Singlet Ground State of the Localized dElectrons Coupled with Conduction Electrons in Metals
Progress of Theoretical Physics
Singlet Ground State of the Localized dElectrons Coupled with Conduction Electrons in Metals
Isamu OKADA 0
Kei YOSIDA 0
0 Institute for Solid State Physics, University of Tokyo , Roppongi, Tokyo
The theory of the singlet ground state for the sd exchange model is extended to the core of delectrons with orbital degeneracy. The effective sd Hamiltonian is derived from the extended Anderson Hamiltonian by the SchriefferWolff canonical transformation. On the basis of the effective Hamiltonian, the groundstate wave function and the groundstate energy are calculated. The anomalous part of the groundstate.energy is given by Dexp[N/(2l+1)pJ], independently of the delectron number, 'fhen the Hund coupling is neglected compared with the effective sd interaction. This bindirlg energy is much larger than that for a localized selectron because of an extra factor of 1/(2l+1) in the exponent. This large value is caused mainly by the orbital quenching. For a more realistic case in which the Hund coupling is larger than the sd interaction, it reduces to a smaller value of Dexp[(2l+1)N/pJ] for a halffilled shell in which orbital exchange does not exist. This value is due purely to the spin quenching. Qualitative discussion is given about the spin quenching and orbital quenching on the effective Hamiltonian derived for n=2l, n being the number of delectrons.

the impurity atom.
at, and ak, are the creation and annihilation operators for a
conduction electron with wave vector k and spin s. This form of the sd exchange
Hamiltonian describes swave scattering of the conduction electrons by the localized
spin.
Therefore, this Hamiltonian is a correct expression for the local sorbital
with spin the magnitude of which is 1/2.
For local delectrons, dwave scattering IS important and 111 this case dpart
of sd exchange integral J(k', k)
should be taken. In (1· 2), P 1 and Y1m are respectively Legendre polynomials
and the spherical harmonics and !Jk represents the direction of wave vector k.
If lpart of (1· 2) is used in place of J in (1·1), the sd exchange Hamiltonian
for dwave scattering is obtained as
 2JN.t 2.l+11 2R3 "~".kk' "£...".J akt'Lms'tTs'saklms• S .,
3 kk' mss'
where aLms represents the creation operator of the ·electron whose wave function
is given by the spherical l, mwave
R being the radius of the spherical crystal, j 1 the spherical Bessel function, and
it is related to at.
(1·4)
(1·5)
This form of H.a expressed by (1· 3) may be used for discussing the Kondo
effect of the localized dorbitals of irongroup impurities. Since this Hamiltonian
does not express the orbital states of delectrons explicitly, some ambiguities
remain except for a special case of the halffilled shell, for which (1· 3) repre
sents the proper sd Hamiltonian as will be shown in this paper.
In order to derive the sd effective Hamiltonian which describes the orbital
states fordelectrons explicitly, one should start with the Anderson Hamiltonian9l
extended to the case of local degenerate dorbitals. This is done in the next
section for dorbitals by the use of the SchriefferWolff transformation.10l In § 3
the starting groundstate wave function in perturbative approach is investigated
for the case in which an impurity atom has one delectrons. The scattering
tmatrix for this effective Hamiltonian is calculated in the approximation that the
most divergent terms are retained in § 4. In § 5 the energy and the wave
function of the singlet ground state are calculated by the perturbational method
developed so far for the usual sd ·exchange Hamiltonian (1·1) .··· This calculation
is extended to the general case in which an impurity atom has n delectroris
in § 6 without taking into account the Hund coupling and the case in which the
Hund coupling is taken account of is discussed in § 7.
(1·2)
(1·3)
§ 2. Effective Hamiltoniap for local delectrons
Here H, and H"' represent respectively the energy of the conduction electrons
and that of the localized delectrons, and Hmix represents the mixing Hamiltonian
expressing mutual transfer between conduction and localized delectrons. at, and
dJ.. 'are the creation operators for conduction and delectrons, respectively. The
second and the third terms in Hd. express the intraCoulomb and exchange interac
tions between delectrons in the impurity atom. This form of interaction which
satisfies rotational invariance in the real space and the spin space was adopted
by Dworin/6l and it can also be expressed as
where nd., Sd. are, respectively, the number and .spin operators of delectrons.
