The Diagramatical Solution of the Two-Impurity Kondo Problem

byChen Zhou

Dissertation submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in

Physics

APPROVED:'

q >é„,„— 2 Lg /‘«’Ting·KuoLee, Co~chai1person

/Samuel P. Bowen, Co-chaixperson

. Ägg" I

IIaytonD. Williams Richard H. Eau

,_Q-.,'

Tom E. Gilmer, Jr. „/ jäoyceäia

August 24, 1988

Blacksburg, Virginial

The Diagramatical Solution of the Two-Impurity KondoProblemby

Chen Zhou

Ting·Kuo Lee,Co·chairpersonSamuel

P. Bowen, Co-chairperson _ ~ __

Physics

;_(ABSTRACT)P--—:

The problem of the two-impurity Kondo problem is studied via the perturbative di-

agrammatical method. The high temperature magnetic susceptibility _is. calculated to

fourth order in the coupling constant J for different regimes. The integral equations for

the ground state energy are established and solved numerically. The two-stage Kondo

effect and corresponding energy scales are found which agree with the scaling results.

Acknowledgements

This dissertation has profited greatly from the members of my comrnittee : Ting-Guo

Lee, Samuel Bowen, Laynam Chang, Clayton Williams, Richard Zallan, Royce Zia, and

Tom Gilmore jr.. I would like to thank them all for the support and encouragement.

Acknowledgemcnts iii

Table of Contents

Chapter I : Introduction ................................................... I

1.1 Single Impurity Kondo Problem .......................................... 1

1.2 Heavy Fexmion Superconductors and The Concentrated Kondo Problem ............. 3

1.3 The Organization of The Dissertation ....................................... 5

Chapter 2: The Formal Theory of the Perturbative Diagrammatical Method .............. 6

2.1 The Anderson Hamiltonian .............................................. 6

2.2 The Feynman·Go1dstone Diagrams for the Two- ............................. ll

Impurity Anderson Hamiltonian ............................................ ll

Chapter 3 : The High Temperature Perturbative Result ............................ I5

3.1 The Motivation and the Model Hamiltonian ................................. 15

3.2 The High Temperature Regime .......................................... 17

3.3 Kondo Temperature Larger than RKKY Interaction ........................... 20

3.4 RKKY Interaction Larger than Kondo Temperature ........................... 21

Chapter 4 : The Self Energies and the Ground State 25

Table of Contents iv

4.1 The Unperturbed States ................................................ 25

4.2 High Temperature Results and The ürst Kondo Temperature .................... 29

4.3 The Ground State Energy .............................................. 32

4.4 The Numerical Solution of the Ground State Energy ........................... 35

4.5 Variational Results of Ground State ....................................... 36

Chapter 5: Conclusion .................................................... 44

References: ............................................................ 47

Appendix ............................................................. 50

A. A sample of the integral evaluation ...................................... 50

B. Diagrams Contributing to eq.(3.8) ....................................... Sl

C. Diagrams contiibuiting to eq. (3.10) ...................................... 53

D. Diagrams contributiong to eq. (3.11) ..................................... 54

E. Diagrams contributing to eq. (3.12) ...................................... 56

F. Diagrams contributing to eq. (3.16) ...................................... 57

Table of Contents v

List of Illustrations

Fig I. Illustration of diagram rules.

(a). A second order diagram for the odd, spin up, singly occupied state. A

descending spin down B channel conduction electron line coming in at the

bottom vertex gives us the energy denominator z — 6,, — 6,. and the numerator

V1/Q.

(b). A fourth order diagram for the doubly occupied state 2a. The interme-

diate states involve singly occupied states eo , oo and unoccupied state <I>. One

ascending A channel electron and one ascending B channel electron line give

us ViI^L(l -ji—)(l —_/Q;). The factor 4 = („/?)‘ comes from four vertices. One‘

minus sign is from the first vertex because it involves an A channel electron

and a doubly occupied state and the other minus sign is from the crossing.

Fig. 2. The energy band configuration of basis states for the two-impurity Anderson

model. The shadowed areas are fermi seas ofconduction electrons and the two

short solid lines are localized states of impurities on two-sites.

_List of Illustrations vi

a). The unoccupied state in which both impurity states are empty. The char-

acteristic energy of the state ( fermi energy ) is set to be zero.

b). Singly occupied impurity states with one hole in the fermi sea. 6 denotes

the spin of the impurity. The total degeneracy is four because the occupied

impurity can be at either site 1 or site 2 with spin up or spin down. The energy

of the states is s,( a negative value).

c). The doubly occupied impurity states with two holes in the fermi sea. The

energy of the states is 2::, and the total degeneracy is also four.

Fig 3. Leading order non-crossing self-energy diagrams for all nine states. The

shadowed blocks represent the self-energies of the corresponding states which

sandwich the block. The expressions for these self-energies are given in eqs.

(4.1).

Fig. 4 Examples of crossed diagrams for the doubly occupied states. Intermediate

states are not specified. Fig. 4.(a) is of the orderV‘

and Fig. 4.(b) of the order

V'. Each diagram represents a group of diagrams with different intermediate

states. All of them should be included in the perturbative calculation.

Fig. 5 An example ofa vertex function in terms of integral equations. The shadowed

areas represent the vertices. Once again the intermediate states are not speci-

lied.

Fig. 6 A family of crossed diagrams for the unoccupied state. These diagrams are of

leading order in 1/N because there are no restrictions og the summation for

List of Illustrations vii

the intermediate states. They represent dominant terms in the ground state

energy.

Fig. 7. Integral equations for the ground state energy in terms of two·particle func-

tions A(k) (fig.7a) and B(k) (fig.7b). The shadowed areas are vertices. The first

terms are lowest order two~particle functions. The coupling between different

channel electrons gives the factor 3 in eq.(4. 10) while the coupling between the

same channel electrons gives the factor 1.

Fig. 8. The selfenergy of the unoccupied state as a function of z at zero temperature.

Only the part below 26,- I is drawn. z is in units of T}, The constant

26,- I is taken to be as zero on this scale for simplicity. The divergence at

z= -1.26 looks like a simple pole. The solution for the ground state energy is

the crossing point of function f(z)= z and function Z„(z). The parameters are

chosen to be : 6,= -6.5,pV‘

= 1.0, D = 22.22, I = 0.3 and

T,, = D exp( - é-) = 0.0334.

List of lliustmions viii

List of Tables

• • TkTab. 1. The solution for the ground state energy as a function of—7- . The parameters

are chosen to be the same as in Fig. 8. A is the ground state energy in addi-

tion to the constant 2::,- A values in the last column are in units of T,,_.

List of Tables ° ix

Chapter 1 : Introduction

The physical properties of the Kondo and the mixed valence systems have long been of

great interest and a challenge to condensed matter theorists in the past two decades

(1-8). The discovery of the heavy Fermion superconductors (4,9,10,11) recently has

generated a great deal of new interest on the problem since these heavy fermion com-

pounds have shown similar characteristic behaviors to that of dilute and concentrated

Kondo systems.

I.1 Single Impurity Kondo Problem

Many rare earth alloy systems have anomalous but similar properties at low temper-

atures. In these systems the rare earth ions are embedded in a continum of a delocalized

electron band and interact with the conduction electrons. lt is generally believed that

these systems can be described by the degenerate Anderson model. The Anderson model

contains much richer structure than the original Kondo model but they are equivalent

Chapter 1 zlntroduction 1

via a canonical transformation (5) if the parameters are chosen properly. The main dif-

ficulty in solving the Kondo problem is the infrared divergence (6,7) associated with the

physical quantities. For example : the magnetic susceptibility calculated perturbatively

has an expansion form :

—L—’1 JITD JITZ2 —p7( +z> ¤g I +(p ¤g ID) +···)Elly

At the temperature

J.T= Tk = De PJ

all terms in the perturbation series are of order l and the perturbation approximation

breaks down. This instability prevents one from using the usual perturbative techniques

at lower temperatures.

In order to overcome this difficulty , nonperturbative methods such as the numerical

renormalization group (7) and Bethe Ansatz (8) have been developed successfully and

the exact ground·state solution has been found in one dimension. Since these two

methods are very difficult to extend to the concentrated systems the perturbative di-

agrammatical method has been intensively studied in recent years.(12-15) The famous

1/N expansion proposed by Ramakrishnan and Anderson (16) has been applied to the

Anderson model by numerous groups (10,11,12) and the best results are in excellent

agreement with the exact solutions.

The distinctive physics of the Kondo problem is the spin flip scattering of conduction

electrons by the impurity. Due to this scattering the impurity spin is compensated by the

conduction electons at a characteristic temperature T, to form a singlet. This is known

Chapter 1 slntroduction 2

as the Kondo screening effect. The Kondo temperature 1], is the characteristic energy

which determines all of the important physical properties of the system.

1.2 Heavy Fermion Superconductors and The Concerztrated

Kondo Problem

A few years ago a group of rare earth compounds were found to be superconducting at

very low tcmperatures with very unusual properties. These compounds are called heavy

fermion superconductors (HFSC). The characteristic of the HFSC is the anomalous

electron effective mass of 10* - 10* times that of the free electron. This is manifested in

the specific heat measurement at low temperatures. The large specific heat corresponds

to the high density of states at Fermi level which is generally attributed to the effect

known as Kondo resonance. While the HFSC systems show a single characteristic en-

ergy scale just as in the dilute magnetic impurity systems there is also a clear coherent

effect in the resistivity measurements which is believed to come from impurity·impurity

interactions. The problem of explaining this unusual phenomenon has generated new

interest in the dilute and concentrated Kondo systems for both theorists and

experimentalists.

Since the properties of the dilute impurity system are well understood, a lot ofeffort has

been spent on the concentrated systems in the past few years.(l2,14,15,l7) Unlike dilute

impurity systems where the interactions between the impurities can be neglected one has

to consider these intersite interactions between impurities. These intersite interactions

can sometimes be very important and change the whole physical picture. In the lattice

Chapter 1 tlntroduction 3

case, for example, coherent scattering has to be accounted for and is believed to cause

the low temperature resistivity to differ so much from that of dilute systems.

