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Intermediate valence and kondo features of the Anderson model by perturbation theory
0038-1098/85/$ 3.00 + .00 Pergamon Press Ltd.
Solid State Communications, Vol. 54, No. 11, pp. 957-960, 1985. Printed in Great Britain.
INTERMEDIATE VALENCE AND KONDO FEATURES OF THE ANDERSON MODEL BY PERTURBATION THEORY B. Horvatid and V. Zlati6 Institute of Physics of the University of Zagreb, P.O. Box 304, 41 001 Zagreb, Yugoslavia
(Received 9 January 1985 by B. Miihlschlegel) We discuss the LT properties of the nondegenerate Anderson model using the perturbation theory with UlTrA as the expansion parameter. Summing the w-independent part of the self-energy to all orders, we use the Friedel sum rule and Ward identities to express the physical quantities in terms of the remaining w-dependent part of the self-energy, which we evaluate to the second order. Results for the spin and charge susceptibilities and Wilson ratio, which compare very well with Bethe-ansatz, are used to study the LT "phase diagram" of the model.
MANY IMPORTANT properties of the single-impurity non-degenerate Anderson model [1] have already been clarified by the perturbation theory [2-5], the numerical renormalization group (RG) approach [6] and by the Bethe-ansatz (BA) method [7-13]. In particular, it turns out that in the parameter space defined by u = U/ rrA and 7/= ~ + ea/U (where U, ea and A have their usual meaning and 77 measures the asymmetry of the model) one obtains the following low-temperature "phase-diagram". For u ~< 1 the many-body (MB) effects are not important and the behaviour of the model is Hartree-Fock-like irrespective of the value of r/. For u >> 1, however, the (u, 7/) parameter space comprises two regions which are believed to model real physical systems: the localized moment (spin fluctuation, Kondo) region for 0 < 171 <~ rl*(u) and the intermediate valence (charge fluctuation) region for r/*(u) <~ Ir/I <~ rff(u) + 1/lru. The "phase boundary"
~l*(u) = ½ -- ln(Tr2eu/4)/n2u
(1)
which separates the two regions should be understood only as the centre of a wide and fuzzy transition region between them. It is connected with the well known scaling parameter e~ by the relation e,] = ( 7 / - r/*)U and corresponds to the condition e~ = 0 [8]. For Ir~l> r~*(u) + 1/Tru (or ea*> A) the intermediate valence (IV) region decays gradually into the "large asymmetry region" which is essentially Hartree-Fock-like, with no significant effects of MB fluctuations. Although the "phase diagram" outlined above is rather simple, it has been obtained by means of fairly complicated methods [6-13]. It might therefore still be of interest to employ a method which is simple and yet accurate enough in the whole part of the parameter
space relevant for the explanation of experimental results, and which also allows the calculation of some important quantities (e.g. density of states) not accessible to either the BA or the RG method. In this paper we propose such a method, compare it with the BA and discuss its accuracy. We evaluate the spin and charge susceptibilities and use them to obtain more detailed information about the "phase diagram" of the model. In a subsequent publication we are going to present the results for the density of states and discuss the effects of finite temperature. The method is summarized in the following four points: (i) Following Yosida and Yamada [2, 3] as well as our previous work [4], we divide the nondegenerate (single-orbital) Anderson Hamiltonian [1] into the unperturbed part Ho, equal to the nonmagnetic HartreeFock (HF) approximation of the full Hamiltonian, and the perturbation U(ndt--(ndt) ) (nd4,--(ndl.)), where () denotes thermal averaging over the eigenstates of rio and (ndO= (na~). The self-energy of the exact oneparticle d-electron Green function is then expanded in the power series of the intraionic Coulomb integral U, with the coefficients given in terms of the imaginarytime integrals of the determinants built from the oneparticle temperature Green functions for Ho. (ii) We note that the irreducible (proper)d-electron self-energy ~do(W) can be separated into two parts as follows,
957
d
o
(2)
958
THE ANDERSON MODEL
where Y~ao(co) is the reducible self-energy and G°cr(¢o) = (/co + iAsgn co -- ea -- ( n d , - o ) U ) -1
(3)
is the unperturbed (HF) d-electron Green function. The first part is co-independent and represents the sum of all "one-legged" diagrams, while ~do(co) is the sum of all the remaining diagrams, which are co-dependent. The co-independent term is recognized as U(<
~da(co)]-l.
