The phase boundary of the random site Ising model

TL;DR

This paper introduces a new combinatorial approach to determine the phase boundary of the disordered 2D site-diluted Ising model, accurately mapping the transition from pure Ising to percolation limits.

Contribution

It extends the combinatorial solution to randomized supercells, enabling precise calculation of the phase boundary in disordered Ising models.

Findings

The phase boundary $T_c(p)$ is accurately mapped from the pure Ising point to the percolation limit.

The critical eigenvalue interpolates linearly between Ising and percolation endpoints.

Near the percolation threshold, the crossover exponent is confirmed as $oxed{1}$.

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 $\phi_{\rm RSIM}=1$ and extract the nonuniversal amplitude ${\alpha_{\rm RSIM}\simeq 1.616}$.