Re: Derivation for the Heisenberg model for interaction between spins in a crystal

On 2007-01-29, dhaile@xxxxxxxxxxx <dhaile@xxxxxxxxxxx> wrote:
Hi every one, I am wondering if there is a derivation for the
Heisenberg model for interaction between spins in a crystal. I will
appreciate your help

Ashcroft & Mermin, _Solid State Physics_, Chapter 32.

Consider only two spins (electrons, nuclei, or what not). Somehow you
take their total Hamiltonian and find its energy eigenstates. Freeze all
degrees of freedom except the different spin states. What remains is a
4-dimensional Hilbert space (assuming that both spins have magnitude
1/2). There are two mutually commuting operators, S_1 and S_2 (these are
actually vector valued operators, whose components obey the usual
angular momentum commutation relations), representing the separate
spins. There is also the total spin operator S = S_1 + S_2.

When squared, the spin operators become

S_i^2 = 1/2 (1/2+1) = 3/4
S^2 = S_1^2 + S_2^2 + 2 S_1 S_2
= 3/2 + 2 S_1 S_2

One possible basis for the spin Hilbert space consists of

the singlet: |^v> + |v^>, and
the triplet: |^^>, |^v> - |v^>, |vv> .

Rotational invariance insists that each of these states is an energy
eigenstate and that each state from the triplet has the same energy
eigenvalue. But the operator S^2 has very similar spectral properties.
It has eigenvalue 0*(0+1) = 0 on the singler subspace and eigenvalue
1*(1+2) = 3 on the triplet subspace. In other words, we can write the
effective Hamiltonian on the spin Hilbert space by shifting and scaling
the operator S^2, or S_1 S_2, since they can be expressed in terms of
each other. When you do the arithmetic, you find that

H_spin = (E_s + 3 E_t)/4 - (E_s - E_t) S_1 S_2,

where E_s and E_t are the singlet and triplet energy eigenvalues,
respectively.

Note that S_1 S_2 is precisely the Heisenberg interaction term. The
above derivation was only for a pair of spins. When there are many
spins, it is reasonable to generalize to

H_spin = sum_<ij> J_<ij> S_i S_j + const,

where <ij> are the pairs of spins whose interaction cannot be neglected.
There are approximations that are necessary to be assumed for this
generalization to hold. A&M discuss them only very briefly and give a
reference to a more in depth one. I think the following article
discusses the same topic:

Conyers Herring, Critique of the Heitler-London Method
of Calculating Spin Couplings at Large Distances
Rev. Mod. Phys. 34, 631 - 645 (1962)
URL: http://link.aps.org/abstract/RMP/v34/p631
DOI: 10.1103/RevModPhys.34.631

Hope this helps.

Igor

.

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