Due to the mathematical difficulties in solving the lattice model the system with just two

impurities is of great interest because it contains the simplest and most important inter-

site interaction known as Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. The

interplay between the RKKY interaction and the Kondo effect can give one a great deal

of insight into the difference between the dilute and concentrated systems.

Jayaprakash et al.(l7) have done pioneer work on the two-impurity Kondo problem by

using the thermodynamic scaling method. The main conclusion is that in the limit that

the RKKY interaction is larger than the Kondo temperature there will be a two-stage

Kondo effect and two corresponding energy scales. Rasul and Hewson (I4) have calcu-

lated the lowest order intersite contributions to the ground state energy. Abraham and

Varma (I9) also calculated intersite terms diagramrnatically and pointed to the possible

existence of a new divergence.

In our article we will use the diagrarnmatical method developed by Keiter and Kimball

and Grewer and Keiter (I2) to do the calculations on the high temperature magnetic

susceptibility perturbatively and to do infinite order surnmation for the ground state

energy. Also we will compare our results with other groups.

Chapter l : Introduction 4

1.3 The Organization of The Dissertation

The dissertation is organized as follows: In chapter 2 we will outline the formal theories

of the Feynman and Goldstone diagram approach to the Anderson model. We will first

perform a canonical transformation to the two-impurity Anderson model and write the

Hamiltonian in terms of projection operators. The diagrammatical rules can then be

derived from the time ordered perturbative expansion method. In chapter 3 we will ex-

amine the high temperature limit perturbatively through the magnetic susceptibility cal-

culations in different regimes. The results will be compared with the scaling theory result.

We will derive and solve the self energy equations at high temperature to obtain two

Kondo temperatures in chapter 4. The integral equations for the ground state energy

will also be derived and solved numerically. Some variational calculations will be done

to verify our results. The conclusions will be given in chapter 5.

Chapter 1 zlntroductiou s

Chapter 2: The Formal Theory of the Perturbative

Diagrammatical Method

2.1 The Anderson Hamiltonian

The Hamiltonian describing a system in which magnetic impurities located on a lattice

interact with a continuum ofconduction electrons can be written in the following form,

known as the Anderson model:

H = Ho ‘l'Hmlx

UH¤=2¤,;,,,€,T,„C,;„,+§;¤;„„f”r„f«„„+7 E mm. mogm lm !.n,m¢¤

Chapter 2: The Formal Theory of the Perturbative Diagrammatical Method 6

11„,,_„_§;[V,;,„c,§,,,5,„6"'? 7) + 61.6.] (2.16)

11761

where 7]’s are the positions of impurities located on a lattice, m and n are the magnetic

qauntum numbers and /7 is the conduction band wave vector. C;*m(C;·m) is a creation (

annihilation ) operator for the conduction electrons and f§„(/§,„) is a creation (annihi-

lation) operator for an impurity at site i . ez; is the conduction electron energy in the

range - D < 1:; < D where D is the band width ofa halffilled band. The band is assumed

to be S-like which means that no orbital degeneracy will be considered. 1:,,,, is the energy

difference between the occupied and unoccupied f state. U is the strong coulomb

repulsive energy which we will take to be infinitely large. This will prevent the f-state on

each site from being doubly occupied. H,„,, describes the mixing of strength V between

the localized f states and the conduction electrons. We will be working in the config-

uration in which the H, part of the Hamiltonian is diagonalized and we will treat the

H,„,, part as perturbation.

As indicated before, the difference between the lattice and the single impurity model is

the intersite interaction induced by the hybridization H„,,,. The simplest Hamiltonian

that can produce this interaction is the two~impurity model. ln the limit U ·-» oo the two

impurity Anderson model with the impurities located at i 7/2 can be written in the fol-

lowing form :

H = Ho + H, ·

Chapter 2: The Formal Theory of the Perturbative Diagrammatical Method 7

”¤ = Z*=Z„„CZ„.C1:„„ + Z=x»·(”m» + num)km

m

H1 = [(V;·mC2'm(f}„„ exp 2 exp 2 )) + H.C.] (2.2)

Em

V;} varies very slowly as a function of I; and m we can neglect its k vector and m de-

pendence. This will sirnplify calculations considerablely while having no significant effect

on the final results.

Furthermore it is more convenient to study the two·impurity Anderson model in certain

regimes by defming two orthogonal conduction electron channels as well as two f

electron operators which are even and odd with respect to the midpoint of the impurities.

We will treat impurities as spin one-half for the simplicity and the results can be gener-

alized to the higher spin cases. Thus we define

1 I dm Z .rak = -——— —— cos- C·· (2.3a)° /}_+(k) 41t 2 kv

1 __

bko = IT sm -*5*- Ck} (2.3b)where uChapter 2: The Formal Theory of the Perturbative Diagrammatical Method 8

_L sin krÄi(/<) — 2 (1ZI:I

fg; = -/*7- (/E., M3,) (Z5)

nec :/:1-fee

noeThen

the Hamiltonian becomes

H = Z£kc(aI:raIcc + bl-ävbko) + 2012:7 + ”oa)cf¤ka V

(2.6)ko ka

where

Vi(k) = ,/li(l<) v= , /é-(1i%) v (2.7)

The unperturbed basis states are :

I <1> > = I 0 >

|e6>=%-(;;;+y;f,)|<1>> I

Chapter 2: The Formal Theory of the Perturbative Diagrammatical Method 9

1„„>=-J? (f§°;—j§,) 1 6 >

|2¤>=ß,;;;,|<1>> (2.8)

I2e>=-JI? (/]°'}f;l+j]°l/Q) |<1> >

<1> >

where the state I <1> > is unoccupied by the f electrons ( _/Q ), states I ea > and I OG >

are singly occupied ( j} ) and the states I 26 > , I 2e > and I 20 > are doubly occupied

( j; ) We can see now that the Hamiltonian (2.6) contains two one-dimensional con-

duction electron bands A and B. The surnrnation with respect to the solid angle oflt.

has been performed in the definitions of new operators. The conduction and the f"

electron operators and basis states are now all defined as even or odd instead of being

labelled with site 1 and 2.

In terms of projection operators the mixing part of the Hamiltonian can be written as :

Hl={Z„/2V+a};[ I <D ><e6I — Io6><26I --1- I0—6><2eIka Ji

+ 7%- Ie —«><20|1} + {11.6.}

I

(DChapter2: The Formal Theory of the Perturbative Diagrammatical Method 10

——x/‘fi=-Io-—o><2eI]}+{H.C.} (2.9)

We can see that each term in the Hamiltonian (2.9) lowers thef“

state occupation number

by one by creating either an A channel or a B channel conduction electron. The

Hermitian conjugate part of the Hamiltonian represents the inverse processes. These

processes are the elementary excitations of the system. The diagrams we are going to

use are just a representation of these excitations. The Hamiltonian has many terms with

different coeflicients and the Feynman rules will be much more complicated than the

ones in the single impurity case.

2.2 The Feynman-Goldstone Diagrams for the Two-

Impurity Anderson Hamiltonian

The perturbative diagrammatic method which we are going to use was first developed

by Keiter and Kimball and by Grewe and Keiter (12) and later revised by Lee and Zhang

(13). The diagrammatic rules can be obtained by doing the perturbative expansion

formally and calculating aH possible contractions of time ordered operators of both

conduction andf“

electrons. We do not intend to go through the derivation procedure in

detail here since it is rather tedious. We will only surnmarize the f'mal rules. There are

two ways to classify the diagrams for the two-impurity Anderson model. One way is to

arrange the interaction terms according to the sites. Each site has its own time ordering

Chapter 2: The Formal Theory of the Perturbative Diagrammatical Method ll

or its complex frequency. The onsite interactions are treated on the same footing as in

the single impurity case. Then the onsite Feynman-Goldstone diagrams on different sites

are connected via the band Green's functions. This method has fewer number of dia-

grams and it is easier to identify the diagrams with the physical processes. The results in

section 3.3 are obtained with this method. It is more convenient to do this when one

wants to do the finite order intersite calculation while including the onsite self energies.

However, if we want to sum the intersite terms to infinite order this method becomes

very inconvenient since the intersite terms contain more than one complex variable and

it is very difficult to write them as self energies. The second way is to use only one time

ordering parameter i.e one complex frequency, so that we can treat the intersite terms

in the same way we treat the onsite terms. The price we pay is that there are more dia-

grams and each diagram may not correspond to an identiliable physical process. The

major part of our calculation is done in second way. Thus we will give diagram rules for

this method only. The diagram rules of the first method can be found in ref (12).

The diagram rules can be surmnarized as follows :

For a diagram which is 2n-th order in Vi, draw a dashed vertical line and distribute 2n

interacton vertices on it. Specify the initial state and its occupation probability. Rep-

resent a singly occupied state jj by a wiggly line , a doubly occupied stateß by a double

wiggly line and an unoccupied state jß by blank. Distribute these lines between the

vertices in all possible ways. Each vertex changes the f electron occupation number by

one and only one. Connect all vertices by either an A channel or a B channel con-

duction electron each represented by a full line labeled by a or b respectively. All full

lines have directions attached to them. Each descending (ascending) A [B] channel line

contributes a fermi statistical factor )§·_(l — and a matrix element V1[V2]. Each

Chapter 2: The Formal Theory of the Perturbative Diagrammatical Method IZ

vertex can have only one full line entering or leaving it. Each full line carries an energy

6,,, and each wiggly line an energy 6/, . When doing this we have to take the Pauli ex-

clusion principle into account, i.e. no two electron lines from the same channel with the

same spin enter two successive vertices. Furthemore, a vertex involving the states j; or

_/Q contributes a factor Ä. The contribution from the diagram to the partition function

is given by :

n-11:%; Ldze‘”' (2.10)

1:0

where s, is the excitation after the i-th vertex calculated by subtracting the energies of

outgoing lines and adding the energies of incoming lines. C encloses all poles of the

integrand. The sign of a diagram can be determined as follows : a) each crossing of the

full lines gives a negative sign; b) the vertices involving an A channel electron and doubly

occupied states each give a negative sign: c) the vertices involving the state I2o > and

conduction electrons with down spin each contribute an extra negative sign. Finally the

product is taken over the initial occupation probability, the matrix elements, the integral

and the Fermi factors and the summation is performed over all initial and internal f and

conduction electron quantum numbers. For an illustration of the rules two diagrams

are drawn in figure 1 and the selllenergy value of each diagram is given below :

