Equivalence of the perturbative and Bethe-Ansatz solution of the symmetric Anderson Hamiltonian

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Equivalence of the perturbative and Bethe-Ansatz solution of the symmetric Anderson Hamiltonian

B. Horvatié, V. Zlatié

To cite this version:

B. Horvatié, V. Zlatié. Equivalence of the perturbative and Bethe-Ansatz solution of the symmetric Anderson Hamiltonian. Journal de Physique, 1985, 46 (9), pp.1459-1467.

�10.1051/jphys:019850046090145900�. �jpa-00210091�

Equivalence ^{of the} perturbative ^and Bethe-Ansatz solution of the symmetric ^Anderson Hamiltonian

B. Horvatié and V. Zlatié

Institute of Physics ^{of the} University ^of Zagreb, ^{P.O. Box} 304,41 001 Zagreb, Yugoslavia

(Reçu ^le 12 février ^1985, accepté le 23 avril 1985)

Résumé.

Nous démontrons que les résultats ^exacts obtenus par l’ansatz de Bethe _{pour les} susceptibilités ^de spin

et de charge ainsi que pour la chaleur spécifique ^{dans le modèle} d’Anderson symétrique peuvent ^être développés

dans des séries de Taylor qui convergent absolument pour ^toute valeur finie du paramètre ^de développement U/03C00394

et coincident avec les développements perturbatifs ^{de Yosida et Yamada} pour ^ces mêmes quantités. ^{Nous trouvons}

que les coefficients de ces développements ^{satisfont la} simple relation de récurrence Cn = (2 n 2014 1 ) Cn-1 - (03C0/2)2 Cn-2 ^et qu’ils décroissent rapidement ^{en fonction} de l’ordre. Par conséquent, ^un petit ^{nombre de termes}

suffit pour donner ^une description précise ^du système, ^{même dans le} régime de corrélations fortes (U/03C00394 ~ 2).

Enfin, ^{nous discutons une} tentative de construire une solution perturbative ^{de l’état} fondamental de l’Hamiltonien d’Anderson non symétrique.

Abstract.

We show that the exact Bethe-Ansatz results for the spin ^and charge susceptibilities ^{and the} specific

heat for the symmetric Anderson model can be expanded in power series which converge absolutely for any finite value of the expansion parameter U/03C00394 and which coincide with Yosida and Yamada’s perturbative expansions

for the same quantities. ^The coefficients of these expansions ^{are found to} satisfy ^the simple recursion relation

Cn

(2 n 2014 1) Cn-1 - (03C0/2)2 Cn-2 ^{and to decrease} rapidly ^with increasing order. Hence a small number of terms proves sufficient for ^{an accurate} description ^{of the} system ^{even in the} strong correlation regime (U/03C00394 ~ 2).

Finally, ^{we discuss an} attempt ^{to construct a} perturbative solution for the ground ^{state of the} asymmetric ^Anderson

Hamiltonian.

Classification

Physics ^Abstracts

75.20H

1. Introduction.

Ever since Anderson proposed his model and Hamil- tonian for the description ^of magnetic impurities ⁱⁿ

metals [1], it has been the subject ^{of numerous theore-}

tical investigations using ^diverse approaches ^and

methods. It has been only recently, ^{however, that the} application of the Bethe-Ansatz (BA) ^{method has} finally ^{led to the exact} diagonalization of the Anderson Hamiltonian [2-7]. ^The ground-state properties ^{of the}

model have been examined in detail and the equili-

brium thermodynamic equations ^{have been obtained}

for finite Coulomb repulsion ^U and under the condi- tion of linear dispersion of the conduction-electron spectrum and infinite conduction-electron bandwidth.

In particular, ^{for the} non-degenerate Anderson model with electron-hole symmetry, 8d ^{= -} U/2, ^{the BA} expressions for various ground-state properties ^like

X., Xc and y ^were obtained in the closed form.

