753
Progress of Theoretical Physics, Vol. 42, No. 4, October 1969
Local Electron Distribution in the Singlet
Ground State Due to the s-d Exchange Interac1tion
Kei YOSIDA and Akio YOSHIMORI
Institute for Solid State Physics, University of Tol?yo, Roppongi, Tokyo
(Received May 16, 1969)
On the basis of the theory of singlet ground state for a localized spin developed so far,
the total charge and spin localized around the impurity are calculated in detail. It is shown
that for ¢a-component associated with the localized up-spin state half of a down-spin electron
and half of an up-spin hole are trapped by the impurity. This leads to a conclusion with
the aid of the Friedel sum rule that a phase shift of the conduction electrons at the Fermi
level is ±n/2. It is further shown in general that the localized charge in the ground state
completely vanishes for the present s-d exchange Hamiltonian.
§I.
Introduction
In a senes of papers 1>..... s> it has been shown that a localized spin embedded
111 non-magnetic metals forms a singlet (non-degenerate) state coupled with the
conduction electrons by the s-d exchange interaction. In this singlet state, the
spins of the. conduction electrons are localized around the impurity spin, and
form a singlet bound state with it.
The energy of the singlet ground state is lower by the binding energy IE I
than the normal-state energy JE which can be obtained by the usual perturbation
calculation which starts from the doubly degenerate free state of a localized spin
and the conduction electrons.
Our theory dealing with the present system consisting of a localized spin
and the conduction electrons is based on the perturbation method.*> It starts
from a singlet state in which one electron (or hole) excited above (or below)
the Fermi sea is coupled with the localized spin instead of starting with the
doubly degenerate free state, and calculates the ground-state energy and the wave
function (and also other quantities) perturbed by the s-d exchange interaction
in a power series of the exchange coupling among which the most divergent
terms are retained. As particularly shown in a previous paper by the present
authors,7) the charge density at the impurity center which is :finite in the starting approximation completely vanishes in the :final stage of pertu:rbation. This
fact indicates that the localized charge around the impurity vanishes and only
the spin-correlation density remains to be localized.
The main purpose of this paper is to calculate the total localized charge
*> Our method is not variational, though it is often cited so in some literatures.
754
J(. Yosida and A. Yoshimori
and the total localized spin-correlation and to show how the former vanishes,
while the latter remains unchanged by actual calculations. Further, the electronic
structure of our singlet ground state is more clarified. In the next section, a
brief summary of our approach developed so far is given. In § 3, the charge
density and spin-correlation density at the impurity site are discussed and the
total localized charge and spin-correlation are calculated in § 4. In § 5, several
relations which generally hold for the singlet ground state are derived and these
relations are shown to be satisfied by the present approximation in which only
the most divergent terms are retained. In the final s~ction general discussion is
given for the singlet ground state of the present system.
§ 2.
Main results obtained in previous papers
The following simple 1-Iamiltonian is adopted for the system consisting of
the conduction electrons and a localized spin situated at the origin which are
coupled with the s-d exchange interaction (J<O):
(1)
1
where the same notations as in previous papers )~s) are used.
of the localized spin is assumed to be one-half.
We solve the Schrodinger equation for this Hamiltonian
The magnitude
(I-I -E) <jJ = 0
(2)
by perturbation theory, starting with a singlet wave function
where a and /3 represent up- and down-spin states for the localized spin and </Jv
denotes the wave function of the Fermi sea. For this purpose, we expand the
singlet wave function in a series of the states with zero, one and two excited
electron-hole pairs besides one electron excited above the F'ermi level as
(3)
(4)
</Jfl =expression derived from </Ja by exchanging up- and down-spins
in the suffixes of annihilation and creation operators only,
where coefficients T1~; 3 and T1~i!4 5 , etc., are defmed in the same way as in Okiji's
paper. 2) cr1~;3 defined in reference 7) is given by - r2~;3 in the present definition.) On account of the condition for singlet, the following relations hold between
a~
d 12, 3, an d [123J, [4.5J' ra~t
d 1[23J, ['J5J·
r[12J,
3 an
[12J.l, 45 ctn
rat
raa
'
ratt
.
