What is the Difference Between Ising and Heisenberg Model?
🆚 Go to Comparative Table 🆚The main difference between the Ising and Heisenberg models lies in the way they treat spin variables and their symmetry properties. Both models describe spin lattices with interactions between neighboring spins, but they differ in the nature of the spins and their interactions.
Ising Model:
- Spins are treated as binary variables (Ising variables), taking values of +1 or -1, representing up or down spins.
- The Hamiltonian for the Ising model is given by $$H\text{Ising} = -J \sum{\langle i,j \rangle} s^zi s^zj$$, where $J$ is the coupling constant, $s^z_i$ represents the spin at site $i$ in the $z$ direction, and the summation is over first neighbors.
- The Ising model has a discrete symmetry (Z2 symmetry), where flipping every spin in the system does not change the energy of the configuration.
Heisenberg Model:
- Spins are treated as quantum operators with magnitude equal to the reduced Planck constant divided by two (h/2).
- The Hamiltonian for the Heisenberg model is given by $$H\text{Heisenberg} = -J \sum{\langle i,j \rangle} \hat{S}i \cdot \hat{S}j$$, where $\hat{S}_i$ is the spin operator at site $i$.
- The Heisenberg model has a continuous symmetry (SU(2) symmetry), where applying the same rotation around the unit vector does not change the energy of the configuration.
Treating spins as operators in the Heisenberg model allows for more general interactions, including anisotropic couplings in the spatial directions. This results in different physical properties and phenomena compared to the Ising model, which has only discrete Z2 symmetry.
Comparative Table: Ising vs Heisenberg Model
The Ising and Heisenberg models are both mathematical models used in statistical physics to describe the behavior of magnetic systems. Here is a table comparing the key differences between the two models:
| Feature | Ising Model | Heisenberg Model |
|---|---|---|
| Spin variables | Discrete (±1) | Continuous (quantum operators) |
| Energy invariance | Invariant under flipping every spin in the system | Invariant under applying the same rotation around the unit sphere |
| Discrete symmetry | Z2 symmetry | None (continuous symmetry) |
| Hamiltonian | $$H\text{Ising} = -J \sum{\langle ij\rangle} s^zi s^zj$$ | $$H\text{Heisenberg} = -J \sum{\langle ij\rangle} \hat{S}i \cdot \hat{S}j$$ |
| Phase transition | Described by mean-field theory when the number of dimensions is above four | No specific method for phase transition |
In summary, the main differences between the Ising and Heisenberg models are the treatment of spin variables (discrete vs. continuous), energy invariance, and the presence or absence of discrete symmetry. The Ising model is a semi-classical model that treats spins as discrete variables, while the Heisenberg model treats spins as quantum operators. The latter model is able to account for the spin in every direction, as opposed to the Ising model, which is limited to parallel or antiparallel orientations.
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