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A variational theory of the ground state of Anderson lattice
Solid State Communications, Vol. 72, No. 7, pp. 711-716, 1989. Printed in Great Britain.
0038-1098/89 $3.00 + .00 Pergamon Press plc
A V A R I A T I O N A L THEORY OF THE G R O U N D STATE OF ANDERSON LATTICE S. Panwar and I. Singh Physics Department, University of Roorkee, Roorkee (U.P.), India
(Received 30 May 1989 by B. Miihlschlegel) We use a variational method for studying the ground state properties of orbitally nondegenerate Periodic Anderson Model. The two-peak valence X-ray photoemission spectroscopy (XPS) structure of Ce and its compounds indicates the finiteness of the intra-atomic interaction U. Therefore we study ground state properties with finite as well as infinite U. We calculate ground state energy, quasi-particle spectrum, conduction-band and f-band occupation number distribution, etc.
lattice is
1. I N T R O D U C T I O N FROM the low temperature behaviour of electronic specific heat and spin susceptibility, it has been concluded that heavy fermion systems [1-3] (e.g., cerium and uranium compounds like CeSn 3, CeB6, CeA12, CeCu2Si2, UPt3, UBe~3, UAI2, etc.) in the low temperature limit are Fermi liquids [4, 5]. Also from the observation of band gaps and of a sharp Fermi surface in some of the heavy fermions [6], (e.g., CeSn3, SmB6) it became clear that the coherent hybridization within the sites of the periodic lattice is very significant. The periodic Anderson model [7] has been successfully used recently to show various mixed-valence-, Kondoand heavy fermion-characteristics of these compounds [8]. This model has been studied using various approaches and approximations, e.g., 1/Nr expansion methods [9], perturbation expansion methods [10] and variational methods [11-15]. Recently Oguchi [14], Brandow [11], Rice and Ueda [12] and Verma et al. [13] have used variational method with infinite interaction U between f-electrons, to discuss ground state properties of Anderson lattice. However two-peak valence X-ray photoemission spectroscopy (XPS) structure of Ce and its compounds [16] indicates the finiteness of the interaction U (i.e., U < oo). In this paper we use a variational method for describing the ground state properties of the Anderson lattice with finite as well as infinite U. In Section 2 we develop the basic formulations for ground state energy, and for number off-electrons per site. In Section 3 we present results for the variational parameter, momentum distribution functions, etc.
H = ~ k C ~ C k ~ + ~f~b~;bk~ ka
ka
- Z (V~C4obk~ t t + + h.c.) + -2U~ n~oni-,,
(1)
ka
where n£ = b+bj,, bk,(bL) and Cko(CL) are f- and conduction-electron annihilation (creation) operators for state ka. Vk is the hybridization matrix element between f and conduction electrons, which we shall take to be k-independent.
2. BASIC F O R M U L A T I O N
2.1. Weak interaction ( U small) case The variational wave functions for the Anderson lattice used by various authors are simply extensions of the Yafet-Verma wave function for single-impurity [11]. For weak interaction case the wave function may be written as 'lkI'J0)
~--- T~ja [1 + ~ ak]~bj + Ck,][F
k
711
),
(2)
A
where I F ) = I-Ik<,~.° CL [0) is the Fermi sea of conduction electrons in which all the conduction states with [k[ < k r are occupied. The akja are the variational parameters. The state Iq'0 ) allows all c o n f i g u r a t i o n s f ° , f 1 a n d f z. For homogenous mixedvalence the system realizes complete site-equivalence given by akj, -
~ 1 Ak, e~k.nj•
(3)
With this, the wave function [~o) may be written in Block representation [tPo) =
The Hamiltonian for the single-orbital Anderson
k
~[1
+ Ak~b~Ck~][F).
(4)
A VARIATIONAL THEORY OF ANDERSON LATTICE
712
Vol. 72, No. 7
The energy expectation value can be expressed as
E =
X
,_~
(1 + A~)
The optimization A~.~ -
OE/OA~.~ =
0 gives
1
2V~ [(ek -- ~./- Un~.)