At first sight the BA results seemed to invalidate the earlier attempts [8-11] ^to solve the Anderson Hamil- tonian by perturbation theory. They created the

impression ^{that «the} ground ^{state of the} impurity ^is essentially non-perturbative » [7] ^{and, moreover, that}

« the perturbation expansion by ^{U can} be considered to be an asymptotic ^{one and not to be a correct} approach ^to the solution of this Hamiltonian » [6].

Specifically, Wiegmann and Tsvelick [2] ^{and Kawa-}

kami and Okiji [3-5] ^have found that in the case of electron-hole symmetry ^{and for U} > 0 the zero-

temperature ^static spin susceptibility ^{of the non-} degenerate ^Anderson impurity ^can be written as a sum

of two terms,

The exponential ^term

which is dominant for me 1, ^{has an essential} singu- larity ^{at u = 0 and} obviously ^{can not be} expanded

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019850046090145900

1460

into a power series of u. The second term, I. ,(u), ^domi-

nates the spin susceptibility ^{for u 1} and decreases

rapidly with the increase of u. For small u they manag- ed to expand Is ^{into a power series of u [3],} neglecting

some exponentially ^small (for ^u 1) corrections :

The coefficients Cn ^{were obtained in a} form which is inconvenient for the investigation ^{of the} convergence of the series. One was thus tempted ^to conclude that the _power series in (2) represents ^the asymptotic expansion ^{of the} spin susceptibility ^{for u - 0 and that} consequently ^{it could never} describe the Kondo behaviour (1) ^for u > ^1.

We have shown [12], however, ^{that the} neglected exponential corrections in (2) (which ^{are not small}

unless _u 1) ^sum up exactly ^{to -} X’Kondo, ^{so that}

and therefore

We proved ^{that this} power series _converges absolutely

for any I u I ^oo and, ^since Is(u) ^{decreases rapidly}

to zero with increasing ^u,

for u > ^2. Moreover, since the convergence of the series happens ^{to be} very quick, ^even the first few terms are found to give ^a surprisingly good approxima-

tion of xs ^not only ^{in the} weak-correlation region (u 1) ^{but in the} strong-correlation region (u > 1)

as well.

In this _paper we present ^{a more detailed derivation} of the results outlined in [12], this time for both u > 0 and u 0. We _compare several finite-order approxi-

mations of the series expansions for various ground-

state quantities ^{with the exact} BA results to demons- trate the quick convergence of these expansions. ^We

also give ^{a brief account of an} attempt to construct a

perturbative solution for the ground ^{state of the} asymmetric (gd 0 - U/2) ^Anderson Hamiltonian.

2. Calculations.

2.1 BETHE-ANSATZ RESULTS. - We start with the BA results [2-6] ^{for the} zero-temperature spin ^and charge susceptibilities, Xr and _Xr , and the linear coefficient _y of the low-temperature specific ^heat C, ⁼ yT ⁺ é>(T3) ^{of the} non-degenerate ^Anderson impurity with electron-hole symmetry :

where

and xgondo(u) ^is given by (1). Strictly speaking, ^the integrals Ig(u) ^and Ic(u) ^{are not} defined for ^{u =} 0, but since they both have the same limiting ^{value of 1}

as u - 0 +, ^{we will} dispense ^with treating ^{the u = 0}

case separately.

We mention in passing that the BA result for

X’(u > 0) ^can be transformed to assume the form

(see Appendix)

which makes it 6bvious that X’(u) ^{assumes very} quickly

its universal form X’Kondo(u) ^{as u increases above - 1.}

Indeed, evaluating ^the integral ⁱⁿ (8), ^{which is} equal ^to

we find this relative difference diminishing ^{from 7.21} % for u = 1 to 0.8 % for u = 2 and only 0.13 % for u = ^3.

