The **key difference** between Ising and Heisenberg model is that** in the Ising model, the energy of a configuration of spins is invariant under flipping every spin in the system from to or vice versa, whereas in the Heisenberg model, the energy of a configuration of spins is invariant to applying the same rotation around the unit sphere to every spin in the system. **

Ising model was developed and was named after the physicist Ernst Ising. Heisenberg model was developed by Werner Heisenberg, a famous physicist.

### CONTENTS

1. Overview and Key Difference

2. What is Ising Model

3. What is Heisenberg Model

4. Side by Side Comparison – Ising vs Heisenberg Model in Tabular Form

5. Summary

## What is Ising Model?

Ising model is a mathematical model of ferromagnetism in statistical mechanics. It was named after the physicist Ernst Ising. There are discrete variables in this model representing the magnetic dipole moments of atomic “spins” that can occur in one of two states, +1 and -1. In this model, we usually arrange the spins in a lattice to allow each spin to interact with its neighbours. This model allows us to identify the phase transitions as a simplified model of reality. Ising model is one of the simplest statistical models to show a phase transition.

When considering the history of this model, it was invented by the physicist Wilhelm Lenz in 1920. He gave this model as a problem to his student; Ernst Ising in 1925 where he solved the model. But his solution had no phase transition in it. The 2-dimensional square lattice Ising model is a much difficult one that was given an analytical description by Lars Onsager in 1944. Usually, this model is solved using the transfer-matrix method though there are some different approaches existing as well. When the number of dimensions is above four, the phase transition of the Ising model can be described by “mean field theory”.

## What is Heisenberg Model?

Heisenberg model is a mathematical model in statistical physics and is important in the study of critical points and phase transitions of magnetic systems. In this model, we treat the spins of the magnetic systems, quantum mechanically. This model was developed by Werner Heisenberg, a famous physicist. This model is related to the prototypical Ising model.

In quantum mechanics, the dominant coupling between two dipoles can cause nearest-neighbours to have the lowest energy when they are aligned. Taking this as an assumption, we can develop mathematical formulas for the Heisenberg model.

There are some important applications of the Heisenberg model. It provides an important and tractable theoretical example for applying density matrix renormalization. We can solve the six-vertex model using the Heisenberg spin chain. Further, the half-filled Hubbard model can be mapped onto a Heisenberg model with coupling constant that is less than 0, representing the strength of the super-exchange interaction.

## What is the Difference Between Ising and Heisenberg Model?

Ising model and the Heisenberg model are discussed mainly under statistical physics. The key difference between Ising and Heisenberg model is that in the Ising model, the energy of a configuration of spins is invariant under flipping every spin in the system from to or vice versa whereas, in the Heisenberg model, the energy of a configuration of spins is invariant to applying the same rotation around the unit sphere to every spin in the system.

Below is a summary of the difference between Ising and Heisenberg model in tabular form.

## Summary – Ising vs Heisenberg Model

Ising model was developed and was named after the physicist Ernst Ising while Heisenberg model was developed by Werner Heisenberg. The key difference between Ising and Heisenberg model is that in the Ising model, the energy of a configuration of spins is invariant under flipping every spin in the system from to or vice versa whereas, in the Heisenberg model, the energy of a configuration of spins is invariant to applying the same rotation around the unit sphere to every spin in the system.

##### Reference:

1. “Ising Model.” *Ising Model – an Overview | ScienceDirect Topics*, Available here.

##### Image Courtesy:

1. “Heisenberg,W. Wigner,E. 1928” By GFHund – Own work (CC BY 3.0) via Commons Wikimedia

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