The generalized density of states in a one-dimensional Ising model with ferrromagnetic and antiferromagnetic interactions

TL;DR

This paper derives the generalized density of states for a 1D Ising model with mixed interactions, enabling deeper understanding of its properties and revealing spontaneous magnetization phenomena at finite temperatures.

Contribution

It provides explicit expressions for the joint density of states $D_N(E,m)$ in a 1D Ising chain, enhancing analysis of magnetization dynamics and phase behavior.

Findings

Spontaneous magnetization occurs at non-zero temperature.

Magnetization can randomly flip sign over time.

Long-term average magnetization tends to zero.

Abstract

Expressions for the density of states $D(E)$, where $D(E)$ is the number of states of energy $E$, are well known. The present paper offers the expressions for generalized density of states $D_N(E,m)$, where $D_N(E,m)$ is the number of states with energy $E$ and magnetization $m$ in a one-dimensional $N$-spin chain. The expressions obtained here can be considered as reference ones, since all the main characteristics were obtained without them: using the transfer matrix technique or using well-known expressions for the density of states $D(E)=\sum_m{D_N(E,m)}$. Nevertheless, the knowledge of quantity $D_N(E,m)$ helps to understand the model properties and allows the analysis of the temporal behavior of magnetization $m=m(\tau)$. In particular, we demonstrate that in a one-dimensional model spontaneous magnetization can be observed at a non-zero temperature. However, the spontaneous magnetization can randomly change its sign, which results in the magnetization averaged over a very long observation period becoming zero $\langle m(\tau)\rangle$.