(See also the inset of Fig. 3 in Sect. 3.) ^{This is not}

surprising since for u > 1 X’ Kondo ^{increases exponen-}

tially, while I. ^{decreases as}

r , , ,

For the integral Ic ^we obtain the analogous asymptotic expansion

(which, ^{however, does not} imply ^that Ic(u) ⁼ Is( - u)).

Thus, ^{while for} u > 1 the spin susceptibility ^increases exponentially according ^{to the} scaling ^law (1), ^the charge susceptibility ^{decreases as} (2/nU)2, ^{which is}

not recognizable ^as anything ^universal.

2.2 SERIES EXPANSION.

In order to expand ^the

BA expression (3) ^for xs ^{into a power series of u, we}

first rewrite the integral Is in the form

where z’ ^{= -} 1 + ixu/2, ^and integrate along ^the

closed contour in the complex plane ^{shown in} figure ^1.

Since the contribution of vertical segments ^{of the}

contour vanish as Re z I --* oo, ^we obtain

where

The term - xgondo originates from half the residue at the pole ^{in z = z’ and} represents ^{the exact sum of} the « exponential corrections » which come out if one

tries to expand ^the integral I. ^{in its original form (6).}

This term obviously cancels the « anomalous » term

I ⁱⁿ X’s(u >, 0). Expression (7) ^for I,, ^{can also}

XKondo in Xs(u > 0)’ Expression (7) ^for Ic(u) ^{can also}

be transformed into a more convenient form

by simply shifting ^the variable of integration by xu/2.

Thus

where the second factor _proves to be an analytic

function for I u I ^{oo, as} shown below.

In order to evaluate the integrals (13) ^and (15), ^we

Fig. ^1.

^{Closed contour in the} complex plane ^{used to}

evaluate the integral ⁱⁿ equation (11). The contribution of the small semi-circle around ^z’ equals - X’Kondo ^{as its} radius goes to zero, while the contributions of vertical segments ^vanish as Re z --+ ^oo.

multiply the series expansion

where

by ^{the series} expansion

and recollect the terms of the same order in x to obtain

with the coefficients Pn given by

In the same way ^one obtains

Noting that Pn happens ^{to be equal to the n-th partial}

sum of the McLaurin expansion ^{of cos} (xx/2) ^{for x = 1,}

one finds

which is just the necessary condition for the ^conver- gence of the series (19) ^and (21). Furthermore, writing equation (20) ^as

and taking ^{into account} equation (22), ^one obtains Pn

in the form

1462

with

A brief inspection ^{of the numerical} series in (24)

shows that

which means that Pn ^{- -} an + 1 ^{as n - oo,} and con-

sequently,

Relations (22) ^and (26) ^{show that the series} expansions (19) ^and (21) converge absolutely ^{for any x I oo,}

and can therefore be integrated (and differentiated)

term by ^term. Inserting ^these expansions ^into (13)

and (15), ^and taking ^{into account that}

for u > 0, ^{we find}

with the coefficients ’n given by

The properties ^of Cn’s follow from those of Pn’s ^via

relation (29). Whiting equation (20) ^as Pn ⁼ Pn-l ⁺ an and using equation ^{(29), we find that} ’n’s satisfy ^the

recursion relation

for n > 1, with ’0 ^{= 1.} Furthermore, using equa- tion (23) ^{one can} write C,, ^as

and it follows immediately ^that limo ’n ^{= 0 and}

n-+ 00

lim I ’n + 1 I ’n = 0, which makes the _power series in

n-+oo

(27) ^and (28) absolutely convergent for u I ^oo.

Thus Jg(u > ^{0) and} Jc(u > 0) define the function

which is analytic ^for any finite value of u, and we can

now write equation (16) ^as

The convergence of the series (32) ^is very quick : ^for

the first few coefficients one has Co ^{= 1,} C, ^{= 0.2337,} C2 ⁼ 0.0599, C3 ⁼ 0.0134, ^and for n >> ¹ they ^diminish

as C’, , I /C" (rc2/8)/(n ⁺ 2). While for small n it is convenient to evaluate C,,’s ^{using relations (20) and (29)}

or the recursion relation (30), for n Z 10 it is easier to obtain them from equation (31) since it suffices to retain only ^{the first few terms} of the series Bn.