+ x/(ek - ~ r -
U n ~ ) 2 + 4V~?].
Equation (12) may be written as
i
!
+
(13)
(6)
The number of conduction electrons is
nk. -
(12)
(5)
(7)
A2
IV)
exp
=
- ~1 ~ Ak~ b• ~. b+kl ka
o"
bkI
a Ck~
}
kI
ka
The number off-electrons is
"~ -- ~ . ' ~
=
l +A~./
2.2. High interaction ( U = oo) case For high intra-atomic interaction U between /°-electrons the probability of f2 configuration is negligible. This fact can be realized by introducing the projection operator P in the trial ground state wave function
x
,IJ)
=
[ l + ~A,.~b:~
('
× l-~Zb;,~b,,_o
FI(, - ,/, ,/i) lq x [l+
= U[I+
)
&
kl
+ l
~akj~b)+Ck~][F).~
(9)
The introduction of P guarantees that the state IV) allows o n l y f ° a n d f ~configurations on each site. State (9) may be written as wV)
(14)
Expanding wavefunction (12)
IV> = PI~Po) =
bL C~ 1 I%).
1
+l{
}3
+
] If)
~a,jobf(1-~ n)Q~)Ck~],F). (lO)
One may easily check that the role of projection operator P (viz. to project out f2 configuration) may be realized if, instead, bi~ is replaced by
b;,,
---
bi~(l
-
b+s o bj_~)
=
bi~(l
-
n/~,)
everywhere. Equation (10) may be expressed as
(11) because
IT)
{bj~+ (1 -
"~1 ~ ) C - o } " ,
=
0
= exp ) ' - -, ZAk~,k.,,b/a+ I - nf ~)Ck~ } I F )
+ . . . ] IF>.
(15)
713
A V A R I A T I O N A L THEORY OF A N D E R S O N LATTICE
Vol. 72, No. 7
5,0 5.0l
.
.
.
.
(b) ~
(0)
U=O 4.0
4,0
T
U=O
3.0
3.0 U=I
U= l
AKcr
AKa-
2.C
2.0
U=oo
1.0
l.[
-0.6
-I,0
- 0.2
0.6 ~F
0.2
,0
-,0
4,
-o!2
D_
0'2
0,
•
Fig. 1. (a) Variational parameters hks as a function o f g k ( = D_ + 1 - cos k) with the bottom of conduction b a n d D = - 1 , V = 0.25.(b) Ak~(ate i = 0.0) as a function of D_, V = 0.25. We see in equation (15) that Ako is accompanied by a factor which may be (1 - O )
or
(1 - 1 )
or
(1 - 2
)
The correlated Hamiltonian may be obtained from the Hamiltonian (1) by simply replacing bjo everywhere by bj~ = (1 - nfD. One gets
H = Z ~,C~;C.+~sZ b:s bks ks
...or
(1
ks
N - I- ) . N
We make an ansatz here and replace these different factors everywhere by the same factor [1 - (nZ_s/N)]. Therefore the correlated wave function (l 3) becomes [W>
=
kiWI[1 + A , ~ ( I -
1(17)
nf_s)b~sCks],F>. (16)
5.0
I
Now the energy expectation value may be obtained as i
(o)
i
i
(b)
U=0 4.0
/.,0
3.0
3.[ U=O
AKCr
U=I
21
2,( U=I U=eo
u=oo
1.0
-1.0
_/.E
_01.2
01.2
, I (~F 0.6
1.0
-1.0
-7.6
-01.2
D_ "----~
01.2
0.6
Fig. 2. (a) Ako as a function ofek = {(2/n) [Ik[ + D_ (n/2)]} with D_ = - 1, V = 0.25. (b) Ak~ (at ef ---- 0.0) as a function o f D , V = 0.25.
