Scaling analysis of the site-diluted Ising model in two dimensions

A combination of recent numerical and theoretical advances are applied to analyze the scaling behavior of the site-diluted Ising model in two dimensions, paying special attention to the implications for multiplicative logarithmic corrections. The analysis focuses primarily on the odd sector of the model (i.e., that associated with magnetic exponents), and in particular on its Lee-Yang zeros, which are determined to high accuracy. Scaling relations are used to connect to the even (thermal) sector, and a first analysis of the density of zeros yields information on the specific heat and its corrections. The analysis is fully supportive of the strong scaling hypothesis and of the scaling relations for logarithmic corrections.

Figure 1

(a) FSS plot for ${\chi }_L$ at $p_c=0.88889$. The slope gives an estimate for $\gamma ∕\nu$ of 1.747(7). (b) Plot of $\ln {\chi }_L-7∕4\ln L$ against $\ln \left(\ln L\right)$ giving slope $-0.01\left(3\right)$, indicating no multiplicative logarithmic corrections to the FSS of the susceptibility.

Figure 2

(a) FSS plot for the lowest Lee-Yang zero at $p_c=0.88889$. The slope gives an estimate for $\Delta ∕\nu$ of 1.879(4). (b) Plot of $\ln r_1+15∕8\ln L$ against $\ln \left(\ln L\right)$ giving slope $-0.01\left(2\right)$, indicating the absence of multiplicative logarithmic corrections to the FSS of $r_1$.

Figure 3

(a) The integrated density of zeros at $p=0.88889$ (diamonds), $p=0.75$ (crosses) and $p=0.66661$ (circles). The excellent data collapse (four data points for each of seven lattice sizes are involved at each value of $p$) and zero intercept indicate the correct transition point and the fitted curves give $a_2$ compatible with the expected value $16∕15$ $(\alpha =0)$. (b) The absence of multiplicative logarithmic corrections in the integrated density of zeros indicates that the specific heat logarithm exponent $\widehat{\alpha }=0$ independent of the degree of dilution.