Local Electron Distribution in the Singlet Ground State
--rat
.r•at
[12], 3 12, 3 r[~~h ['15] =
ra~ ~rat
21, 3 12, 3 -
t [ crc~H3,
755
1-.at
21, 3 '
(5)
r[~~j3, 54)
+ crc~M1, 45 ~ rc~M1, G4)
+ crc~Bz, 45 ~ rc~Bz, 54) J = t r[~~h ['15] '
45-
att [45] ~
~ ratt
ratt
.r·au
r 1[23],
[32]1, 45 ~
[32]1, 54~
[321], [45]
(6)
(7)
'
where the coefficients with [123], etc., denote the antisymmetrized sum and the
suffixes which come before a comma denote excited electrons and those after a
comma denote excited holes.
By inserting the expanded wave function (3) into Eq. (2), the following
Schrodinger equation for each coefficient is derived among which those for
r:;_;3 and rc~~j3,45 are written down. The others can be derived from them by the
relations (5), (6) and (7).
rb
(8)
at ("c.1 + c.z~c.3~
" ,. . E)
r 12,3
L
T
4
+ 4N~
J "--, c r a
t
rat
r'at
14,34z,a+. 12,1
JN (2r1--I r)
2
~ 2.f'at
2rat
)
J "-, ( ra~t
[41], 3 +
21,4 + 4 ii 7r 1 [24]1, 53 +
"(ra~t
.['at!
+ 4JN 4
[G2]3, 45 +
[16]3, 40 -
ratt
2[14], [53]~
-
at 4)
2r23,
+
a~t
2J'att
2.f'at'i'
.2raH
)
r[12]3,
46 ~
[63]1, 54 +
[63]2, 54 ~
3[12], [4G]
r1~;3 included in the Schrodinger equation for
raH
1 [12]0, 40 -
+
2]-.att
) -- 0
[41]2, 35 -
'
(9)
rat~
[12]3, 65
' .'
_
~
0
(10)
•
can be eliminated by iteration
with the use of the equation for higher-order coefficients. Then, the equation
for r1 can be brought into the following integral equation :6)
r(s) (s~E)
~
= -
3 (Jpf
16 N!
r1
3 JP IJ_)r(s')ds'
4N Jo
In ds' r(s') (log ~-:t~_'-_E_~) \(1- Jp log~ +_s'~-~E~•) Jo
D
N
D
1
(11)
'
where only the most divergent terms are retained in the series In Jpj N which
appears in the integration kernel, and the ground state energy E is separated
into a normal part and an anomalous part
E=iJE+E.
(12)
The solution of this integral equation is given by
1 [(·J_Jpl s--K)-114_1
/~(")-" - s -li
. N og - D .
3
(1-4
x
)112(l~Jpl
N
s~E\-3!4]
og --D~)
'
(13)
756
K. Yosida and A. Yoshimori
where x 1s defined by
Jp
-E
N
D
4x=-log~-=1.
(14)
It should be remarked here that for later calculations the second term in
T (e) cannot be omitted in spite of the factor of (1- 4xYI 2 as can be seen later.
In reference 7), only the first term of Eq. (13) is used for calculating the kinetic
energy. As a result, a redundant factor of 4/5 appeared in - E/ (- Jpj N).
In deriving Eq. (11) and also for later calculations, the following expansion
.a: .
r·at
d raJ.t
d.
1 2, 3 an
[1~Js, 45 are use .
f·orms f-or t h e coeHICients
r
+ __1 __ {?£.1 ~ 5T5_ +
Dl24
D154
•
7r2 --:-4r5 + 14r1 +~3L'21
Dl25
D254
J
_ _!_ {1!_4_~ r5_
D24s
D453
2T5 + (2 + 4I'4 + 5T2 ~ + __1__ ~-=-2.£2 -_!'~ + ~ 4-T1- 2T4
D523
D245 J
D2ms l
D245
Dl45
+ ?_[1 :- 5('~ + ~_[1 + 7T2__ -=li~!_4_- 5T2} J+ ... ,
_
1
Dl43
aH _
r [12]3,45-
§ 3.
-
Dl25
(15)
D243
JND 1 -- (Tat
]-'at
I rat
1- 2J'at
2]'at)
[12J,402,5-31,:;10,-123,-1 -1- ···,
412345
(16)
Charge density and spin-correlation density at the impurity site
In § 2, the wave function of the singlet ground state was described in a
power series with respect to Jp/LV. In this and next sections, we consider the
charge density and the spin-correlation density (or spin density) in this singlet
state.