(4) (i/i) At T = 0 the number of localized electrons is given by the Friedel sum rule [14, 15], which now represents a transcendental equation for nd: nd=
1 -- (2/n) tan -1 {[e a q- (nd/2)U + ~do(0÷)]/A}. (5)
It is solved numerically, given l~da(0 +) in some approximation, and the obtained value of na is substituted into equation (4) for Gad(w). It is, however, more convenient to solve the equivalent equation ( f / d / A ) + U tan -i (flu~A) = rrrlu + ~da(O+)/A
(6)
for the quantity fig = ed + (nd/2)U + ~ao(O +) which appears explicitly in the expressions for various physical quantities. In the HF approximation equation (6) reduces to (Ed/A) + U tan - I ( E d / A ) = Z/'I?U,
(7)
where E d = e d + (nd -o)U represents the HF position of the virtual bound state (VBS). It is clear from the above stated division of the Hamilton/an into H0 and H ' that the calculated quantities, ~da in particular, will appear as functions of u and E d / A instead of u and 7/. In order to establish the dependence of these quantities on 77, one solves equation (7) and finds Ed/A for given values of r/and u. (iv) The ground-state thermodynamic quantities are calculated as follows: The charge susceptibility, defined as Xc = - - 3 n a / ed, is obtained by differentiating the Friedel sum rule (5) with respect to ed and solving the resulting expression for a nd/~ ed. One obtains X'~ =- ~ rrA×c
1 + ~do(O+)/Oea = 1 +u +(Ea/A):
(S)
Since ,~acr(0 +) depends on ed through Ed,
-
l+l+(E-d/A)2
t
]'
(9)
Vol. 54, No. 11
where the first factor represents 3Ed/~e d and is obtained by differentiating equation (7) with respect to ea (and noting that 7rrlu = (ed + ~ U)/A). The expression for the T-linear coefficient 3' of the impurity specific heat is standard [16], 7'-
3A 21rk~ 7 = ~/[1 + (ffa/A)2],
(10)
where = 1 -- [a2ao(co)/8(ico)] w=o
(1 1)
since Y'do(CO) -- ~ao(CO) is co-independent. The impurity spin susceptibility X, can then be written as , 2~rA Xs = (gUB)2 Xs = 27' -- X c
(12)
due to the exact relation which has been proved to hold [16] among these three quantities in the Anderson model. Finally, the Wilson ratio Rw is defined as Rw = ×',/7' = 2/[1 + (X'o/X;)].
(13)
As regards the approximation, for Y,aa(co) we take the first nonvanishing perturbative term
<2(co)'= Zh2:(co) =
(14)
which is of the 2nd order in u = UlTrA, and neglect aU higher-order corrections. This approximation finds its justification in the quick convergence of the perturbation expansion (which has been proved explicitly [5] for ~/= O) and, above all, in the very good agreement of the results with the exact ones obtained by the BA method. The diagram (14) as well as [8~, ~(co)/aco] w=o has been calculated before [4]. In Fig. 1 we compare the result for ×c in the case of electron-hole symmetry (77 = 0), obtained by the above described method and approximation (dashed line), with the exact BA result [7] (solid line) and several finite-order approximations of the direct Yosida and Yamada perturbation expansion [3, 5] (dotted line). We see that for u = 2, which is already in the strong correlation (SC) regime, our present result is only 21.5% off and that it is as good as the 6th-order perturbative result. The Wilson ratio Rw shows even better agreement with the BA results: for 17 = 0 and 0 ~< u <~ 4 the maxim u m relative error of R w - 1 amounts to only 1.3% (around u = 1, and is even less for other values of u). For other physical quantities (7, Xs, na) the improvement (with respect to the direct perturbation theory) is somewhat less spectacular, but still significant. Since the results for Xs(u;i?) obtained by the pres-
Vol. 54, No. 11 1.0
THE ANDERSON MODEL
i
I
i
I
0.6
i
I
X'o(U;n =0)
0.6
0.2
'",".
