Transfer-matrix scaling from disorder-averaged correlation lengths for diluted Ising systems

TL;DR

This paper develops a transfer matrix scaling method for diluted Ising models, enabling efficient analysis of phase boundaries and critical exponents near the percolation threshold.

Contribution

A novel transfer matrix scaling technique for diluted systems is introduced, allowing accurate disorder-averaged correlation length calculations and phase boundary analysis.

Findings

Reproduces the pure-system slope near $p_c$ with good accuracy.

Identifies the crossover of the critical exponent $ta$ from pure to percolation values.

Demonstrates exponential behavior of the $T_c imes p$ curve over two decades.

Abstract

A transfer matrix scaling technique is developed for randomly diluted systems, and applied to the site-diluted Ising model on a square lattice in two dimensions. For each allowed disorder configuration between two adjacent columns, the contribution of the respective transfer matrix to the decay of correlations is considered only as far as the ratio of its two largest eigenvalues, allowing an economical calculation of a configuration-averaged correlation length. Standard phenomenological-renormalisation procedures are then used to analyse aspects of the phase boundary which are difficult to assess accurately by alternative methods. For magnetic site concentration $p$ close to $p_c$, the extent of exponential behaviour of the $T_c \times p$ curve is clearly seen for over two decades of variation of $p - p_c$. Close to the pure-system limit, the exactly-known reduced slope is reproduced to a very good approximation, though with non-monotonic convergence. The averaged correlation lengths are inserted into the exponent-amplitude relationship predicted by conformal invariance to hold at criticality. The resulting exponent $\eta$ remains near the pure value (1/4) for all intermediate concentrations until it crosses over to the percolation value at the threshold.