Magnetic properties of the lattice Anderson model - Department of ...

FIG. 1. The ground state energy per site as function of U for the symmetric FIG. 2. Square of the f orbital local moment as function of U for the same
Anderson lattice model with r=l, V=OS. The dashed line is the weak case as in Fig. 1.
coupling perturbation result from Eq. (7), and the solid line is the strong
coupling perturbation result from Eq. (9). The squares are the exact diagonalixation
results and the uncertainties in the infinite system extrapolation
are within the size of the symbol. 2-1
mfz-2
4u
+JqT (12)
According to perturbation theory,““’ for the symmetric
case in the weak coupling limit where U/A is small, the
ground state energy per site is
with
E(U)=; 2 E;-;
k
-N3
U2
ui=$l+ei/(ei+4V2)1’2] ,
vi=gl-ei/(ei+4V2)“2] .
In the strong coupling limit where U/A is large, the ground
state energy per site is
E(U)=-; +; c e&e,,-? T ;;;:;j 7
k
(10)
with f(ek) being the zero temperature Fermi factor.
We show weak coupling results as the dashed line and
strong coupling results as the solid line in Fig. 1. Note that
there is a singularity in strong coupling limit as ZJ+O
[roughly In(U)] but it is hard to see from Fig. 1.
As U increases from 0 to ~0, electrons tend to localize in
the f orbitals and form local moments. We measure these
moments by m& defined by
4z=((nfT-nf1)2) , (11)
which varies from 1 when U =O, to 1 when U=m.
coupling we have
(7)
(8)
(9)
For weak
while for strong coupling we have
l-f(ek)
m +-q c
k (U/2+ek)’ *
(13)
We show these results as the dashed line and solid line in
Fig. 2, along with data points obtained from our exact diagonalization
studies. Again, estimated errors in extrapolations
are within the size of the symbol.
From Fig. 2, it is evident that weak coupling perturbation
is invalid for U/A>8 and strong coupling perturbation
gives bad answers for U/A