I
4
6 ",,. " ~ "
8
2
3
Fig. 1. Charge susceptibility for r / = O (e a = - - U / 2 ) ; solid line - exact BA result [7], dashed line - present method, dotted line - several finite-order approximations of the direct Yosida and Yamada perturbation expansion [3, 5]. ent method are indeed more accurate than the already reported perturbative ones [17], but do not show any qualitative differences, we will here just quote the main conclusions regarding the "phase diagram". Recalling that Xs measures the local spin fluctuations (LSF), we can see that they reach their maximum in the case of '
'
'
'
I
'
'
'
'
I
'
'
'
'
959
electron-hole symmetry (half-filled VBS) and are quickly suppressed as the asymmetry is increased. Also, the increase of Coulomb correlation enhances the LSF only for O ~ 171 <~ rT*(u), while suppressing them for Ir~l >~ ½. Thus it is only in the local moment (LM) region that the increase of u drives the system into the SC regime with the scaling laws appropriate to the Kondo behaviour. On the other hand, the charge susceptibility ×e, which is the measure of local charge fluctuations, exhibits a clearly defined maximum in the IV part of the "phase diagram". As shown in Fig. 2, the peak of Xe0/) appears for u >~ 1 and becomes better pronounced as u increases. The position of the peak r/M(u) (indicated by vertical arrows) moves to higher values of 171 with increasing u, and as far as we can tell (both from our and BA results), it falls halfway between ~/* and ~7" + 1/rru for u >~ 1.5. We consider it very important to find out the exact dependence of r/M on u, since ~ ( u ) might be regarded as the centre of the IV region just as the r? = 0 line represents the centre of the LM region. Further on, the height of the peak, X'eO1M), is found to decrease and approach a constant value with the increase of u. For u = 1.0, 1.5, 2.0 and 2.5 we have X'c(~lu)= 0.419, 0.353, 0.325 and 0.311. (The comparison with BA results [11] shows the relative error of 1.5% for u = 4/rr.) The large u saturation value is easily obtained by differentiating the relation
n~ = 2 - v ~ - 0 r / 1 2 x / 7 )
(e~/a) + ~ [ ( ~ / a ) ~ l
(15)
of Wiegmann and Tsvelick [8, 9] with respect to e~ at e~ = 0 ( r / = 7/* ~ ~M), since a/be~ = O/aea. We find , rl~{u )
t
\
: 2Z o.o
o.5
Iql
t
1.o
(16)
ts
Fig. 2 Charge susceptibility as function of asymmetry for five fixed values of u. The horizontal arrows on the r / = 0 axis show the exact BA values of Xc(U;r/= 0). The vertical arrows indicate the positions of the maxima, t
~M(u).
~
xc(ea = O) = Xc(U > > 1; ~TM) = rr2/24"v/~ = 0.2908 in conformity with the numbers given above. We see also that the approach to saturation is rather quick: the value (16) is almost reached for u >~ 2. The Wilson ratio Rto (u;r/) is plotted in Fig. 3. As a function of u for various fixed values of asymmetry, R w exhibits two types of behaviour: (i) For It/I K½ it increases monotonously and approaches the saturation value of 2 for large enough u. As Ir/I increases from 0 to ½, the onset of saturation moves to higher values of u. (ii) For t r / l > ~, R w has a maximum at some finite u and drops back to 1 as u -* ~. As 177[decreases, the m ; x i m u m increases in height and its position shifts to higher u values. Considering R w as a function of D?I for various fixed values of u, we see that for u > > 1 it has the value of ~ 2 in the LM region, decreases rapidly in the IV region and attains the weak correlation (WC) value of 1 in the large asymmetry region. If we, however, approach 171 = ½ along the line
THE ANDERSON MODEL
960
Rw _
24.
