Understanding the Differences Between Ising and Heisenberg Models

In summary: It has the interesting property that the energy depends only on the total number of sites occupied, not on the individual site occupancy. In summary, the Heisenberg model is a more general model that includes anisotropic couplings in the spatial directions. This allows for more complicated energy dependences than the Ising model.
  • #1
rafizzi
2
0
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?
 
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  • #2
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.
 
  • #3
can anyone explain briefly about the Heisenberg model and is it possible to simulate it with monte carlo simulation?
 
  • #4
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.
 
  • #5
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.
 
  • #6
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.
 
  • #7
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.
 
  • #8
nnnm4 said:
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
 
  • #9
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.
 

1. What is the Ising and Heisenberg model?

The Ising and Heisenberg model is a mathematical model used in statistical mechanics to describe the behavior of a collection of interacting particles. It was developed to study the properties of magnetic materials, but has also been applied to other systems such as social networks and neural networks.

2. What are the main differences between the Ising and Heisenberg model?

The Ising model only takes into account the spin of particles, while the Heisenberg model considers both spin and angular momentum. In the Ising model, particles can only have two states (up or down), while in the Heisenberg model, particles can have an infinite number of states.

3. What is the significance of the Ising and Heisenberg model in physics?

The Ising and Heisenberg models are important because they provide a simplified yet powerful framework for understanding the behavior of many complex systems. They have been used to study phase transitions, critical phenomena, and other phenomena in condensed matter physics and beyond.

4. How are the Ising and Heisenberg models solved?

The Ising and Heisenberg models can be solved analytically for simple, idealized systems. However, for more complex systems, numerical methods such as Monte Carlo simulations are used to approximate the behavior of the models. In recent years, machine learning techniques have also been applied to solve these models.

5. What are the real-life applications of the Ising and Heisenberg models?

The Ising and Heisenberg models have been applied in various fields, including materials science, social sciences, and computer science. They have been used to study the properties of magnets, analyze social networks, and develop algorithms for optimization problems, to name a few examples.

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