The phase boundary of the random site Ising model

Abstract

We introduce a new approach to disordered two-dimensional Ising models based on the extension of the combinatorial solution to randomized supercells. Applying it to the site-diluted Ising model on the square lattice, we resolve the full phase boundary $T_c(p)$ from the pure-Ising point to the percolation limit $T_c(p_c)=0$ with, in principle, arbitrary precision. The critical eigenvalue governing the transition is found to follow a remarkably accurate linear interpolation between the Ising and percolation endpoints, whose small but systematic deviations reveal the nontrivial fine structure of the phase boundary. Near the percolation threshold, we confirm the crossover exponent $φ_{\rm RSIM}=1$ and extract the nonuniversal amplitude ${α_{\rm RSIM}\simeq 1.616}$.

Figures (8)

• Figure 1: Phase boundary $T_c(p)$ of the site-diluted Ising model computed for increasing supercell size $L$. The inset magnifies the critical region near the percolation threshold $p_c$. The $L \to \infty$ extrapolation is also shown. Statistical errors are smaller than the symbol size.
• Figure 2: Distributions of the FV eigenvalue $\lambda_c$ for increasing supercell size $L$ at three representative dilutions: (a) $p=0.9$, (b) $p=0.6$, and (c) $p=p_c$. For high occupancy ($p=0.9$) the distributions rapidly converge, indicating complete self-averaging of $\lambda_c$, while at intermediate dilutions ($p=0.6$), convergence is slower and requires larger supercells. At the percolation threshold $p_c$ distributions remain size-dependent, indicating that convergence is not yet attained.
• Figure 3: Mean FV eigenvalue $\overline{\lambda}_c(L,p)$ of the FV matrix as a function of the site occupation probability $p$, for increasing supercell sizes $L$. Horizontal lines indicate the bounds $\lambda_c(p_c) = 1$ and $\lambda_c(1) = 1+\sqrt{2}$, while the vertical dashed line indicates the site-percolation threshold $p_c \simeq 0.592746...$. Inset: enlargement of the near-critical region. The green line shows the linear approximation $\lambda^{\rm{lin}}_c(p)$ defined in Eq. \ref{['pseudolambda']}.
• Figure 4: Scaling of the critical temperature near the percolation threshold. The plot shows $\log(p-p_c)$ versus $1/{\overline{T}}_c$ for increasing supercell sizes $L$. Near the percolation threshold (before the breakdown of the approximation) the data show that $T_c(p)$ approaches the expected asymptotic behavior \ref{['asymp']} with $\phi_{\rm RSIM}=1$ and slope $-2$. The intercept from the fit (dashed line) gives the estimate $-\log \alpha_{\rm RSIM} = - 0.48$.
• Figure 5: Deviation of the mean FV eigenvalue from the linear approximation, $\Delta\lambda_c(L,p) = \overline{\lambda}_c(L,p) - \lambda_c^{\mathrm{lin}}(p)$ for increasing supercell sizes $L$. The vertical dashed line marks $p_c$; the horizontal line is the zero baseline. Inset: magnification of the near-critical region $p \in [0.59,\,0.60]$.