The matrix element Vkm in H,a is given ·by
Vkm = Jv Setkryimp(r)Rd.(r)Y1m(SJ.),r 2drdSJ,
where the impurity pQtential is assumed to be spherically symmetric, anq Rd.(r)
· represents the radial part of the delectron wave function. If one introduces the
mixing matrix element between the sp~erical l, mwave given by (1·4) and the
localized dorbital
Vkm 1s given m terms of vkz as
(2·1)
(2·2)
(2·3)
(2·4)
(2·5)
(2·6)
(2·7)
I. Okada and 1(. Yosida
Here and also in the following, q denotes the degree of degeneracy of delectrons,
namely, q=2(2l+1) =10.
Now, the delectron number in the most stable unperturbed state is assumed
to be nd = n. Then we have
JEn=sd+ nU>sF,
where ep is the Fermi energy and is taken as the origin of energy.
With the use of matrix S obtained by (2 ·12) ~ (2 ·15), the transformed
Hamiltonian is given to second order in vk by
H=H.+Hd+H.d,
H ld.  J "".J [ akt'm's'akms'dmtid.m/s' 1ak/ mts a..kmsUmIt/s'd m'l'J,
2N ••~,:;;,., q
Here those parts which are constant and which change the number of delectrons
(2·9)
(2·10)
(2·11)
(2·12)
(2·13)
(2·14)
(2·15)
(2·16)
(2·17)
(2·18)
Singlet Ground State of tAe Localized dElectrons
are omitted, and the ekdependence of J is also neglected. The effective sd
interaction of (2·17) is chosen so as to exclude the part due to potential scatter
ing which is diagonal with respect to the dstate by subtracting the second
terms.lo), 16)
When the intraexchange integral Ji between delectrons is taken into account,
the calculation becomes somewhat complicated, though practicable, because the
spin multiplicity should be taken care of. So, here, the effective Hamiltonian
for the case in which the impurity atom has one delectron in the unperturbed
state is only written down: as
, H=H.+Ha+H.a,
H att·:
"kJ {Jakl'mt 's'akmsdmt•d m's' + Ja2k'mt/sakmBdmt•'d m's'
k' km'm•'• 2N 2N
 _!i_al'ms'akmsdJ.,,dm'•' _!i_al'msakmsdJ.,,,dm's'},
2N 2N
~=.lvkFI2
2N ·
J2 1 VkF 12
2N
U(ea+ U) Ji'
ea(ea+ UJi) (ea+ U+Ji)
Ji ,
(ea+ UJi) (ea+ U+Ji)
<O'
<O ,
_!i_ = _!_ (~ (2l1) _!i_).
2N q 2N 2N
(2·19)
(2·20)
The first term of the effective sd interaction of (2·19) exchanges both orbital
and spin states, the second only orbital states and the third only spin states.
The second and third terms. seem to be important in distinguishing between
orbital and spin states for the case in which the impurity atom has one localized
delectron, but, for simplicity, we shall neglect Ji compared with U in the sd
effective interaction and take account of it only in Ha in this paper.
§ 3. Zeroapproximation of the groundstate wave function
for the effective Hamiltonian
In this section, we calculate the ground.,$tate wave function of zeroapproxi
mation for the effective Hamiltonian for one' delectron given by (2 · 17), which
will be taken as a starting wave function of the groundstate wave function by
perturbative approach. The zeroapproximate wave function denotes the ground
state eigenfunction for the effective Hamiltonian in the subspace in which only
extra oneelectron is excited above the Fermi sea or extra onehole is excited
below the Fermi sea. In the present case, these two cases give different results
as we. shall see later.
First ~e shall consider the case in which onehole is added to the Fermi
sea. Since the effective Hamiltonian conserves the total angular momentum L.
and spin S., we seek for the solution in the subspace of L.=O and S.=O. Such
a wave function 1s written as
where JF) represents the wave function of the Fermi sea and amplitude'S T~cms
are determined by the Schrodinger equation
Inserting (3 ·1) into (3 · 2) and cutting the states with electronhole pair excitations
besides one hole excitation, we obtain the following eigenvalue equation with
reSpeCt tO amplitudeS TkmB;
This solution expresses the bound state whose binding energy, is given by
E= Dexp [
2N J.
pJ(q'':1/q) . .