In our starting wave function
(17)
an extra electron is added to the Fermi sea and
the impurity site. This electron has down-spin
spin for the component ¢ 13 as shown in Fig. 1.
(1) commutes with the total spin and the total
this electron 1s localized around
for the component </Ja and upSince the present Hamiltonian
number of electrons, the final
Local Electron Distribution in the Singlet Ground State
757
singlet state also should have one extra electron. However, the number of localized electrons is not necessarily conserved. What we
want to show here is that there appears no
localized charge and only the spin-correlation
density is localized in our final state. This
means that the final state is really locally
singlet.
t/10
1/Jo
fJ
In order to calculate the charge and spin
"'
Fig. 1. Electron distribution in ¢a0 and
density, we must calculate <cfJa lai'takt IC/Ja) and
¢ l of the initial singlet state.
<cfJa la%,takt IC/Ja)· These quantities have different
expressions according ash' or k state is electron-state (ck' >O) or hole-state (ck<O):
""
L...J
+ 41 1234
crau
raH
+ 4ratt
ratt
+ ratt
ratt
)
[k'l2], [34] [k12], [34]
[lk']2, 341 [l k]2, 34
lc' [12], [q4] k[12], [34]
_I
T
' ' ' '
(18)
(19)
-
1 "
"
{)L..J
(TaU
_
[123], [ik'] rau
[123], [4k] + 3TaH
[12]3, k'4 ra~t
[12]3, k4 ) T
1
' •• ,
(20)
1234
+ 21
"
"
L..J
1234
cratt
[12]k'. 34 ra{,t
[l2]k, 34 + ratt
1[2k'], [34] ratt
l [2k], [34] )
+ .••
'
(21)
(22)
1 "
"
-2
L..J
1234
(Tatt
[12]~.4k' raH
[12]3,4k + ratt
1[23],[4k'] ratt
l[23],[4k] )
+ ....
(23)
In deriving Eqs. (20) and (23), the contributions from the Fermi vacuum are
omitted and only deviations from it are given.
With the use of Eqs. (18)
(23), the charge density at the impurity site can
be calculated as
rJ
758
K. Yosida and A. Yoshimori
(24)
As wns shown m reference 7), ·with the use of Eq. (15) this can be calculated
as
The spin-correlation density or the spin density for the component
the impurity site can also be calculated as
0a
at
(26)
On the other hand, if we sum Eq. (8) multiplied by T'1 over 1, Eq. (9) by T'1~;3
and the Schrodinger equation for T'[lb, 3 by T'1~; 3 over 1, 2 and 3 and so on, we
obtain the following relation:
(27)
By usmg this relation, together '.vith Eq. (12) nnd the relation
the first series is found to be identical with
Thus, with the use of the expression for
can be evaluated as
Ekin
obtained in reference 7), Eq. (26)
Local Electron Distribution in the Singlet Ground State
1
1
-E
[
]
759
(29)
=2V (3J/4N) -JEpot+ (-Jpj1V) '
where JEpot represents the normal part of the exchange energy.
From Eqs. (29) and (25), aside from the normal part, the up-spin and downspin densities at the impurity site for the component 0a can be obtained as
~ ~ <<Pa\a%',taktl0a)/<<Pai</Ja)
=-
-
~ H<<Pala~'J.akti</Ja)I<<Pai</Ja)
~
3
2_
----=E;__
+1.
2
----/-
(30)
V (Jp/NY
Therefore, the electron distribution around
the impurity in each of two-component states
</Ja and ¢ 13 for the singlet ground state becomes
as shown in Fig. 2. We have here calculated
only the electron density at the impurity site
for </Ja and ¢ 13 • How many up-spin or downspm electrons are localized in </Ja and ¢ 13 is the
next problem.
+j_
I
/
Fig. 2.
2
Electron distribution in ¢a and
¢ fJ of the final singlet state.