2(2.1la)

k

Chapter 2: The Formal Theory of the Perturbative Diagrammatical Method I3

Z2 (Z) (2 ub)b (z + %— sk)(z + 2% — sk — sk„)(z + %

— sk) °Ick'

Chapter 2: The Formal Theory of the Perturbative Diagrammatical Method I4

Chapter 3 : The High Temperature Perturbative

Result

3.1 The Motivation and the Model Hamiltonian

The problem of the two-impurity Kondo system had only been studied by a few groups

because of its difliculty (14,15,17). Five years ago Jayaprakash et al. (17) used a gener-

alized poor man's scaling method (4) to study this model. They have given a very inter-

esting conclusion that in the limit where the RKKY interaction I is much larger than the

Kondo temperature there is a two-stage Kondo effect and the ground state is a spin

singlet formed from the screened impurity spin triplet. Recently Abraham and Varma

(19) have reported ( in contradiction to the results ofJayaprakash et al. ) a possible new

divergence from the intersite interaction in the perturbative calculation of the two-

impurity model. They claim that the two impurities will pair up coherently no matter

how small the initial RKKY interaction is. This controversy has brought the need for a

careful examination of the perturbative expansion to the attention of the theoretical

Chapter 3 : The High Temperature Perturbative Result I5

' community. In this chapter we will derive the magnetic susceptibility to fourth order in

the coupling constant J for three different regimes. In the first regime we assume that the

temperature is greater than all the relevant interactions. In this regime we have found

the well known divergent terms due to the Kondo effect. One of these terms is the one

pointed out by Abraham and Varma. Depending on the ratio of the two relevant energy

scales T?]-, we will have two different ways to introduce the renormalization as tem-

perature lowers. In the regime where T, is greater than II I and the temperature T, we

have recalculated the magnetic susceptibility by including the Kondo screening in the self

energy. The susceptibility then has no divergent terms at low temperatures as the mag-

netic moments are quenched. The most interesting regime occurs when II I is greater

than T, and the temperature T. To make the perturbation calculation more efficient we

transform the conduction electrons into even and odd states with respect to the midpoint

of the impurities as discussed above. The Hamiltonian is then written in terms of the

total spin of the impurities. The magnetic susceptibility is calculated by treating I as a

self energy for both cases of I being positive or negative.

As indicated before, the Anderson Hamiltonian in the Kondo regime, i.e. — a,> > pl",

has the same behavior as the Kondo exchange model. Making the Schrieffer-Wolff

transformation (I) to the two-impurity Anderson Hamiltonian of eq. (2.2) we obtain the

two-irnpurity Kondo Hamiltonian

2 2H„.= Z=,;„~,;, -%Ztsm + sm - é Zp..»·,. 6-1>Ea 1-1

‘“"’

where J = < 0. The spin and the density fluctuation operators are given by

Chapter 3 : The High Temperature Perturbative Result I6

+(—) ..-3) —ßi<1>Ü1<r> (3-2)

+(-) _ zu? - P)- E _°‘ ” Er 'Cl3)t(1>Ck'1<t) (3-3)

E1?

R' Cä„Ci·„ (3-4)E1?

Here we have neglected the f electron hopping term but retained the potential scattering

term which we include in the third term in eq. (3.1). This is the only difference between

our model and the model studied by Jayaprakash et al. lt is well knovm that the poten-

tial scattering terms do not change the low temperature properties of the Kondo sys-

tems, so the outcomes from the two models should be the same. The magnetic

susceptibility for the l-lamiltonian H„ will be calculated by using the Goldstone di-

agrammatic method devoloped by Keiter et al. (9) and revised by Lee and Zhang (10).

3.2 The High Temperature Regime

The magnetic susceptibility due to impurities is calculated in a straightforward

perturbative expansion. This expansion is valid only in the high temperature regime. For

convenience of discussion we shall separate the results into three parts, i.e.

Chapter 3 : The High Temperature Perturbative Result 17

T

8 #3

Here X, involves only the on-site processes, X, includes contributions from intersite po-

tential scattering and on—site spin flip terms, and X, represents a new process involving

the spin flip scatterings on both sites while electrons are exchanged. In our calculaton

we will always neglect terms of order ln 6,/D because 6, and D are of the same order.

The leading and the next leading logarithmically divergent terms obtained for one im-

purity are given by

xi=j,··pJ[l+y+y +y + pJ( /4+4y+TY)] ( -6)

where

y = pJln T/D (3.7)

and p is the density of states at the ferrni level which is assumed to be constant. This

result agrees with the perturbative result given by Wilson (3) and other groups. The

second term for the susceptibility X, is given by

l I ° 2rx. =; {ü—+ <p1>’«·—% yi (6.8)

where

G = kFR

and I is the famous RKKY interaction, given by

Chapter 3 : The High Temperature Perturhative Result 18

_ 2 /i.(1—fP) 1(1-Ä—1?)•E,__ n 2 cos 2aI-] 8 =7(pJ) (3.9)zur

in the limit of large k,R (12). The first term in eq. (3.8) is from the lowest order intersite

interaction. Its proportionality to 1/T suggests this term is the lowest order intersite

self-energy. The logarithmic divergence in the second term is essentially from the re-

normalization effect of Kondo screening on the intersite interactions. This will become

clearer later on. The third term, x,, represents a purely intersite term first obtained by

Abraham and Varma (16), given by

I 2 sin a 2Tx; (PJ) (T) l¤(D/7) (3-10)

At first glance it seems that this divergent term will result in a new energy scale at which

the intersite interaction becomes infmite and the two-impurity pairing will become in-

evitable. However, since the energy scale associated with this divergence is much smaller

than the Kondo temperature, we have to consider the renormalization effect of other

interactions, i.e. we must recalculate this term by taking either Kondo screening or

RKKY interaction as selfenergy. We will see that this divergence disappears and the

important energy scales are still T, and I.

Chapter 3 : The High Temperature Perturbative Result 19

3.3 Kondo Temperature Larger than RKKY Interaction

ln the regime where T, > II I and T, we use Kondo screening selfenergy in just the or-

der of the leading l/N expansion for simplicity to renormalize the intersite coupling.

The result is that the magnetic moments of the impurities are quenched at temperature

T, = D exp? and there are no longer any divergent terms left since the intersite coupling

or the RKKY interaction is now strongly renormalized due to the moment-quenching.

As a consequence the magnetic susceptibility for the single impurity is proportional to

%- (3). The contribution from leading intersite coupling terms to the susceptibilityk

X', + X', gives us

, _ L sin 2o: LX2'- 8 agTk1

sin oz 2 Ix';=—(i) —· (112)4 °‘T:

Eq. (3.11) was f1rst derived by Rasul and Hewson (14). Notice that the Kondo temper-

ature T, is now effectively cutoff energy. The new intersite contribution X', is essentially

due to the overlapping of the wave functions of the conduction electrons participating

in screening the two Kondo impurities. The divergent term in eq. (3.9) is now quenched

by the Kondo screening and the system can be described as two screened Kondo singlets

with a weak residual interaction. They are not likely to pair up in the ground state.

Chapter 3 : The High Temperature Perturbative Result 20

3.4 RKKY Interaction Larger than Kondo Temperature

In the limit that the RKKY interacton has much greater influence than the Kondo ef-

fect, the two impurities will pair up and behave either like a spin-one triplet or spin-zero

singlet depending on whether I is positive or negative. If I < 0 , the total spin S= O state

(singlet) has lower energy ( by an amount -1) than the S= l (triplet) state. At temper-

ature T < < -1 the two impurities will be locked up antiferromagnetically to form an

effective spin-zero singlet and no Kondo screening will occur. In this case the suscepti-

bility can be calculated by a second order perturbation theory easily . If I > T, and

temperature, we will have an effectively S= 1 Kondo problem. The divergent terms dis-

cussed by Abraham and Varma once again are renormalized away after we take the

RKKY interacton as self energy. The divergent terms left are all due to the Kondo effect.

In this limit we will choose the triplet and singlet states of the total spin operator

$ = $, + .3, as the basis states for the sake of convenience. Performing a canonical

transformation on the Hamiltonian (3.1) as we did in chapter 2, we have

Hu = ;81;e(°1i«“1«« + biebze) + 2;**;./'e

— ($+0; + $-0:) — ($+0; + S-0;)

zur zur _

Chapter 3 : The High Tempemure Perturbative Result 21

J (kk') _ _+ ($«T¤„1„ + So Gb) — ZJ+(/</<')¤„1>„.„

6l¢k’1:16

— 2J.(/</<’)¤„ß„., + Z!„.(/</<')¤o„1>„„„ (3-13)6kk' 6kk'

where a,, and b,, are defined as in chapter 2 and

Ji(kk’) = ,/).i(k)Äi(k') J ä ).i(k,.-)J (3.14:1)

J+_(kk') = ./l+(k))._(k') J ä ,/).+(kF))._(kF) J (3.14b)

The operators S* , 6;,,, etc. are defined as follows :

lSi = (S + S5':)

Ü:.= aläakl

aj, = :1,;:),,. + b;,a,,.

paba = "1;b1:·« + bk"6ak6

E = -1- (E fs )0 2 l 2

rr =L (rt + rz ) -6 2 16 2Ü

Chapter 3 : The High Temperature Perturhative Result 22

and

1"Oa =

_"Z6)

The structure of the transformed Hamiltonian shows that there are two 0ne·dimensionalconduction electron bands ( a and b) coupling to the S = l and S = 0 impurity states with

the strength J, and J_ respectively. These coplings do not change the total spin S of the

irnpurities. The transition between the singlet and the triplet is represented by the inter-

action between the bands with the strength J,_. Now we effectively have only one im-

purity with complicated internal degrees of freedom. The diagrammatic technique of the

single impurity Anderson model can now be applied. The self-energies due to the lowest

order intersite interaction for the triplet and the singlet are I/2 and -I/2 respectively.

