Heisenberg model: Difference between revisions

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:<math>H = -J{\sum}_{\langle i,j\rangle}\mathbf{S}_i \cdot \mathbf{S}_{j}</math>
:<math>H = -J{\sum}_{\langle i,j\rangle}\mathbf{S}_i \cdot \mathbf{S}_{j}</math>


where the sum runs over all pairs of nearest neighbour spins, <math>\mathbf{S}</math>, and where <math>J</math> is the coupling constant.
where the sum runs over all pairs of nearest neighbour spins, <math>\mathbf{S}</math>, and where <math>J</math> is the coupling constant.  
The classical model is known to have a phase transition in three or higher spacial dimensions, and the ferromagnetic (<math>J>0</math>) and antiferromagnetic (<math>J<0</math>) 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 <math>S</math> and the sign of <math>J</math>.
 
==See also==
==See also==
*[[Ising Models]] (n=1)
*[[Ising Models]] (n=1)

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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 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

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)