For this solution, amplitudes T~cms are i11dE)pendentof m and s ;mdso it repre~ents
a singlet state with respect to both orbital and spin states, th~t is, in this state
orbital and spin moments are quenched.
*> p is connected with the state density Pp for plane waves by p= (3/2kF2R 2)Pp· On the other
hand, lv~cFI 2 =ikF2R2(Jv~cp.ml 2) by (2·8). Therefore, the relation pJ=ppJp holds if one defines ·'p by
replacing Jv~cFI 2 in the expression of J with (v~cFmJ2). ·
We assume that the statedensity for lwave conduction electrohs is a constant
p*> between D and D and zero outside, and introduce Gm, arid X by
Then we have
 1~ 1
2N 1c  e~c E
~ p
2N
l o gE_ 1 .
D X
{  X  J)Gms+J~ Gm's'=O,
\ q m'•'
which leads to the following eigenvalue equation:
Only the first factor of (3 · 7) gives rise to a negativevalue solution for X, ·
(3·1)
(3·2)
(3·3)
(3·4)
(3·5)
(3·6)
(3·7)
(3·8)
(3·9)
Singlet Ground State of .the Localized dElectrons
As shown above, W('! have obtained the singlet bound state by making an
extra hole excited below the Fermi sea coupled with a localized delectron, while
the state with an extra electron added to the Fermi sea coupled with the lo~alized
delectron does not give any singlet bound state. For this case, the zeroapproxi
mate wave function is given by
Inserting this wave function into the Schrodinger equation, we obtain the follow
ing equation for amplitudes Fkms:
Corresponding to (3·4) and (3·5) we introduce Gms and X in which the sum
'illation is now taken over the states above the Fermi energy. Then (3·11) gives
which leads to the eigenvalue equation
The boundstate solution is given by
The amplitudes for this solution satisfy
(x: J)Gms+JG_m•=0,
(3·10)
(3·11)
(3·12)
(3·13)
(3·14)
(3·15)
This state has (2l+ 1)fold degeneracy. If one introduces the intraexchange Ji,
this level splits into two levels having lfold and (l+ 1)fold degeneracies. These
belong to the states of (S, L) = (
1, 1
), (
1, 3
) and those of (S, L) = (0, 0), (
0, 2
)
and (
0, 4
) for l = 2. Thus: in the zeroapproximation, the degeneracy of the ground
state is not removed by the sd interaction in contrast to the case in which a
hole is coupled to the d~electron. This is quite natural, because our Hamiltonian
has not particlehole symmetry from the beginning for' the present case where
one of ten local dstates is occupied.
For the usual sd exchange model, the fact is recognized that if the bound
state solution in its zeroapproximation is degenerate, this boundstate solution
disappears in its final stage in which particlehole excitations are taken into
account by the perturbation calculation starting from the zeroapproximate state.7h 17>
For the present case, also, the bound state solutions having degeneracy in the
starting approximation disappear as particlehole excitations are taken into account
and only the nondegenerate solution survives in the final stage. Before we show
this fact by taking into account higherorder effects, we shall calculate the scat
tering tmatrix for the present effective Hamiltonian (2 ·17) in the approximation
in which the most divergent terms are taken into account.
§ 4.
Scattering t·matrix
In this section we calculate the scattering tmatrix for one occupied delectron
m the most divergent approximation after the Abrikosov theory18> for the usual
sd model. This calculation can be done in a way quite parallel to what Abrikosov
has done for the sd model and we have the following integral equation for the
scattering matrix r (w):
(m1s1o massir(t) imzsa, m,s,)
J
=  tJm1miJmamaiJslsiJszsa +J 1tJmlmz?Jmam,(Jslsztlsas,
2N 2N q
 Jrotds :E {(m!Sb masair(s) im5ss, msSa)(msSs, mssair(s) lm2s2, m,s4)
m 5m 8
B5Be
In the matrix element (m1s1o mass Ir (t) Im2s2, m 4s,), m 1s1 and m2s2 denote orbital and
spin states of the conduction electron and m 3s8 and m 4s4 denote those of the local
delectron. t is defined by p log(D/Iwl). The first and the second terms are
nothing but the matrix element of the effective sd interaction, and the third term
consists of two parts AI and A2• AI represents the contributions from process
(a) and A2 those from process (b) shown in Fig. 1. The matrix element of r
can be put as
(m1s1o masalrlm2sz, m4s4)
(4·1)
(4·2)
ibecause it is expected that r has the same structure as the sd interaction. If
one inserts this form of r into (4 ·1), the following integral equations for r 1, and
·m,s.
