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Infrared divergences associated with heavy particles in metals
Physica 132B (1985) 303-309 North-Holland, Amsterdam
INFRARED DIVERGENCES
ASSOCIATED
WITH HEAVY PARTICLES
IN METALS
Jun KONDO Electrotechnical
Received
Laboratory,
12 April
Tsukuba
Research
Center, Zbarakiken, Japan
198.5
We consider heavy particles placed in the sea of metallic electrons. Both systems are represented by plane waves and interaction between them is considered. We pay special attention to the limit where the mass of the heavy particles tends to infinity. Even in this limit, the heavy particles are not equivalent to fixed particles, but represent a system with internal degrees of freedom. As a result of it, infrared divergences occur in physical parameters. Scattering matrix of the electrons on the heavy particles involves a factor (D/kT)B, where D is the cut-off energy for the electron and g is a coupling constant. The RKKY-type effective interaction between two heavy particles involves the square of the above factor. Furthermore, a new effective interaction which is always negative results from higher-order terms.
1. Introduction
Consider heavy particles (muons, protons or nuclei) with mass A4 placed in the sea of metallic electrons. The electrons will be represented by a jellium model. Then, the particles will see a constant mean potential and will be represented by a plane wave. Our question is what happens when A4 tends to infinity. One might consider that in that limit the particles cannot move, so they will be equivalent to fixed particles. Scattering of the electrons by the particles may, then, be treated in the frame of a one-body problem. Namely, one sets up a potential arising from randomly placed particles, and solves the onebody Schrijdinger equation for the electron. That the above expectation does not hold was implicit in our previous paper [l], in which we treated a problem of a heavy ion in liquid helium 3. We calculated the vertex correction for the interaction between the ion and the quasi-particle of liquid helium 3. We found that the correction involves a large factor, which comes from a fermi-surface effect; a manifestation of the fact that the scattering of the quasi-particle on the ion cannot be considered in the frame of the one-body scheme, but must be treated by taking account of the presence of other quasi-particles. This result holds even when the mass of the ion 0378-4363/85/$03.30
(North-Holland
tends to infinity. This contradicts the expectation mentioned before. Let us argue in the following way. In order to fix the heavy particle in space, it is necessary to introduce a potential, in which there may occur several discrete levels for the particle. When the separation between the levels is large, the particle in the ground level may be considered to represent a system without internal degrees of freedom. It will exert a static potential on the electrons. Then, the scattering of the electron on the particle will be a one-body problem. Such a situation cannot be reached for a particle represented by the plane wave, even when its mass tends to infinity. It remains to be a system with many internal degrees of freedom even in that limit. Its interaction with degenerate metallic electrons will, then, give rise to infrared divergences, the vertex correction mentioned being an example. Our task in this paper is to investigate problems associated with heavy particles placed in metallic electrons. In section 2 we will find electrical resistivity of metals containing free heavy particles. In section 3 we will derive an effective interaction between two heavy particles placed in the metallic electrons. One particle will excite an electron-hole pair, which will be absorbed by the other. If the particles are fixed and the energy of
@ Elsevier Science Publishers B.V. Physics Publishing Division)
304
J. Kondo
/ Infrared
divergences
the electrons is calculated as a function of the distance between the particles, one will find a Friedel-type or RKKY-type effective interaction. Instead, if the particles are described in terms of a plane wave, we will find an important correction arisi,ng from the fermi-surface effect.
associated
with heavy particles in metals
q %
4’
lo;
Fig. 1. The vertex. The thick line is for the heavy particle the thin line for the electron.
and
2. Scattering of metallic electrons on heavy particles We consider Hamiltonian:
a
system
represented
by
&j
a Fig. 2. Logarithmic
vertex
neglected the energy lowed when + c
V,(k’ - k)a,*.,a,b&,.b,
,
correction
in the lowest order.
of the particle.