Thus by taking into account these self-energies the triplet splits away from the singlet.

Transitions from the singlet to the triplet require an energy I. Therefore, in the temper-

ature range T, < < T< < I, each triplet has occupation probability l/3 while the singlet

has zero. Hence we only need to consider those diagrams with the triplet as initial and

final states. The magnetic susceptibility obtained is separated into two parts, i.e.,

Tx 2 „ „(3.15)

8 #3

The constant 2/3 is due to the fact that only two out of three states in the triplet have

magnetic moment. The second part of susceptibility is given by

Ty, = {pJ+[l + Y, + Yi + Yi] + p1_[1 + Y_ + Yi + Yi]} (2.16)where lChapter 3 : The High Temperature Perturbative Result 23

Yi = pJi ln T/D + pl; ln I/D. (3.17)

Here we have listed only the leading divergent terms in X,". It is straightforward to show

that in the limit a = k,R —> eo the leading divergent terms in eq. (3.6) are identical to

those in TX", given above. Thus X", represents the usual Kondo spin·flip scattering be-

tween a S=1 impurity and two conduction electron channels with two coupling con·

stants J, and J_. The series of Y, and Y_ in eq. (3.16) will lead to two energy scales 7],,

and T,,_. Their values are determined by the equations Yi(T,,t) = 1. Explicitly, they are

of the form :

ln 0/1 + 47 T TT = 1 -4- = 1 4 T 3.18ki expI

) t ( )

These results are in complete agreement with those of the scaling theory (10). The pure

intersite contribution in X, of equation (3.8) is essentially the same as what we obtain forX,”,

given by

Txf =%<pJ>’@ Y 13.19)a

As compared to X, of eq. (3.8), the renormalization by the RKKY interacton only

changes the number of degenerate states from 4 to 3 in X,". The divergent term X, in eq.

(3.10), also obtained by Abraham and Varrna, has once again been renormalized away

as suggested ( to the cogniscenti) by its proportionality to I.

Chapter 3 : The High Temperature Perturbative Result 24

Chapter 4 : The Self Energies and the Ground State

4.1 The Unpertarbed States

Using the Kondo Hamiltonian is easier when one wants to do high temperature calcu-

lations and examine the temperature dependence of physical quantities qualitatively.

However, if one wants to study the low temperature behaviour of the dilute and con-

centrated impurity systems quantitatively the Anderson Hamiltonian would be more

convenient and gives more information about the physical properties of the system due

to its richness in structure. That is why we are going to use the Anderson Hamiltonian

in our self energy calculation in this chapter.

As discussed in chapter 2 the unperturbed eigenstates of the two-impurity Anderson

Hamiltonian contain one unoccupied (;f,), 4 singly occupied (/Q) and 4 doubly occupied

(/Q) magnetic states in the conduction electron Fermi sea. (see fig. 2.) The energy of the

ji, state is the fermi energy which is taken to be zero. The states j} andß have energies

6, and 26, relative to the fermi energy respectively. In the Kondo Aregime the magnetic

Chapter 4 : The Self Energies and the Ground State 25

states are very deep in the fermi sea so that they are always occupied in the absence of

the perturbation. Furthermore the energy of the f states 6, has a very large negative

value that satisfies the condition — 6,> > A, where A = pV‘ is the characteristic transi-

tion strength between the magnetic states and p the density of states at Fermi energy.

Thus at temperature T< — 6, the magnetic states are still occupied. However, due to the

strong temperature dependence of the hybridization we could have cross·overs between

the various states at some temperature. We are going to examine this possibility through

the self energy calculation below.

In the single impurity Anderson model it has been shown (13) that the crossed diagrams

have little contribution to the ground state energy so that one can get two coupled in-

tegral equations for the occupied and unoccupied states after neglecting the crossed di-

agrams. However this is not the case with the two-impurity model; the crossed diagrams

are important, in fact they can be leading order diagrams as far as the 1/N expanson is

concemed. Therefore we must be careful throughout the self energy calculation in de-

termining whether a crossed diagram should be kept or thrown away.

Following the diagrammatical rules in chapter 2 we can draw the non-crossing part of

the Dyson self energy diagrams easily as shown in figure 3. The integral equations for

these diagrams are written down below along with the formal crossed diagram contrib-

ution :

2 2(4.la)

ka ka

Chapter 4 : The Self Energies and the Ground State 26

E (z)_ wia-a> + wia.k k

Via Via .k k

E(Z):°°Z·—•£k+80—Zok

k

Via Via (4.lc)k k

2VÄ(1 #2) 2V2(l #2) „= l + *·—·—· + 4.1¤~<=> Z.-.,,..,-., -.,, ¤».<=> < «>

k k

Chapter 4 : The Self Energies and the Ground State 27

V30 -6.) Vio-1;.)ko

ka

V10 —ßZ) V30 -.6Z) ZZi- Zk + Z, - z„„ 4 Z - Zk + Z, - z„„ 4 *‘=«>(Z) (4-*/)ka ka

where the superindices c stand for crossing.

The energy level of each state now can be determined by solving the self energy

equations

Zo "" 2(Zo) =

Owherethe self energy Z(z) can be any of the self energies in eqs. (4.1). Here we have in-

cluded all the irreducible non·crossing diagrams. We don't give the expressions for the

crossed diagrams because they are very complicated. Some examples have been given in

figure 4 which are important to the lowest order intersite interactions and to the mag-

netic susceptibility as shown in Chapter 3.

Chapter 4 : The Self Energies and the Ground State 28

4.2 High Temperature Results und The first Kondo

Temperature

At high temperatures the doubly occupied states have lowest energy. The fluctuations

between the ß, , _/Q and fQ will make the energy levels of all these states shift downwards

with different rates as the temperature is lowered. Ifwe only consider the second order

self energies we will get exactly same results as in the single impurity case since I comes

into play only in 4th or higher orders. At temperature T; 7} the state fg will cross both

j] and f} states and acquire a lower energy. This is the well known Kondo screening eil

fect. This is only natural because we did not consider the intersite interactions which

only come in at or above 4th order in V.

As discussed before we have to include the crossed diagrams when we calculate the high

orderer diagrams. The direct calculation in chapter 3 has shown in eq. (3.16) that the

contribution to the magnetic susceptibility from the crossed diagrams has leading order

logarithmic divergences. These crossed diagrams exist in all orders in W and their con-

tributions to the self energies can be obtained in forms of vertex functions. The vertex

functions satisfy certain integral equations themselves. A diagrammatic example is

shown in figure 5. The integral equations are very complicated and very diflicult to

solve. However there is a way to avoid calculating these vertex integral equations di-

rectly. First if we expand the selfenergies of eq. (4.ld-I) to the fourth order in Vi for the

f; states and separate the Kondo screening terms from the intersitenterms we will get

Chapter 4 r The Self Energie; and the Ground State 29

21/i<1—16.> ZV30 -17:) 1 (4.30)

1: 1:

ZVi0 -17:) ZV30 —ß:) 1X20 = + + (4.3b)1: 1:

where

%0 —f1:)(Vi — V3)2](z) = (4.4)

(Z+¢1:‘%) (Z+cIc_£l¢')l¢k'

is the RKKY interaction. We can see now that there is a difference I between the theV

triplet (eq. 4.3a ) and the singlet (eq. 4.3b) along with an identical Kondo screening part.

This interaction will cause the two impurity spins to be correlated if it is strong enough.

In the limit I > > T, and when the temperature is in the range I > > T > > T, ,the

impurity spins will be paired ferromagnetically and the triplet has the lowest energy

26,- Here the renormalization part of the self energy (first two terms in eqs. (4.3) )

has been combined with 6, to form an 6; ( and we still call it 6, for simplicity ) while the

singlet has an energy 26, + I/2. As the temperature lowers the energy levels of the singly

occupied states ea and oa will shift downwards with different rates and eventually will

cross the triplet. The temperatures at which the crossovers happen can be determined

by solving the self energy equations for the even and odd singly occupied states respec-

Chapter 4 : The Self Energies and the Ground State 30

tively : Letting the energy of the _/Q states be equal to the that of the triplet in the eq.

(2.7) we have

0 (4.sa)

28 -L—[2(2 -—i+ -0 461;f_2oc sf

2)Inorder to avoid the complicated intersite vertex function in eqs. (4.5) we are going to

do what we did for the j} states. Taking the hint from the perturbative calculation in

chapter 3 we combine the potential scattering part (half of the second term in eq. (4.lb)

and in eq. (4.lc)) of the self energies with the spin flip part ( the third term) to eliminate

the sumxnation restrictions on the a'. We use the result as effective selfenergy along with

the lowest order intersite term which is - I/2 for all]; states. It can be shown that the rest

of the noncrossing part of the self energy will effectively cancel the crossed terms

through fourth order. Substituting these effective self energies into eq. (4.5) we have

V1! Vj;°"

‘“ = °""6"’

ka ka

cf- 27:4; + = O (4.6b)ka kv

The solutions for the eq. (4.6) are given by :U

Chapter 4 : The Self Energies and the Ground State 3l

ef- 2p V; ln(I/D) — 2p V;1¤(T,,i}o) = 0 (4.7)

or

.1.ln 0/1+ PJ Tk LTki = Icxp I(T ) Ai (4.8)

We can see that these two Kondo temperatures are exactly the same as the results ob-

tained in chapter 3 in eq. (3.18) The values of Tu depend on the sign and values of' k R . . .-%, which for now is assumed to be a negative number, so T,,_ 1S larger than T,,+

F

which means that the even states will cross the triplet first and acquire a lower energy

as the temperature is lowered. This provides our second energy scale ( the first is I ).

At this temperature the spin triplet of the combined impurities is screened by the even

channel conduction electrons and becomes an effective spin one-half magnetic moment.

The excitation energy of the system will have a magnitude on the order of T,,_ below this

temperature.