714
A V A R I A T I O N A L T H E O R Y OF A N D E R S O N
LATTICE
Vol. 72, N o . 7
(a)
(b)
j
U= 0 U=l
U=O U=I
0.0
0.0
l"
0.5
U=~
n~
n[,o0.4
0.4
U=~
0.2
-0.6
-1.0
-0.2
0.2
0.5 ~JF --
-10
1.0
-°%
-d~ D_
Fig. 3. (a) The m o m e n t u m distribution function n/, as a function o f e~ ( = D V = 0.25. (b) n[~ (at ey = 0.0) as a function o f D_, V = 0.25.
E _
0,
+ 1 - cos k) with D_ = - 1,
(u?[ Hl U?> (q~lqJ)
The number o f conduction electrons and f-electrons are given by
~. e, - 2Vk Ak~(1 -- ni_,) 2 + as A~,(1 - ni_,) 3 ~. [1 + A / ~ ( 1 - nf_,.) 2] (18)
1 nk,~ --= 1 + A~,,(I
The optimization
Ak~ =
0'~ :"
OE/OAk~=
1 2V,(1 - nf~) [{ek
+ d{~k
--
el(1
--
-
0 gives
nf
~f(1 - n~o)}
n/~)} 2 + 4V~(1
--
=
(20) -
-
r/f,,) 2
2 Ak~(1 - - n fL ) 3 1 + A ~ ( 1 -- nf~) 2
(21)
3. R E S U L T S A N D D I S C U S S I O N For calculating various quantities like variational parameters A ~ m o m e n t u m distribution functions nk~
nf~)2]. (19)
"7
1~1
(e)
I
I
I (b)
1[-
U=O U= 0
U= 1
0.9 U=I
l
n'.~,
0.5
f nka, U=~ 0.4
U=oo 0.2
-10
-0,6
-0.2
0,2
i (~F 0,5
1.0
-I0
-~,
-d2
~2
0.5
D _ --------~
Fig. 4. (a) nk~ as a function o f e, [ = (2/n)(Ikl + D_ n/2)] with D a function o f D _ , V = 0.25.
= - 1, V = 0.25. (b) n{~ (at ei = 0.0) as
Vol. 72, N o . 7
A VARIATIONAL
THEORY
OF ANDERSON 1.0
(a)
LATTICE
715
I
(b)
0.8 U=O
l
I
O.E
0.6 U=O
n"
n~
U=I
0.41
0.4
0.2
~2
I
-1.4
-1.0
-0.6
-0.2
0.2
U=I
r
-14
0.6
I
-1.0
D_
Fig. 5. n{ as a function o f D
T
¢
I
-0.2
0.2
0,6
w
[for dispersion ek = (D_ + 1 -- cos k)] with (a) V = 0.0 and (b) V -- 0.25.
and n ~ and number o f f - e l e c t r o n s n{, we have chosen (i) a one-dimensional system with c o n d u c t i o n band dispersion ek = (D_ + 1 - c o s k ) , (ii) Vk to be k-independent, (iii) a paramagnetic state, i.e., n[~ = n~ ~ = ~nklr, n{ = n r , = ½nr and Ak, = Ak_,. For c o m p a r i s o n we have also repeated calculations for fiat c o n d u c t i o n band ek = (21re) [Ikl + D_(~/2)]. We have taken total number o f electrons per site to be 1.5, and ~-r = 0.0. D_ is the b o t t o m o f the c o n d u c t i o n band, the c o n d u c t i o n bandwidth being 2 eV. In Fig. l(a) we have s h o w n variation o f the variational parameters Ak, as a function o f the band energy ek (for fixed b o t t o m o f the band). In Fig. l(b) 1.2
I
-0.6
I
~ - -
Ak~ (at energy O = 0.0) is s h o w n as a function o f the bottom o f the band D . In Figs. l(a) and (b) we have taken conduction band asek = ( D + 1 -- cos k). In Figs. 2(a) and (b) we have s h o w n Ak, as in Figs. l(a) and (b) but a linear dispersion e~ = (2/rt) [[kl -tD 0t/2)]. Figures 3 and 4 give variation o f the m o m e n t u m distribution function n[~ as a function o f Sk and o f D _ . Figures 5 and 6 s h o w variation o f j:electrons n~ as one m o v e s ef with respect to the b o t t o m o f conduction band. One observes that the variational parameter Ak, increases with Sk for the fixed value o f V~ = V. Also Ak~ decreases as U increases for a fixed ek and V~.. Ak~ 1.2
I
I
[
(o)
(b) 1.0
0.8
U=0
T
U=0
0.6:
O,G
n"
n~
U= 1
U=I
0.2
-1.4
-1.0
- O.G
D_
-0 2
0.6
I
-1,4
I
-1,0
~
Fig. 6. n~ as a function o f D (b) V = 0.25.
