Magnetic properties of the lattice Anderson model

Abstract

We perform exact diagonalization studies of the one‐dimensional lattice Anderson model for various clusters. The ground state energy and spin–spin correlation functions are calculated as functions of Hubbard U, hybridization V, and f level occupancies. For the symmetric case, we compare our results with weak and strong coupling perturbation theory.

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Magnetic properties of the lattice Anderson model
H. Q. tin
Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
H. Chen
Cray Research, Inc., Minneapolis, Minnesota 55121
J. Callaway
Department of Physics, Louisiana State University, Baton Rouge, Louisiana 70803-4001
We perform exact diagonalization studies of the one-dimensional lattice Anderson model for various
clusters. The ground state energy and spin-spin correlation functions are calculated as functions of
Hubbard U, hybridization V, and f level occupancies. For the symmetric case, we compare our
results with weak and strong coupling perturbation theory.
We study the lattice Anderson model on finite, one-
dimensional (lD), lattices in this paper. The model is defined
by the Hamiltonian
lattices are large enough. As we will show in the following,
fortunately, we are able to achieve accurate extrapolation
based on our finite lattice results.
One can show, without much difficulty, that, for a 1D
tight binding band, the ground state energy per site on an N
site lattice with total electron number N,=4n
H=-tC (c~&+,,+hc)+EfC ntia+uC rlfiplfil
io
ia
i
is
+vC (cTJic+hc) P
(1)
ia
$=i q e(k) = -G
sinj T) tanzNj ,
(3)
where ci&ci,)
erators for electrons in d and f orbitals on site i with spin a;
and npg=ficfiw. The d electrons hop with amplitude t, thus
forming a conduction band ek. The f electrons hybridize
with d electrons with amplitude V, and have the usual Hub-
bard interaction U between spin up and down f electrons on
the same site. The site energy E, defines the relative position
of f states with respect to the Fermi energy of the d elec-
trons. An important special case is the so-called symmetric
Anderson model in which Ef= - U/2.
When U=O, one can diagonalize Eq. (1) easily. Two
bands are obtained with energies
and fio(fi,) are creation (annihilation) op-
while for the infinite system with density NJN it is
E,
-=-;
N
2t 2t
T
2nrr
-
N
sin kf=--
sin
( i
’
(4)
where kf= (7r/2)(N,/N)
finite lattice results for N, = 4n give an upper bound.
With total electron number N, = 4n + 2, the ground state
energy per site is
is the Fermi wave number. Thus,
E4n+2
-=-
N
2
2 e(k)
N,
Ei=gek+Ef+. \I(ek-E,)2+4vZ] ,
(2)
in which ek = - 2t cos(k) is the d electron band energy. A
characteristic quantity in the Anderson lattice model is the
gap A that separates the two bands. This is given for U=O by
2A=E&,-Ei=
,r= dm’-2t.
ratio U/A defines weak and strong coupling regions.
This model has been studied for many years in connec-
tion with valence fluctuation problems and an extensive lit-
erature exists.lm3 Several approximation techniques have
been applied to deal with many-body aspects of this model
including renormalization group analysis,4 the Gutzwiller
variational approach,5-7
expansion,‘p9
and perturbation theory.“‘”
tions have also been carried out.11-r9
In this paper we use an exact diagonalization method to
study the ground state properties of model of Eq. (1) for
lattices with N =2,4, 6, and 8. We are restricted by computer
memories so that we are unable to go beyond N=8. In order
to obtain reliable answers in the thermodynamic limit, finite
size extrapolation is needed and one may wonder if these
The magnitude of the
2t
T
2nrr
N
n-
i i
TIN
=-- sin -
+z sin(r/N) ’
(5)
while for the infinite system with density N,IN it is
E,
x=-T
2t 2n7r 7r
sin F+g
i
.
i
(6)
the large-orbital-degeneracy l/N
Numerical calcula-
Thus finite lattice results for N, = 4n + 2 give a lower bound.
For the symmetric Anderson lattice model, the oscilla-
tion between lattice size N= 4n and N=4n
present, as we found numerically. Therefore, we can use data
obtained from N= 4n and 4n + 2 to set up upper and lower
bounds for infinite system results. This approach can also be
applied to other quantities to be discussed below. Hence we
are able to obtain results for infinite systems with reasonable
accuracy. We also tried other extrapolation schemes by as-
suming asymptotic forms of calculated quantities.
We have performed exact diagonalization calculations
for the symmetric case with t=l.O, V=OS (hence A=O.12),
and U varying from 0 to 4, i.e., from weak coupling to strong
coupling, in the l/2-filled sector. Our results for the ground
state energy per site are shown in Fig. 1. The estimated errors
in extrapolations are within the size of the symbol.
+ 2 is still
J. Appl. Phys. 75 (lo), 15 May 1994
0021-8979/94/75(10)/7041/3/$8.00
0 1994 American institute of Physics 7041
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Page 2