4.3 The Ground State Energy

For temperatures below T,_ one can do the perturbation calculation by taking into ac-

count the renormalization due to the first stage-Kondo effect. Then one can examine the

temperature dependence of the physical quantities to get some ideas about the ground

state. This calculation is rather complicated mathematically. Therefore we use another

approach instead. We solve the self energy equations at zero temperature and find the

Chapter 4 : The Self Energies and the Ground State 32

ground state energy directly. Solving equation (4.2) with the self energies of single oc-

cupied states at zero temperature we get the energies of thej] states E,,_E,, as follows:

Eu = 26,- I/2 — Tk_ (4.961)

Elo = 26,-- I/2 — TH (4.9b)

They are lower than energies of the j} states which are 26,- é and 26,+ -3 respectively.

Now we only need to calculate the energy of the _/Q state and compare with that of the

j] states to get the ground state energy. The importance of the crossed diagrams is es-

pecially manifested for the unoccupied state because they are of the leading order in 1/N,

the same as the uncrossed diagrams. So here we have to keep these crossed diagrams at

all times. The leading order selfenergy diagrams for the _/Q state are drawn in figure 6.

These diagrams exist in all orders of V2 and their contributions to the self energy are of

the same order of magnitude. We can do the infinite order summation by using the

two-particle vertex functions as do Read et al. (17). The diagrams in the Dyson

equations for the vertex functions are drawn in figure 7, and these equations for the

two-particle functions A and B can be expressed as integral equations as follows :

° d ·A< ' >1*"'

‘° _Dz+6+6'-E--26,

° d 'ßt · >+ 3p 1/if -42;--1 (4.l0a)

Chapter 4 : The Self Energie; and the Ground State 33

0d 'B ',[1+pVEZsk S"

°" _Dz+s+e'--5--2::]

0dc'A(s’ z)+ 3p :/+2 +2 (4.106)

_Dz+s +6+-;-2sf

where

0 0I I

Z£(z)=3pVi @·%·+pV; ———£·lT (4.11)

°°are the leading order self energies for the _/{ states. We have changed the summations

over k into the integrations over 6,, and the fermi statistical factors _/Q become step func-

tions since the temperature is zero now. We again assume a constant density of states

p for the conduction electrons and take them out of the integrations. We use the lowest

order self energies for the statesj] and j}, i.e. we keep the lowest order intersite terms for

the states _/Q( i-é-) and the leading order terms in l/N for the states _/Q. The extemal

magnetic field is set to be zero. The factors of 3 in the above equations come from the

transitions between states j] and the triplet of the ß states, whereas the other integral

terms come from the transitions between the j} states and the singlet of the _/Q states

naturely. Here we can see that we have two coupled integral equations coming from the

coupling between the A channel and B channel conducton electrons and from inter-

Chapter 4 : The Self Energie: and the Ground State 34

actions among themselves. In the limit R —» oo the two equations in (4.7) will become

the same and essentially identical to the integral equation obtained by Read et al. The

uncrossed-term contributions to the self energy are only the first terms in the brackets

in equations (4.l0a,b) and they are small compared to the last two terms which come

from the crossed diagrams. The selfenergy of the unoccupied state can be obtained

easily as

ZZ0(z) = 4Vj_ZA(z, 6,,) + 4ViZB(z, 6k) (4.12)1: 1:

4.4 The Numerical Solution of the Ground State Energy

Attempts to solve the equations (4.10) analytically would encounter great mathematical

difliculties if it is possible at all. The trial function method can be used in the limit

R —> oo where V,. = V_ = and l= 0 (14) but is not very promising in the general case.

Therefore we are forced to use numerical methods for the solution. Naturally the iter-

ation method was first used but it didn't work. The reason for this failure was that the

large factors (kemels) with the A's and B's in the equations caused the iterated solutions

to converge very slowly if at all. The remaining alternative is to come back to the

nonperturbative matrix method. We discretized the energy 6 and converted the integral

equations to matrix equations. When doing this we have to be careful in choosing the

discrete values of6 because the kemels are not linear functions of6. They have maximum

values at 6 = 6' =0 and fall off rapidly. In fact these maxima can diverge for certain

values of z . The column vectors A and B also have this property. These divergences

Chapter 4 : The Self Energie; aha the Ground State ss

have to be treated with great care. From the above considerations we chose the discrete

s values more densely around s = 0 and cancel the positive divergences with the negative

ones in the integration beforehand to reduce the error in the numerical calculation.

Then the equations can be solved by the standard matrix convertion method. The sol-

utions for the A's and B's are in the form of discrete vectors. They are calculated for

various values of z so that we can find the shape of2„(z) as a function of z and thus the

solution of the equation (4.12). The solutions for the ground state energy E,, are listed

in table l for different values of the RKKY interaction. We can see that for larger values

of I the solutions differ more from the single impurity result which has the ground state

energy - 2T,. The energies in last two rows are very close to those of scaling results which

assume I > > T,. A plot of2,, as function of z is drawn in figure 8. The solution for the

ground state energy deterrnined by the equation z - Z„(z) = 0 is just the crossing point

of the two functions z and Z„(z).

4.5 Variational Results of Ground State

ln the previous sections we have found through the selfenergy calculations that the state

fg has lowest energy at zero temperature. Thus the ground state is a spin singlet. In order

to ensure that we have included the right group of diagrams for the selfenergy we are

going to do some variational calculations and compare with the diagrammatic results.

It is well known that the variational calculation give upper limits for the energy levels

at zero temperature. How close these energies are to the exact results heavily depend

on the choice of the trial wave functions. It is just as difficult to choose an exact trial

wave function as to sum all irreducible diagrams in the diagrammatic calculation. In fact

Chapter 4 : The Self Energiea and the Ground State 36

these two methods are equivalent at lowest order as can be seen below. In constructing

the trial wave functions, the method we are going to use is similar to that of Varma et

al. (22) . The wave function and the Hamiltonian will all be in the second quantized

form. The trial wave functions will be constructed for both spin doublet and spin singlet

states and the energy levels of these states will be calculated accordingly via the vari-

ational eigenvalue equations. Starting from the basis states of eq. (2.8) the trial wave

functions for the doublet are constructed to be of form

I<1>,,>„= aoIe6> +ZaIkbk„I2a> +Za2kbk_„I2 e> 0> (4.13a)It 1: 1:

|o„>„=ß, |o¤> +Zß2ka,„I2a> +Zß2kak_„I2e> +Zß,,,b,,__,|2o> (4.121,)1: 1: k

where the cz 's and ß's are variational parameters and the b's and a’s are the conduction

electron operators defined in eqs. (2.3). We can see that the first terms in eqs. (4.13) are

just the unperturbed doublet basis states and the rest of the wave function comes from

the coupling between the spin·half conduction electrons and doubly occupied irnpurity

states. Let z„ be the energy of the even state, where (cf eq. (2.9))

< ¢• IH I<D > (4_ I4)< <1>„I <D„ >

Imposing the condition

Özw = 0

we obtain the variational equations

Chapter 4 : The Self Energies and the Ground State 37

(6,, - E,,)¤6„=-16 16 16

(2,,, + 6,, - 15,,,)::6,,, = „/-2 V_ao (4. 15b)

(Zee + 816 ‘ Ez6)°‘216 = V-ao (4· 15C)

(Z6, + @16 — Erolm = — V+¤o (4- 15dJ

The energy levels L,, L, and L, of the doubly occupied states would split from their

original value 26, due to the RKKY interaction at high temperatures as discussed earlier.

This effect has to be accounted for when one wants to calculate the low temperature

properties. Therefore as before in this chapter we shall set L, and L, to 26,-% and

L, to 26,+ -é- ( we shall do the same for the singlet state wave function calculation later

on ). This is equivalent to neglecting all high order contributions while adding an ef-

fective interaction I(§, • S, -—ä-) to the model I-Iamiltonian of eq. (2.2). Substituting eqs.

(4.15b) (4.l5c) and (4.15d) into eq. (4.15a) we get the eigenvalue equation

sv} v}2,,, — s,·= ··'—··——·—i'T· + ····—·—··—··l·i· (4.16a)k

z„+s,,—26,+·i·R

2,,,+6,,-26,--E

Similarly for the state Ioa > we have

Chapter 4 2 The Self Energie: and the Ground State 38

3:/* :/Z(4.16b)

IC Zoa'l"8k"'28f'l"'i'k

200+8}:*28/*7

We can see that these equations are identical to eq. (4.1 1) obtained through the di-

agrammatical self-energy calculation.

The trial wave function for the singlet is constructed as

a a

ka ka

(iakobya I 2(7 > +“'L (aktbkal Ze >)„/7 ., „/Y

Ick'

(ßzkkvaktakrl "' ßgkk-bktbk-1) I 20 >(4-Ick'

where the first term is the unperturbed Fermi sea singlet, the second and the third terms

come from the antiferromagnetic pairing of the spin-half impurity doublet and the spin-

half conduction electrons and the last two terms come from the coupling between the

Chapter 4 : The Self Energies and the Ground State 39

doubly occupied impurity states and the two conduction electron channels. The vari-

ational equations are

Zcac (4.kaka

(zo + ek 21:10 V+ (4.18b)16 16

(Zc + Bk ZacV__1616

( 26 i — „/T V V 4 1820+%+816* j+2)ß11u6—* (-*11:+ +*216) (· dl

I(80 + 81. + 816 - 28;- 3· >/*21.6 = — —/7 V+(alI¢ + 8116) (4-188)

I(zo +6f+ sk.- 2sf-7)ß3„.= -„/?V_(a„+a2k.) (4.18])

Deline

„ =JL „,,_*82«„V+ 2* 261,, V_

and

/;:1616 -ßllclv " Zac

Chapter 4 : The Self Energiu and the Ground State 40

where i runs from l to 3, and substitute eq. (4.18d) (4.18e) and (4.181) into eq. (4.18a)

(4.18b) and (4.18c); we get

II, = 22 Vialk + 22Viazk (4.19a)ka ka

+am +°k flll

Zo+€k+8k4"28f+?

++ vii--E-’¥-i¥’——T) (4.1%)ll

Z0+8k+€k:"‘%8f*'i'

1 oz + oz«z~=——.„+..-.,<¤+3Vä2l-‘””‘

Ikl Zo+€k+€k·*28f+'i

+ (4.19c)

lllHerethe u,,, and (1,,, are exactly the same as the A and B in eqs. (4.10). We can see this

by moving the (1,,,, (1,,, terms on the right hand sides of eq. (4.19b) and (4.l9c) to the left

Chapter 4 : The Self Energie: and the Ground State 4l

hand sides. Then those terms are seen to be just the lowest order self-energies of eq.

