Exact solution for one-dimensional spin models with Markov property

TL;DR

The paper addresses exact thermodynamics of one-dimensional spin/pseudospin systems that admit a Markov-chain description by expressing the finite-temperature free energy as a function of bond concentrations and minimizing that function. It connects the transfer-matrix formalism to Markov-chain representations, with the largest eigenvalue $\lambda_1$ of the non-negative matrix $\mathbf{W}$ determining the free energy via $f = - \frac{1}{\beta} \ln(\lambda_1)$ and mapping to a Markov transition matrix $P_{ab} = \frac{W_{ab} v_b}{\lambda_1 v_a}$. The key contribution is demonstrating that, for the dilute Ising chain with interacting impurities, solving the conditional extremum problem for $f(x_{ab})$ yields exact thermodynamics, including explicit bond-concentration solutions in special cases (e.g., $n=0$ and $h=0$) and the corresponding observables such as magnetization; the approach offers a practical alternative to transfer-matrix diagonalization and is applicable to other 1D anisotropic models with Markov property, such as decorated Ising chains and 1D Blume-Capel, Blume-Emery-Griffiths, and Potts models. The results provide a framework that can yield full thermodynamic characterizations from bond concentrations without requiring the largest eigenvalue calculation. Formulas presented include $f = - \frac{1}{\beta} \ln(\lambda_1)$, $P_{ab} = \frac{W_{ab} v_b}{\lambda_1 v_a}$, and the explicit Hamiltonian $\hat{H} = -J \sum_i \hat{S}_{z,i}\hat{S}_{z,i+1} + V \sum_i \hat{\delta}_{0,i}\hat{\delta}_{0,i+1} - h \sum_i \hat{S}_{z,i}$ for the dilute Ising chain.

Abstract

For one-dimensional spin and pseudospin models that allow mapping to a Markov chain, the free energy of the system at a finite temperature can be expressed in terms of bond concentrations. Minimizing the free energy function makes it possible to obtain an exact solution of a statistical model. A dilute Ising chain with interacting impurities is considered as an example.