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Heisenberg model

2021-04-09T06:27:10Z

60.111.245.26: small correction.

{{Stub-general}}
The '''Heisenberg model''' is the n=3 case of the ''n''-vector model. The [[Hamiltonian]] is given by

:<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.
The classical model is known to have a finite-temperature 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==
*[[Ising Models]] (n=1)
*[[XY model]] (n=2)
*[[Mermin-Wagner theorem]]
==References==
#[http://dx.doi.org/10.1016/0031-8914(64)90223-X A. C. Hewson and D. Ter Haar "On the theory of the Heisenberg ferromagnet", Physica '''30''' pp. 271-276 (1964)]
#[http://dx.doi.org/10.1063/1.335131 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)]
#[http://dx.doi.org/10.1103/PhysRevLett.80.3535 F. Lado and E. Lomba "Heisenberg Spin Fluid in an External Magnetic Field ", Physical Review Letters '''80''' pp. 3535-3538 (1998)]
#[http://dx.doi.org/10.1140/epjb/e2003-00246-8 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)]
[[Category: Magnetism]]
[[Category: Models]]

Heisenberg model

2021-04-09T06:26:15Z

60.111.245.26: Added a very minimal ad brief explanation of the features of the model.

{{Stub-general}}
The '''Heisenberg model''' is the n=3 case of the ''n''-vector model. The [[Hamiltonian]] is given by

:<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.
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==
*[[Ising Models]] (n=1)
*[[XY model]] (n=2)
*[[Mermin-Wagner theorem]]
==References==
#[http://dx.doi.org/10.1016/0031-8914(64)90223-X A. C. Hewson and D. Ter Haar "On the theory of the Heisenberg ferromagnet", Physica '''30''' pp. 271-276 (1964)]
#[http://dx.doi.org/10.1063/1.335131 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)]
#[http://dx.doi.org/10.1103/PhysRevLett.80.3535 F. Lado and E. Lomba "Heisenberg Spin Fluid in an External Magnetic Field ", Physical Review Letters '''80''' pp. 3535-3538 (1998)]
#[http://dx.doi.org/10.1140/epjb/e2003-00246-8 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)]
[[Category: Magnetism]]
[[Category: Models]]