Set of supplementary notes.

9.2. MAGNETISM 141
Multiplying the Curie law for local moments with this fraction k B T/E F transforms it, up to a
constant of order one, into the Pauli expression for metals.
Now let us include in a very simple fashion the effect of interactions. The Stoner-Hubbard
model, which provides arguably the simplest way forward, includes an energy penalty U for
lattice sites which are doubly occupied, i.e. they hold both an up- and a down-spin electron.
Ĥ int = ∑ sitesi
Un i↑ n i↓ , (9.27)
If we treat this interaction in a mean-field approximation, it leads to a shift of the energies
of the two spin bands
ɛ k↑ = ɛ k + U ¯n ↓ − µ 0 µ B H
ɛ k↓ = ɛ k + U ¯n ↑ + µ 0 µ B H (9.28)
We see that the presence of spin-down electrons increases the energy of the spin-up electrons
in the same way as a magnetic field pointing down would. Conversely, spin-up electrons cause
the energy of spin-down electrons to increase in the same way as a magnetic field pointing
up. The interactions between the electrons appear formally in the same way as an additional
magnetic field. This so-called exchange field is not physical in the sense that it could deflect
a compass needle, it is a book-keeping device to handle the effects of the Coulomb interaction
between the electrons.
With the same approximation as before - that the density of states can be taken to be a
constant, we can then self-consistently determine the average spin density
N
V (¯n ↑ − ¯n ↓ ) = [U(¯n ↑ − ¯n ↓ ) + 2µ 0 µ B H] 1 2 g v(E F ) . (9.29)
The magnetisation is M = µ B (n ↑ − n ↓ ) which then gives us the static spin susceptibility
χ σ = µ 0
µ 2 B g(E F )
1 − Ug(E F )
2
. (9.30)
Here, g denotes the density of states per atom, in contrast to g v = N g, which is the density
V
of states per unit volume. In comparison to the non-interacting case, the magnetic susceptibility
is enhanced, and will diverge if U is large enough that the Stoner criterion is satisfied
Ug(E F )
2
which marks the onset of ferromagnetism in this model.
> 1 , (9.31)
The Stoner criterion for ferromagnetic order has a very fundamental interpretation: because
the density of states per atom is of order 1
F
, the Stoner criterion expresses the balance between
interaction energy U and kinetic energy E F . If the kinetic energy of the electrons is high, then
they will not form a magnetically ordered state. If, on the other hand, the interaction strength
is higher than the kinetic energy, then the electron system can lower its energy by aligning its
spins. Variations on this criterion surface in many other areas of correlated electron physics.