§ 4. Total localized up- and down-spin electrons in each of two
components ¢a and c/JfJ
In this section we calculate the total localized up- and down-spin electrons
m each of two components 0a and ¢ 13 ;
(31)
(32)
As mentioned before, nt + nt and nt- nt consist of three parts, an electronelectron part, an electron-hole part and a hole-hole part. Among them, the electron-electron and hole-hole parts are continuous as lz' tends to k. Therefore,
these parts can be obtained from Eqs. (18), (20), (21) and (23) as
Cnt + nDee + Cnt + nt)hh = _<_} --~--> ~ I2 r12+ ~ ( rc~g, 3+ r1~;~)
1
</Ja </Ja
_L
I
-
l
l
123
1
2
1 "'
, rattz
+ 3Tattz
)
.L..J (raaz
~ [12:1],[45] T 6
[12]3,45
1[23],[45]
J_ 2
12345
+ . • • l( -_--L
)
(3'))
0
760
K. Yosida and A. YoshiJnori
and
(nt- nt)ee + Cnt- n~)hh = - (nt + n~)ee- (nt + n~)hh = -1 .
(34)
The question is the electron-hole part to which some cares are needed.
This part gives a crucial contribution. If (n 1 ± nt)~~k is expressed by
(35)
(n± =:finite)
we have
=
~ lim O(e~c/)~ O(s~c)n±
k
ck'->ck
Ck'-
Clc
(36)
where
O(:x:)=l
for
:x:>O
= 0
for
:x:<O.
Therefore, we have to investigate the following quantity:
lim (sTc'-sTc) (n 1 ±n~)~~k.
(37)
&k'->0-1&k-->0--
The sum and the difference of Eqs. (19) and (22) yield
Cnt + nt)~~k<<Pa I<Pa> = ~
I
~
-j- L.J
In
rl (T/t~. k- 2Tf;t k)
rat
( ,.aH
1-- ,.att
+ ,.att
I ratt
) + ...
21,8
[k'l2],[3k][lk']2,k3
[12]k',3k -1[2k'],[3k]
1
1
1
'
(38)
and
In order to draw the factor (e~c'- 8Tc) out of the right-hand sides of these
equations, we use the Schrodinger equations for
T1~!a and Tc~Hs,45, etc. For
example, for T1 T 1'1c1',k' we subtract Eq. (9) multiplied by T1 from Eq. (8) multiplied by T1'1ct,k· Then, we obtain T 1 Tl'1c\~c(8Tc/-s~c). By this procedure, we have,
for the :first term of Eq. (38),
rb
Cnt + nt)~~.k(l-3) (ek/- e~c)
<<Pa I<Pa> =
~ rl
]
(T,'1c\ l c - 2Tf\ k) (ek/- ek)
Local Electron Distribution zn the Singlet Ground State
- -J- "
L..J r1 (raH
. [k'2]1,3k
4N
761
+ ratt
2 pa~t
. k'[12],[3k][2l]k',k3
1
123
-
a~t 3 k - 2ratt
+ 4raH
)
2r[12]k',
1[k'2], [3k]
[2k']l, k3
(40)
•
For the second term of Eq. (38), the same procedure gives
3J "L..J rat
12,3 crat
l k ' , k - 2rat
k'1,k )
4N
~-
123
(rat
+rat
2rat
+rat
)
JN ".L..J r 1 [TaH
- 3J
N "L..J r,at
21,3
lk',s
k'2,32k',s
12, k -+
c~c'lJ2,:3k
4
4
1~
1~
2T'aH
+ r,aH
+ 2 T'a~t
+ 4TaH
J+ ....
[1k']2,k3[12]k',3k
[12]k',k3
[k'2]1,3k
[2k']l,k3
+ 2pa~t
1
1
1
(41)
Adding Eq. (40) to Eq. (41) and using the relation
pa~t
1 [12]3, 45
+ r,aH
+ pa~t
+ r,a~t
+ patt
+ PaH
. [12]3, 54
[23]1' 45
[23]1, 54
[31]2, 45
[31]2,
1
1
1
fl4
= 0,
(42)
we obtain
Cnt + n~)~~k (ck'- 8k) =
1
<</Jai</Ja)
(-~) [rk' (~ rl) + ~ rl cr]~\2- T\~;k)
4N
l
12
+ ~ r2~;3c2r2~~.3-r1~1',3-r:1,3-rl~;k) + ... J = -
(/-!J) (Mk'-Mk),
(43)
\vhere Jl..;[k' and Mk are defined by
Mk' = ~ (</Jaia~',ta~t-ai,~a~~I</Ja)/(</Jai</Ja),
Mk= ~ (</Jala0akt-ai'1,ak~I</Ja)/(</Jai</Ja) ·
l
(44)
The relation ( 43) can be proved generally as will be shown m the following
section.