This
(1)
is al-
(4)
where aks and b4 are the operators for the electron and the heavy particle, respectively. Em is h2k2/2m, and E, is h2q2/2A4.If the heavy particles were fixed randomly in the space, and exerted the same potential V,(k - k’) on the electrons, the electrical resistivity would be expressed by
where k, is the fermi wave number of the electrons [l]. In the limit M+cQ, expression (4) is always satisfied. Expression (3) is calculated to the logarithmic accuracy as
R = (2z-m/ne2h)V$.wzi,
where D is the order of the cut-off energy for the electrons. When (4) is not satisfied, kT in the logarithm will be replaced by -h2ks/2h4. Examples of important higher-order corrections are shown in fig. 3. In VF+’ order, there are n ! such diagrams. n vertexes are placed to the left of the central vertex, and another n vertexes to the right of it. There are II ! ways of connecting both through n electron-hole bubbles. All of them contribute a power of log(D/kT). Summing all of them, we find the corrected vertex to be
(2)
where IZ is the density of the electrons, ni is that of the particles and p is the density of states of the electrons. V, is assumed to be independent of k - k’. Going back to the case of eq. (l), let us consider the vertex correction to the interaction V,. We reproduce the result of ref. [l]. The bare vertex is represented in fig. 1 and is expressed by V,. Let us consider the case w = Ed, OJ’= +, and .sk+Eq= wO= E,, w;, = E,,., k+q=k’+q’ first important correction is Ed,+ E,,. The represented by fig. 2, and is expressed by
2v3
kkr
Vo-W)
(5)
3
fk(l -fv)
c
O
2V$‘log(D/kT),
(ek -
The factor
fk =f(~)
ek. +
iS)2
’
2 comes from the electron spin, and Here we have is th e f ermi function.
Fig. 3. Examples rection.
of
important
higher-order
vertex
cor-
J. Kondo
I Infrared divergences
where
associated
with heavy particles in metals
u
(Q)
=
2
vow- k)2f,(l- fk,)
2
la kk’
g=2v&2.
Ed -
tskp+
3. Effective interaction particles
k’-k
.
(6) (7)
v,,(R) + V,,(R) = 4 2
VO(k’-
(8)
k)2fk(1 - fti)
ck -
kk’
Thus, it is ever increasing is satisfied.
sQ,
ia
That of fig. 4(b), UiJQ), is obtained by replacing By Fourier-transforming, we ‘Q,k’-k by ‘Q,k-k’. have
(Note that 2 defined here is the inverse of Z in ref. [l].) From (5), we may assume that the electrical resistivity of our system is given by (2) multiplied by Z(T)2: R = (2mn/ne2h)V$niZ(T)2.
30.5
sk.
+
if3
k).R.
x cos(k’-
(10)
This is the RKKY-type interaction between the particles. It is convenient to add contributions of fig. 5, which are independent of R. The sum of these will be denoted by V,(R):
as T --) 0 as long as (4)
(11) F(k, k’) = 4) V,(k’ -
between two heavy
k)[‘f,(1 - fks)
x [1 + cos(k’
Let two heavy particles be in the momentum states q and q’. We will calculate an effective matrix element for the scattering into the states q - Q and q’+ Q. It will be denoted by U(Q), and its Fourier transform V(R) = c e-iQ’RU(Q)
(9)
Q
may be interpreted as an effective interaction between the two particles. The lowest order process is shown in fig. 4(a) and (b). The contribution of fig. 4(a) to U(Q) is expressed by
- k) - R] .
(12)
In higher order terms, we take those diagrams where each electron-hole bubble has only two vertexes. Thus, fig. 6 is an example of diagrams taken and fig. 7 is that of those abandoned. This procedure gives us the most strongly divergent logarithmic contributions. For each kept diagram with n bubbles, there are.2n vertexes. They may be connected either to the q-line or to the q’line. So, there are 2’” diagrams for a fixed configuration of the bubbles. For example, there are 16 ways of connecting 4 vertexes to the
qFL“‘G-l 4
q
Fig. 5.
4-Q
9
(a)
Fig. 4. The lowest-order
q
9-Q
(b) particle-particle
a
scattering.
Fig. 6. Higher-order
b effective
interaction
with two bubbles.
J. Kondo
306
Fig. 7. An example
of unimportant
/ Infrared divergences
effective
particle lines for fig. 6(a). The contributions to V(R) is expressed
V,(R)
= c
with heavy particles
of
D
-
their
Fig. 9. Possible
configurations
of three
bubbles.