(b) Aa
ro are obtained:
Equation (4 · 3) can easily be solved and leads to
rz.(t) =  J+ q st dsr~.(s),
2N o
ro(t) =J 1
 2N q
dsd.(s).
st
o
and
rz.{w) =  J {1 + qpJ l o gD} 1
2N 2N lwl
ro (w) =  _!_rz. (w).
q
rz.(w) and r0 (w) have a pole at lwl =wk=D exp[2NjqpJ] =Tk. This value of Tk
is very big compared with the value of D exp[N/pJ] on account of an extra
factor q = 10 for l = 2 in the denominator in the exponent. With the use of these
results A1 and A2 of the third term on the righthand side of (4·1) are calculated
as
These two A's play the role of integration kernel of integral equation which the
the amplitude for the groundstate wave function satisfies.19l
§ 5. The groundstate wave function. and its energy eigenvalue·
for one local delectron
After the method developed for the sd model, we expand the groundstate
wave function in the perturbation series, starting with the zeroapproximate wave
function (3 ·1) or (3 ·10). For the case in which a hole is coupled with the
=  (__!___) { pJ log _!!____/ (1 +.q pJ log _!!____)}
2N 2N. lwl 2N . lwl
X { ('1 + q1')' iJm1m2iJmamiJs1slJsas• ;;_iJm1m.iJm2maiJs1s.iJs2sa} ,
2
= (_!_) {pJ log _!!____/ ( 1 + q pJ log _!!____J}
2N 2N lwl , 2N lwl'
X { : 2 iJm1m2iJmam.iJs1s2iJsas• + (q : ) iJm1m,iJm2maiJs1s,iJs2sa}.
delectron, this expansion becomes
+ ···.
By inserting (5 ·1) into the Schrodinger equation, the following relations between
the amplitudes r are obtained:
(5·1)
(5·2a)
In (5 · 2a) and (5 · 2b), a single suffix m is used for ms and [ · · ·] m T indicates
the antisymmetrized form:
(5·2b)
(5·3)
From the set of equations (5 · 2a), (5 · 2b), ·· ·, we' derive the equation for the
lowest amplitudes rkmB• This equation for the boundstate solution. in which
Tkms=Tk holds is calculated as
Singl~t Ground, State of the Localized dElectrons
T(ea) {,eaE+ (__{__) 2.E(_!_  q )1 + ···}
, 2N 4,o q D4,~6
( ekE)Tkm= .E rk'm'{(m', miH.alm, m')
1 k'm'
 (1  1) 1 (qJ) 2p ""''rk' 1og
e=ke"k',E
l q2 2N k' D
x [1+ qpJ log· ekek,E + ( qpJ log ekek'E) 2 + ... J.
2N D 2N D
If we here identify the series in pJ/2N in the integration kernel as the geometric
series, (5 · 5) can be written as
·
(ekE)T(ek)=(
1 1,
) qJp so r(e')de'(1 q12 ) 1qa(qJp)a
, _q 2N » 2N
x f»r(e') {log ek~'E /(1 ~~ log ek~'E)}ae'.
E is defined by
E=E.JE'
where LIE is called the normal part of the energy which is obtained by the usual
perturbation calculation. I
The integral equation (5 · 5) can be more generally written as
  1r. }l  11{1r1   1r 2
D2,u · q Ds,2s Ds,1s D1,2s
+···,
where the suffixies k, are abbreviated simply to i, and D 1, 28 is used for energy
denominator (e1  e2 _: e8  E). Carrying out the summations in (5 · 4) with
l~garithmic accuracy or in the most divergent approxim~tion, we obtain
(5·4)
(5·5)
(5·6)
I. Okada and K. Yoshida
where (mh m 8 l A21 m2, m4) is given by (4 · 8). With the use of (4 · 8) for A2 and
relation (3. 15) which says rkm±B=  TkmB =  rk> (5. 8) can be written as
It is noted· that the binding energy IEl 1s enlarged by a factor (2l + 1) in the
denominator of the exponent.
we obtain a boundstate solution for (5 · 5) as
G (e)= G( D) [P(e) ]a[P(O) ]<aP>[P(e)]P.
