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Ising's Roots and the Transfer-Matrix Eigenvalues.

Reinhard Folk¹, Yurij Holovatch^2,3,4,5

¹Institute for Theoretical Physics, Johannes Kepler University Linz, 4040 Linz, Austria.

Entropy (Basel, Switzerland)

|June 26, 2024

Summary

This summary is machine-generated.

Ernst Ising

Keywords:

Ernst Ising Ising model Potts model transfer matrix

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Area of Science:

Statistical Physics
Condensed Matter Physics

Background:

The Ising model is a fundamental concept in statistical physics for understanding collective ordering.
Ernst Ising's 1924 thesis explored problems beyond the classical 1D Ising model.

Purpose of the Study:

To examine lesser-known aspects of Ernst Ising's 1924 thesis.
To highlight the combinatorial methods used and their connection to modern concepts.

Main Methods:

Analysis of Ernst Ising's combinatorial method for calculating partition functions.
Comparison of Ising's combinatorial results with transfer matrix eigenvalues.
Investigation of a generalized three-state model from Ising's thesis.

Main Results:

Ising's combinatorial approach yields partition functions determined by polynomial roots ('Ising's roots').
These roots are equivalent to the eigenvalues of the transfer matrix, a later developed concept.
A three-state model in the thesis predates modern many-component order parameter models.

Conclusions:

Ernst Ising's thesis contained significant contributions beyond the 1D Ising model solution.
The combinatorial methods and generalized models foreshadowed later developments in statistical physics.