(13)
where
Furthermore, fig. 10 gives us a denominator l/A:(A, + A,)A:. The sum of the two gives us
V,,(R) F2 and A, are defined
similarly.
= 2 m. 1
Fig. 6(b) gives us Finally,
V,,(R)
in metals
interaction.
sum by
FIFz A,(A, + AJA,’
associated
= 2
F1F2 A,(A,+AJA,’
The sum of the two is expressed
2
3
we have a term from fig. 11:
(14) F,F2F3
V,,(R) = 2
A,(A,+A,M:
by =~------
(15)
FlF2F3
’
A,A,A:
2
F,F,F, c----
A,A;A;’
The sum of the two results where F = F (k, k’) and A = sk - Ed.+ ia. We do not take those diagrams where there exists no bubble at an intermediate instant. An example is fig. 8. These diagrams represent multiple scattering in terms of the effective interaction, and will not be discussed here. Then, there are 10 different configurations for three bubbles. They are distinguished by energy denominators. The sum of some configurations results in a simplified denominator. For example, the sum of 4 configurations shown in fig. 9 gives us an energy denominator l/A ,A,(A 1+ A,)A :. When 2” ways of connecting 6 vertexes are considered, we have a contribution to V(R):
(16) -
-
-
G
Fig. 10. Possible
configurations
-C Fig. 8.
becomes
Fig. 11. Possible
of three
bubbles.
-YT-
2 configurations
of three
bubbles.
J. Kondo / Infrared divergencesassociated with heavy particlesin metals
We define
p(R),
q(R)
307
and r(R):
(17)
Fig. 12. An example
p,, and q,, come from 1 in F 0: 1 + cos(k’- k) -R, and p1 and q1 from cos(k’- k) - R. r(R) is not logarithmically divergent, and will be neglected. The sum of (ll), (15) and (16) with neglect of r(R) results in
VW = p(R)[l We assume exponential V(R) = p(R)
+ q(R)+
;dW*l
.
(20)
that (20) is the first three terms of the of q. Thus, eqCR).
(21)
We are able to prove (21) in the limit of R + m. In this limit, both p, and q, is small, and in the first order of them we have V(R) = p. eqO+ eqo(P1f P&d
(R+m).
(22)
Now in that limit, we may take only those diagrams where only one of the bubbles is connected to the q- and q’-lines, and all the others are connected to one of the two lines. An example is shown in fig. 12. When we sum all such diagrams, we obtain the second term of (22). (The first term is unimportant.) The method of the proof is similar to that used in ref. [l], and will not be described here. Assuming that (21) is generally valid, we now analyse it. We only have to calculate po, pl, q. and q,. We find p. = -2c0, where e. is the gain of the energy when a heavy particle is introduced into the metallic electrons and is fixed. p, is the RKKY-type interaction as we mentioned before: p,(R)
= grrgc,F(2@),
F(x)
= $ (cos x - i sin x) ,
(23)
of leading
contributions
at R + =.
where Ed is the fermi energy of the electron, we assumed that V,(k’- k) is a constant. essentially equal to (3): q. = 2g
log(D/kT)
.
Finally
we find for k,R 4 D/kT [2]
91=2g
sin* k,R D k2R2 l”gE F
(25)
(26)
*
Using these results and adding a constant so that V(R)-, 0 as R + 00, we have V(R) = 8mg~,Z(T)*.
and is
q.
Z(T)2Si”2kFR’kbRZ.
term
F(2k,R)
- ~E,~(T)*[Z(T)*“‘“~~~~‘~~~* - l] .
(27)
If one sets Z = 1, we have the RKKY interaction V(R) = p,(R). For R + a, the leading terms in R-’ give us V(R) = &rg&,Z(T)*F(2k,R) -4e,Z(T)**logZ(T)*
sin* k,R k2R2 .