1 ((3/a) [P(O)]<aP>
The eigenvalue E is determined by the relation qbtained by putting e=  D in
(5·13)'
This relation is satisfied by P(O) = 0, namely,
1=
1 [P(O)]<aP>
1((3/a) [P(O)]'aP>
E= D exp[(2l+~)pJ].
This integral equation can also be derived by direct calculation. The integral
equations (5 · 5) and (5 · 9) · can be solved with logarithmic accuracy by applying
Yoshimori's method.6> Calculation is straightforward. Putting
G(e)= fr(e')de',
P(e) =1~qpJlog
eE
2N D
1
a=1~
q2 '
(5·9)
(5·10)
(5·11)
(5·12)
(5·13)
(5 ·14)
(5·15)
On the other hand, for (5 · 9) we have no boundstate solution except for
l = 0. Defining, as before,
we have the relation, corresponding to (5 · 14), which determines the eigenvalue
E as
Since a' {3' = t >O for l = 0, the binding energy is obtained as
but for l>1, a' {3' becomes negative, and therefore, we have P' (0) = oo which
means E= 0. Thus, we have no boundstate solution from (5 · 9).
§ 6. Ground state for n delectrons
In the preceding section we have derived the ground singlet wave function
for one localized delectron coupled with the conduction electrons by the sd
effective interaction. In this and the next sections we consider the case in which
the impurity atom carries n dcelectrons (l<n<10). For this case, the intra
exchange interaction between delectrons may be neglected in the effective sd
Hamiltonian, but it may not be neglected in Ha. On account of this Hund
coupling included in Ha, the situation becomes .. complicated. Therefore, to avoid
this difficulty, we take two extreme cases; in one case the intraexchange coupling
J, is assumed to be zero, and in the other case J~, is much greater than H•a·
We begin with the consideration for the first case. By inferring from the
results obtained for one delectron, a starting wave function for the ground singlet
state may be put as
(5·16)
(5·17)
(5·18)
(6·1)
where n is assumed n<2l+l. For n>2l+1, electrons and holes are exchanged.
Inserting (6 ·1) into the Schrodinger equation and cutting the higherorder states
with particlehole excitations, we obtain the equation for amplitudes T as
where the summation over ms is taken over the unoccupied states and misi. If
rkms are independent of ms and satisfy the equation
( e~~: E)rk +L{q (n1) ~} :E r~~:,=O,
n 2N q k'
E .= nD exp [
2N
{q (n1)n/q}pJ
Eq. (6·3) 1s satisfied by such T~~:. From (6·4) the energy eigenvalue is
calculated as
In order to take into account the higherorder effects due to particlehole pair
excitations, we expand the groundstate wave function as
:E
(mi'Bi')=F(mJBJ)
1=1,2,···'1l,i+l,···n
For the suffixes possessed by the second amplitudes,
n+l n
:E mimn+2=:E m/,
l=l i=l
n+l n
:E Si Sn+2 = :E S/
i=l i=l
are satisfied. Also in the present case, we can derive an equation for the first
amplitudes T[k1k2···k,] that are independent of (misi) by the same iterative
procedure as has been done in § 5. To do this needs a somewhat lengthy calcu
lation but its principle is so simple that we shall here write down only the
(6·2)
(6·3)
(6·4)
(6·5)
where e£ is used for ek£ for simplicity, and a,. and /3,. are respectively defined by
This equation (6 · 7) has been obtained within logarithmic accuracy. In deriving
the integration kernel those scattering processes in which morethanone holes
participate have not been taken into account. Their contributions are always
lessdivergent.
As has been shown before, the solution for the equation of zero approximation
obtained by omitting the term with the integration kernel can be solved in a
form of the products of oneparticle amplitudes. ·For the complete equation of
(6 · 7), a solution cannot be obtained in such a simple product form, but it can
be obtained with logarithmic accuracy in the following form:
result without describing detailed processes of calculation.
where suffix n is attached to amplitude T to signify explicitly the number of d
electrons. If we. introduce T,. of this form in (6·7), assume /(log(( eE)/D))/
( e E) for T 1(e; E), which will be confirmed later and neglect lessdivergent
quantities, then (6· 7) is reduced to the following oneparticle equation:
( eE,~.)T1(e; EN.,.)= a,.qpJ so T1(e'; E~.,.) de' a,.f3n (qpJ)2
. 2N D 2N
(6·7)
(6·8)
(6·9)
x fJlog e~En /(1 ~~log e~En )}r1 (e';En)de'.
