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 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== | ==See also== | ||
*[[Ising Models]] (n=1) | *[[Ising Models]] (n=1) | ||
Latest revision as of 07:27, 9 April 2021
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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 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)