Finally, ^we multiply the series expansion ^of exp(n2 u/8) ^{and the} power series (32) together ^and

recollect the terms of the same order in u to obtain xs

in the form

where

By ^{virtue of} (4) ^and (5) ^{we also have}

and

Using the recursion relation (30) ^for ’n’s ^{and the} defining ^relation (35), ^{it is} straightforward ^{to show}

that the coefficients Cn satisfy the recursion relation

for n > 2, ^with Co ⁼ C1 ^{= 1. And once one has this}

recursion relation for Cn’s, ^{one can dispense with}

their connection with C.’s ^and fl.’s, ^since equation (38)

alone enables one to generate ^{all the} coefficients C,,

up ^{to n = oo} by iteration. For the first few of them

we find

Already the coefficients Co - C5 suggest ^{that both}

Cn ^and Cn+ 1 /Cn decrease rather quickly ^with increasing

n. However, ^{in order to} really investigate ^{the con-}

vergence of the series (34), (36) ^and (37), ^{one should}

have a general expression ^for C. ^{of the form} analogous

to equation (31) ^for ’n° ^{It is easy to} verify ^{that the}

expression

gives Co ⁼ C1 ^{= 1} and satisfies the recursion relation

(38) ^for any n > ^{2. It can} also be written as

where Pn denotes the numerical series

analogous ^to Bn. ^{A brief} analysis of the series in (42)

reveals that Pn+ 1 > Pn ^for any n > ^{0 and that}

Moreover, P,, --+ ^{1 rather} quickly ^with increasing n, as shown in figure ^{2. This means that} Cn acquires

rather quickly ^its large - n form, given by the square bracket in (41), ^while Cn+ 1/Cn (n/2)’/(2 n ⁺ 3). ^Thus

both Cn ^and Cn+ l/Cn ^{go to 0 as n - oo and conse-} quently ^the power series (34), (36) ^and (37) converge

absolutely ^for any u I ^{oo. That is to} say, ^we have shown that the exact BA expressions ^{for the} ground-

state quantities X8, X. and y ^{can be} represented by ^the

power series which _converge for any finite value of the

expansion parameter ^{u =} UlnA. ^{As a} straight ^con-

sequence of this, X8’ Xc and y prove ^to be analytic

functions of u four I u I ^oo.

2. 3 PERTURBATION EXPANSION.

In the perturbative approach of Yosida and Yamada [8-10] ^to the sym- metric Anderson model the Hamiltonian is divided into the unperturbed part

equal ^{to the} nonmagnetic Hartree-Fock (HF) approxi-

mation to the full Hamiltonian, ^{and the} perturbation

Fig. ^2.

Pn, ^{the sum} of the numerical series (42); it repre- sents the ratio of C. ^{and its} large-n ^form.

where 1/2 represents ^{the HF} average of nda. ^In such

an approach ^the mean-field part ^{of the} problem (HO)

is solved exactly and what remains to be treated

perturbatively ^{is the} effect of fluctuations, i.e., ^the deviations from the HF solution. The macroscopic quantities ^{are then} expanded ^{in the} power series of u, with the coefficients given ^{in terms of the} imaginary-

time integrals ^{of the} determinants built from the one-

particle temperature ^Green functions for the _unper- turbed Hamiltonian Ho. Specially, ^for X’, X’ ^and y’

at T ⁼ 0 one obtains the expansions of the form (34), (36) ^and (37), with the coefficients given by (n ⁺ 2)-

dimensional imaginary-time integrals or, after the Fourier transformation, by (n ⁺ I)-dimensional ^inte- grals ^over frequencies, ^{with the} products ^of 2(n ⁺ 1) unperturbed ^{Green functions as the} integrands.