A boundstate solution for this equation is given by (5 · 13) modified by replacing
a and {3 by an and f3n, and En is given by
E~n= nDexp
N
(2l+ 1)pJ
.
We can see from this result that the singlet ground state for the ndelectron
system has a binding energy whose exponent is the same as that for onedelectron
system.
§ 7. Effect of a strong Hund coupling
In the actual case, a strong Hund coupling will change the results obtained
above. In the limit of a strong Hund coupling, the delectrons are mutually
coupled and they are in the state of the lowest multiplet specified by the total
angular momentum L and the total spin S.
For the halffilled shell with delectron number n equal to 2l + 1, the effective
sd Hamiltonian can easily be written down by taking only the terms of m = m'
in (2 · 17), because in this special case delectrons cannot change their orbital
states, and by projecting them into the subspace of S = 5/2, that is, by replacing
d~,dms' by (2/q) :Em d~,dmr'· The result is
This can be written in the following form:
H.a =
J L; at'ms'o,,,. Sakms.
2 (2l + 1) N kk'm,.'
This is nothing but the form given by {1· 3) and has first been used for discus
sing the Kondo effect of degenerate dorbitals by Schrieffer.14l However, this
form is allowable only for the halffilled shell.
For the sd exchange Hamiltonian given by (7 · 2), 2l + 1 conduction electrons
or holes can be coupled with the localized spin S, forming a singlet state on
account of 2l + 1 dwave components. The situation is quite similar to the case
 for the sd model with S = 1/2, and the energy lowering is given by
This has a factor 2l + 1 in the numerator of the exponent, and is therefore very
small compared with the value given by (6·11). This is due to the absence of
(6·10)
(6·11)
(7 ·1)
(7 ·2)
(7 ·3)
orbital exchange in this case.
For the case in which delectron number is less than the halffilled value of
2l +1, the lowest multiplet has an orbital angular momentum L as well as a spin
momentum S. Therefore, for such cases, the sd Hamiltonian should be projected
onto the subspace made by L and S. This process can most easily be done for
the case in which one electron is removed from the halffilled shell. For this
case of n = 2l, there exists one multiplet with the maximum value of S.
For this case of n = 2l, the effective Hamiltonian obtained by the projection
process mentioned above can be written down as
+ ~~J (1Yt ~ (  1)M'MaktM'Bak'Ms'T(M'.( M)
2N 2 kk~.'M,,M'
X[&,,,+~ (s·S),.,].
(7·4)
In this expression the state of the localized delectrons is specified by zcomponents
M and M, of the orbital and spin moments L and S whose magnitudes are both
equal to l. T (M' ~ M) denotes the ope:t:ator of changing the orbital state of d
electrons from M to M'. The second term of (7 · 4) exchanges spins between
conduction electrons and the localized delectrons, and the third term consists
of the part which exchanges only orbital states and that which exchanges both
orbitals and spins.
For n = 1, the effective Hamiltonian (2 · 17) can be written in the represen
tation in which the delectron state is specified by M and M, as
As shown in the previous sections, this Hamiltonian has an effect of quenching
spin and orbital moments of the lodttlized delectron simultaneously. In contrast
to this special case, the effective Hamiltonian given by (7 · 4) has different effects
on spin quenching and on orbital quenching through the process of projection to
the LS subspace. This can be seen from the fact that the Abrikosovtype integral
equations which are satisfied by three parts of the scattering tmatrix, correspond
ing to spin exchange, orbital exchange and orbital and spin exchange terms of
(7 · 4) give rise to different wdependence for these three parts. Therefore, in
the following we shall give qualitative discussion on the quenching of spin and
orbital moments on the basis of the effective Hamiltonian (7 · 4) for n = 2l.
Hamiltonian (7 · 4) can also he written as
+ 2JN c2w~ Tclc"£'.M..'JM's c 1)M'MaktM'sak'M•TCM' ~ M )·
+2JN ( 18Y1'lcTMc"'£=M.F..MJ'M'B'B' (  1)M'MaktM'Iak'Ms'
X T(M'~M) (s·S),,,.