I
02
1
- 0.6
-01,2
1.2
0.6
D_ "---~
(for linear dispersion ek = (2/g)[]kl + D
re/2)] with (a) V = 0.0. and
716
A VARIATIONAL THEORY OF ANDERSON LATTICE
increases as the bottom of the conduction band shifts upwards with respect to f-level. We have done calculations for finite U (allowing all three configurations f0, fJ and f2) as well as for infinite U (allowing only two configurations f0 and f ) . As U increases, the hybridization Vk~ reduces because of the occupation of - a state. Also the variational parameter Ak~ decreases as U increases. In the extreme case of U ~ ~ , where a site can have only one electron at a time in f-state, the space off-band is contracted. The contraction factor is (1 - ni_~), effective Ak, is reduced to A~,(I - n f . ) , effective f-level position ef is changed to e.f(1 - n1~). Our results using the variational wavefunction [equation (16)] are close to Yanagisawa's [15] higher order results. We are calculating other ground state properties, e.g., static magnetic susceptibility etc. using our variational wavefunction equation (16).
Acknowledgement - O n e of us (S. P.) is thankful to the University Grants Commission (India) for the financial support. REFERENCES 1. 2.
3. 4.
G.R. Stewart, Rev. Mod. Phys. 56, 755 (1984). P. Fulde, J. Keller & G. Zwicknagl, Solid State Physics (Edited by H. Ehrenreich & D. Turnbull), Vol. 41, p. 1, Academic, New York (1988). Theory of Heavy Fermions and Valence Fluctuations (Edited by T. Kasuya & T. Saso), Springer, New York (1985). D . M . Newns & A.C. Hewson, J. Phys. F10, 2429 (1980).
5. 6.
7. 8. 9.
10. 11. 12. 13. 14. 15. 16.
Vol. 72, No. 7
R.M. Martin, Phys. Rev. Lett. 48, 362 (1982). T. Kasuya, K. Takegahara, T. Fujita, T. Tanaka & E. Bannai, J. Phys. (Paris) Colloq. 40, C5-30 (1980); P. Haen, F. Lapierre, J.M. Mignot, R. Tournier & F. Holtzenberg, Phys. Rev. Lett. 43, 304 (1979); G.W. Crabtree, W.R. Johanson, A.S. Edelstein & O.D. McMasters, Valence Fluctuations in Solids (Edited by L.M. Falicov, W. Hanke & M.P. Maple), p. 93, North Holland (1981). S. Doniach, Physica 91B, 231 (1977); H.J. Leder & B. Muhlschlegel, Z. Phys. B29, 341 (1978). C.M. Verma, Ref. [3], p. 277; S. Doniach, Phys. Rev. B35, 1814 (1987); Z. Tesanovic & O.T. Valls, Phys. Rev. B34, 1918 (1986). T.V. Ramakrishnan & K. Sur, Phys. Rev. B26, 1798 (1982); P. Coleman, Phys. Rev. B29, 3035 (1984); N. Read & D.M. Newns, Solid State Commun. 52, 993 (1984). K. Yamada & K. Yosida, J. Mag. Mag. Mat. 31-34, 461 (1983), in Ref [3] p. 183. B . H . Brandow, Phys. Rev. B33, 215 (1986); J. Mag. Mag. Mat. 63-64, 264 (1987). T.M. Rice & K. Ueda, Phys. Rev. Lett. 55, 995 (1985); Phys. Rev. B34, 6420 (1986). C.M. Verma, W. Weber & L.J. Randall, Phys. Rev. B33, 1015 (1986). A. Oguchi, Prog. Theor. Phys. 77, 278 (1987). T. Yanagisawa, Phys. Rev. B37, 2050 (1988); P. Fazekas, SolidState Commun. 60, 431 (1986). J.C. Fuggle, F.U. Hillebrecht, Z. Zolnierek, R. Lasser, C. Freiburg, O. Gunnarsson & K. Schonhammer, Phys. Rev. B27, 7330 (1983); O. Gunnarson & K. Schonhammer, Phys. Rev. B31, 4815 (1985).
