Heisenberg model

The Heisenberg model is the n=3 case of the n-vector model. The Hamiltonian is given by

$H=-J{\sum }_{\langle i,j\rangle }\mathbf {S} _{i}\cdot \mathbf {S} _{j}$

where the sum runs over all pairs of nearest neighbour spins, $\mathbf {S}$, and where $J$ is the coupling constant. The classical model is known to have a finite-temperature phase transition in three or higher spacial dimensions, and the ferromagnetic ($J>0$) and antiferromagnetic ($J<0$) share essentially the same physics. The quantum version differs greatly, and even the one-dimensional case has a rich variety of phenomena depending on the spin number $S$ and the sign of $J$.

See also

• Ising Models (n=1)
• XY model (n=2)
• Mermin-Wagner theorem

References

1. A. C. Hewson and D. Ter Haar "On the theory of the Heisenberg ferromagnet", Physica 30 pp. 271-276 (1964)
2. T. M. Giebultowicz and J. K. Furdyna "Monte Carlo simulation of fcc Heisenberg antiferromagnet with nearest- and next-nearest-neighbor interactions", Journal of Applied Physics 57 pp. 3312-3314 (1985)
3. F. Lado and E. Lomba "Heisenberg Spin Fluid in an External Magnetic Field ", Physical Review Letters 80 pp. 3535-3538 (1998)
4. E. Lomba, C. Martín and N.G. Almarza "Theory and simulation of positionally frozen Heisenberg spin systems", The European Physical Journal B 34 pp. 473-478 (2003)