(28)
F
The first term is the RKKY-type interaction with the enhanced vertex part V,Z(T) as (5) might suggest. The second term is new. It is a real manifestation of the fermi-surface effect. This term does not oscillate, but is always negative and long-ranged. We now go back to U(Q). From (27) we find U(Q)
= V-’ j V(R) ei9” d V
= -1z(T)2[gH(~.Z(T))+~D(~~Z(T))I P F
3
(29)
J. Kondo
308
/ Infrared
divergences
where @z
WC&Z)=-1 yqcos2R-ygq 0 x
z2sinZ
R/R2 do,
(30)
_ D(q, z) = i j- R “‘4”qR (Z2sin2R’R2- 1) dR .
Finally, let us analyse U(0) in another way. Let q’= -4. The particle states will be changed from q, -q into q- Q, -q + Q. Let both q and q - Q have the same magnitude qo. Then JQI = qod2(1 - cos 13), where 0 is the angle between q and q - Q. U(O) is then a function of 8, which will be expanded as U (Q) = c
In the approximation calculated analytically:
of
(28),
H=lf$l)logpl
H
and
)
D
are
(21+ l)A,P,(cos
The coefficient
0) .
A, will be written
(34)
as
(32) (35)
logZ.1 q<2, 4
0
(33)
q>2.
D represents a long-range interaction. In the general case, H and D are calculated numerically. Some results are shown in fig. 13. Up to this point, we have neglected the spin of the particle. When we take account of it, we have exchange scattering as well. By including it, we find the effective interaction between the particles in the second quantized form:
; c
with heavy particles in metals
(31)
0
D=
associated
C, and B, are the contributions from H and D, respectively. Some results of the numerical calculation are shown in fig. 14. They are all attrac-
-
co a BO
.
U(Q)b;+,,,br,,b,B,,,
b
----________~'~~---__----, OO
1
2
3
q Fig. 13. H(q, Z) and D(q, Z) vs. q for several values of Z.
Fig. 14. (a) &(a, Z), Co(a, Z) vs. for several values of Z. (b) Bl(a, Z), C,(cr, Z) vs. (Y for several values of Z.
J. Kondo
/ Infrared divergences
associated
tive. This is because the energy of the particle is much smaller than that of the screening system (conduction electrons). This is analogous to the electron-phonon case, where the attraction between electrons results when the electron energy is smaller than the Debye energy.
4. Discussion Impurity atoms in metals are bound to their sites by atomic forces, and are not free at all. Scattering of the electrons on them is correctly treated in the frame of one-body theory, if the static potential arising from them is properly assumed, If the atoms are loosely bound as may be the case in amorphous metals, however, one would observe the effect predicted in section 2. A simplified model of such a situation is the two-level system. In the original two-level model [3], the atomic configuration was not specified. We introduced a model, in which an atom is jumping between two sites [4], and showed that the resistivity of metals involving such a model has a logarithmic term. This is consistent with the result of section 2. It would be important to analyse empirical data on amorphous metals in detail. Are the results of section 3 and the mass enhancement found in refs. [2] and [S] relevant
with heavy particles
in metals
309
to the heavy fermion superconductors [6]? At present we have no definite answer. We need more analysis of our result. First, the long-range interaction, 4-l in (33), must be screened. Second, intermediate states of heavy fermions must also be restricted by the Pauli principle, because they are degenerate in the heavy fermion systems. Third, we have calculated only the matrix elements between plane-wave states. When the particles form Cooper pairs, their motion is restricted spatially and it is not immediately allowed to use our result. Fourth, the direct repulsive interaction is also renormalized by the 2 factor. Finally, it is not clear whether the high density of states reflects charge fluctuation, so that the simple interaction such as (1) is applicable to this system.
References
111J. 121J.
Kondo and T. Soda, J. Low Temp. Phys. 50 (1983) 21. Kondo, Physica 84B (1976) 40. B.I. Halperin and C.M. Varma, Phil. 131 P.W. Anderson, Mag. 25 (1972) 1. [41 J. Kondo, Physica 84B (1976) 207. 151J. Kondo, Physica 123B (1984) 175. 161F. Steglich, J. Aarts, C.D. Bred], W. Lieke, D. Meschede, W. Franz and J. Schlfer, Phys. Rev. Lett. 43 (l!Z9) 1892. For a review, see G.R. Stewart, Rev. Mod. Phys. 56 (1984) 755.