If we use the expansion forms, Eqs. (15) and (16), obtained by iteration
for r1~; 3 and rc~~~ 3 ,4 5 in Eq. (40) or Eq. (43), we obtain a geometric series with
respect to (Jpj N) log { (8 1 + 8k'- E)/ D} for the most divergent contribution, and
Eq. ( 43) is calculated as
Cnt+ n~)~~k (ck' -8k)
!
=~<-- 1~------></Ja
+ rk' z;= rl
=
-~~---
</Ja
ir
(log
3J [ 3 rk'
(</Jai</Ja) 4N 4
3
NJ [rk'
4
I: r1
1
ir
~1 ±-jj ~-'4_) (1- log~~ j;--=-!) -]
2: r1 + l__rk' I: rl (1- Jp log 81 + 8k'- E) ~l].
1
4
1
1V
D
(45)
It is to be noted here that there remains no most divergent contribution from
Eq. (41).
762
K. Yosida and A. Yoshinwri
The first term can be calculated as
and the summation m the second term as
With the use of the normalization integral Eq. (28),
. ( nt+nt )lc'k
( CJc'-8k )
1llTI
eh
n+=
Bk'• s~c->0
\VC
obtain
1.
= --
p
(46)
From this result, combined with Eq. (36), we can conclude that the contribution of the electron-hole part (n1 + nt)eh to the total localized charge is equal to
-1 and cancels exactly the contribution from the electron-electron and hole-hole
parts.
The procedure used in deriving Eqs. (40) and (41) from Eq. (38) gives
rise to
(nt- nD:~,k(l-3) (elc'- 87c) <rf;a 1</Ja)
=-!_Tlc'
4N
~ T1
(4 7 ·1)
1
+__.[__ ~ T1 2
2N
-
(47. 2)
1
~ (~ T1) ~ T,~ct,~c
3
- 41V 'f5
£"
(47. 3)
Tlt;_t
.. r]a2r",..
"· , ...
(47·4)
(47. 5)
J "r
+ -L-J
1 era+~
[k'2]1, 3k + ralt
k'[J2], [37c]- 2ra+i
[2l]k', k3 ) '
4N 12s
<
(nt- nt)~;:'c3-5) (eTc'- e~c) r/Ja frf;a)
=-
J __
4N
2:: rl~!3Tl~t,r.,--- JN_ I:: r2~!3CTc~h3-r'},~!3)
U3
2 .
In
(47. 6)
Local Electron Distribution in the Singlet Ground State
+ 4J
- -N~
-
763
"L..J rat
21,3 (2rat
zk',s- rat
k'2,3- rat
lk',s- 2rat
7c'l,3 + 3rat
. 12,.1;-- 2Tat
. 21,1c )
123
r 1 [-rat~
2ratt
ratt
[lk']2, k8 +
[1 k']2, 3lc +
[2k']1, 3k
J "L..J
~-N-
4
123
+
alt
pall
r [2k']1,k8+1
[12]k',37c J +ooo.
(48)
Here again, the most divergent terms cancel with each other in Eq. ( 48).
The first term gives a most-divergent contribution, but the second term does
not in Eq. (47). Contributions of other terms of Eq. (47) to most-divergent
terms are calculated as follows:
_ -r~./
J
,
(47 ·3)-(~ 1 r)
N
r
Jt-1 J--log---+P
- E 3 (J
p
- E)
-log----4 N
1
- E)
Jp
7 1
+ 128 (J.V log- D -=-
J r1c, ( ~ rr)
N
1
(4 7 · 4) = __.!_ rlc'
4-N
+
J rlc' ( 6 rr) (1- 4.rY14 ,
N
1
2
+ 12218 (~)slogs el DJf
9
(Jp)
3
( 4 7. 5) =
+ i~~ (~r log
(4 7 · 6) =
(J
-
i¥s (~)slog _-:-J
3
8
1
+ ~~ (~)
3
log
3
log2 el; J1__
D
(50)
2
Jog2
32 1V
~~ 15~ + 000},
~-1_=-.!
