Heisenberg model

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{S}} , and where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J} is the coupling constant. The classical model is known to have a phase transition in three or higher spacial dimensions, and the ferromagnetic (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J>0} ) and antiferromagnetic (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} and the sign of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J} .

See also

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

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)