INFRARED DIVERGENCES
ASSOCIATED
WITH HEAVY PARTICLES
IN METALS
Jun KONDO Electrotechnical
Received
Laboratory,
12 April
Tsukuba
Research
Center, Zbarakiken, Japan
198.5
We consider heavy particles placed in the sea of metallic electrons. Both systems are represented by plane waves and interaction between them is considered. We pay special attention to the limit where the mass of the heavy particles tends to infinity. Even in this limit, the heavy particles are not equivalent to fixed particles, but represent a system with internal degrees of freedom. As a result of it, infrared divergences occur in physical parameters. Scattering matrix of the electrons on the heavy particles involves a factor (D/kT)B, where D is the cut-off energy for the electron and g is a coupling constant. The RKKY-type effective interaction between two heavy particles involves the square of the above factor. Furthermore, a new effective interaction which is always negative results from higher-order terms.
1. Introduction
Consider heavy particles (muons, protons or nuclei) with mass A4 placed in the sea of metallic electrons. The electrons will be represented by a jellium model. Then, the particles will see a constant mean potential and will be represented by a plane wave. Our question is what happens when A4 tends to infinity. One might consider that in that limit the particles cannot move, so they will be equivalent to fixed particles. Scattering of the electrons by the particles may, then, be treated in the frame of a one-body problem. Namely, one sets up a potential arising from randomly placed particles, and solves the onebody Schrijdinger equation for the electron. That the above expectation does not hold was implicit in our previous paper [l], in which we treated a problem of a heavy ion in liquid helium 3. We calculated the vertex correction for the interaction between the ion and the quasi-particle of liquid helium 3. We found that the correction involves a large factor, which comes from a fermi-surface effect; a manifestation of the fact that the scattering of the quasi-particle on the ion cannot be considered in the frame of the one-body scheme, but must be treated by taking account of the presence of other quasi-particles. This result holds even when the mass of the ion 0378-4363/85/$03.30
(North-Holland
tends to infinity. This contradicts the expectation mentioned before. Let us argue in the following way. In order to fix the heavy particle in space, it is necessary to introduce a potential, in which there may occur several discrete levels for the particle. When the separation between the levels is large, the particle in the ground level may be considered to represent a system without internal degrees of freedom. It will exert a static potential on the electrons. Then, the scattering of the electron on the particle will be a one-body problem. Such a situation cannot be reached for a particle represented by the plane wave, even when its mass tends to infinity. It remains to be a system with many internal degrees of freedom even in that limit. Its interaction with degenerate metallic electrons will, then, give rise to infrared divergences, the vertex correction mentioned being an example. Our task in this paper is to investigate problems associated with heavy particles placed in metallic electrons. In section 2 we will find electrical resistivity of metals containing free heavy particles. In section 3 we will derive an effective interaction between two heavy particles placed in the metallic electrons. One particle will excite an electron-hole pair, which will be absorbed by the other. If the particles are fixed and the energy of
@ Elsevier Science Publishers B.V. Physics Publishing Division)
304
J. Kondo
/ Infrared
divergences
the electrons is calculated as a function of the distance between the particles, one will find a Friedel-type or RKKY-type effective interaction. Instead, if the particles are described in terms of a plane wave, we will find an important correction arisi,ng from the fermi-surface effect.
associated
with heavy particles in metals
q %
4’
lo;
Fig. 1. The vertex. The thick line is for the heavy particle the thin line for the electron.
and
2. Scattering of metallic electrons on heavy particles We consider Hamiltonian:
a
system
represented
by
&j
a Fig. 2. Logarithmic
vertex
neglected the energy lowed when + c
V,(k’ - k)a,*.,a,b&,.b,
,
correction
in the lowest order.
of the particle.
