm 0.06$. (e) The variation of the maximum value ${\chi }_{max}$ of the susceptibility $\chi$ with $L$. The solid line is a best fit to the data with slope $\gamma /\nu =1.76\pm 0.05$. (f) The variation of the order parameter $Q_{\mathrm{sl}}$ at the critical point with $L$. The solid line is a best fit to the data with slope $-\beta /\nu =-0.113\pm 0.015$. (g) Variation of ${\mu }_c$ for different $L$ with $L^{-1/\nu }$. The straight line intersects the $\mu$ axis at ${\mu }_c=5.066\pm 0.01$. (h) The variation of compressibility $\kappa$ at the critical point ${\mu }_c$ with $L$. The solid line is a best fit to the data with slope $\alpha /\nu =-0.34\pm 0.08$.

Figure 9

Examples of one- and two-defect configurations on sublattices 1 and 3 for an ordered state where particles are on sublattice 0. Black circles are particles, and empty and lightly shaded circles are excluded sites. Exclusions only on sublattice 0 are shown in (a)–(c) and (e). $0,1,2,3$ denote the diagonals that belong to sublattices $0,1,2,3$. (a) A single defect on sublattice 1. (b) Two defects on sublattice 1 but on neighboring diagonals. A and B are two other lattice sites where the defect may be placed. (c) Two defects on sublattice 1 but on the same diagonal. (d) A defect on sublattice 1 and a defect on sublattice 3 at one of the four closest positions. The empty circles are all the excluded sites due to defect on sublattice 1 while the lightly shaded circles are excluded sites on sublattice 0 due to defect on sublattice 3. (e) A defect on sublattice 1 and a defect on sublattice 3 positioned such that they exclude the same two sites on the diagonal 0 separating them.

Figure 10

Example of a rodlike defect on sublattice 2 and defects on sublattice 1. Black circles are particles on sublattice 2. Empty circles are excluded sites on sublattice 0 or sublattice 1 (all not shown) due to defect on sublattice 2. Lightly shaded circles are excluded sites due to defects at $A$ or $C$. Circles with a pattern are excluded sites due to defect at $B$.

Figure 11

Variation of the densities of particles in different sublattices, truncated at order $z^{-3/2}$ [see Eqs. (33), (34), (35)], with $1/z$.

Figure 12

One of the configurations of maximum density for different $k$. Particles are shown by large filled circles. The excluded sites are shown by small circles. A circle that encloses the excluded sites due to a particle are drawn around the leftmost and topmost particles. The empty large circles denote sites that may be occupied by sliding all particles on that diagonal, keeping other particles fixed. The examples are for (a) 6-NN, (b) 7-NN, (c) 8-NN, (d) 9-NN, (e) 10-NN, and (f) 11-NN.

Figure 13

The sublattices of the 6-NN model. (a) All sites on a diagonal oriented in the ${tan}^{-1}(1/3)$ direction belong to the same sublattice. (b) All sites on a diagonal oriented in the ${tan}^{-1}\left(3\right)$ direction belong to the same sublattice.

Figure 14

The sublattices of the 7-NN model. (a) All sites on a diagonal oriented in the ${tan}^{-1}(2/3)$ direction belong to the same sublattice. (b) All sites on a diagonal oriented in the ${tan}^{-1}(3/2)$ direction belong to the same sublattice.

Figure 15

The sublattices of the 8-NN and the 9-NN models. (a) All sites on a diagonal oriented in the ${tan}^{-1}(1/4)$ direction belong to the same sublattice. (b) All sites on a diagonal oriented in the ${tan}^{-1}(-4)$ direction belong to the same sublattice.

Figure 16

The probability density function for density $\rho$ (left panels) and order parameter $Q_k$ (right panels) for $\mu \approx {\mu }_c$, the critical chemical potential, for two different system sizes. The data are for the $k$-NN model with $k=6$ [(a) and (b)], $k=7$ [(c) and (d)], $k=8$ [(e) and (f)], and $k=9$ [(g) and (h)].

Figure 17

The sublattices of the 10-NN model. (a) All sites on a diagonal oriented in the $\pi /4$ direction belong to the same sublattice. (b) All sites on a diagonal oriented in the $3\pi /4$ direction belong to the same sublattice.

Figure 18

An example of rodlike defects of length 3 on sublattices 2 (black circles) or 4 (gray circles) in the 10-NN model. Exclusions only on sublattice 0 are shown by empty circles. $0,1,2,3,4,5$ on the edge of the box denote diagonals that belong to sublattices $0,1,2,3,4,5$, respectively.

Figure 19

The data for cumulants of $Q_1$ for the 10-NN model near the first transition from the disordered phase to sublattice phase. (a) The Binder cumulant $U_1$ for different L crosses at ${\mu }_c=5.30$. The data for (b) $U_1$, (c) $Q_1$, and (d) ${\chi }_1^{\left(2\right)}$ for different $L$ collapse onto a single curve when scaled as in Eq. (4) with the Ising exponents $\beta /\nu =1/8,\gamma /\nu =7/4$, and $\nu =1$.

Figure 20

The probability density function for (a) density $\rho$ and (b) order parameter $Q_{10}$ near the transition point $\left({\mu }_c\approx 6.0\right)$ for three different system sizes. The data are for the 10-NN model.

Figure 21

The sublattices of the 11-NN model. (a) All sites on a diagonal oriented in the ${tan}^{-1}(1/2)$ direction belong to the same sublattice. (b) All sites on a diagonal oriented in the ${tan}^{-1}(-1/2)$ direction belong to the same sublattice. (c) All sites on a diagonal oriented in the ${tan}^{-1}\left(2\right)$ direction belong to the same sublattice. (b) All sites on a diagonal oriented in the ${tan}^{-1}(-2)$ direction belong to the same sublattice.

Figure 22

The probability distribution function for density $\rho$ near the transition point for three different system sizes. The data are for the 11-NN model.

Figure 23

The variation of (a) density $\rho$ and (b) order parameter $Q_{11}$ with time $t$ for the 11-NN model when $\mu \gtrsim {\mu }_c$. The data are for two different initial conditions. In the first, labeled “ordered,” all particles are in sublattice 0. In the second, labeled “random,” particles are initially deposited at random.