# References and papers to distinguish between the Heisenberg and Ising Model

Does anybody have any good papers or references to explain the differences between the Heisenberg model and Ising model?

To the best of my knowledge, I am aware that the Hamiltonians are similar, however the Heisenberg model represents the spins with Pauli operators.

I would like some solid article which I could reference please.

• I doubt you can find modern articles that cover the basics of these models. Your best bet is textbooks. I would open the wiki on these models and try to see in the references if there is a book that appears in both articles. Besides the wiki on this models should be pretty good too.
– oleg
Jul 18, 2019 at 16:07
• Both models represent spin-spin interactions with Paulis. The difference is that for Ising it is only spin-z that interacts while Heisenberg involves all 3 axis x,y,z .
– oleg
Jul 18, 2019 at 16:09

Does anybody have any good papers or references to explain the differences between the Heisenberg model and Ising model?

As the comments already suggest, this is more introductory textbook material. Parkinson and Farnell's "An Introduction to Quantum Spin Systems" (Springer Lecture Notes in Physics 816, 2010) might be a good choice for you. It's an overall accessible introduction to quantum spin systems, particularly in one and two dimensions, covering models and common techniques. It also be available as an e-book through many university libraries. (But you should be able to find the distinction between Ising and Heisenberg models in many other books on magnetism or statistical physics too.) From p. 17:

Depending on the types of atom involved and the environment in which they exist the exchange interaction may have different forms. Examples are:

• Heisenberg $$J\mathbf{S}_1\cdot\mathbf{S}_2$$ (as before)
• Ising $$JS_1^zS_2^z$$
• Anisotropic (a combination of the above) $$J[\Delta S_1^z S_2^z + (S_1^xS_2^x + S_1^y S_2^y)]$$
• Biquadratic $$J\left( \mathbf{S}_1 \cdot \mathbf{S}_2\right)^2$$

To the best of my knowledge, I am aware that the Hamiltonians are similar, however the Heisenberg model represents the spins with Pauli operators.

Both models are generally defined in terms of spin operators, as shown above. In the important case of spin-1/2 spins, the spin operator can be represented in terms of Pauli matrices. In fact, there's an equality $$\mathbf{S}=\hbar \vec{\sigma}/2$$, where $$\vec{\sigma}=(\sigma^x, \sigma^y, \sigma^z)$$ is the vector of Pauli matrices. Often we just redefine the $$J$$ to avoid the factor of $$\hbar/2$$, which is why you might have seen an Heisenberg model written $$J\, \vec{\sigma}_1\cdot\vec{\sigma}_2$$, but it can make a difference when comparing observables calculated using different notations.