# Cusp singularity in mean field Ising model

## TL;DR

This paper derives the entropy of the mean field Ising model using Hamilton-Jacobi formalism, revealing a cusp at the critical point that offers new geometric insights and educational value for understanding phase transitions.

## Contribution

It introduces a novel geometric perspective on the Ising model's phase transition by identifying a cusp in the entropy surface using Hamilton-Jacobi formalism.

## Key findings

- Identifies a cusp at the critical point in the entropy surface.
- Provides a new geometric interpretation of phase transitions.
- Enhances educational understanding of statistical phase transitions.

## Abstract

An entropy of the Ising model in the mean field approximation is derived by the Hamilton-Jacobi formalism. We consider a grand canonical ensemble with respect to the temperature and the external magnetic field. A cusp arises at the critical point, which shows a simple and new geometrical aspect of this model. In educational sense, this curve with a cusp helps students acquire a more intuitive view on statistical phase transitions.