D
(51)
(Jp) 2log ~-=~E
log e.l=_J!_
D
D
2
log 2 ~-1-=!J- ~
N1
D
16 1V
~~-}jlf _!is(~)
3
Jog
~If-log 2 81 ; E
--1~58(~rlog2 =-:r:-logel;E:} + .. o.
Summing up Eqs.
(50)
+ (51)
=
(50)~
(52)
(52), \Ve obtain
+ (52)
1
ifrh"'
2f
r
~
~log
~
jj!+ ~ (~)
4
1[
D
-
Lrk, ~ r)l 3 (Jp)
4.N
~_1__-:- E_
s- E
}
--D-E log~+
ooo '
4 1V
1
(49)
2
log 2 ~ 1 -=~li- 3
P\ log ---=-'§; lov
D
16 N/
D
_,
rl Jl~ Jp log 81 - E + ?3 (JP)
_!__ rk' 6
4J.V
log2
D
+ 000}
32 J.V
-128 J.V
32 N
3
~ r1 {2 (Jp)
1
D
2
2
2 1
log
~ ;E;
764
K. Yosida and A. Yoshi?nori
-
~ (Jp)
3
16 N
log - E_log 2 0:_- E- 2 (Jp) log 2 ~ E log
D
D
16 N
D
3
~1_--=-fl +
D
· · ·].
(53)
It is not so easy to find out a function of (Jpj N) log { (c 1- E)/ D} and (Jpj N)
X log (- E/D) whose expansion form coincides with Eq. (53) up to third order
in Jp/ N. However, if it is noticed that Eq. (53) can be rewritten as
_l_ Jplog-=Ji(_§_ Jplog~l~E + 45(Jp)2log2~1-_fr_+ ... )
4 N
D
-2 (Jp)
32 N
2
4 N
D
log 2 ~--fJ__ (~ Jp log
D
4 N
32 N;
D
I
0_- E + · · · \ - · · ·} J
D
)
(54)
'
it would be easy to identify the series of (54) with
(53)= LTk'
4N
+
1
~ T1[ 3 -2. (1- Jp log ~1_=-F;_) -
~ (1 -
4
1
ir
log
N
4
--=!-)
114
D
514
{(1
-~log 81-jf) -
-1}
J.
(55)
By summing up Eqs. (55), (49) and (47 ·1), Eq. (47) is calculated as
+ 3 J (1- 4xY 14To
5N
~ T1 +I_
1
J (1- 4x) 114To
5N
E) D
~ T1 (1- Jp log 81 1
N
514
'
(56)
where T 0 is defined by To= (Tx)sk=o· The summations of the second and the
fourth terms of this equation can be performed and the results are as follows:
Local Electron Distribution in the Singlet Ground Stale
2 Jp
- - (1-4x) 114T 0
5N
i
D
o
E)
I
Jp
C' T1l 1-- -log~---
N
\
D
=_!To (1-4x) 114 - _! (1-4x)- 1 1 2 --~~
5
9
- E
765
-5/4
ds1
•
Therefore, noticing
we find that Eq. (56) completely vanishes, namely
lim Cnt- n~)~~k (Bk'- Bk) = n_ = 0
(57)
e.7c/, C!c->0
and there is no contribution from the electron-hole part to Cnt- n~).
we obtain
Therefore,
(58)
Thus, we see that the localized spin correlation IS preserved
and this state is locally singlet.
Since nt + n~ = 0, we have the following relations:
111
the final state
(59)
(60)
This result means that in the ¢a-component of the singlet ground state half an
electron with down-spin and half a hole with up-spin are trapped by the impurity,
and this can be regarded as a physical basis for the unitarity limit of t-matrix
obtained by other theories, 9l-l 2 ) although these theories failed in leading to a
singlet ground state.
If we assume that the localized down-spin-electron or up-spin-hole density in
¢a is constant and keeps its value at the impurity center, Eq. (30), a minimum
volume v needed for !-electron or !-hole to be trapped by the impurity is given
by
4 v
-Ep
3 V (JpjN) 2
_1 _
From this equation, we obtain an upper bound of the impurity copcentration 1n
which the impurity spins are compensated as
4 p
Ccri.t
=
-It
3 1V --(]pj }1)2
-It
-·
1
-·-
·-·-~--
(Jp/ N)2
(61)
This value 1s still larger by (Jp/ N)- 1 than that given by Anderson's argument. 13 l
766
K. Yosida and A. Yoshimori
~
5.
