Landau free energy and the absence of spontaneous magnetization of the one-dimensional Ising model

TL;DR

This paper analyzes the one-dimensional Ising model using Landau free energy and density of states, providing a rigorous proof that it does not exhibit spontaneous magnetization at any finite temperature.

Contribution

It introduces a density of states approach and employs the Landau free energy framework to rigorously prove the absence of spontaneous magnetization in the 1D Ising model.

Findings

Density of states shows monotonicity, implying no spontaneous magnetization.

Landau free energy increases with |m| and is non-analytic at zero magnetization.

Exact solution confirms no spontaneous magnetization at finite temperature.

Abstract

We revisit the problem of spontaneous magnetization of the one-dimensional Ising model from the Landau free energy perspective. To this end, we define and calculate the density of states of the one-dimensional Ising model following a technique introduced by Ising. The observed monotonicity property of the density of states suggests heuristically that the model does not exhibit spontaneous magnetization at any finite temperature. Subsequently, we solve the model exactly in the thermodynamic limit by employing the maximum-term approximation, which is feasible due to the simple analytical expression of the density of states. We also show that the Landau free energy is an increasing function of $|m|$ and its second derivative at $m=0$ is positive and non-analytic in temperature, proving rigorously the absence of spontaneous magnetization of the model at any finite temperature.