TOPICS
Search

Quantum Heisenberg Ferromagnet

A quantum Heisenberg ferromagnet is a quantum spin model with isotropic nonnegative exchange couplings. For a finite set Lambda subset Z^d and spin S=1/2, 1, 3/2, ..., its Hilbert space is the tensor product of copies of C^(2S+1) indexed by Lambda. The spin operators S_x=(S_x^1,S_x^2,S_x^3) act on the factor at x, satisfy the commutator relations [S_x^1,S_x^2]=iS_x^3 and its cyclic permutations, and have (S_x^1)^2+(S_x^2)^2+(S_x^3)^2=S(S+1)I. Operators at distinct sites commute. A finite-volume quantum Hamiltonian is

H_(Lambda,h)=-1/2sum_(x,y in Lambda)J(y-x)S_x·S_y-hsum_(x in Lambda)S_x^3,

where J(z)=J(-z)>=0, J(0)=0, and h is an external field. The model is distinct from the classical continuum Heisenberg ferromagnet equation.

Bloch's law (Bloch 1930) predicts a leading magnetization deficit proportional to T^(3/2) in three dimensions.

See also

Haldane Conjecture, Heisenberg Ferromagnet Equation, KMS Condition, Quantum Hamiltonian, Tensor Product

Explore with Wolfram|Alpha

References

Bloch, F. "Zur Theorie des Ferromagnetismus." Z. Phys. 61, 206-219, 1930. https://doi.org/10.1007/BF01339661.

Cite this as:

Weisstein, Eric W. "Quantum Heisenberg Ferromagnet." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/QuantumHeisenbergFerromagnet.html

Subject classifications