Two-Dimensional Melting: From Liquid-Hexatic Coexistence to Continuous Transitions

The phase diagram of two-dimensional continuous particle systems is studied using the event-chain Monte Carlo algorithm. For soft disks with repulsive power-law interactions $\propto r^{-n}$ with $n\gtrsim 6$, the recently established hard-disk melting scenario ($n\to \infty$) holds: a first-order liquid-hexatic and a continuous hexatic-solid transition are identified. Close to $n=6$, the coexisting liquid exhibits very long orientational correlations, and positional correlations in the hexatic are extremely short. For $n\lesssim 6$, the liquid-hexatic transition is continuous, with correlations consistent with the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) scenario. To illustrate the generality of these results, we demonstrate that Yukawa particles likewise may follow either the KTHNY or the hard-disk melting scenario, depending on the Debye-Hückel screening length as well as on the temperature.

Figure 1

Event-chain algorithm for continuous pair interactions. (a)–(d) Evolution of the algorithm for four particles through three subsequent collision events. (e) Probability for soft disks that a collision event takes place at a distance $r$ larger than the cutoff $r_c$. The vertical dashed line is the cutoff chosen in this work. (f) Equation of state with different cutoffs ($N=6.5\times {10}^4$, $n=6$).

Figure 2

Phase behavior of $r^{-n}$ soft disks for $n\ge 6$. (a) Phase diagram as a function of density $\varphi$ relative to the density ${\varphi }_{\mathrm{hex}}$ of the pure hexatic at coexistence. The nonmonotonic liquid-hexatic coexistence interval vanishes around $n=6$. Symbols match the following figures: bullets are liquid states; filled triangles hexatics, of which downward filled triangles are the hexatic at ${\varphi }_{\mathrm{hex}}$; empty triangles are solids. Center: Local orientational order parameter ${\mathrm{\Psi }}_6$ in $N=2.6\times {10}^5$ particles, the color code is illustrated to the right. Upper row: (b) Liquid phase at $(n,\varphi )=(8,1.193)$ (subset of a $1.0\times {10}^6$ configuration); (c) hexatic; (d) solid. Lower row: Coexistence in the $n=64$ system: (e) pure liquid close to coexistence. (f) At $\varphi =0.889$, the hexatic and the liquid form stripes. (g) At $\varphi =0.898$, a small bubble of liquid remains on a hexatic background of uniform orientation. (h) Equations of state with $n$ from 6 through 1024 (solid lines correspond to $N=6.5\times {10}^4$ soft disks, symbols correspond to $N=2.6\times {10}^5$, ${\varphi }_{\text{liq}}$ is the liquid density at coexistence, dashed lines are for Yukawa particles).

Figure 3

(a) Pair correlation function $g(x,y)$ along the $x$ axis in the pure hexatic phase, showing exponential decay for large $x/d$ (ensemble average of $N=2.6\times {10}^5$ configurations after aligning their global orientational order parameters ${\mathrm{\Psi }}_6$, as in Ref. [9]). (b) Two-dimensional pair correlation function $g(x,y)$ for single configurations at the same parameters. The solid line is the $x$ axis for the left-hand plot. (c) Square boxes of side length $40d$ extracted from the $2.6\times {10}^5$ configurations: Orientational order is preserved as the positional order is lost [color code for ${\mathrm{\Psi }}_6$ as in Figs. 2].

Figure 4

(a) Orientational correlations $g_6(r)$ close to the liquid-hexatic transition, for several exponents $n$. (b) Scaling of the orientational order parameter in sub-blocks of linear size $L_B$ (see Ref. [23] for details). The KTHNY hexatic is stable below the bold line of slope $-1/4$; short-range order corresponds to the steep bold line of slope $-2$. Bullets are liquid states, triangles hexatics (symbols match those in Fig. 2).

Figure 5

Yukawa interactions exhibiting first-order (top) and continuous liquid-hexatic transitions. (a) Pair correlation function along the $x$ axis, as in Fig. 3. (b),(c) Interaction potential of the soft disks and the Yukawa particles around $r=\sigma$.