Local distributions of the 1D dilute Ising model

TL;DR

This paper investigates the local distributions and correlation behaviors of the one-dimensional dilute Ising model with impurities, deriving explicit formulas and analyzing critical phenomena at low temperatures.

Contribution

It provides explicit expressions for pair distributions and correlation lengths, and characterizes ordering processes through Markov chain analysis in the dilute Ising model.

Findings

Correlation lengths diverge only at zero temperature.

Ordering processes influence the maximum of specific heat at finite temperature.

Different types of Markov chains correspond to ordered and non-ordered states.

Abstract

The local distributions of the one-dimensional dilute annealed Ising model with charged impurities are studied. Explicit expressions are obtained for the pair distribution functions and correlation lengths, and their low-temperature asymptotic behavior is explored depending on the concentration of impurities. For a more detailed consideration of the ordering processes, we study local distributions. Based on the Markov property of the dilute Ising chain, we obtain an explicit expression for the probability of any finite sequence and find a geometric probability distribution for the lengths of sequences consisting of repeating blocks. An analysis of distributions shows that the critical behavior of the spin correlation length is defined by ferromagnetic or antiferromagnetic sequences, while the critical behavior of the impurity correlation length is defined by the sequences of impurities or by the charge-ordered sequences. For the dilute Ising chain, there are no other repeating sequences whose mean length diverges at zero temperature. While both the spin correlation and the impurity correlation lengths can diverge only at zero temperature, the ordering processes result in a maximum of the specific heat at finite temperature defined by the maximum rate of change of the impurity-spin pairs concentration. A simple approximate equation is found for this temperature. We show that the non-ordered dilute Ising chains correspond to the regular Markov chains, while various orderings generate the irregular Markov chains of different types.