0038-1098/89 $3.00 + .00 Pergamon Press plc
A V A R I A T I O N A L THEORY OF THE G R O U N D STATE OF ANDERSON LATTICE S. Panwar and I. Singh Physics Department, University of Roorkee, Roorkee (U.P.), India
(Received 30 May 1989 by B. Miihlschlegel) We use a variational method for studying the ground state properties of orbitally nondegenerate Periodic Anderson Model. The two-peak valence X-ray photoemission spectroscopy (XPS) structure of Ce and its compounds indicates the finiteness of the intra-atomic interaction U. Therefore we study ground state properties with finite as well as infinite U. We calculate ground state energy, quasi-particle spectrum, conduction-band and f-band occupation number distribution, etc.
lattice is
1. I N T R O D U C T I O N FROM the low temperature behaviour of electronic specific heat and spin susceptibility, it has been concluded that heavy fermion systems [1-3] (e.g., cerium and uranium compounds like CeSn 3, CeB6, CeA12, CeCu2Si2, UPt3, UBe~3, UAI2, etc.) in the low temperature limit are Fermi liquids [4, 5]. Also from the observation of band gaps and of a sharp Fermi surface in some of the heavy fermions [6], (e.g., CeSn3, SmB6) it became clear that the coherent hybridization within the sites of the periodic lattice is very significant. The periodic Anderson model [7] has been successfully used recently to show various mixed-valence-, Kondoand heavy fermion-characteristics of these compounds [8]. This model has been studied using various approaches and approximations, e.g., 1/Nr expansion methods [9], perturbation expansion methods [10] and variational methods [11-15]. Recently Oguchi [14], Brandow [11], Rice and Ueda [12] and Verma et al. [13] have used variational method with infinite interaction U between f-electrons, to discuss ground state properties of Anderson lattice. However two-peak valence X-ray photoemission spectroscopy (XPS) structure of Ce and its compounds [16] indicates the finiteness of the interaction U (i.e., U < oo). In this paper we use a variational method for describing the ground state properties of the Anderson lattice with finite as well as infinite U. In Section 2 we develop the basic formulations for ground state energy, and for number off-electrons per site. In Section 3 we present results for the variational parameter, momentum distribution functions, etc.
H = ~ k C ~ C k ~ + ~f~b~;bk~ ka
ka
- Z (V~C4obk~ t t + + h.c.) + -2U~ n~oni-,,
(1)
ka
where n£ = b+bj,, bk,(bL) and Cko(CL) are f- and conduction-electron annihilation (creation) operators for state ka. Vk is the hybridization matrix element between f and conduction electrons, which we shall take to be k-independent.