Yamada calculated these coefficients _up to the third order analytically and the fourth-order one numeri-

cally [9], while the higher-order ^ones proved ^too complicated ^to be evaluated directly ^even by ^numerical

methods.

The first five perturbational coefficients for x§, x§

and y’ ^{are found to} be identical with the Cn’s ^obtained by ^the expansion ^{of BA} expressions ^{for the same} quantities. Assuming that these quantities ^have unique series expansions ⁱⁿ powers of _u, which follows from

their analyticity for u I oo, ^one has no reason

whatsoever to believe that the higher-order pertur- bational coefficients would start to differ from the

Cn’s. And with this observation we arrive at the main

point ^{of this} paper, advertised in the title : the expan- sions (34), (36) ^and (37) ^{of the BA} expressions ^{and the} corresponding perturbative series of Yosida and Yamada are one and the _same, and anything ^{that has}

been said about the former holds for the latter as well.

That is, ^{for the} ground-state quantities Xs, Xc’ ^and y’

at least, the series expansions ^obtained by ^{the deter-}

minantal perturbation theory of Yosida and Yamada

are absolutely convergent ^for any finite value of the

expansion parameter ^{u and are} fully equivalent ^to

the BA results.

Thus, although it is neither evident nor easy ^to prove, the (n ⁺ I)-dimensional integrals ^{for the} per- turbational coefficients must satisfy the recursion relation (38). ^{This means that} i) ^{due to the} high sym- metry ^{of the} integrands, ^the (n ⁺ I)-dimensional integral ^for C. ^{can be reduced to} the linear combi- nation of an n-dimensional and an (n - I)-dimensional integral ^{of the same} form, ^{which is a} highly unlikely thing ^to expect a priori, ^and ii) instead of the tedious calculation of the higher-order perturbational ^coef- ficients, ^{one can} generate ^them up ^to the arbitrary

order simply by ^iteration.

3. Comparison ^{with BA.}

In figures ^{3, 4 and 5 we} compare the ^exact BA results for xg, Rw ⁼ /g// ^and X’ ^{with several} finite-order truncations (partial sums) ^{of the} perturbation ^series

for the same quantities. ^{To start} with, figure ^{3 is meant}

1464

Fig. ^3.

The BA result for X’(u) (full line) and the finite- order approximations (partial sums) ^{of the} expansion series,

N

Y Cn ^{u", for N}

^{1, 2,..., 5} (dashed lines). ^The inset shows

n=0

xs decomposed ^into xKondo and Is.

to illustrate two points : i) The inset shows that xs

assumes its universal form xKondo ^very quickly ^{as u}

increases above unity, indicating ^{that the} strong- correlation (SC) region, ^{where the behaviour of the} system ^is governed by ^the scaling laws, appears

already ^{for u} between 1 and 2, ^{and not for} u > 1

(as ^{one would} expect ^{from the} Schrieffer-Wolf trans-

formation). ii) On the other hand, ^the convergence of the perturbation expansion for xs ^{is seen to be very} quick ^as well, ^that is,

for relatively large ^{u with} relatively ^{small N.} Speci- fically, ^{the third order is} quite satisfactory ^{for u 1.5}

and the fourth order for u is ^2. (E.g. ^{for u = 2 and}

N ⁼ 4 the relative error is 8.2 %). ^Put together, ^this

shows that the low-order perturbative results for

X’s ^{are able to describe} correctly the transition from

the weak-correlation (WC) ^{to the SC} regime, ^{as well}

as to give ^{a rather} accurate account of the « lower

edge » of the latter.

