A new look at Ernst Ising’s thesis

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 \(T=0\). 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.