Quenched disorder: Demixing thermal and disorder fluctuations

We present a finite-size scaling study of the phase transition in the two-dimensional Potts model modified by random bond disorder, in which the average is taken over quantities calculated at quasicritical temperatures of individual disorder configurations. We used the recently proposed equilibrium-like invaded cluster algorithm, which allows us to examine separately the fluctuations in the thermodynamical ensemble from the fluctuations induced by changing the disorder configuration. We point out the crucial role of the spatial inhomogeneities on all scales for the critical behavior of the system. These inhomogeneities are formed by “dressing” of the disorder via critical correlations and are demonstrated to exist for any system size despite the critical fluctuations in thermodynamical ensemble. Such inhomogeneities were previously not thought to be relevant in disorder-altered classical systems when critical temperature is finite. However, we confirm that only by averaging at quasicritical temperatures of each disorder configuration is the thermal critical exponent $y_{\tau }$ not obscured by the influence of the aforementioned spatial inhomogeneities.

Figure 1

The distribution of spatial deviations of the order parameter ${\mu }_{\mathit{\alpha }}\left(\delta {\overline{m}}_{\mathit{\alpha }}\left(\vec{r}\right)\right)$ for a single-disorder configuration $\mathit{\alpha }$ in the case $q=1$ $c=25\%$ for two different statistics of MC steps in a system of size $L=256$. The small peak on the left is a contribution from the spins that are never in a percolating cluster.

Figure 2

Collapsing fits of distributions of spatial deviations of the order parameter, ${\mu }_{\mathit{\alpha }}\left(\delta {\overline{m}}_{\mathit{\alpha }}\left(\vec{r}\right)\right)$ rescaled with the exponent $\beta /\nu =5/48$ and averaged over several disorder configurations (number given in parentheses for every $L$) for $q=1$ and $c=25\%$. In the inset is shown the nonrescaled distributions for lattice sizes $L=128$ and 768, to illustrate its narrowing consistent with the approach to a $\delta$ function in the limit $L\to \infty$.

Figure 3

Distributions of spatial deviations of the order parameter ${\mu }_{\mathit{\alpha }}\left(\delta {\overline{m}}_{\mathit{\alpha }}\left(\vec{r}\right)\right)$ averaged over several disorder configurations (in parentheses) for $q=3$ and concentration $c=25\%$. Ordinate axis is on logarithmic scale. The changes in the distribution resulting from increasing the system size are made more obvious by the circles.

Figure 4

Comparison of three quantities connected with magnetization in case when disorder is irrelevant [percolation, shown in (a)] and when it is relevant [three-state Potts, shown in (b)]. The convergence of $\left[{\overline{m}}_{\mathit{\alpha }}\right]$ and ${\delta }_{T}m$ is determined by the same exponent in both cases ($-\frac{\beta }{\nu }$); however, when disorder is relevant, spatial fluctuations $\delta {\overline{m}}_{\mathit{\alpha }}\left(r\right)$ are stronger than the thermal fluctuations and have an increasing influence for the order parameter with increasing $L$.

Figure 5

A plausible form of the distribution of spatial inhomogeneities of order parameter at the critical point in systems where the bond disorder is relevant.

Figure 6

If the inequality between critical exponents $y_{\tau }>{\tilde{y}}_{\tau }$ holds, the free energy (14) describes a phase transition displaying inhomogeneities at all scales. If variations of local quasicritical temperatures of subsystems of sizes $L$ exist, and are larger than respective rounding areas, one can find a subsystem of size $L$ that can spontaneously form a diffuse domain wall, leaving neighboring subsystems predominantly in different phases.

Figure 7

Log-log plot of values $\sqrt{[\overline{p_{\mathit{\alpha }}^2}-{\overline{p}}_{\mathit{\alpha }}^2]}$ for three concentrations of disorder. Full line represents the given EIC constraint, which is narrower than the original RC distribution and thus ensures the canonical ensemble.

Figure 8

Log-log plot of values ${\delta }_{T}m$ for the three concentrations of disorder. Dashed lines represent linear fits [to a power law dependence $f_1\left(L\right)=a{L}^{-\frac{\beta }{\nu }}$], and full lines represent a fit to $f_2\left(L\right)=a{L}^{-\frac{\beta }{\nu }}+c{L}^{-\frac{\beta }{\nu }+{\omega }_1}$. The values of ${\delta }_{T}m$ are determined up to five significant digits, which makes error bars much smaller than the symbol sizes.

Figure 9

Collapsing fits of distributions of thermal fluctuations of the order parameter density $\left[{\mu }_{\mathit{\alpha }}\left([m_{\mathit{\alpha }}-{\overline{m}}_{\mathit{\alpha }}]\right)\right]$ for $q=3$ with $25\%$ of disorder, rescaled with the exponent $y=0.132\left(3\right)$. The discrepancies from the full collapse are visible for the smallest lattice sizes, signaling scaling corrections.

Figure 10

Log-log plot of values $\left[{\overline{m}}_{\mathit{\alpha }}\right]$ for three concentrations of disorder. Dashed lines represent the power law dependence $f_2\left(L\right)=a{L}^{-\frac{{\beta }^{\prime }}{\nu }}+c{L}^{-\frac{{\beta }^{\prime }}{\nu }+{\omega }_1}$. The statistical errors are on the fifth significant digit and are much smaller than the symbol sizes.

Figure 11

Running averages of $U_{\mathit{\alpha }}^{*me}$ for the concentration of $25\%$ of disorder, for several lattice sizes. N is the length of the running average over disorder. The horizontal lines designate the values $0.001$ and $-0.001$.

Figure 12

Size dependence of magnetization-energy cumulants $U_{\mathit{\alpha }}^{*me}$ in (a) and $U_{\mathit{\alpha }}^{me}$ in (b). Full and dashed lines represent the best fits to the forms $f_1\left(L\right)=a{L}^{y_{\tau }-d/2}$ or $f_2\left(L\right)=a{L}^{y_{\tau }-d/2}+c{L}^{y_{\tau }-d/2+{\omega }_2}$ (for details see Table ) in all cases except for $U_{\mathit{\alpha }}^{me}$ for $37.5\%$. In the latter case, we draw a function $f_1$ through the largest three lattice sizes, which gives the exponent $y_{\tau }\approx 1$.

Figure 13

Log-log plot of values $L^{d/2}{\delta }_{T}e$. Lines represent best fits either to the form $f_3\left(L\right)=\sqrt{a{L}^{2(y_{\tau }-d/2)}+c}$ or $f_1\left(L\right)=a{L}^{y_{\tau }-d/2}$ (see Table ).

Figure 14

Log-log plot of values ${\delta }_{\text{dis}}\overline{p}$ for three concentrations of disorder. Dashed lines represent the power law dependence $f_1\left(L\right)=a{L}^{-{\tilde{y}}_{\tau }}$.

Figure 15

Log-log plot of values ${\delta }_{\text{dis}}\overline{e}$ for three concentrations of disorder. Dashed lines represent the power law dependence $f_1\left(L\right)=a{L}^{-\tilde{x}}$.

Figure 16

Convergence of self-averaging ratios $R_m\left(L\right)$. Dashed lines represent fits most robust fits to the form $f_1^{\prime }\left(L\right)=a+b{L}^c$, where $c$ is taken to be positive.