Derivation of the Heisenberg Hamiltonian

Obtaining the approximated energy spectrum for an N=2 hydrogen "crystal", by solving the Schrodinger equation which includes electronic momentum operators and electrostatic interactions only, and demanding the antisymmetry of the wave functions, is rather simple compared with the case of a real crystal of N=O(1023) atoms. This is the reason why the exchange Hamiltonian that was derived for the hydrogen molecule is introduced. It can provide a simpler mathematical tool in the case of real crystals, rather then dealing with the original Hamiltonian. The new mathematical tool should be some kind of a compact operator that will lead to a similar energy spectrum that is obtained by the original Hamiltonian.
We assume that a crystal is formed in a similar way to the hydrogen molecule i.e. it consists of individual atoms, each of which contains one electron in the ground state. As we have seen in the case of the hydrogen molecule, the original Hamiltonian is equivalent to the exchange Hamiltonian, (under the appropriate considerations, namely for low-lying energy levels T<<TC). The assumption is that one should obtain the correct physical representation of the energy spectrum of a magnetically ordered crystal near the ground state, if the original Hamiltonian is replaced by the sum of the exchange hamiltonians for all the pairs of atoms:

This operator is called the Heisenberg Hamiltonian (in the old literature it is known as the Heisenberg-Dirac Hamiltonian), where Si and Sj are the atomic spins at the ith and jth lattice sites, rij is the radius vector joining the ith and jth sites, and J(rij) is called the exchange integral (or coupling constant) for the ith and jth atoms. The sum is multiplyed by 1/2 in order to avoid double summation, but some write it without the 1/2 factor; it's alright as long as one remembers that the summation is over i and j so that i<j.

One may note that this Hamiltonian was derived from the exchange Hamiltonian for the hydrogen molecule i.e. for Si=1/2 only. But since it predicts correctly the ground state of any ferromagnet model and provides a good description of the energy spectrum near the ground state, the Heisenberg Hamiltonian is considered as leading to physically reasonable results at sufficiently low temperatures also in the case of any arbitrary S. By sufficiently low temperature we mean at T much smaller than the critical temperature (which above it the magnrtic order vanishes).

We note that J(rij) falls off exponentially with increasing distance between the ith and jth atoms, since it depends on the degree of overlap of the atomic wave functions. Thus it is of appreciable size only for ith and jth atoms that are nearest neighbours. Then the Heisenberg Hamiltonian is simply reduced to summation over nearest neighbours and J(rij) ≈ ξe2/a , where a is the lattice constant and ξ is a numerical parameter so that
ξ = O(0.1). ξ is determined by the degree of overlap of the neighbouring atoms' wave functions.

In general, the exchange integral- J - has different magnitudes in different directions, as a consequence of variation of overlap between assymetric atomic wave functions in different orientations. Hence the atomic moments may be coupled more strongly in some directions than the others.

Since the Heisenberg Hamiltonian is the sum of spin scalar products, if J(rij)>0 for every i and j then, under energy considerations, all the atomic spins will favore parallel alignment. If however J(rij)<0 for every i and j then, all the atomic spins will favore antiparallel alignment. Hence for ferromagnets J(rij)>0 and for antiferromagnets J(rij)<0 .
One may note that in the case of ferromagnets the above Heisenberg Hamiltonian predicts the parallel alignment of atomic spins, but dosen't specify a preferential direction of alignment. Thus in such case the Heisenberg Hamiltonian is called the isotropic Heisenberg Hamiltonian. However, in real crystals the isotropy is broken by other magnetic effects that were neglected in the original Hamiltonian, like the dipolar interactions and spin-orbit coupling. Also an external magnetic field can be applied so that the isotropy is broken by introducing a certain direction (the direction of the field). Then the anisotropic Heisenberg Hamiltonian is written as:

where B is the external field.
[The interaction energy of a magnetic moment m with a magnetic field B is U=-m·B . If the magnetic moment is due to spin effect, then m=-gμBS.] Usually B is chosen so that it defines the z-axis direction. Then the Heisenberg Hamiltonian may be written as:

The magnetic effects that were neglected at the original Hamiltonian can be taken into account by suitable definition of B.