General relations
In this section, we show that the relation n 1 + nt = 0 holds rigorously.
Let us denote the ground state wave function by <jJ which is normalized and
non-degenerate for the present Hamiltonian given by Eq. (1), then the following
relation generally holds:
(62)
From this we can derive
If we put
Eq. (63) can be written as
(64)
where Uc is defined by Eq. ( 4t1).
identical with Eq. (43). Therefore,
lim
lllc/• olc->0
When
81c/
>O and
s~c<O,
this equation is
<<Pa la~l'/taTct + a~'/taJct 1</Ja)ch (8Jc/- c~e) / <<Pa 1</Ja)
3J
=tt_\-= --~---[M(
4N
+0) -M( -0)].
(65)
From Eq. (64) for e1c/, s~c>O and 8~c/, s~c<O, we can calculate (n1 + nt)ee + (n 1 + n~)nh
as
= _ 3J_ L; c!_l\ljJ_c_ = - 3 JP { (M(D) -lid(-- D)) - (NJ( + 0) - M(- 0))}.
4N
1c
de~c
4N
(66)
Since, for a symmetrical band, the relation JV!(D) = M(- D) holds, Eq. (66)
just cancels the contribution from the electron-hole part. Thus, the total localized
charge generally vanishes for the present Hamiltonian.
The kinetic energy can be expressed with the use of Eq. (64) as
Ekiu
= ~ 81c
lc
lim
lc/->lc
<<Pa lczr/talct + aZ/talc~I</Ja)/ <<Pa 1</Ja)
Local Electron Distribution zn the Singlet Ground State
767
Therefore, the total energy E is given by
3J:
- +E t = --~PDM(D).
E=E1uu
N
po
2
(67)
It might be interesting to notice that the total energy is determined by the value
of Mk at the band edge.
In the previous section, we have shown that the general relation n 1 + nt = 0
is satisfied by the leading-logarithmic approximation. We can easily check other
general relations including Eq. (67) by the same approximation.
§ 6.
Conclusion and discussion
The singlet ground state of a localized spm m metals 1s examined in detail
111 the approximation of the most divergent terms.
The Fourier components of
the charge density of conduction electrons and of the spin-correlation density
between conduction electrons and a localized spin are calculated in the limit of
vanishing wave vector. Relations between amplitudes in the expansion of the
ground state wave-function and some general relations between expectation values
are presented and discussed. Together with the previous results/) almost all
the important quantities in the singlet ground state, which are independent of
the details of the band structure, seem to have been exhausted.
On the basis of the obtained results, we reach the following picture to the
singlet collective bound state. The state </Ja (¢ 13 ) has half of a down-spin-electron
(hole) and half of an up-spin-hole (electron) localized around the impurity in
total, so that in this state we have the localized spin density of the total S = 1/2
and Sz = -1/2 (1/2). Therefore, we expect that the down-spin-electron has the
phase shift n/2 ( -n/2) and the up-spin-electron -n/2 (n/2) or the t-matrix reaches
the unitarity limit at the Fermi level, assuming the Friedel sum :rule. Thus, we
can interpret the J-independent value of the resistivity at the absolute zero as
due to the localized spin density. The picture given above is closely related
to that by Anderson in his ground state calculation13 ) . based on the Anderson
model in the magnetic limit, which is equivalent to the s-d exchange model.
Thus, we can conclude that all the calculated physical quantities such as the
kinetic energy, the exchange energy, the spin-correlation density and the charge
density behave reasonably to give a satisfactory physical picture in spite of the
divergence of the normalization integral. The singlet collective bound state has
now a firm basis as the ground state of a localized spin in metals.
It was suggested by Suhl that because the s-d exchange rnodel leads to strong
coupling effects, it is necessary to return to fundamentals. 14 ) However, in the
limit of the strong local Coulomb interaction and the weak s-d mixing, for example, in the Anderson model, the s-d exchange model should give the same
result in the essential properties as in the Anderson model even in the singlet
.K. Yosida and A. Yoshimorz
768
ground state. Since the s-d model is believed to contain the essential feature
in the dilute magnetic alloy problem and also it is very simple, this model would
be worth investigating thoroughly. T'o do this would shed light on the whole
problem of the dilute magnetic alloy including the local spin fluctuation model. 15 )
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