(4.11) for the singly occupied states. So once again the two methods give us the same

results. The calculation in this section confirms that the _/Q state has lower energy than

the _/{ state. The difference is of order Tx,. Thus we can confidently conclude that the

ground state is indeed a singlet.

Calculations on the two-impurity Anderson model have also been done by other groups

(18,23,29) using different techniques. Jones and Varma (23) have used the numerical

renormalization group method to study the behaviour of the two-impurity Kondo sys-

tem. By examining the "flow" of the model 1-Iamiltonian they derive an effective

Hamiltonian at low temperature in different regimes. In the limit II I > 7, their results

are essentially the same as that ofscaling and ours. In contrast to general beliefhowever,

they found that the system still behaves as a correlated Kondo system even in the range

II I < T,. This means that the ground state would be quite different from that of the

single impurity even if the initial intersite interaction is small at high temperature, be-

cause it will become bigger and eventually diverges at very low temperatures. In our

calculation we find that the originally existing divergences for the intersite interactions

disappear after taking renormalization from Kondo effect into account as discussed in

chapter 3 and in refi (24). The residual interaction between the impurities is still small

so that the correlations are negligible. Hirsch and Fye (29) have calculated the magnetic

susceptibility and the spin-spin correlation function by using the Monte Carlo method.

They show that the two impurities are not likely to form a spin triplet in the limit

I S T,, which is in disagreement with Varma's results. This difference is yet to be un-

derstood.l

Chapter 4 : The selr Energics und the Ground state 42

Diagrammatical techniques have also been utilized by other people (14,18) in ap-

proaching the two-impurity Anderson model. Coleman (18,26) solved the problem first

at the mean field level and then considered the fluctuations by using the l/N expansion

method first developed for the single impurity model. He proceeded to calculate the dy-

narnic and static magnetic susceptibility at the 1/N level. He also derived the Fermi liq-

uid propertities of the system which were first discussed by Nozieres (25). Since the

RKKY interaction for the impurities is of order of l/N2, its role is not manifested in

Coleman’s calculation. More discussions on the two-impurity model can be found in

two recently published review articles (27, 28) and references therein.

Chapter 4 : The Self Energien and the Ground State 43

Chapter 5: Conclusion

We have successfully applied the perturbative diagrammatic technique to the two-

impurity Anderson model in the Kondo regime. Naturally the complexity of the

Hamiltonian makes the diagrammatic calculations more complicated than in the single

impurity case. The emergence of the intersite interactions, mainly the RKKY interaction

gives one new physics and makes the system more interesting. The system behave dif-

ferently depending on the strength and the sign ofthe interaction between the impurities.

If the magnitude of the RKKY interaction I is smaller than the Kondo temperature T,

of the single impurity system it has little effect on the properties of the system since the

impurities will be screened by the conduction electrons first and the intersite interaction

only gives a negligible contribution. If I is negative and the magnitude is large, i.e., if

the interaction is antiferromagnetic, then the two impurities will pair up into a singlet

and Kondo screening by conduction electrons will not happen. The reason for this is that

the magnetic interactions between the impurities and the conduction electrons which are

responsible for the Kondo screening are absent now because the impurities themselves

have already formed a singlet and have no mametic moment.A

Chapter S: Conclusion 44

The most interesting case is when the RKKY interaction is strong and positive. In this

situation the competition between the ferromagnetic coupling of impurities and the

antiferromagnetic screening of conduction electrons is in favor of ferromagnetic pairing

at high temperatures, so the two impurities will form a spin-one triplet first. As temper-

ature lowers the impurities will be screened gradually, first by the odd-channel electrons

to form an effective spin one-half particle with complicated intemal structure and then

by the even channel electrons to become a spin singlet. This is the so—called”

two-stage

" Kondo effect. There are two energy scales besides the RKKY interaction associated

with the two-stage Kondo screening. The first energy scale can be obtained from the

high temperature perturbative expansion series for magnetic susceptibility as discussed

in chapter 3. lt is just the temperature at which the susceptibility diverges. The second

energy scale and the ground state energy are obtained through the zero temperature

selfenergy calculations. A set of integral equations is found for the ground state energy

by including the l/N leading order diagrams of the unoccupied state and the renormal-

ization due to the first stage Kondo screening. Unlike the single impurity case the

crossed diagrams are important and result in two coupled integral equations. After

solving the equations numerically we found that the unoccupied state has lower energy

than the singly and doubly occupied states at zero temperature, which means the ground

state is a singlet, i.e. the spins of the impurities are totally screened by the conduction

electrons. We have also constructed trial wave functions for each of the states and cal-

culated the corresponding energies. This variational calculation confirms to the di-

agrammatic results and shows that the ground state is indeed a singlet.

Further applications of the diagrammatic methods to the two-impurity Anderson model

for the calculations of various physical quantitiess like magnetic susceptibility, specific

heat and resistivity at low temperatures would be very interesting and the calculations

Chapter 5: Conclusion 45

should be straightforward. The intersite contributions to the properties ofheavy fermion

superconductor systems can then be compared with the experimental results. The ex-

tension to arbitary spin system should be practical although the calculations are more

complicated due to the multiplicity of the tota1·spin magnetic states resulting from the

coupling between the impurities. instead splitting between the triplet and the singlet due

to the RKKY interaction in the spin-half case one has to exarnine the energy differences

between all the states due to the RKKY interaction and consider the screening from

there. One of the drawbacks of this diagrammatic method is that it is diflicult to extend

to the lattice case because of its complexity.

Chapter 5: Conclusion 46

References:

(1) "Valence Fluctuation in Solids " ed. by L. M. Falicov, W. Hanke and M. B. Maple

( North Holland, Amsterdam, 1981)

(2) Proceedings of The Intemational Conference on Valence Instabilities”

ed. by P.

Wachter ( North Holland, Amsterdam, 1982)

(3) G. P. Stewart, Rev. Mod. Phys. 56 , 755 (1984) and references therein.

(4) P. W. Anderson, Phys. Rev. 126 , 41 (1961); B. Coqblin and J. R. Shrieffer, Phys.

Rev. 185 , 847 (1969)

(5) J. R. Shrieffer and P. A. Wolff] Phy. Rev. 149 , 491 (1966)

(6) J. Kondo, in "Solid State Physics" Vol. 23, eds. F. Seitz, D. Tumbull, and H.

Ehrenreich (Academic Press, New York, 1969) p. 183

(7) K.G. Wilson, Rev. Mod. Phys. 47 , 773 (1975)

(8) N. Andrei, K. Furuga and J. H. Lowenstein, Rev. Mod. Phys. 55 , 331 (1983)

(9) F. Steglich, J. Aarts, C. D. Bredl, W. Lieke, D. Meshede, W. Franz, and H. Schafer,

Phys. Rev. Lett. 43 , 1892 (1979)

(10) G. R. Stewart, Z. Fisk, J. O. Willis and J. L. Smith, Phys. Rev. Lett., 52 , 697

(1984)A

Rermnm: 47

(11) H. R. Ott, H. Rudigier, Z. Fisk and J. L. Smith, Phys. Rev. Lett., S0 , 1595 (1983)

(12) H. Keiter and J.C. Kimball, Int. J. Magn. 1 , 233 (1971); H. Keiter and N. Grewe,

Phys. Rev. B24 , 4420 (1981)

(13) F. C. Zhang and T.K. Lee, Phys. Rev. B28 , 33 (1983). F. C. Zhang and T. K. Lee,

Phys. Rev. B30 , 1556 (1984).

(14) J.W. Rasul and H.C. Hewson, Solid State Comm. 52 , 217 (1984)

(15) P. Colman, Phys. Rev. B28 , 5255 (1983)

(16) T.V. Ramakrishnan p. 13 Refi (1); P.W. Anderson p. 451 Re£ (1)

(17) C. Jayaprakash, H.R. Krishnan-murthy and J.W. Wilkins, Phys. Rev. Lett. 47 ,

737 (1981).

(18) P. Colman, Phys. Rev., B29 , 3035 (1984) P. Coleman, J Magn. Magn. mat. 52 ,

223 (1985b)

(19) E. Abraham and C.M. Varma, The Theory of Fluctuating Valance State, ed. by T.

Kaysuya (S pringer-Verlag, New York, 1985)

(20) N. Read, K. Dharamvir, J. W. Rasul and D. M. Newns, J. Phys. C, 19 , 1597

(1986)

(21) P.W. Anderson, J. Phys. C3 , 2436 (1970)

(22) C.M. Varma and Y. Yafet, Phys. Rev. B13 , 295 (1975)

(23)B. A. Jones and C.M. Varma, Phys. Rev. Lett. 58 , 843 (1987). B. Jones and C.

M. Varma, J. Magn. Magn. Mat. 63&64 , 251 (1987).

(24) C. Zhou and T. K. Lee, J. Magn. Magn. Mat. 63&64 , 248 (1987).

(25) P. Nozieres, J. Low Temp. Phys. 17 , 31 (1974).

(26) P. Coleman, in " Theory of Heavy Fermions and Valence Fluctuations " eds. T.

Kasuga and T. Saso ( Springer Verlag, Berlin, 1985c) p. 163.

(27) P. Fulde, J. Keller and G. Zwicknagl, Preprint.1

References: 48

(28) P. A. Lee, T. M. Rice., J. w. Serene, L. J. Sham and J. W. Wilkins, Comments on

Condensed Matter Physics, 12 , 99 (1986) _

(29) J. E. Hirsch and R. M. Fye, Phys. Rev. Lett., 56 , 2521 (1986)

References: 49

Appcrzdix

A. A sample of the integral evaluation6 +0

I sdrkk' k:

{Pik @:29 fdkläßMM TM mlév F)/W (/2-/2,;)}:2

E

zzpj MU! (k+k,=)(9v~lzR¢¥¤2„ÄpR+C<¢2]zl?9<#~]2,„l?)