This
(1)
is al-
(4)
where aks and b4 are the operators for the electron and the heavy particle, respectively. Em is h2k2/2m, and E, is h2q2/2A4.If the heavy particles were fixed randomly in the space, and exerted the same potential V,(k - k’) on the electrons, the electrical resistivity would be expressed by
where k, is the fermi wave number of the electrons [l]. In the limit M+cQ, expression (4) is always satisfied. Expression (3) is calculated to the logarithmic accuracy as
R = (2z-m/ne2h)V$.wzi,
where D is the order of the cut-off energy for the electrons. When (4) is not satisfied, kT in the logarithm will be replaced by -h2ks/2h4. Examples of important higher-order corrections are shown in fig. 3. In VF+’ order, there are n ! such diagrams. n vertexes are placed to the left of the central vertex, and another n vertexes to the right of it. There are II ! ways of connecting both through n electron-hole bubbles. All of them contribute a power of log(D/kT). Summing all of them, we find the corrected vertex to be
(2)
where IZ is the density of the electrons, ni is that of the particles and p is the density of states of the electrons. V, is assumed to be independent of k - k’. Going back to the case of eq. (l), let us consider the vertex correction to the interaction V,. We reproduce the result of ref. [l]. The bare vertex is represented in fig. 1 and is expressed by V,. Let us consider the case w = Ed, OJ’= +, and .sk+Eq= wO= E,, w;, = E,,., k+q=k’+q’ first important correction is Ed,+ E,,. The represented by fig. 2, and is expressed by
2v3
kkr
Vo-W)
(5)
3
fk(l -fv)
c
O
2V$‘log(D/kT),
(ek -
The factor
fk =f(~)
ek. +
iS)2
’
2 comes from the electron spin, and Here we have is th e f ermi function.
Fig. 3. Examples rection.
of
important
higher-order
vertex
cor-
J. Kondo
I Infrared divergences
where
associated
with heavy particles in metals
u
(Q)
=
2
vow- k)2f,(l- fk,)
2
la kk’
g=2v&2.
Ed -
tskp+
3. Effective interaction particles
k’-k
.
(6) (7)
v,,(R) + V,,(R) = 4 2
VO(k’-
(8)
k)2fk(1 - fti)
ck -
kk’
Thus, it is ever increasing is satisfied.
sQ,
ia
That of fig. 4(b), UiJQ), is obtained by replacing By Fourier-transforming, we ‘Q,k’-k by ‘Q,k-k’. have
(Note that 2 defined here is the inverse of Z in ref. [l].) From (5), we may assume that the electrical resistivity of our system is given by (2) multiplied by Z(T)2: R = (2mn/ne2h)V$niZ(T)2.
30.5
sk.
+
if3
k).R.
x cos(k’-
(10)
This is the RKKY-type interaction between the particles. It is convenient to add contributions of fig. 5, which are independent of R. The sum of these will be denoted by V,(R):
as T --) 0 as long as (4)
(11) F(k, k’) = 4) V,(k’ -
between two heavy
k)[‘f,(1 - fks)
x [1 + cos(k’
Let two heavy particles be in the momentum states q and q’. We will calculate an effective matrix element for the scattering into the states q - Q and q’+ Q. It will be denoted by U(Q), and its Fourier transform V(R) = c e-iQ’RU(Q)
(9)
Q
may be interpreted as an effective interaction between the two particles. The lowest order process is shown in fig. 4(a) and (b). The contribution of fig. 4(a) to U(Q) is expressed by
- k) - R] .
(12)
In higher order terms, we take those diagrams where each electron-hole bubble has only two vertexes. Thus, fig. 6 is an example of diagrams taken and fig. 7 is that of those abandoned. This procedure gives us the most strongly divergent logarithmic contributions. For each kept diagram with n bubbles, there are.2n vertexes. They may be connected either to the q-line or to the q’line. So, there are 2’” diagrams for a fixed configuration of the bubbles. For example, there are 16 ways of connecting 4 vertexes to the
qFL“‘G-l 4
q
Fig. 5.
4-Q
9
(a)
Fig. 4. The lowest-order
q
9-Q
(b) particle-particle
a
scattering.
Fig. 6. Higher-order
b effective
interaction
with two bubbles.
J. Kondo
306
Fig. 7. An example
of unimportant
/ Infrared divergences
effective
particle lines for fig. 6(a). The contributions to V(R) is expressed
V,(R)
= c
with heavy particles
of
D
-
their
Fig. 9. Possible
configurations
of three
bubbles.
