Understanding the Differences Between Ising and Heisenberg Models

[rafizzi]

I've a problem to understand the Ising model and the Heisenberg model.
Can anyone explain to me what is the different between ising model and the Heisenberg model?

 

[Jezuz]

In the Ising model the spins are allowed to be only -1 or 1 and in a given direction. In the Heisenberg model the spins are allowed to point in any direction.

 

[rafizzi]

can anyone explain briefly about the Heisenberg model and is it possible to simulate it with monte carlo simulation?

 

[salparadise]

Hi,

I'm having the same doubt here.

I think that both models share in common that only two spin states are allowed for each position in the lattice. But while in the Ising model only nearest neighbours interactions are allowed, in the Heisenberg model this restriction does not apply.


Could anyone confirm this please.

 

[nnnm4]

No that's incorrect. The Ising model is semi-classical and treats spin in only parallel or antiparallel to a given direction (defined by the external field if included), and thus the spin is represented as a scalar. The Heisenberg model uses the appropriate quantum mechanical spin operators.

 

[DrDu]

Alternatively you can say that in the Ising model, only the z-components of the spin interact while in the Heisenberg model, there is equal interaction of all components of spin.

 

[correlation]

The order parameter of Ising model is 1 dimensional, i.e. scalar, but that of Heisenberg model is 3, i.e. vector. So the critical behavior (scaling law) of these two models are different. They have different scalings.

 

[mavipranav]

The Ising model is semi-classical and treats spin in only parallel or antiparallel to a given direction (defined by the external field if included), and thus the spin is represented as a scalar.

That is not entirely correct. There are classical and quantum versions of the Ising model.

Mavi

 

[thebobguy]

Also, we can extend both models over whatever range of lattice sites we like with differing couplings for nearest-neighbour, next-nearest-neighbour, and so on...

As was already said, the Ising model is a good model for when only the z components of the spins interact. Unfortunately, there isn't much in nature that can be modeled very well by it. The most general form of the Heisenberg model includes anisotropic couplings in the spatial directions, usually called the XYZ model.

The Ising model can then be viewed as a limit of the Heisenberg model when the couplings in the x-y plane vanish. If we let the coupling in the z-direction vanish, then we have the XY model, which is really quite an interesting model.