The same holds for the specific ^heat coefficient y’,

which is not shown here separately ^{as it looks very}

much like figure ^{3 for} X’. ^{Instead, we} present ⁱⁿ figure ⁴

the BA and perturbative results for the Wilson ratio

Rw X’ly’. ^The transition between its WC value of 1 and SC value of 2 is seen to take place ^{at u N 1}

and the SC value is almost fully ^{reached for u N 2.}

u

Fig. ^4.

The Wilson ratio Rw

xs/y’. ^{Full line}

^{the BA} result; dashed lines

the finite-order approximations, obtained as the ratio of the corresponding finite-order

approximations (partial sums) ^of X’ and y’.

As for the finite-order approximations, ^the tendency

of Rw towards saturation is clearly ^{seen even in the}

second-order approximation (although ^the exact satu- ration value remains a guess), while the inclusion of the fourth order enables also a quite ^{reliable estimate}

of the saturation value. The sixth-order result is

practically ^{exact for u} 2.5, ^{which covers all the} values of ^u that are relevant for the discussion of model

properties.

Thus, ^{as far as} Xs, y and Rw ^are concerned, ^{u ~ 2} represents ^{a «} practically ^{infinite u » and} nothing qualitatively ^{new sets in if u} is increased further.

Due to this, ^{as well as to the} quick convergence of the

perturbation expansions, the low-order perturbative

results are seen to be able to reach the ^« lower edge »

of the SC region, ^{where the} scaling ^laws (e.g. Eq. (1))

take over. And once this point ^{has been attained, the} scaling ^{laws can be used to} extrapolate the results to an arbitrary point ^{in the SC} region.

The determinantal perturbation theory ^{has been}

used [13-15] ^{to show that the} low-temperature ^beha-

viour of the Anderson model obeys scaling ^{laws in}

the SC regime. ^That is, ^{both the} transport ^{and the} thermodynamic quantities become universal functions when written in terms of the reduced variables T/0

and PB Hlku 0, ^where kB ^{8 =} A /ay’ ^{in the} symmetric

case. By ^« universal » we mean that the coefficients of

(T/0)" ^and (PB HlkB 8)ft in ^the appropriate ^low- temperature and/or ^low-field expressions ^{attain cons-}

tant values for u > 2, ^{while the} characteristic tem-

perature 0 becomes proportional ^to 1 /X’Kondo. ^This

transition to the SC regime ^{is described rather accu-} rately by the finite-order perturbation theory. ^For example, ^the impurity contribution to the low-tem- perature electrical resistivity ^is given by

with xp ^{= [1 + 2(Rw -} 1)2]/3 ^{in the} symmetric ^case.

The coefficient xo, ^which equals ^{1 in the u - oo} limit,

attains 95 % ^{of this} saturation value at u ⁼ 2, ^while

the finite-order approximations give K(p4) ^{= 0.860,}

Fig. ^5.

Charge susceptibility X’(u) ; ^{the BA result} (full line)

and several finite-order approximations (dashed lines).

Kp(6) ^{= 0.938 and} Kp(8) ⁼ 0.949 for the same value of u.

The same can be said about the coefficient Ks of the

low-temperature thermoelectric _power

S(7) ⁼ So[(TIO) - Ks(TIO)3 ⁺ ...] .

The BA and perturbative results for the charge susceptibility ^{are shown in} figure ^{5. One can see} that X, ^{does not converge so} quickly ^as X,, y and Rw, namely, ^{one needs the 3rd or 4th order to reach u - 1}

and the 7th or even 8th order to reach u ~ 2. Also, X’ c ^{does not assume any} scaling ^{form for} u Z 2;

instead, ^it attains the asymptotic ^form (10) only ^for

u > 1. The « slow » convergence should be ascribed

to the smallness of x§ ^{in the} symmetric ^{case. While}

the absolute error of the Nth-order approximation of X’ ^is always less than that of xs, ^the corresponding

relative error may be large for X’c ^although it is small for xs. ^{And as} regards ^the scaling, ^{one should remember}

that _Xc is the measure of localized charge fluctuations,

which for u > 0.7 reach their maximum in the asym- metric case - ed U, ^while being minimal in the

symmetric ^{case which is} discussed here. The scaling

behaviour of Xc ^{can thus be} expected ^{to show} up in the

asymmetric ^{case, where the} dynamics ^{of the} system is predominantly determined by ^localized charge fluctuations.