2. BASIC F O R M U L A T I O N
2.1. Weak interaction ( U small) case The variational wave functions for the Anderson lattice used by various authors are simply extensions of the Yafet-Verma wave function for single-impurity [11]. For weak interaction case the wave function may be written as 'lkI'J0)
~--- T~ja [1 + ~ ak]~bj + Ck,][F
k
711
),
(2)
A
where I F ) = I-Ik<,~.° CL [0) is the Fermi sea of conduction electrons in which all the conduction states with [k[ < k r are occupied. The akja are the variational parameters. The state Iq'0 ) allows all c o n f i g u r a t i o n s f ° , f 1 a n d f z. For homogenous mixedvalence the system realizes complete site-equivalence given by akj, -
~ 1 Ak, e~k.nj•
(3)
With this, the wave function [~o) may be written in Block representation [tPo) =
The Hamiltonian for the single-orbital Anderson
k
~[1
+ Ak~b~Ck~][F).
(4)
A VARIATIONAL THEORY OF ANDERSON LATTICE
712
Vol. 72, No. 7
The energy expectation value can be expressed as
E =
X
,_~
(1 + A~)
The optimization A~.~ -
OE/OA~.~ =
0 gives
1
2V~ [(ek -- ~./- Un~.)
+ x/(ek - ~ r -
U n ~ ) 2 + 4V~?].
Equation (12) may be written as
i
!
+
(13)
(6)
The number of conduction electrons is
nk. -
(12)
(5)
(7)
A2
IV)
exp
=
- ~1 ~ Ak~ b• ~. b+kl ka
o"
bkI
a Ck~
}
kI
ka
The number off-electrons is
"~ -- ~ . ' ~
=
l +A~./
2.2. High interaction ( U = oo) case For high intra-atomic interaction U between /°-electrons the probability of f2 configuration is negligible. This fact can be realized by introducing the projection operator P in the trial ground state wave function
x
,IJ)
=
[ l + ~A,.~b:~
('
× l-~Zb;,~b,,_o
FI(, - ,/, ,/i) lq x [l+
= U[I+
)
&
kl
+ l
~akj~b)+Ck~][F).~
(9)
The introduction of P guarantees that the state IV) allows o n l y f ° a n d f ~configurations on each site. State (9) may be written as wV)
(14)
Expanding wavefunction (12)
IV> = PI~Po) =
bL C~ 1 I%).
1
+l{
}3
+
] If)
~a,jobf(1-~ n)Q~)Ck~],F). (lO)
One may easily check that the role of projection operator P (viz. to project out f2 configuration) may be realized if, instead, bi~ is replaced by
b;,,
---
bi~(l
-
b+s o bj_~)
=
bi~(l
-
n/~,)
everywhere. Equation (10) may be expressed as
(11) because
IT)
{bj~+ (1 -
"~1 ~ ) C - o } " ,
=
0
= exp ) ' - -, ZAk~,k.,,b/a+ I - nf ~)Ck~ } I F )
+ . . . ] IF>.
(15)
713
A V A R I A T I O N A L THEORY OF A N D E R S O N LATTICE
Vol. 72, No. 7
5,0 5.0l
.
.
.
.
(b) ~
(0)
U=O 4.0
4,0
T
U=O
3.0
3.0 U=I
U= l
AKcr
AKa-
2.C
2.0
U=oo
1.0
l.[
-0.6
-I,0
- 0.2
0.6 ~F
0.2
,0
-,0
4,
-o!2
D_
0'2
0,
•
Fig. 1. (a) Variational parameters hks as a function o f g k ( = D_ + 1 - cos k) with the bottom of conduction b a n d D = - 1 , V = 0.25.(b) Ak~(ate i = 0.0) as a function of D_, V = 0.25. We see in equation (15) that Ako is accompanied by a factor which may be (1 - O )
or
(1 - 1 )
or
(1 - 2
)
The correlated Hamiltonian may be obtained from the Hamiltonian (1) by simply replacing bjo everywhere by bj~ = (1 - nfD. One gets
H = Z ~,C~;C.+~sZ b:s bks ks
...or
(1
ks
N - I- ) . N
We make an ansatz here and replace these different factors everywhere by the same factor [1 - (nZ_s/N)]. Therefore the correlated wave function (l 3) becomes [W>
=
kiWI[1 + A , ~ ( I -
1(17)
nf_s)b~sCks],F>. (16)
5.0
I
Now the energy expectation value may be obtained as i
(o)
i
i
(b)
U=0 4.0
/.,0
3.0
3.[ U=O
AKCr
U=I
21
2,( U=I U=eo
u=oo
1.0
-1.0
_/.E
_01.2
01.2
, I (~F 0.6
1.0
-1.0
-7.6
-01.2
D_ "----~
01.2
0.6
Fig. 2. (a) Ako as a function ofek = {(2/n) [Ik[ + D_ (n/2)]} with D_ = - 1, V = 0.25. (b) Ak~ (at ef ---- 0.0) as a function o f D , V = 0.25.