Im é'(l2'+Zkp)(k'+l@,=)R'<„=

where

Expanding around kbä 0 and keeping the lowest and next lowest order terms only

_ IT: _Q {dk, cakpklwhk Mswkpßiwzkk'“·=Lg I2 2 hR

Sirnil l : . , Tar;_ Z

"Fhg‘^kR= Ä [-lT.caz,kpR+9‘^”}?Pk—Q«“?¥]

= ‘ nz geh-eppmz '<~" 2References: 50

B. Diagrams Contributing to eq.(3.8)

0- Q" (T O”

(1 ) (Z )

The diagrammatical rules used here are given by Grewer and Keiter in reference (12)

U) _ 1 _2 1 ‘PQ-?} 1

2.111_

1 z I _ 2 I:— 2_11,*(逗,'1J,,) O' 2 I

where —"_E _ ;_

) Z,,1- €"‘

Z 1 vihz vII

Il ·1·I11 ··€‘(,_

Ia kk (1% ‘·‘I‘1·)

I ZT 2f,. vid, vbw?1' 1 (6,-.11,)* 61,* M 16,-6,111, O1,

Here we have used the condition I¤,I > > iw„ for small n and a,=k,R

·'* 1 .>‘2"° 1 — I 2*11} 1 - = =: — —-~ =—

-~ —— : ... I‘\ O'Z (5;+1*PiiuamI

I 8T‘ °

where

Appendix 51

~— , Q} 0*o· H Z e.p6;, 2 { 4-Q']

$5t2> .. 2 *2- *'•w _·f [d LL.i..Z #1%, '*rÄ;7JV€

‘5, k ( 5.+€·«·-F~*„)2* };+¢r'“*¤*‘°k‘6„9

__ __ , , - Mna=·2nv 2 9%-J-) ·2<+6,.-„ 2 zi 6, ,-+6,-,u,,) ( ,6}-+6-,- „)

6‘ ‘

.. - -. 2 V { wol wol —=·ß L) **52.lcm-

lzjzak [6,-6,) 6{ (€k·£;+L~f—6’,«)

The leading order contribution to the magnetic susceptibility comes from taking the first

order derivative with respect to n, (rr,.) and 6, (6,) once each so the contribution from the

second term disappears.

2 zflfk 9«}td„9«Äro*,_ Vqi g' *’ M-u' -6* >'·*

*1*-___1 6 '-.. zf,{ £‘«Ä¤¤l,$<;w* Va;. —I

kl{

( | l) y 2 Ü éß-°é'+éa,-é·d„'

lntegrating by parts in the integral respect to 6, and using

9+L _ 9 I _6?ä‘¥-·—‘,„--— ‘ *’“

**Appendix 52

we have6

‘ x

T _ 1 >._2PV1Cp$<^d;9<^6I{Za: Z0-n'

hk. ¤(;<>?2 (-*2) *k 6;

T: —— Ln°*p

C. Diagrams contribuiting to eq. (3.10)

WGJ 0-I :7 °_;0-(T

U- U- O"

_ 2 “’ro" nz ° 3*

[TE Y } 3~( 54-6;·¢‘M)($+*« -”·%*'Y*’¢"*’s)(6**«"’¢""""*}"/*)

: li (n°°"nO'92ä ¢ 2. — ‘ 2.2 4-o·' fl 6;; ß (#‘V:""‘(2"éo’+éo".)

The factor 1/2 come from the rotation symmetry. It is a good approximation to neglect

(w, — cu,) in the delta function since it is very small ( of the order (6, — 6,).)

Appendix 53

Z= -26 2. (^6·^6') 2. *-6*L6-6·'fl 6]; Iz,k2 <>€•<>¢2(‘é,+62+6„·<;·)

2. ¢I·‘ I

W, _ V Sut¤l.$‘·^¤Zt z_ 26: 1 z (nz@)6-6*" 6; mz, 6; <>l»°‘z (6z‘6«)

’I .. A20} gl/4 __ 8 I:-2,- gp

0,2

. „ _

. .-.1 é——IßI(PI)l$“I"ß»t?2 6,2 6. 2*12*; éz 2 oz; D

D. Diagrams contributiong to eq. (3.11)

- 1°:zVz.- $°<2>= 1-%—6

— I ·* ;g°I”g,?„)) fol}-J¥' )Z

... 2 -· *¤’ r

n (I-S' )I(E„-66+***01

where E„ is the solution of the equation E., — S"(E,,) = 0 Since we are in the regime T < <

T, we can neglect the poles at iw„+ E„—-6, because IE.,-6,] ; T, > > T.

Appendix 54

T 6*0T

V0

k,Thepartion function Z has to be divided by Ä = (n„e·”¤)' which is the single site parti-

tion function renormalization due to S,. Using

„ v’

I

QIE u aw L - - - »-i’ :-— —— Ä ,*6 — I

wehave

: — —- Z — 2"‘

{5hlck

ol; (•—$°°)z 2H} E¤‘°e6·

‘ 2.

¤’ °lpl(l*$°')z (F.*6•·)l (E6*¢¢)$

: 5- .l|Hz )

Appendix 55

E. Diagrams contributing to eq. (3.12)

Wu

0' 3, °“‘

.ä 2

WuO‘ <r'

..-’*

- ,· - .. w -141 x2-; Z-[lrnfnß

(4):,+ 3 0-)

0’d"~ ../93-,

;E(ui;) Ä"}, ($,+~‘r„--fw, -$°)( 3*,+*;-I‘W¤+"W;"~’=¢')(}¤+€'¢"'”•+"*(z“'“§)

€~ß}·2.X },(},+£‘;'-wh ··$°)(3'1+*t'q-""‘)z‘*'*~)¤ éo·)(Ü•"'*¤” "‘)¢" u'

z +wI

"—z

E. '(I.1„),‘+

I gie! céud

vo" n EI IA!. <>l,¤I, (I-SI

· - Z Ro E. I *j;·—Z “ (Id; <E„—€„)(E,·é4·')

The approximation made here in frequency summation is the same- as in appendix C.

Appendix 56

~ I 7.x_1a*?_2l'$¢^¤lTx Zylß <}H* Z, 0' 4- cl 1 (_E,—F„-) o·' >H* E°..(.U,,

„ 1 1 z zEa é

2 ol z (E.*€‘«)v' 1E,—é-,)* (E.—é,)’

_ 1 mit TQ14a*(2.-6,)* 4¤<* TK

F. Diagrams contributing to eq. (3.16)

The diagrammatic rules used here are derived from the symmetrized two impurity

Kondo Hamiltonian eq. (3.13). The rules are essentially the same as those described in

section 2.2. The only difference is that all singly occupied stats are now eliminated, i.e.

their contributions to the energy denominators are taken to be constants 6, which com-

bine with 2V* and become coupling constant J. lt is illustrated diagrammatically as

Iä X "é; +

Taking into account the renormalizatoins due to the RKKY interaction the energies of

the four doubly occupied states are E,, = E,, = 26,- é and E,, = 26,+ é- . For the sake

of simplicity we will represent state 21 as I, state2e as 2, state 21 as 3 and state 20 as

0. Here are some rules regarding the factors and the signs to a diagram we want to state

again. a) Each potential scattering vertex (p) gives a factor l while each spin flip scat-

tering (8*) gives a factor -1. b) States 2a ( 1 and 3 ) give a factor 1 while states 2e andAppendix 57

2o ( 2 and O ) gives a factor 1/2. c) Each crossing gives a factor -1 and each p,„ gives a

factor -1. ( All diagrams obtained through the exchage a = b and 1 = 3 from diagrams

drawn explicitely below are included in the calculation.)

The third order diagrams

5+2G-I

‘I' 'I‘ + ___ I6· <r* -— a+¤I<~6*+6*+6*r -2 W

| 2 xx x/

I ID+ + + I I

° : J- a,bII>I6*+6M°6r = 3 6*+I 2 Q-.?I PM CL?

II; I ( Ia alb bIa, ala bI6I,6*IIb+)2 -

2x·— 0+ L 6 I°’

6 ¢+ - xx xx Qggzx

I I

= J. ( 06 W )2 6% + al

bl, bt

I + + + + IGII af)

| + + ..- .. a a + 6*+6*+0+0 W - 6* -6*I

I

Appendix 58

I 6*+ 6%0 --L +6161+Cl a+b +b a+ I>+0,, =iO

at 6%

6% ‘ß 6-% 3 7° ‘I° {3:-Linie -—-— )·.,.j;3[.2..S_..2 ,7- r (7-FL Ö,3·

( + )(3·+L~z—é,}{}+é,-61-207-1)

- , · )l/· ) ..= - -5*6;* ,-¥6H)Ü"€* ,'6°I·•) {}+&—61—Z°’*‘}(«?+¢ä·6;·2¤I+)

Yg

++-§·(:—I$)f2.‘)c3_L

~f, (1-1%)(/~‘I°s)

fl -L) Z (H62+, -26/+ ·1)(3+62—67‘2'H-p

_n- PZ{ s -3 zi,-15)+21% -= -3=—— c +1- X2 (Ä

2 2 (I-15)f2f; l(6 ‘67)(6;+7-26H *1) (6,-6*,-za/~1)(lg-6,-2;H—I)

Here the second term and the third term in the first braket are canceled which is appar-

ent after making the exchage 6, —> — 6,. Similarly, we have combined the second and the

third term and canceled the fourth and the fifth term in the second braket. The leading

order contribution to the magnetic susceptibility comes from taking first order derivative

with respect to nf and -2oH in the energy denominators once each _as we did before.

Appendix _ 59

2 = ——;pe : J 2 ;

°_ " Q 1 E 2. ‘@ - 2.BMC

%*+M€ 2 SMCJSI/‘

1 .. -1 ..1zPM 4-· äß V J JJ

(él-6;) (-6-, -1) 6, (6-3 -1)

: gi H]‘,J+]’j),Q„% +2Ö-+ )°.+)‘.1X»)ßn—g—L«%—+(I,‘JZ+]'jI)ß-^l-g-]

Fourth order diagrams

a) Diagrams contributing to 11+ Ji.

