# Ising-Heisenberg model

1. Jul 17, 2013

### LagrangeEuler

Hamiltonian
$H=-J\sum_{\vec{n},\vec{m}}(S_{\vec{n}}^xS_{\vec{m}}^x+S_{\vec{n}}^yS_{\vec{m}}^y+aS_{\vec{n}}^zS_{\vec{m}}^z)$
If $a>2$ is that behaves like Ising model? For example in $2d$ lattice is critical temperature of that system $T_c \approx 2.269$?

2. Jul 17, 2013

### Jolb

Ising and Heisenberg models only have nearest-neighbor interactions. Your model doesn't restrict long-range interactions and thus the model is quite different.

3. Jul 17, 2013

### LagrangeEuler

Ok and for nearest neighbours? I thought of that case when I right.

4. Jul 18, 2013

### Andy Resnick

I don't recognize that Hamiltonian- The Heisenberg model of ferromagnetism is H = -2J $\sum s_{n}\cdot s_{m}$, where the sum is over nearest neighbors. Any Hamiltonian that is invariant under Z_2 has Ising symmetry (the Ising model is equivalent to a Z_2 clock model), but the primary difference between Heisenberg and Ising models is that the Ising model treats spin classically (the spin variables do not obey quantum commutation relations). Both models describe a high-temperature paramagnetic phase and a low-temperature ferromagnetic phase.

Another link is that the classical Heisenberg model is invariant under O_3, while the Ising model is invariant under O_1 (with the caveat that rotations in 1D are not defined).