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Monte Carlo simulations of two-dimensional hard core lattice gases
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10.1063/1.2539141
/content/aip/journal/jcp/126/11/10.1063/1.2539141
http://aip.metastore.ingenta.com/content/aip/journal/jcp/126/11/10.1063/1.2539141

Figures

Image of FIG. 1.
FIG. 1.

Lattice division for the (a) 1NN and (b) 2NN exclusions: two and four sublattices are needed, respectively. The 1NN case is equivalent to tilted, nonoverlapping hard squares of length , while in the 2NN case the squares are not tilted and have . The equivalent hard squares are shown in gray.

Image of FIG. 2.
FIG. 2.

Two possible labelings for the sublattices in the 3NN case—panels (a) and (b). For the same high density configuration in (a) all particles are on the same sublattice (five in this example), while in (b) all sublattices are equally populated. For the labeling (b), panel (c) shows another configuration in which particles occupy only one sublattice. Note that the ground states shown in panels (b) and (c) are chiral, in the sense that the two are the reflections of one another about the left-to-right body diagonal. The exclusion problem can be formulated either in terms of the symmetric cross-shaped pentamers (shown in gray) or the tilted hard-squares of side length (d).

Image of FIG. 3.
FIG. 3.

Sublattice division for the exclusion of up to (a) 4NN and (b) 5NN. These are equivalent to hard squares with and 3, respectively. Notice that in both cases the ordered phase is columnar (as in the 2NN case), but in the former, the columns are along the diagonals.

Image of FIG. 4.
FIG. 4.

Density as a function of the chemical potential for the 1NN exclusion for several lattice sizes. The inflection close to the transition point is very small, being noticeable only for larger system sizes. Inset: compressibility as a function of the chemical potential.

Image of FIG. 5.
FIG. 5.

Order parameter , Eq. (5), as a function of the chemical potential , for the 1NN exclusion for several lattice sizes, along with the corresponding data collapse (inset). The critical exponents used are those of the two dimensional Ising model, and .

Image of FIG. 6.
FIG. 6.

Order parameter fluctuations , Eq. (2), as a function of the chemical potential for the 1NN model using several lattice sizes. In the inset, the corresponding data collapse is shown using the critical exponents of the bidimensional Ising model, and .

Image of FIG. 7.
FIG. 7.

Density as a function of chemical potential for several lattice sizes for the gas of 2NN exclusion region where the inflection point appears (noticeable only for the larger system sizes). Inset: compressibility in the region where the transition is found. Only for the largest lattice simulated the maximum is prominent.

Image of FIG. 8.
FIG. 8.

Order parameter for 2NN [Eq. (7)] as a function of the chemical potential for different lattice sizes. Small values of signal that the system is disordered. For closer to unity, the system is in a columnar phase. Inset: data collapse onto a universal curve for the larger system sizes , and 360. A very good collapse is obtained for , , and , in close agreement with the exact Ising values. The collapse remains good if we use the Ising value .

Image of FIG. 9.
FIG. 9.

Several estimates of the exponent (2NN): the position of the maxima of Eq. (4) (top curve) and the susceptibility (bottom), that shift as , as a function of . The solid lines are power-law fits neglecting the smaller sizes, from which we get and , top and bottom, respectively. These values are very close to the Ising , as can be seen by the fits fixing the exponent to this value (dashed lines). Also from the fit of the location of the maximum of the susceptibility we have an independent estimate of the transition: . Inset: the height of the maximum of Eq. (4), increasing as , as a function of . Again the dashed line is a fit with , while the solid line is the best fit with .

Image of FIG. 10.
FIG. 10.

The staggered susceptibility as a function of for several lattice sizes in the 2NN case. Inset: data collapse (top) onto a universal curve with exponent obtained from fitting the height of the maximum of as a function of (bottom), , along with .

Image of FIG. 11.
FIG. 11.

Density as a function of chemical potential for the 3NN exclusion model. At low densities the system behaves as a fluid and undergoes a first order phase transition to an ordered phase as the density increases. The vertical dashed line at is the infinite size extrapolation obtained with matrix methods (Ref. 80) and agrees well with the crossing point of the curves. Inset: the order parameter as a function of the chemical potential.

Image of FIG. 12.
FIG. 12.

Compressibility as a function of the chemical potential for the 3NN exclusion for several lattice sizes. Inset: after rescaling the height and the width of the curves by and , respectively, an excellent collapse is observed for the two largest systems with .

Image of FIG. 13.
FIG. 13.

Staggered susceptibility as a function of the chemical potential for the 3NN exclusion case for several lattice sizes. Inset: after rescaling the height and the width of the curves by and , respectively, an excellent collapse is observed with .

Image of FIG. 14.
FIG. 14.

Several estimates of the exponent (4NN): the position of the maxima of Eq. (4) and the susceptibility (bottom), that shift as , as a function of . The solid lines are power-law fits neglecting the smaller sizes, from which we get and , top and bottom curves, respectively. These values are indeed very close to the Ising one, (notice also that the peaks of Eq. (4) are broader than those of , and their positions are less reliable). Also from the fit of the location of the maximum of the susceptibility we have an independent estimate of the transition point: . Inset: the height of the maximum of Eq. (4), increasing as , as a function of . The solid line is a fit with .

Image of FIG. 15.
FIG. 15.

Order parameter for the 4NN exclusion [Eq. (5)] as a function of the chemical potential for different lattice sizes. For large values of , the system is in a columnar phase with columns aligned along the diagonals. Inset: collapse of data onto a universal curve with , , and . Equally good collapse is obtained with a broad range of .

Image of FIG. 16.
FIG. 16.

The staggered susceptibility as a function of for several lattice sizes for 4NN exclusion. Insets: (top) data collapse onto a universal curve with exponent obtained from fitting the height of the maximum of as a function of shown in the bottom inset; and .

Image of FIG. 17.
FIG. 17.

Density as a function of the chemical potential for 5NN exclusion for different lattice sizes. The steepness increases as the system develops a discontinuity. Inset: order parameter as a function of for the same system sizes.

Image of FIG. 18.
FIG. 18.

Staggered susceptibility as a function of the chemical potential for the 5NN exclusion for several lattice sizes. Inset: after rescaling the height and the width of the curves by and , respectively, a good collapse is observed for the larger simulated sizes with .

Tables

Generic image for table
Table I.

Chemical potential and the density at the order-disorder transition along with the technique used and the putative order of the transition for a lattice gas with nearest and next-nearest neighbors. The symbol “?” is used when some uncertainty is acknowledged by the authors. The several techniques used are: transfer matrix (TM), cluster variational method (CVM), low and high series expansion, generalizations of the Bethe method, interface method (Ref. 75), density functional theory (DFT), and Monte Carlo. Notice the wide range of values found for the critical chemical potential.

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/content/aip/journal/jcp/126/11/10.1063/1.2539141
2007-03-21
2014-04-19
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Monte Carlo simulations of two-dimensional hard core lattice gases
http://aip.metastore.ingenta.com/content/aip/journal/jcp/126/11/10.1063/1.2539141
10.1063/1.2539141
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