[ O,- QIIZ 6l?z„<·<> PM a„*3+=%I

1* Q, rII

IPp (1I„ 1*

cr"I

aw+-1 .L * : L L8 ¤+

$6+ (lg,M

Appendix 60

‘-2 6

111@1

· 61* )

81 61*

I2 1 öfäzévß

·

8“1“6¢‘,1¢11 61411 1

Zxl 1 +6 %„%ß1„¤141I41,41111*

1l·=-L

I1 1

4 1-

1

1443

L"2J1

& 4* 4*"

I4

I

3.1

1% 1 61611)

I :...L(_4‘ 1

4—

L

1 **

1 )

4 11 I1

Avpendjx61

I IW +8-2 _ Q 2- 8- 4 -4 ;@_ de_Q__ (I··I°I)I°zI°?I4

{E-

2 Z I(ä+<*I·€)I3+é;—¢7)(ä+¢·1—q—2¤n)II PX g_ _ ·-fQ __ 1 -4 e E(e—'f;)(I'f1)+$ I.—— 3* - .> I —— ä2

ébndÜ)—•

‘-‘l*+

_Z_

{I-f}Q(/**7c2)1‘e"f4 Ä(€'e‘§)[<-‘u**¢;)(¢‘«·¢‘;+¢·u·¢*1·Z<'*')

zig 1} __ß JW go

JßézfJ +

)I>(6>‘¢*e)(é;·¢3)l·¢3) (6;·lv)(£—g—é)c—„

_ {eff (l'°f$)

:—§P(3’+I+7.“)

(1-%- -¤)bag

I2 Fg —pe

P6- Z Y IL IQ'Z=——-· I 7. --0-* —————-28I

+I,Hä

I3-+¢—¢$,·2¢I·I) (}+¢7·¢§)(3·+¢‘e—¢¢·Z°'H)

___@ ’~'+ -[I [(I—h)I<1'I;s'I¢ flI,-TLM;-iiI(e-f)— ‘r EIIMUNL)

(61*%-4/+)I‘>·‘*·)(‘*«“%*Z"*}+f~'¤—%·Z«Ie)I¢T¢;X6¢‘«4;H)I‘

, _ -f„)3‘13° P Il 4 2 $7-I‘* ésäm °ägä! "#”*I-°II@% ‘%"’? *’·)"? ”’^

6

Appendix 62

I2:

__I

2 __2¢I=

1( *+ß 5

I(61%)(¢‘z·l‘z*¢’+} (* Ö;)

- zß *+ ‘*1 1 1 * ..

' Im2 € — V V—V=—( - —: d

I

JVAppeudix 63

X‘Fs‘Vs¢ __ lv-f;)(!—f1)7°a7°¢

(64; ‘6'Z4'H)(6•;'6z ·2¤'H) (63 ·¢‘, -Z¤‘H)(6§-6,+6;-«g -20/-I)(@„—é,·zo*I·;)]

'* _"P aw

56s6;(¢·;—¢·1) (6s·6z)Ié«—<*1) 64 I

ziß ‘* —‘* 1-; ;-;

IZ Q 1 4; -

_§__ ..._l_.._....

ÄZ: °;I¥m~(7++);4)(‘£)j-dä

ä [(}+65'6»'Z°”H)(3+6v·6·Z¤H)

1 X I](3+63 -6-, +6·«;-63 —4<rH)

2I;-1c!)

)( 6;+44 -6. );-. %3CT+°+T-4) (-;)l„

ZZ

The total contribution to J1 + J1 is

31:2; (T+a+T.u) bg

Appendix 64

b) diagrams Comributing to 121. + ./21+

bw 0% v

„+I Ib 4,. 6%

I

I = 2% {aiaw/aiai/alßyßlw2l

_ Iat

_Lab

- lLu

+4+ 6%+bwb+ bv

I I

I II I by Q4.

Q4: 2+ QI awQ4.

al- I>+

I

6 ‘ 3+I2

Appcndix 65

1 I 1 1 1 10 0 0 2 o 01 + 1 + 2 + 6 + 6 + OI 2 I Z I 0

1/ - +1Z; = 2Z

gi 5*

I3+6• -Z4-)( 3+ ‘Ü

.1..1.(3--1--1) 2(3+é-61,-24-H)

(3+ 6;-60 —2r1·{-l)(3·+§~6,·1·#) 2 (2+6-6;-

-1-2( 3+6, —@ -261013+6, -6, +6; -61, -261-1-I)( 2+6, —6„ -2611) 2( 3+6,-63 -;,11 -1)

X / + -1 -1(3+61-6;+63 -6,, 'ZTH -1)( 2+61- 6-1/)

.1.-1)

=_ß _ 1 18% 3 2- _ 3

Appcndix 66

2+ 2

-I ) 2 -26/1)

-3 -12 $11* Q (1*1+1* L1

ff-6-;)

X -1-

-/ -IX 1- 2( 6%-6;-2«H)(61-6; +61-% -21/+*1)/% % ·>-6/U 2/61-61

Z(6,-6*, -2r/-1 -1)/61-6;-2611 -1) {6-,-6-6)

.. 2 _) '

4/6-,- -2vH -1)

21*6 2 if "*

“‘

4ß- Z6· 2 (6-,-1)/62-1) (6,-1) 1 + Y

[6,-1)

2(-6-;)Z

'I) Z

U1Appcudix 67

-~’1ß— -2 -2- 1 f

’ ’ >‘

1 3 1 21 , 3 1 3

The total contribution to 111_ + 111+ is

1 3I7 Z gä (7+;]-+I.%I+)($·(I«^% + M6)

c) diagrams contributing to 1111

I 4+—I·

+ +

Z : 7;- 0.,¢l„Q7;I>IbIb+bv“¢ = {b1;.71 W ‘°

I Ö1,

Appcndix 68

L Ä — {>¢b+“~»“1ß»“ ÖLWa + s —- zxl [ E-

°‘2 O - 4 + + fb bf

1 M2, =2*1

{43 * 2 { { * 4

b1

O{*Z)I

ZX O6

b4 a6 bf ac*91

Appcndix 69

Z * "' ·· +

a1~ ab ag, M

1

äo

“+ L ab ag,bg, b¢ a¢ bg, bt

= O

*~ 1“+ 3

géub 1 1%3 = _g_ { éézäß ) :LZi

1 M

Appendix 70

I 66

;> 8 61, BL 6 I**6

I° 4 * * * 6*61 4*6 6*4 *6 6*612:8oI 0,

4 0:66

6 _ 8

bw bf 4 QV T 171L

: 2 -— 4* 4* =

Ä 8 1 .L1%

V? 4I 1

I Ä- J.

6 *1 .. 6:6,10,:68 bzßgtß ) 0, 8*6

Appendix 71

I I I 1g .1 f 1 g + g5:l

0 0 0 7*

1 1 1 1

- 3fd ii {

I ÄÄ-1)X

-1) I)

E * M ** -*1* ++11-111-1) ·/3 1 sf —L.+ 4¤;11nf€(FL U-

Z ((;+¢1·l;·Z1H·Y)

X3+ 61- ¢·4· -14/1J)

2- T $11, 6 (1,1-+7. J”+){,I Tüll 1‘s)(1 1%)

../3 __ ; PIA. _; 1. ,).-7-)

Z?

I)

——-—————————' 1Q; : l Ä X 4

+11—1‘z)l1—1‘s)

P’"’

° 3

°’Appendix 72

f1f;I1·f<1) _ 2/3 ..2-2 -2 ingI 2

/4..

L’

1 *2 LZlp·‘I

I 1 1Z 0 0 00 *I’ Z -I· 3 + O2 1 2 o1 I I 1

_ px -,53E

I)(4

I

{IT“1'

g" g

„„fh -1,1

-1*1) ("1*‘?·2¤I·I—I)I

3Appeudix 73

EL flfz5rh (6,—(?)(¢‘z—(‘g·!)‘z (~,(ét-J9((·1‘(’¢)

+gf, («·f;)(/—fu)

+2 {, I1-f;)(wf¢)

+ fi fdwf:)(-6,) ( Cu -¢ä ·(·¢ )(· ég ·I) (6, ·I) ( 6-*;*%) ((2*660 (4*9 @(4*1)

+ ·-{ f,£(»—1‘¢> 4 -g fn"; (wfg)}(6:% éwéw -1)/6-1- éwl) ((—,—<;—1)(6.-I)( (rl')

1. 1 _2. 1 [ 1 l * __gk Z(ékZs

Ä 6 Ä* 2 Ä Ä 3 Ä

ggg - g ijg}

The total contribution to .1}.11 is

ZP -1 1 -2-1 LIjk: ···§··

()+)-+J-J+)XAppendix 74

66*

|¤,1 6

<#ZE #,6*

b QV b

O1 26*

(M (b)

lF{ä_ 1

é é

EF'O

O'

(a) (b)

é

—•- —•- efo' o"

Z76

20 kz,0 ho_

77

0 rw 2cr

2

a2.0* b

2o' €O’

2920 Iz,-cr

26 b2.0 Q

20* 2o'

Fig. am78

hl4) T za

UL

b20*

OO"

ZE Wb

OG" OO"

Pr;. acc}

eo.k,0* Iz,0

00*

20* *7 00* Q

20* 2.0’

*:*9- BM)80

Ä -0* }z,·o*

eo- I 00*

60' 00*b 61

26 29

Pfg. am81

26*k,—6* °°’ lz,—o*

26 aoo*

2o

b

2o

95.61+)

82

~

~

0,//

O_|

,0-/ ‘l" o·'

o‘<r'

b0*

O"

qb 4>

F,‘9_6

85

h,0·kp-

= -|—A

a

A + n§¤' A

A

b

VFj.7(«)86

A

bb

hy . }z,o*

b a

FF};. °!U>)87

II

I

II

III I

I

I II II II II II II II I fII)= 3·

III

I

III II II

88

II

6.,6 6.6666-,,.6,6 -,.66,,6, .,.66,6 ,,.66,6 -66,, -,.660.4 0084 5_0008I 0.0054 -060084 -1.6l

D=22.22

.2 .6 . 664k,a O T" 00

Tama I,

89