(13)
where
Furthermore, fig. 10 gives us a denominator l/A:(A, + A,)A:. The sum of the two gives us
V,,(R) F2 and A, are defined
similarly.
= 2 m. 1
Fig. 6(b) gives us Finally,
V,,(R)
in metals
interaction.
sum by
FIFz A,(A, + AJA,’
associated
= 2
F1F2 A,(A,+AJA,’
The sum of the two is expressed
2
3
we have a term from fig. 11:
(14) F,F2F3
V,,(R) = 2
A,(A,+A,M:
by =~------
(15)
FlF2F3
’
A,A,A:
2
F,F,F, c----
A,A;A;’
The sum of the two results where F = F (k, k’) and A = sk - Ed.+ ia. We do not take those diagrams where there exists no bubble at an intermediate instant. An example is fig. 8. These diagrams represent multiple scattering in terms of the effective interaction, and will not be discussed here. Then, there are 10 different configurations for three bubbles. They are distinguished by energy denominators. The sum of some configurations results in a simplified denominator. For example, the sum of 4 configurations shown in fig. 9 gives us an energy denominator l/A ,A,(A 1+ A,)A :. When 2” ways of connecting 6 vertexes are considered, we have a contribution to V(R):
(16) -
-
-
G
Fig. 10. Possible
configurations
-C Fig. 8.
becomes
Fig. 11. Possible
of three
bubbles.
-YT-
2 configurations
of three
bubbles.
J. Kondo / Infrared divergencesassociated with heavy particlesin metals
We define
p(R),
q(R)
307
and r(R):
(17)
Fig. 12. An example
p,, and q,, come from 1 in F 0: 1 + cos(k’- k) -R, and p1 and q1 from cos(k’- k) - R. r(R) is not logarithmically divergent, and will be neglected. The sum of (ll), (15) and (16) with neglect of r(R) results in
VW = p(R)[l We assume exponential V(R) = p(R)
+ q(R)+
;dW*l
.
(20)
that (20) is the first three terms of the of q. Thus, eqCR).
(21)
We are able to prove (21) in the limit of R + m. In this limit, both p, and q, is small, and in the first order of them we have V(R) = p. eqO+ eqo(P1f P&d
(R+m).
(22)
Now in that limit, we may take only those diagrams where only one of the bubbles is connected to the q- and q’-lines, and all the others are connected to one of the two lines. An example is shown in fig. 12. When we sum all such diagrams, we obtain the second term of (22). (The first term is unimportant.) The method of the proof is similar to that used in ref. [l], and will not be described here. Assuming that (21) is generally valid, we now analyse it. We only have to calculate po, pl, q. and q,. We find p. = -2c0, where e. is the gain of the energy when a heavy particle is introduced into the metallic electrons and is fixed. p, is the RKKY-type interaction as we mentioned before: p,(R)
= grrgc,F(2@),
F(x)
= $ (cos x - i sin x) ,
(23)
of leading
contributions
at R + =.
where Ed is the fermi energy of the electron, we assumed that V,(k’- k) is a constant. essentially equal to (3): q. = 2g
log(D/kT)
.
Finally
we find for k,R 4 D/kT [2]
91=2g
sin* k,R D k2R2 l”gE F
(25)
(26)
*
Using these results and adding a constant so that V(R)-, 0 as R + 00, we have V(R) = 8mg~,Z(T)*.
and is
q.
Z(T)2Si”2kFR’kbRZ.
term
F(2k,R)
- ~E,~(T)*[Z(T)*“‘“~~~~‘~~~* - l] .
(27)
If one sets Z = 1, we have the RKKY interaction V(R) = p,(R). For R + a, the leading terms in R-’ give us V(R) = &rg&,Z(T)*F(2k,R) -4e,Z(T)**logZ(T)*
sin* k,R k2R2 .