714
A V A R I A T I O N A L T H E O R Y OF A N D E R S O N
LATTICE
Vol. 72, N o . 7
(a)
(b)
j
U= 0 U=l
U=O U=I
0.0
0.0
l"
0.5
U=~
n~
n[,o0.4
0.4
U=~
0.2
-0.6
-1.0
-0.2
0.2
0.5 ~JF --
-10
1.0
-°%
-d~ D_
Fig. 3. (a) The m o m e n t u m distribution function n/, as a function o f e~ ( = D V = 0.25. (b) n[~ (at ey = 0.0) as a function o f D_, V = 0.25.
E _
0,
+ 1 - cos k) with D_ = - 1,
(u?[ Hl U?> (q~lqJ)
The number o f conduction electrons and f-electrons are given by
~. e, - 2Vk Ak~(1 -- ni_,) 2 + as A~,(1 - ni_,) 3 ~. [1 + A / ~ ( 1 - nf_,.) 2] (18)
1 nk,~ --= 1 + A~,,(I
The optimization
Ak~ =
0'~ :"
OE/OAk~=
1 2V,(1 - nf~) [{ek
+ d{~k
--
el(1
--
-
0 gives
nf
~f(1 - n~o)}
n/~)} 2 + 4V~(1
--
=
(20) -
-
r/f,,) 2
2 Ak~(1 - - n fL ) 3 1 + A ~ ( 1 -- nf~) 2
(21)
3. R E S U L T S A N D D I S C U S S I O N For calculating various quantities like variational parameters A ~ m o m e n t u m distribution functions nk~
nf~)2]. (19)
"7
1~1
(e)
I
I
I (b)
1[-
U=O U= 0
U= 1
0.9 U=I
l
n'.~,
0.5
f nka, U=~ 0.4
U=oo 0.2
-10
-0,6
-0.2
0,2
i (~F 0,5
1.0
-I0
-~,
-d2
~2
0.5
D _ --------~
Fig. 4. (a) nk~ as a function o f e, [ = (2/n)(Ikl + D_ n/2)] with D a function o f D _ , V = 0.25.
= - 1, V = 0.25. (b) n{~ (at ei = 0.0) as
Vol. 72, N o . 7
A VARIATIONAL
THEORY
OF ANDERSON 1.0
(a)
LATTICE
715
I
(b)
0.8 U=O
l
I
O.E
0.6 U=O
n"
n~
U=I
0.41
0.4
0.2
~2
I
-1.4
-1.0
-0.6
-0.2
0.2
U=I
r
-14
0.6
I
-1.0
D_
Fig. 5. n{ as a function o f D
T
¢
I
-0.2
0.2
0,6
w
[for dispersion ek = (D_ + 1 -- cos k)] with (a) V = 0.0 and (b) V -- 0.25.