4. Asymmetric ^{case :} sd U/2.

The general Anderson model without the electron- hole symmetry, frequently ^{referred to as} the « asym- metric Anderson model », has the impurity energy level Ed different from - U/2. ^{The «} asymmetry » ^is found to be conveniently ^described by ^the parameter

q ⁼ 1/2 ⁺ ed/ U. ^{It is the} original Anderson’s _para- meter ed/ U ^shifted by 1/2, ^which makes ?I ^{= 0 in the}

case of electron-hole symmetry. ^{It can} also be under- stood as (Ed - edsymm)/U, ^where edsymm= - U/2. ^Since

the particle-hole transformation takes the Anderson Hamiltonian to itself with Ed replaced by - (Ed ⁺ U),

that is, with q replaced by - n, it is sufficient to discuss the region n > ^0.

Once one has the exact perturbative solution for 11 ⁼ 0, ^{one can} attempt ^an expansion for n # ⁰

such as the one Yamada [16] did for the special ^case

n ⁼ 1/2 (Ed ⁼ 0). Namely, for n # 0 the full Hamil- tonian can be written as H = Ho ^{+ H’ + H", where} Ho ^{+ H’ is the} symmetric (n ⁼ 0) Hamiltonian, ^with Ho ^{and H’} given by (43) ^and (44), ^and

is the additional ^« asymmetry ^{term ».} The idea of the calculation is then to take Ho ^{as the} unperturbed

Hamiltonian and treat H’ + H" as the perturbation, taking ^{into account that it} really consists of two terms : the many-body ^{term H’ and the} one-body potential

H". As the result of this ^« double » perturbation expansion ^one obtains the ground-state quantities

in the form

where

The coefficients a:,c,e( r) up ^to the fifth order are

presented ^{in table I as well as in} figure ^6. For il ^{= 0} they obviously ^{reduce to the} symmetric coefficients Cn,

as it should be, ^since for q ^{= 0 the} potential ^H"

vanishes and the ^« double » perturbation expansion

reduces to the one outlined in section 2.3.

Aiming ^{at the} description ^{of the} charge fluctuation

effect, which is believed to be modelled by ^{the Ander-}

son Hamiltonian for q = 1/2, ^Yamada [16] ^{tried to}

use the expansions (46) for q ⁼ 1/2. However, ^{a brief} glance ^at figure 6, ^{where we} plot X. and an ^{as functions} of I tj 1, shows that the expansions ^{(46) are «} good » only for I nil I, ^{1, that is, in the symmetric and} nearly-symmetric region, where Kondo effect arises for u > 1. For I r around 1/2 ^{or so} they ^{reduce to} asymptotic expansions ^{for u 1,} which renders them

inappropriate ^{for the} investigation ^{of the} charge

fluctuation effect, ^{known to arise for} large ^u or, at

least, ^{for u} > 1. That is why ^we prefer ^{a different} type of perturbation expansion ^{for the} asymmetric ^Ander-

son model, ^{where the} nonmagnetic ^HF approximation

of the full (asymmetric) Hamiltonian is taken as the

unperturbed Hamiltonian and the deviations from

1466

Table I.

Fig. ^6.