2.0
1.5
Vol. 54, No. 11
Xrc ~ 7r2/24V~. Since both in the LM and IV region the transition from the WC to the SC regime happens to take place at relatively small u values, the present method with the 2nd-order approximation forZaa can be used to study even the SC features in both of these regions. REFERENCES
/ ~ ,
/, /, /, /, /7"/-..,2% L . Y , \ ,~
1.o
1. 2. 3. 4.
3
Fig. 3 Wilson ratio Rw(u;71). The thick line with arrows represents Rw [(u; r/*(u)]. Ir/I = r/*(u), R w [u; r?*(u)] is seen to saturate at a value different from either 2 or 1. Since BA results [8] give X's07*) = n2/2V~ = 12 ×'c(r/*) for u > > 1, one has [via rel. (13)] Rw = ~ = 1.846 at the "phase boundary" between the LM and IV region. The curve Rw [u; r/*(u)] in Fig. 3. seems to confirm this result. To summarize, for 1771~ 0, i.e. in the centre of the LM region, the SC regime (characterized by the scaling laws and Rw ~ 2) sets on at u ~ 1 and is almost fully reached for u ~ 2 . For larger Ir/I the onset of SC is pushed to higher values of u since the asymmetry reduces the effects of LSF. However, for r~* ~< 171 ~<7/* + 1/Tru the correlation effects (seen here as charge fluctuations) become visible again at u ~ 1 and some kind of SC behaviour is established for u >~ 2, characterized by
5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
P.W. Anderson, Phys. Rev. 124, 41 (1961). K. Yosida & K. Yamada, Prog. Theor. Phys. Suppl. 46,244 (1970). K. Yamada, Prog. Theor. Phys. 53,970(1975). B. Horvati6 & V. Zlati6, Phys. Status Solidi (b) 99, 251 (1980); 111,65 (1982). V. Zlati6 & B. Horvati6, Phys. Rev. B28, 6904 (1983). H.R. Krishna-murthy, J.W. Wilkins & K.G. Wilson, Phys. Rev. B21, 1003 (1980); 21, 1044 (1980). P.B. Wiegmann, V.M. Filyov & A.M. Tsvelick, JETP Lett. 35, 92 (1982). P.B. Wiegmann & A.M. Tsvelick, J. Phys. C16, 2281 (1983). A.M. Tsvelick & P.B. Wiegmann, Adv. Phys. 32, 453 (1983). N. Kawakami & A. Okiji, J. Phys. Soc. Jpn 51, 1145 (1982). N. Kawakami & A. Okiji, .L Phys. Soc. Jpn 51, 2043 (1982). A. Okiji & N. Kawakami, J. Phys. Soc. Jpn 51, 3192 (1982). A. Okiji & N. Kawakami, Phys. Rev. Lett. 50, 1157 (1983). D.C. Langreth, Phys. Rev. 150, 516 (1966). H. Shiba, Prog. Theor. Phys. 54, 967 (1975). A. Yoshimori, Prog. Theor. Phys. 55, 67 (1976). B. Horvati6 & V. Zlati6, Phys. Rev. B30, 6717 (1984).
Solid State Communications, Vol. 54, No. 11, pp. 957-960, 1985. Printed in Great Britain.
INTERMEDIATE VALENCE AND KONDO FEATURES OF THE ANDERSON MODEL BY PERTURBATION THEORY B. Horvatid and V. Zlati6 Institute of Physics of the University of Zagreb, P.O. Box 304, 41 001 Zagreb, Yugoslavia
(Received 9 January 1985 by B. Miihlschlegel) We discuss the LT properties of the nondegenerate Anderson model using the perturbation theory with UlTrA as the expansion parameter. Summing the w-independent part of the self-energy to all orders, we use the Friedel sum rule and Ward identities to express the physical quantities in terms of the remaining w-dependent part of the self-energy, which we evaluate to the second order. Results for the spin and charge susceptibilities and Wilson ratio, which compare very well with Bethe-ansatz, are used to study the LT "phase diagram" of the model.