(7·6a)
(7·6b)
(7 · 6c)
bound electron number 1a ( M, s),
bound hole number a (m1 s) (m2 s) (ms s) · ·· (m2z s) lmt+M,
(7·8)
where the number of bound holes a is given by the neutrality condition (1a)=2la,
namely, a=1/(2l+1). These numbers of bound holes and electrons are deter
mined by the Anderson condition20> for the matrix elements of (7 · 6) between
two different Mcomponents not to vanish. The characteristic ··energy for this
process is given by D exp [ (4/2l +1) (N/ pJ)].
The third term represents simultaneous exchange of spin' and orbital states
between conduction electrons and the localized delectrons and will link mutually
independent spin exchange and orbital exchange given by (7 · 6a) and (7 · 6b).
In the ground state for the whole sd effective Hamiltonian (7 · 6), the local
spin and orbital moments are both quenched. The groundstate wave function
would be given by
1/Jcond(M=l, M,=S)~
i/Jcond(M=l1, M,=S)~
electron
.
hole a ((l1)j)((l2)t) .. ·((l1)j)(lj),
(1a) ( n) ( (l1H) ... en), (lt)
electron (1a) cn) .. ·(l~), ((l1)j)
a (lj)((l2)j)· .. (l1j)(lj),
hole
etc.
1/Jground = :L; 1/Jcond (M, M,) ¢ (M, M,),
M,M8
m which the wave function of conduction .electrons in each (M, M,)component
1s characterized by the following nu mIbers of bound electrons and holes:
a is given by (1a)2(l+i)=2al, namely, a=(l+1)/(2l+1).
Although in this paper we cannot derive the two characteristic energy values
T Ks and T Ko, corresponding respectively to spin quenching and orbital quenching,
the former value being much smaller than the latter, we can derive the phase
shifts of the conduction electrons at the Fermi surface from the number of bound
electrons and holes in each (M, M,)component of the groundstate wave function
by the use of the Friedel relation. The phaseshift values thus obtained im
mediately lead us to the following residual resistivity for l = 2:
p=5p0 sin2 (;0 n),
(n=4)
(7 ·9)
(7·11)
(7·12)
where p0 denotes the resistivity value of sd model (S = 1/2). This result coincides
with the result obtained by the HartreeFock approximation for the case in which
local spin and orbital moment are quenched. It is expected that this coincidence
is generally valid. It is easy to see that the same result as Hartree and Fock's
is obtained for general values of n when the Hund coupling is neglected. At
high temperatures at which only orbital moment is quenched, the resistivity value
can be evaluated by (7 · 8) as
p=25Po sm.2(Snn' ) .
§8. Summary
Starting from the Anderson Hamiltonian generalized to the case in which
the orbital degree of freedom of the localized delectron is taken into account,
we have derived the effective sd Hamiltonian for any number of localized d
electrons in the limit of strong correlation, U'P J; On the basis of this effec
tive Hamiltonian, the nature of the singlet ground state of the localized delectrons
has been studied in two limiting cases, one unrealistic in which the Hund cou
pling is completely neglected and the other realistic in which the Hund coupling
is so strong that the state of the local delectrons is restricted into the subspace
of the lowest multiplet of the isolated ion.
In the former case, the spin and the orbital moment are simultaneously
quenched and the energy gain due to the formation of the singlet bound state is
given by D exp [ (NI pJ) (11 (2l + 1))]. This large value of the binding energy
indicates that the orbital moments of the localized delectrons are quenched at
highenough temperatures. The residual resistivity is equal to the value obtained
by the HartreeFock approximation.
For a strong Hund coupling, the effective Hamiltonian for n=2l+ 1 and n=2l
has been derived. For the halffilled shell, the situation is so simple that the
groundstate wave function and its energy are easily obtained by using the results
for sd exchange model. The binding energy or the Kondo temperature is given
by D exp [ (NI pJ) 2S]. For general cases, further investigations are needed, but
here a qualitative discussion has been given for the ground singlet state for
n = 2l and it has been concluded with the aid of the Anderson theorem that
the residual resistivity again coincides with the HartreeFock value.
The authors would like to express their sincere thanks to Professor A.
Yoshimori, Dr. A. Sakurai, Dr. H. Ishii and Mr. K. Yamada for frequent valuable
discussions and comments. One of the authors (1.0.) is also much indebted to
them for their advices and encouragements throughout the course of this work.
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