Statistical Mechanics - Physics at Oregon State University

8.1. INTRODUCTION. 159
In a first approximation the strength of the exchange interaction depends on
the distance between lattice sites only. We denote the lattice sites by Ri and
have
H = −
J(| Ri − Rj|) Si • Sj
(8.3)
i (8.4)
Unfortunately, these are not eigenstates of of the Hamiltonian, since the operators
Si are able to lower and raise the z quantum number. We have
Si • Sj = S +
i S− j + S− i S+ j + SizSjz (8.5)
Because we do not know what to do with the first two terms, we ignore
them. The Hamiltonian is now
H = −
J(| Ri − Rj|)SizSjz
i. The original
energy expression contains the product of spin-components perpendicular to the
quantization axis, but leaving out these terms, like we did above, corresponds
to assuming that they average out to zero. Therefore, the energy eigenvalue of
a given quantum state |σ1, · · · , σN > is
E {σ1, · · · , σN} = −
i