MANY IMPORTANT properties of the single-impurity non-degenerate Anderson model [1] have already been clarified by the perturbation theory [2-5], the numerical renormalization group (RG) approach [6] and by the Bethe-ansatz (BA) method [7-13]. In particular, it turns out that in the parameter space defined by u = U/ rrA and 7/= ~ + ea/U (where U, ea and A have their usual meaning and 77 measures the asymmetry of the model) one obtains the following low-temperature "phase-diagram". For u ~< 1 the many-body (MB) effects are not important and the behaviour of the model is Hartree-Fock-like irrespective of the value of r/. For u >> 1, however, the (u, 7/) parameter space comprises two regions which are believed to model real physical systems: the localized moment (spin fluctuation, Kondo) region for 0 < 171 <~ rl*(u) and the intermediate valence (charge fluctuation) region for r/*(u) <~ Ir/I <~ rff(u) + 1/lru. The "phase boundary"
~l*(u) = ½ -- ln(Tr2eu/4)/n2u
(1)
which separates the two regions should be understood only as the centre of a wide and fuzzy transition region between them. It is connected with the well known scaling parameter e~ by the relation e,] = ( 7 / - r/*)U and corresponds to the condition e~ = 0 [8]. For Ir~l> r~*(u) + 1/Tru (or ea*> A) the intermediate valence (IV) region decays gradually into the "large asymmetry region" which is essentially Hartree-Fock-like, with no significant effects of MB fluctuations. Although the "phase diagram" outlined above is rather simple, it has been obtained by means of fairly complicated methods [6-13]. It might therefore still be of interest to employ a method which is simple and yet accurate enough in the whole part of the parameter
space relevant for the explanation of experimental results, and which also allows the calculation of some important quantities (e.g. density of states) not accessible to either the BA or the RG method. In this paper we propose such a method, compare it with the BA and discuss its accuracy. We evaluate the spin and charge susceptibilities and use them to obtain more detailed information about the "phase diagram" of the model. In a subsequent publication we are going to present the results for the density of states and discuss the effects of finite temperature. The method is summarized in the following four points: (i) Following Yosida and Yamada [2, 3] as well as our previous work [4], we divide the nondegenerate (single-orbital) Anderson Hamiltonian [1] into the unperturbed part Ho, equal to the nonmagnetic HartreeFock (HF) approximation of the full Hamiltonian, and the perturbation U(ndt--(ndt) ) (nd4,--(ndl.)), where () denotes thermal averaging over the eigenstates of rio and (ndO= (na~). The self-energy of the exact oneparticle d-electron Green function is then expanded in the power series of the intraionic Coulomb integral U, with the coefficients given in terms of the imaginarytime integrals of the determinants built from the oneparticle temperature Green functions for Ho. (ii) We note that the irreducible (proper)d-electron self-energy ~do(W) can be separated into two parts as follows,
957
d
o
(2)
958
THE ANDERSON MODEL
where Y~ao(co) is the reducible self-energy and G°cr(¢o) = (/co + iAsgn co -- ea -- ( n d , - o ) U ) -1
(3)
is the unperturbed (HF) d-electron Green function. The first part is co-independent and represents the sum of all "one-legged" diagrams, while ~do(co) is the sum of all the remaining diagrams, which are co-dependent. The co-independent term is recognized as U(<
~da(co)]-l.
