Interfacial adsorption in two-dimensional pure and random-bond Potts models

We use Monte Carlo simulations to study the finite-size scaling behavior of the interfacial adsorption of the two-dimensional square-lattice $q$-states Potts model. We consider the pure and random-bond versions of the Potts model for $q=3,4,5,8$, and 10, thus probing the interfacial properties at the originally continuous, weak, and strong first-order phase transitions. For the pure systems our results support the early scaling predictions for the size dependence of the interfacial adsorption at both first- and second-order phase transitions. For the disordered systems, the interfacial adsorption at the (disordered induced) continuous transitions is discussed, applying standard scaling arguments and invoking findings for bulk critical properties. The self-averaging properties of the interfacial adsorption are also analyzed by studying the infinite limit-size extrapolation of properly defined signal-to-noise ratios.

Figure 1

Typical equilibrium Monte Carlo configurations of an $L=100, q=10$, pure (upper panel) and random-bond $r=1/10$ (lower panel) Potts model at $k_{\mathrm{B}}T/J_1=0.98k_{\mathrm{B}}{T}_{\mathrm{c}}/J_1$. In both cases, red color depicts the $q=1$ states and blue color denotes the $q=2$ states, whereas the nonboundary states ($q\ge 3$) adsorbed at the interface are shown in black. Note that the fixed boundaries $[1:2]$ are also included in these illustrations.

Figure 2

Finite-size scaling of the interfacial adsorption of the pure and random-bond $q=3$ Potts model. The inset illustrates the infinite limit-size extrapolation of the effective exponent $a_{\mathrm{r}}^{\left(\mathrm{eff}\right)}$.

Figure 3

Finite-size scaling of the interfacial adsorption of the pure and random-bond $q=4$ Potts model. The inset shows fitting results for the exponent $a_{\mathrm{p}}$ of the pure system by varying the value of the corrections exponent $\omega$ within the regime $[-1/16,0]$. The dashed line marks the value $a_{\mathrm{p}}=0.875\left(9\right)$ that corresponds to the case $\omega =0$.

Figure 4

Finite-size scaling of the interfacial adsorption of the pure and random-bond Potts model for various values of the internal states $q$ in the originally first-order regime, as indicated.

Figure 5

$R_{\mathrm{W}}$ as a function of the inverse linear system size for two values of $q$, as indicated. The solid lines are second-order polynomial extrapolations to the limit $L\to \infty$.