Local distributions of the 1D dilute Ising model

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.

Unconventional low temperature features in the one-dimensional frustrated $q$-state Potts model

“…(19). It can be seen that the curve is an almost straight line well represented by (20). We can observe also how the sharp boundary between quasi-phase melts smoothly for higher temperature.…”

“…Chaves and Riera [19] investigated a particular case of dilute Potts chain. Recently, a different dilute Ising spin-1 chain [20] was also studied in the framework of the projection operator.…”

Residual entropy of the dilute Ising chain in a magnetic field

“…Phase diagrams at zero temperature of the dilute Ising chain without a magnetic field are presented in Ref. [48] in the "interaction constant"-"chemical potential" planes. Qualitatively, the ground state with accounting for a magnetic field is considered in Ref.…”

“…(1) We use the pseudospin σ = 1 operator, where the spin doublet states and impurity correspond to the pseudospin z-projections σ z = ±1 and σ z = 0, respectively, J is the exchange constant, V > 0 is the effective [48] inter-site interaction for impurities, P 0 = 1 − σ 2 z is the projection operator on the impurity state. We assume that the concentration of non-magnetic charged impurities n = j P 0,j /N is fixed.…”

“…In the present paper, we propose an analytical method for calculating the residual entropy of a dilute Ising chain in a magnetic field for all possible values of the model parameters, which is based on the Markov property of the model [48]. Exact analytical expressions for the residual entropy depending on the concentration of impurities are obtained.…”