(4) (i/i) At T = 0 the number of localized electrons is given by the Friedel sum rule [14, 15], which now represents a transcendental equation for nd: nd=
1 -- (2/n) tan -1 {[e a q- (nd/2)U + ~do(0÷)]/A}. (5)
It is solved numerically, given l~da(0 +) in some approximation, and the obtained value of na is substituted into equation (4) for Gad(w). It is, however, more convenient to solve the equivalent equation ( f / d / A ) + U tan -i (flu~A) = rrrlu + ~da(O+)/A
(6)
for the quantity fig = ed + (nd/2)U + ~ao(O +) which appears explicitly in the expressions for various physical quantities. In the HF approximation equation (6) reduces to (Ed/A) + U tan - I ( E d / A ) = Z/'I?U,
(7)
where E d = e d + (nd -o)U represents the HF position of the virtual bound state (VBS). It is clear from the above stated division of the Hamilton/an into H0 and H ' that the calculated quantities, ~da in particular, will appear as functions of u and E d / A instead of u and 7/. In order to establish the dependence of these quantities on 77, one solves equation (7) and finds Ed/A for given values of r/and u. (iv) The ground-state thermodynamic quantities are calculated as follows: The charge susceptibility, defined as Xc = - - 3 n a / ed, is obtained by differentiating the Friedel sum rule (5) with respect to ed and solving the resulting expression for a nd/~ ed. One obtains X'~ =- ~ rrA×c
1 + ~do(O+)/Oea = 1 +u +(Ea/A):
(S)
Since ,~acr(0 +) depends on ed through Ed,
-
l+l+(E-d/A)2
t
]'
(9)
Vol. 54, No. 11
where the first factor represents 3Ed/~e d and is obtained by differentiating equation (7) with respect to ea (and noting that 7rrlu = (ed + ~ U)/A). The expression for the T-linear coefficient 3' of the impurity specific heat is standard [16], 7'-
3A 21rk~ 7 = ~/[1 + (ffa/A)2],
(10)
where = 1 -- [a2ao(co)/8(ico)] w=o
(1 1)
since Y'do(CO) -- ~ao(CO) is co-independent. The impurity spin susceptibility X, can then be written as , 2~rA Xs = (gUB)2 Xs = 27' -- X c
(12)
due to the exact relation which has been proved to hold [16] among these three quantities in the Anderson model. Finally, the Wilson ratio Rw is defined as Rw = ×',/7' = 2/[1 + (X'o/X;)].
(13)
As regards the approximation, for Y,aa(co) we take the first nonvanishing perturbative term
<2(co)'= Zh2:(co) =
(14)
which is of the 2nd order in u = UlTrA, and neglect aU higher-order corrections. This approximation finds its justification in the quick convergence of the perturbation expansion (which has been proved explicitly [5] for ~/= O) and, above all, in the very good agreement of the results with the exact ones obtained by the BA method. The diagram (14) as well as [8~, ~(co)/aco] w=o has been calculated before [4]. In Fig. 1 we compare the result for ×c in the case of electron-hole symmetry (77 = 0), obtained by the above described method and approximation (dashed line), with the exact BA result [7] (solid line) and several finite-order approximations of the direct Yosida and Yamada perturbation expansion [3, 5] (dotted line). We see that for u = 2, which is already in the strong correlation (SC) regime, our present result is only 21.5% off and that it is as good as the 6th-order perturbative result. The Wilson ratio Rw shows even better agreement with the BA results: for 17 = 0 and 0 ~< u <~ 4 the maxim u m relative error of R w - 1 amounts to only 1.3% (around u = 1, and is even less for other values of u). For other physical quantities (7, Xs, na) the improvement (with respect to the direct perturbation theory) is somewhat less spectacular, but still significant. Since the results for Xs(u;i?) obtained by the pres-
Vol. 54, No. 11 1.0
THE ANDERSON MODEL
i
I
i
I
0.6
i
I
X'o(U;n =0)
0.6
0.2
'",".
