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arXiv:2405.05703 · Entropy

Ising’s Roots and the Transfer-Matrix Eigenvalues

R. Folk, Yurij V. Holovatch

Published May 28, 2024• 5 citations
arXivHTMLPDF

Abstract

Today, the Ising model is an archetype describing collective ordering processes. 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 method Ernst Ising used to calculate the partition function for a chain of elementary magnets. In the thermodynamic limit, this method leads to the result that the partition function is given by the roots of a certain polynomial. We explicitly show that ‘Ising’s roots’ that arise within the combinatorial treatment are also recovered by the eigenvalues of the transfer matrix, a concept that was introduced much later. Moreover, we discuss the generalization of the two-state model to a three-state one presented in Ising’s thesis, which is not included in his famous paper of 1925 (E. Ising, Z. Physik 31 (1925) 253). The latter model can be considered as a forerunner of the now-abundant models with many-component order parameters.

DOI
10.3390/e26060459

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Today, the Ising model is an archetype describing collective ordering processes. 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 method Ernst Ising used to calculate the partition function for a chain of elementary magnets. In the thermodynamic limit, this method leads to the result that the partition function is given by the roots of a certain polynomial. We explicitly show that ‘Ising’s roots’ that arise within the combinatorial treatment are also recovered by the eigenvalues of the transfer matrix, a concept that was introduced much later. Moreover, we discuss the generalization of the two-state model to a three-state one presented in Ising’s thesis, which is not included in his famous paper of 1925 (E. Ising, Z. Physik 31 (1925) 253). The latter model can be considered as a forerunner of the now-abundant models with many-component order parameters.
Abstract The famous paper (Ising in Z Phys 31: 253, A translation to English is by Jane Ising, Tom Cummings, Ulrich Harsch is found Bibliotheca Augustina 2001/02 https://www.hs-augsburg.de/~harsch/anglica/Chronology/20thC/Ising/isi_fm00.html , 1928) contains only a greatly reduced version of the material presented in his thesis. Looking at it therefore provides new insights into the Ising model in terms of its later implications. The method of calculating the partition function from the configuration of the system succeeds by finding a polynomial whose largest root defines the result in the thermodynamic limit and which coincides with the characteristic polynomial of the transfer matrix. This matrix works with the states of the elementary elements of the model and was found later in 1941 by Kramers and Wannier. Ising used his method for several one-dimensional models—variants of the Ising chain: a chain with three instead of two states (a forerunner of the Potts model formulated 1952), the Ising ladder given by two interacting chains and the Ising chain with next nearest-neighbor interaction published by Montroll in 1941. The largest root could not be found in all cases and approximations had to be made. Although no phase transition at finite temperature was found for the Ising chain, it turned out in the following decades that there is a critical point at T=0T=0T=0 T = 0 . Thus, the simple Ising chain can be used to study all properties approaching a transition point in this exactly solvable model. The historical introduction to the problem, given by Ising in his dissertation, goes back to Richard Kirwan (1733–1812), who was the first to associate ferromagnetism with the ordering of interacting magnetic elements similar to crystallization. Graphical abstract
Abstract The Ising model is a well known statistical model which can be solved exactly by various methods. The most familiar one is the transfer matrix method. Sometimes it can be difficult to approach the open boundary case rather than periodic boundary ones in higher dimensions. But physically it is more intuitive to study the open boundary case, as it gives a closer view of the real system. We have introduced a new method called the pairing method to determine the exact partition function for the simplest case, a 1D Ising lattice. This method simplifies the problem's complexities and reduces it to a pure combinatorial problem. The study also reveals that it is possible to apply this pairing method in the case of a 2D square lattice. The obtained results agree perfectly with the values in the literature and this new approach provides an algorithmic insight to deal with such problems.
Using a method suggested by Montroll, we extend the well-known matrix formulation of the nearest-neighbor one-dimensional Ising problem to allow for interactions with an arbitrary finite range n, general spin l, and an applied magnetic field B. We exhibit the relevant matrix element explicitly and hence formally obtain the partition function via an eigenvalue problem of order (2l + 1)n. For the case B = 0, l = ½ we introduce a change of variable which simplifies the partition function while still allowing a matrix formulation. Using this approach we have computed specific-heat curves for infinite, ferromagnetic Ising chains with interactions of range n (n ≤ 7). We prove in an appendix that open and cyclic boundary conditions are equivalent for the system under consideration.
Ising’s one-dimensional model is a model whose constituent elements are magnetic moments of spin ±1/2. Spins interact with short-range forces limited only to the first neighbours. The partition function of this model is easy to calculate using transfer matrix (section 4.1). From the known partition function, free energy, magnetization and other thermodynamic functions can be easily obtained. It shows that the magnetization is analytical function of the temperature for all T > 0, which shows that in this Ising model, there is no phase transition for T > 0.
We present an alternative procedure for solving the eigenvalue problem of replicated transfer matrices describing disordered spin systems with (random) 1D nearest neighbour bonds and/or random fields, possibly in combination with (random) long range bonds. Our method is based on transforming the original eigenvalue problem for a 2 n × 2 n matrix (where n → 0) into an eigenvalue problem for integral operators. We first develop our formalism for the Ising chain with random bonds and fields, where we recover known results. We then apply our methods to models of spins which interact simultaneously via a one-dimensional ring and via more complex long-range connectivity structures, e.g., (1 + ∞ )-dimensional neural networks and 'small-world' magnets. Numerical simulations confirm our predictions satisfactorily.
Abstract The aim of this work is to present a formulation to solve the one-dimensional Ising model using the elementary technique of mathematical induction. This formulation is physically clear and leads to the same partition function form as the transfer matrix method, which is a common subject in the introductory courses of statistical mechanics. In this way our formulation is a useful tool to complement the traditional more abstract transfer matrix method. The method can be straightforwardly generalised to other short-range chains, coupled chains and is also computationally friendly. These two approaches provide a more complete understanding of the system, and therefore our work can be of broad interest for undergraduate teaching in statistical mechanics.