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Heisenberg model

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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[edit]


  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)