(28)
F
The first term is the RKKY-type interaction with the enhanced vertex part V,Z(T) as (5) might suggest. The second term is new. It is a real manifestation of the fermi-surface effect. This term does not oscillate, but is always negative and long-ranged. We now go back to U(Q). From (27) we find U(Q)
= V-’ j V(R) ei9” d V
= -1z(T)2[gH(~.Z(T))+~D(~~Z(T))I P F
3
(29)
J. Kondo
308
/ Infrared
divergences
where @z
WC&Z)=-1 yqcos2R-ygq 0 x
z2sinZ
R/R2 do,
(30)
_ D(q, z) = i j- R “‘4”qR (Z2sin2R’R2- 1) dR .
Finally, let us analyse U(0) in another way. Let q’= -4. The particle states will be changed from q, -q into q- Q, -q + Q. Let both q and q - Q have the same magnitude qo. Then JQI = qod2(1 - cos 13), where 0 is the angle between q and q - Q. U(O) is then a function of 8, which will be expanded as U (Q) = c
In the approximation calculated analytically:
of
(28),
H=lf$l)logpl
H
and
)
D
are
(21+ l)A,P,(cos
The coefficient
0) .
A, will be written
(34)
as
(32) (35)
logZ.1 q<2, 4
0
(33)
q>2.
D represents a long-range interaction. In the general case, H and D are calculated numerically. Some results are shown in fig. 13. Up to this point, we have neglected the spin of the particle. When we take account of it, we have exchange scattering as well. By including it, we find the effective interaction between the particles in the second quantized form:
; c
with heavy particles in metals
(31)
0
D=
associated
C, and B, are the contributions from H and D, respectively. Some results of the numerical calculation are shown in fig. 14. They are all attrac-
-
co a BO
.
U(Q)b;+,,,br,,b,B,,,
b
----________~'~~---__----, OO
1
2
3
q Fig. 13. H(q, Z) and D(q, Z) vs. q for several values of Z.
Fig. 14. (a) &(a, Z), Co(a, Z) vs. for several values of Z. (b) Bl(a, Z), C,(cr, Z) vs. (Y for several values of Z.
J. Kondo
/ Infrared divergences
associated
tive. This is because the energy of the particle is much smaller than that of the screening system (conduction electrons). This is analogous to the electron-phonon case, where the attraction between electrons results when the electron energy is smaller than the Debye energy.
4. Discussion Impurity atoms in metals are bound to their sites by atomic forces, and are not free at all. Scattering of the electrons on them is correctly treated in the frame of one-body theory, if the static potential arising from them is properly assumed, If the atoms are loosely bound as may be the case in amorphous metals, however, one would observe the effect predicted in section 2. A simplified model of such a situation is the two-level system. In the original two-level model [3], the atomic configuration was not specified. We introduced a model, in which an atom is jumping between two sites [4], and showed that the resistivity of metals involving such a model has a logarithmic term. This is consistent with the result of section 2. It would be important to analyse empirical data on amorphous metals in detail. Are the results of section 3 and the mass enhancement found in refs. [2] and [S] relevant
with heavy particles
in metals
309
to the heavy fermion superconductors [6]? At present we have no definite answer. We need more analysis of our result. First, the long-range interaction, 4-l in (33), must be screened. Second, intermediate states of heavy fermions must also be restricted by the Pauli principle, because they are degenerate in the heavy fermion systems. Third, we have calculated only the matrix elements between plane-wave states. When the particles form Cooper pairs, their motion is restricted spatially and it is not immediately allowed to use our result. Fourth, the direct repulsive interaction is also renormalized by the 2 factor. Finally, it is not clear whether the high density of states reflects charge fluctuation, so that the simple interaction such as (1) is applicable to this system.
References
111J. 121J.
Kondo and T. Soda, J. Low Temp. Phys. 50 (1983) 21. Kondo, Physica 84B (1976) 40. B.I. Halperin and C.M. Varma, Phil. 131 P.W. Anderson, Mag. 25 (1972) 1. [41 J. Kondo, Physica 84B (1976) 207. 151J. Kondo, Physica 123B (1984) 175. 161F. Steglich, J. Aarts, C.D. Bred], W. Lieke, D. Meschede, W. Franz and J. Schlfer, Phys. Rev. Lett. 43 (l!Z9) 1892. For a review, see G.R. Stewart, Rev. Mod. Phys. 56 (1984) 755.