FIG. 1. The ground state energy per site as function of U for the symmetric
Anderson lattice model with r=l, V=OS. The dashed line is the weak
coupling perturbation result from Eq. (7), and the solid line is the strong
coupling perturbation result from Eq. (9). The squares are the exact diago-
nalixation results and the uncertainties in the infinite system extrapolation
are within the size of the symbol.
FIG. 2. Square of the f orbital local moment as function of U for the same
case as in Fig. 1.
2-1 4u
mfz-2 +JqT
(12)
According to perturbation theory,““’
case in the weak coupling limit where U/A is small, the
ground state energy per site is
for the symmetric
E(U)=; 2 E;-;
k
with
U2
-N3
(7)
ui=$l+ei/(ei+4V2)1’2] ,
vi=gl-ei/(ei+4V2)“2] .
(8)
(9)
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. For weak
coupling we have
while for strong coupling we have
l-f(ek)
(U/2+ek)’
m +-q
c
k *
(13)
We show these results as the dashed line and solid line in
Fig. 2, along with data points obtained from our exact diago-
nalization studies. Again, estimated errors in extrapolations
are within the size of the symbol.
From Fig. 2, it is evident that weak coupling perturba-
tion is invalid for U/A>8 and strong coupling perturbation
gives bad answers for U/A<20.
ergy calculations, both perturbation results are quite good
over a much wider range of U/A. This behavior occurs in
many highly correlated systems and our results show that a
good ground state energy is not a sufficient criterion for the
validity of an approximation scheme.
As U increases from zero, the effective hybridization is
expected to be reduced due to correlations. To measure this
reduction we calculate the matrix element (c~rfi,+hc)
plot the ratio with respect to its U=O value in Fig. 3. The
weak and strong coupling results can be obtained from
$E/W.
We see that weak coupling perturbation is pretty
good for UJAh<8 again, but strong coupling perturbation is
not good although qualitatively reasonable. This can be un-
derstood by observing that in weak coupling the perturbation
was carried out to second order in U, while in strong cou-
pling the perturbation was only carried out to first order in
l/U.
Now we consider the nonsymmetric
EfZ -U/2. As Ef varies, one enters different physical re-
gions. Basically, there are five regions.14 (1) Maximally oc-
cupied f states, where the f orbitals are doubly occupied
(nf-2,
This region is characterized roughly by
Ef<-U-2t. (2) The
first
-U-2t<Ef<-U-t.
The f occupation in this region
One may note that for en-
and
case in which
n, -0).
mixed valence region,
7042
J. Appl. Phys., Vol. 75, No. 10, 15 May 1994
Lin, Chen, and Callaway
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Page 3

