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# Heisenberg model

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

where the sum runs over all pairs of nearest neighbour spins, , and where 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 () and antiferromagnetic () 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 and the sign of .

## See also[edit]

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

## References[edit]

- A. C. Hewson and D. Ter Haar "On the theory of the Heisenberg ferromagnet", Physica
**30**pp. 271-276 (1964) - 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) - F. Lado and E. Lomba "Heisenberg Spin Fluid in an External Magnetic Field ", Physical Review Letters
**80**pp. 3535-3538 (1998) - 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)