Coefficients an and a" ^as functions of the asymme- try parameter tj ^{for n}

0, 1,..., ^5.

the HF are dealt with as the perturbation [11]. ^{It is a} straightforward generalization ^{of the} original ^Yosida

and Yamada’s expansion ^{for the} symmetric ^case [8, 9]

and, ^{in contrast with the} above-described q-expansion

it is not an expansion ^with respect ^to asymmetry ^and is not confined to small asymmetries. ^{For the} ground-

state quantities ^it gives ^{the series of the form}

where x ⁼ x(u, n) is the solution of the HF self-

consistency equation ^{x + u tan-1 x =} ir nu [11]. ^In

the symmetric ^case x(u, n ⁼ 0) ^{= 0} and the coeffi- cients Cn(x) ^{reduce to} symmetric Cn’s, Cn(x ^{= 0) =} Cn,

which guarantees ^the quick convergence of the series

(47) ^{in this} special ^{case. As the} convergence is at least not spoiled, ^{and in most} cases even improved ^with

the increase of asymmetry, ^such expansions ^enable

the discussion of the model properties both in the Kondo and charge fluctuation regions [17]. Also,

in its general formulation, ^this perturbation theory

can be used to calculate the finite-energy properties (density ^of states) [11, 18], inaccessible to the BA method, ^{as well as} the effects of finite magnetic ^fields [ 15] ^and temperatures.

Appendix.

We rewrite xs(u > 0), given by equation (16), ^as

where

For A ⁼ 0 we calculate (A. 2) ^as

Differentiating Y(A) ^with respect ^{to A one} obtains the differential equation

and solving ^this equation with the initial condition

(A. 3), ^{one obtains}

Inserting (A. 5) ^into (A .1), ^one obtains the sought

for form (8) ^of X’(u > 0).

References

[1] ANDERSON, ^P. W., Phys. ^{Rev. 124} (1961) ^41.

[2] ^{WIEGMANN, P.} B., FILYOV, ^{V. M. and} TSVELICK, ^A. M., Pis’maZh. Eksp. Teor. Fiz. 35 (1982) 77/JETP

Lett. 35 (1982) 92.

[3] KAWAKAMI, ^{N. and} OKIJI, A., ^J. Phys. ^Soc. Jpn. ⁵¹ (1982) ^1145.

[4] OKIJI, ^{A. and} KAWAKAMI, N., Solid State Commun.

43 (1982) ^365.

[5] KAWAKAMI, ^{N. and} OKIJI, A., Solid State Commun.

43 (1982) ^467.

[6] OKIJI, ^{A. and} KAWAKAMI, N., ^J. Phys. ^Soc. Jpn. ⁵¹ (1982) ^3192.

[7] WIEGMANN, ^{P. B. and} TSVELICK, ^A. M., J. Phys. ^{C 16} (1983) 2281.

[8] YOSIDA, ^{K. and} YAMADA, K., Prog. ^Theor. Phys. Suppl.

46 (1970) ^244.

[9] YAMADA, K., Prog. ^Theor. Phys. ⁵³ (1975) ^970.

[10] YOSIDA, ^{K. and YAMADA,} K., Prog. ^Theor. Phys. ⁵³ (1975) ^1286.

[11] HORVATI0107, ^{B. and} ZLATI0107, V., Phys. ^{Status Solidi B}

99 (1980) ^{251 and 111} (1982) ^65.

[12] ZLATI0107, ^{V. and} HORVATI0107, B., Phys. ^{Rev. B 28} (1983)

6904.

[13] YAMADA, K., Prog. ^Theor. Phys. ⁵⁵ (1976) ^1345.

[14] ZLATI0107, ^{V. and} HORVATI0107, B., ^J. Phys. ^{F 12} (1982)

3075.

[15] HORVATI0107, ^{B. and} ZLATI0107, V., Phys. ^{Rev. B 30} (1984)

6717.

[16] YAMADA, K., Prog. ^Theor. Phys. ⁶² (1979) ^354.

[17] HORVATI0107, ^{B. and} ZLATI0107, V., ^Solid State Commun.

54 (1985) 957.

[18] ZLATI0107, V., HORVATI0107, ^{B. and} SOK010DEVI0107, D., ^Z. Phys.

B 59 (1985) 151.