The Heisenberg model in magnetism

1 The Heisenberg model 

1.1 Definition of the model 

The model we will focus on is called the Heisenberg model. It has the following Hamiltonian:

 H = 1/ 2 X i,j i6=j JijSi · Sj . (1)

Here i and j refer to sites on a lattice. The model can be defined on any lattice, but for concreteness (and to keep things simple) we will here limit ourselves to a hypercubic lattice in d dimensions (d = 1, 2, 3).1,2 The Si are spin operators which live on the lattice sites. Spin components on the same lattice site obey the standard angular momentum commutation relations

[S α j , Sβ j ] = i X γ αβγS γ j (α, β, γ = x, y, z) (2)

 and spins on different sites commute with each other. The spin operators all have spin S, i.e. the operators S 2 i have eigenvalue S(S +1) where S is an integer or half-integer. The spin interaction in (1), which is of the form Si · Sj , is called an exchange interaction, and the coefficients Jij are called exchange constants. We will make the simplifying assumption (which is often realistic) that the spin interactions are negligible between spins that are not nearest-neighbors, i.e. Jij is nonzero only if i and j are nearest-neighbor lattice sites, in which case we further assume Jij = J, where J is a constant. There are then two different cases to consider, J < 0 and J > 0. For J < 0 the interaction energy of two spins favors them to be parallel; this is the ferromagnetic case. For J > 0 antiparallel orientation is instead favored; this is the antiferromagnetic case.

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