Columnar-disorder phase boundary in a mixture of hard squares and dimers

A mixture of hard squares, dimers, and vacancies on a square lattice is known to undergo a transition from a low-density disordered phase to a high-density columnar ordered phase. Along the fully packed square-dimer line, the system undergoes a Kosterliz-Thouless-type transition to a phase with power law correlations. We estimate the phase boundary separating the ordered and disordered phases by calculating the interfacial tension between two differently ordered phases within two different approximation schemes. The analytically obtained phase boundary is in good agreement with Monte Carlo simulations.

Figure 1

Snapshot of a typical equilibrium configuration of a system where the squares at the left boundary are fixed to be on even rows (green) and the squares at the right boundary are fixed to be on odd rows (blue). At high enough activity $z_s$ (as in the figure), a sharp interface separates the left even phase from the right odd phase.

Figure 2

A schematic of an interface that has no overhangs. The interface is indicated by the red line, and its $x$ coordinates are denoted by ${\eta }_i$. The boundary conditions are periodic in the $y$ direction.

Figure 3

The shape of a generic track of two rows. It is characterized by two lengths $\ell$ and $\mathrm{\Delta }$ and has partition function $\mathrm{\Omega }(\ell ,\mathrm{\Delta })$.

Figure 4

Diagrammatic representation of the recursion relations obeyed by the generating functions $G(y,0)$ and $G(y,1)$ [see Eq. (6) for the definition]. The leftmost column may be occupied by vacancies, dimers, or squares.

Figure 5

Diagrammatic representation of the recursion relation obeyed by $\mathrm{\Omega }(\ell ,\mathrm{\Delta })$, the partition function of a track of two rows as shown in Fig. 3 for $\mathrm{\Delta }\ge 2$. The leftmost column may be occupied by a vacancy or a horizontal dimer.

Figure 6

Phase diagram on (a) the activity $z$ plane and (b) the density $\rho$ plane. $S$ and $D$ represent the state where the lattice is packed fully by squares and dimers, respectively. $V$ represents the empty lattice. The estimates of the phase boundaries obtained from modeling the interface without overhangs is shown by the magenta line whereas that obtained by including overhangs of height one are shown by the blue lines. The data points (red circles) are obtained from Monte Carlo simulations (see Sec. 4c).

Figure 7

Schematic of an interface with overhangs. Overhangs are indicated by yellow lines whereas the rest of the interface is indicated by red lines. Overhangs could be of type right (shown by $A$) or left (shown by $B$). The boundary conditions are periodic in the $y$ direction.

Figure 8

The two kinds of right overhang in which the first downward step is followed by (a) a rightward or (b) a leftward step. Overhangs are indicated by yellow lines.

Figure 9

Diagrammatic representation of the recursion relation satisfied by the generating function $\tilde{\omega }\left(x\right)$ [see Eq. (51) for the definition]. The leftmost column may be occupied by a vacancy or by a horizontal dimer.

Figure 10

For each right overhang, there are four possible configurations (a–d) depending on whether the particles adjacent to the downward steps are squares or vertical dimers.

Figure 11

Schematic of two ways of taking two downward steps before the beginning of a left overhang. The first downward step may be followed by (a) a left or (b) a right step. Overhangs are denoted by the yellow line.

Figure 12

Variation of $\chi /L^{7/4}$ with an activity of squares $z_s^{1/4}$ for a fixed activity of dimer $z_d=0.031$ for different system sizes. The curves cross at $z_c^{1/4}\approx 0.684$. The critical value of $z_0$ may be calculated from Eq. (30).