I
4
6 ",,. " ~ "
8
2
3
Fig. 1. Charge susceptibility for r / = O (e a = - - U / 2 ) ; solid line - exact BA result [7], dashed line - present method, dotted line - several finite-order approximations of the direct Yosida and Yamada perturbation expansion [3, 5]. ent method are indeed more accurate than the already reported perturbative ones [17], but do not show any qualitative differences, we will here just quote the main conclusions regarding the "phase diagram". Recalling that Xs measures the local spin fluctuations (LSF), we can see that they reach their maximum in the case of '
'
'
'
I
'
'
'
'
I
'
'
'
'
959
electron-hole symmetry (half-filled VBS) and are quickly suppressed as the asymmetry is increased. Also, the increase of Coulomb correlation enhances the LSF only for O ~ 171 <~ rT*(u), while suppressing them for Ir~l >~ ½. Thus it is only in the local moment (LM) region that the increase of u drives the system into the SC regime with the scaling laws appropriate to the Kondo behaviour. On the other hand, the charge susceptibility ×e, which is the measure of local charge fluctuations, exhibits a clearly defined maximum in the IV part of the "phase diagram". As shown in Fig. 2, the peak of Xe0/) appears for u >~ 1 and becomes better pronounced as u increases. The position of the peak r/M(u) (indicated by vertical arrows) moves to higher values of 171 with increasing u, and as far as we can tell (both from our and BA results), it falls halfway between ~/* and ~7" + 1/rru for u >~ 1.5. We consider it very important to find out the exact dependence of r/M on u, since ~ ( u ) might be regarded as the centre of the IV region just as the r? = 0 line represents the centre of the LM region. Further on, the height of the peak, X'eO1M), is found to decrease and approach a constant value with the increase of u. For u = 1.0, 1.5, 2.0 and 2.5 we have X'c(~lu)= 0.419, 0.353, 0.325 and 0.311. (The comparison with BA results [11] shows the relative error of 1.5% for u = 4/rr.) The large u saturation value is easily obtained by differentiating the relation
n~ = 2 - v ~ - 0 r / 1 2 x / 7 )
(e~/a) + ~ [ ( ~ / a ) ~ l
(15)
of Wiegmann and Tsvelick [8, 9] with respect to e~ at e~ = 0 ( r / = 7/* ~ ~M), since a/be~ = O/aea. We find , rl~{u )
t
\
: 2Z o.o
o.5
Iql
t
1.o
(16)
ts
Fig. 2 Charge susceptibility as function of asymmetry for five fixed values of u. The horizontal arrows on the r / = 0 axis show the exact BA values of Xc(U;r/= 0). The vertical arrows indicate the positions of the maxima, t
~M(u).
~
xc(ea = O) = Xc(U > > 1; ~TM) = rr2/24"v/~ = 0.2908 in conformity with the numbers given above. We see also that the approach to saturation is rather quick: the value (16) is almost reached for u >~ 2. The Wilson ratio Rto (u;r/) is plotted in Fig. 3. As a function of u for various fixed values of asymmetry, R w exhibits two types of behaviour: (i) For It/I K½ it increases monotonously and approaches the saturation value of 2 for large enough u. As Ir/I increases from 0 to ½, the onset of saturation moves to higher values of u. (ii) For t r / l > ~, R w has a maximum at some finite u and drops back to 1 as u -* ~. As 177[decreases, the m ; x i m u m increases in height and its position shifts to higher u values. Considering R w as a function of D?I for various fixed values of u, we see that for u > > 1 it has the value of ~ 2 in the LM region, decreases rapidly in the IV region and attains the weak correlation (WC) value of 1 in the large asymmetry region. If we, however, approach 171 = ½ along the line
THE ANDERSON MODEL
960
Rw _
24.
2.0
1.5
Vol. 54, No. 11
Xrc ~ 7r2/24V~. Since both in the LM and IV region the transition from the WC to the SC regime happens to take place at relatively small u values, the present method with the 2nd-order approximation forZaa can be used to study even the SC features in both of these regions. REFERENCES
/ ~ ,
/, /, /, /, /7"/-..,2% L . Y , \ ,~
1.o
1. 2. 3. 4.
3
Fig. 3 Wilson ratio Rw(u;71). The thick line with arrows represents Rw [(u; r/*(u)]. Ir/I = r/*(u), R w [u; r?*(u)] is seen to saturate at a value different from either 2 or 1. Since BA results [8] give X's07*) = n2/2V~ = 12 ×'c(r/*) for u > > 1, one has [via rel. (13)] Rw = ~ = 1.846 at the "phase boundary" between the LM and IV region. The curve Rw [u; r/*(u)] in Fig. 3. seems to confirm this result. To summarize, for 1771~ 0, i.e. in the centre of the LM region, the SC regime (characterized by the scaling laws and Rw ~ 2) sets on at u ~ 1 and is almost fully reached for u ~ 2 . For larger Ir/I the onset of SC is pushed to higher values of u since the asymmetry reduces the effects of LSF. However, for r~* ~< 171 ~<7/* + 1/Tru the correlation effects (seen here as charge fluctuations) become visible again at u ~ 1 and some kind of SC behaviour is established for u >~ 2, characterized by
5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
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