Solving the one-dimensional Ising chain via mathematical induction: An intuitive approach to the transfer matrix

We review various derivations of the partition function of the two-dimensional Ising Model of ferromagnetism and proofs of the stability of matter, paying attention to passages where there would appear to be a lacuna between steps or where the structure of the argument is not so straightforward. Authors cannot include all the intermediate steps, but sometimes most readers and especially students will be mystified by such a transition. Moreover, careful consideration of such l...

The purpose of the present work is to apply the method recently developed in reference [chain_m] to the spin-1 Ising chain, showing how to obtain analytical $\beta$-expansions of thermodynamical functions through this formalism. In this method, we do not solve any transfer matrix-like equations. A comparison between the $\beta$-expansions of the specific heat and the magnetic susceptibility for the $s=1/2$ and $s=1$ one-dimensional Ising models is presented. We show that thos...

The complete framework for the $\epsilon$-machine construction of the one dimensional Ising model is presented correcting previous mistakes on the subject. The approach follows the known treatment of the Ising model as a Markov random field, where usually the local characteristic are obtained from the stochastic matrix, the problem at hand needs the inverse relation, or how to obtain the stochastic matrix from the local characteristics, which are given via the transfer matrix...

We propose a method for generalizing the Ising model in magnetic fields and calculating the partition function (exact solution) for the Ising model of an arbitrary shape. Specifically, the partition function is calculated using matrices that are created automatically based on the structure of the system. By generalizing this method, it becomes possible to calculate the partition function of various crystal systems (network shapes) in magnetic fields when N (scale) is infinite...

A new and efficient algorithm is presented for the calculation of the partition function in the $S=\pm 1$ Ising model. As an example, we use the algorithm to obtain the thermal dependence of the magnetic spin susceptibility of an Ising antiferromagnet for a $8\times 8$ square lattice with open boundary conditions. The results agree qualitatively with the prediction of the Monte Carlo simulations and with experimental data and they are better than the mean field approach resul...

Today, the Ising model is an archetype describing collective ordering processes. And, as such, it is widely known in physics and far beyond. Less known is the fact that the thesis defended by Ernst Ising 100 years ago (in 1924) contained not only the solution of what we call now the `classical 1D Ising model' but also other problems. Some of these problems, as well as the method of their solution, are the subject of this note. In particular, we discuss the combinatorial metho...

The Ising model was generalized to a system of cells interacting exclusively by presence of shared spins. Within the cells there are interactions of any complexity, the simplest intracell interactions come down to the Ising model. The system may be not only one-dimensional but also two-dimensional, three-dimensional, etc. The purpose of the paper is to develop an approach to constructing the exact matrix model for any considered system in the simplest way. Without this, it ...

There is no an exact solution to three-dimensional (3D) finite-size Ising model (referred to as the Ising model hereafter for simplicity) and even two-dimensional (2D) Ising model with non-zero external field to our knowledge. Here by using an elementary but rigorous method, we obtain an exact solution to the partition function of the Ising model with $N$ lattice sites. It is a sum of $2^N$ exponential functions and holds for $D$-dimensional ($D=1,2,3,...$) Ising model with o...

We present here various techniques to work with clean and disordered quantum Ising chains, for the benefit of students and non-experts. Starting from the Jordan-Wigner transformation, which maps spin-1/2 systems into fermionic ones, we review some of the basic approaches to deal with the superconducting correlations that naturally emerge in this context. In particular, we analyse the form of the ground state and excitations of the model, relating them to the symmetry-breaking...

We apply the real space Renormalisation Group (RNG) technique to a variety of one-dimensional Ising chains. We begin by recapitulating the work of Nauenberg for an ordered Ising chain, namely the decimation approach. We extend this work to certain non-trivial situation namely, the Alternate Ising Chain and Fibonacci Ising chain. Our approach is pedagogical and accessible to undergraduate students who have had a first course in statistical mechanics.