0.2
0
,,,I I I I I I1 I I I I*
3
I I I
0
1
z
4
FIG. 3. The ratio of the hybridization matrix element (c!,fi,+hc)
lJ=O value as a function of U for the same case as in Fig. 1.
to its
changes between 1 and 2. (3) The Kondo region, in which
one has nf- 1, n,-1. Its range is roughly -U-
(4) The second mixed valence region, t<Ef
cupation in this region changes between 1 and 0. (5) The
region of empty f levels (nf-0),
and c electron occupations and clearly demonstrates these
five regions.
t<Ef<t.
< 2t. The f oc-
Ef>2t.
Figure 4 shows f
-10 -5
0
5
El
FIG. 4. The electron occupation off orbitals (squares) and c orbitals (dia-
monds) as a function of E, for U=S.O, t=l.O, and V=O.l.
In summary, we have studied some ground state proper-
ties of the one-dimensional Anderson lattice model by an
exact diagonalization technique. We have performed finite
size analysis and estimated results for infinite systems. Com-
parison with perturbation theory for the symmetric Anderson
lattice model showed that weak coupling calculations are
quite accurate for U/A<8 and strong coupling calculations
are reasonably satisfactory for U/A>20.
metric case, we found that five distinct regions exist depend-
ing on the site energy Ef. The different types of physical
behavior illustrated in Fig. 4 have been found in small sys-
tem calculations for other geometries,14 and are presumably
characteristic of the strong interaction-weak hybridization
sector of the model generally, independent of the restriction
of the present calculations to one dimension. Within the
Kondo regime where n, and nf are close to 1, our finding
that the energy and other physical quantities vary smoothly
between weak and strong coupling limits is probably also
independent of dimensionality.
Some of the computations were performed at the Na-
tional Center for Supercomputing Applications, University of
Illinois at Urbana-Champaign, and at National Energy Re-
search Supercomputer Center, Lawrence Liver-more National
Laboratory. We are grateful for their support. The portion of
the work performed at LSU was supported by the National
Science Foundation under Grant No. DMR 91-20166.
For the nonsym-
‘P Fulde, J. Keller, and G. Zwicknagel, in Solid Srare Physics, edited by H.
Ehrenreich and D. Turnbull (Academic, New York, 1988), Vol. 41, p. 1,
and references therein.
‘D. M. Newns and N. Read, Adv. Phys. 36, 799 (1987).
3N. E. Bickers, Rev. Mod. Phys. 59, 845 (1987).
4H. R. Krishna-Murthy, J. W. Wilkins, and K. G. Wilson, Phys. Rev. B 21,
1003 (1980).
‘B. H. Brandow, Phys. Rev. B 33, 215 (1986).
6T. M. Rice and K. Ueda, Phys. Rev. Lett. 55, 995 (1985); 55, 2093(E)
(1985); Phys. Rev. B 34, 6420 (1986).
7C. M. Varma, W. Weber, and L. J. Randall, Phys. Rev. B 33, 1015 (1986).
*N. Read, D. M. Newns, and S. Doniach, Phys. Rev. B 30, 3841 (1984).
9A. J. Minis and P A. Lee, Phvs. Rev. B 35, 3394 (1987).
._. _. _-
IoK. Yamada and K. Yoshida, in Proceedings of the ‘Third Taniguchi Sym-
posium, Mount Fuji, Japan, 1980, edited by T. Moriya (Springer, Berlin,
1981), p. 210.
“R. Blankenbecler, J. R. Fulco, W. Gill, and D. J. Scalapino, Phys. Rev.
Lett. 58, 411 (1987).
“J. E. Gubematis, J. E. Hirsch, and D. J. Scalapino, Phys. Rev. B 35, 8478
(1987); J. E. Gubematis, ibid. 36, 394 (1987).
r3R M. Fye, J. E. Hirsch, and D. J. Scalapino, Phys. Rev. B 35, 4901
(l-987); 44,7486 (1991).
r4J. Callaway, D. P. Chen, D. G. Kanhere, and P. K. Misra, Phys. Rev. B 38,
2583 (1988); Y. Zhang and J. Callaway, ibid. 38, 641 (1988); J. Callaway,
J. W. Kim, L. Tan, and H. Q. Lin, Phys. Rev. B 48, 11545 (1993).
*‘A. Reich and L. M. Falicov, Phys. Rev. B 34, 6752 (1986).
16J. A. White, Phys. Rev. B 46, 13905 (1992). . ..-. ..-
17P. Santini, L. Andreani, and H. Beck, Phys: Rev. B 47, 1130 (1993).
“K. Ueda J Phys. Sot. Jpn. S&3465 (1989); K. Yamamoto and K. Ueda,
ibid. 59,‘3&4 (1990).
19R. Julhen and R. M. Martin, Phys. Rev. B. 26, 6173 (1982).
J. Appl. Phys., Vol. 75, No. 10, 15 May 1994 Lin, Chen, and Callaway 7043
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