and n ~ and number o f f - e l e c t r o n s n{, we have chosen (i) a one-dimensional system with c o n d u c t i o n band dispersion ek = (D_ + 1 - c o s k ) , (ii) Vk to be k-independent, (iii) a paramagnetic state, i.e., n[~ = n~ ~ = ~nklr, n{ = n r , = ½nr and Ak, = Ak_,. For c o m p a r i s o n we have also repeated calculations for fiat c o n d u c t i o n band ek = (21re) [Ikl + D_(~/2)]. We have taken total number o f electrons per site to be 1.5, and ~-r = 0.0. D_ is the b o t t o m o f the c o n d u c t i o n band, the c o n d u c t i o n bandwidth being 2 eV. In Fig. l(a) we have s h o w n variation o f the variational parameters Ak, as a function o f the band energy ek (for fixed b o t t o m o f the band). In Fig. l(b) 1.2
I
-0.6
I
~ - -
Ak~ (at energy O = 0.0) is s h o w n as a function o f the bottom o f the band D . In Figs. l(a) and (b) we have taken conduction band asek = ( D + 1 -- cos k). In Figs. 2(a) and (b) we have s h o w n Ak, as in Figs. l(a) and (b) but a linear dispersion e~ = (2/rt) [[kl -tD 0t/2)]. Figures 3 and 4 give variation o f the m o m e n t u m distribution function n[~ as a function o f Sk and o f D _ . Figures 5 and 6 s h o w variation o f j:electrons n~ as one m o v e s ef with respect to the b o t t o m o f conduction band. One observes that the variational parameter Ak, increases with Sk for the fixed value o f V~ = V. Also Ak~ decreases as U increases for a fixed ek and V~.. Ak~ 1.2
I
I
[
(o)
(b) 1.0
0.8
U=0
T
U=0
0.6:
O,G
n"
n~
U= 1
U=I
0.2
-1.4
-1.0
- O.G
D_
-0 2
0.6
I
-1,4
I
-1,0
~
Fig. 6. n~ as a function o f D (b) V = 0.25.
I
02
1
- 0.6
-01,2
1.2
0.6
D_ "---~
(for linear dispersion ek = (2/g)[]kl + D
re/2)] with (a) V = 0.0. and
716
A VARIATIONAL THEORY OF ANDERSON LATTICE
increases as the bottom of the conduction band shifts upwards with respect to f-level. We have done calculations for finite U (allowing all three configurations f0, fJ and f2) as well as for infinite U (allowing only two configurations f0 and f ) . As U increases, the hybridization Vk~ reduces because of the occupation of - a state. Also the variational parameter Ak~ decreases as U increases. In the extreme case of U ~ ~ , where a site can have only one electron at a time in f-state, the space off-band is contracted. The contraction factor is (1 - ni_~), effective Ak, is reduced to A~,(I - n f . ) , effective f-level position ef is changed to e.f(1 - n1~). Our results using the variational wavefunction [equation (16)] are close to Yanagisawa's [15] higher order results. We are calculating other ground state properties, e.g., static magnetic susceptibility etc. using our variational wavefunction equation (16).
Acknowledgement - O n e of us (S. P.) is thankful to the University Grants Commission (India) for the financial support. REFERENCES 1. 2.
3. 4.
G.R. Stewart, Rev. Mod. Phys. 56, 755 (1984). P. Fulde, J. Keller & G. Zwicknagl, Solid State Physics (Edited by H. Ehrenreich & D. Turnbull), Vol. 41, p. 1, Academic, New York (1988). Theory of Heavy Fermions and Valence Fluctuations (Edited by T. Kasuya & T. Saso), Springer, New York (1985). D . M . Newns & A.C. Hewson, J. Phys. F10, 2429 (1980).
5. 6.
7. 8. 9.
10. 11. 12. 13. 14. 15. 16.
Vol. 72, No. 7
R.M. Martin, Phys. Rev. Lett. 48, 362 (1982). T. Kasuya, K. Takegahara, T. Fujita, T. Tanaka & E. Bannai, J. Phys. (Paris) Colloq. 40, C5-30 (1980); P. Haen, F. Lapierre, J.M. Mignot, R. Tournier & F. Holtzenberg, Phys. Rev. Lett. 43, 304 (1979); G.W. Crabtree, W.R. Johanson, A.S. Edelstein & O.D. McMasters, Valence Fluctuations in Solids (Edited by L.M. Falicov, W. Hanke & M.P. Maple), p. 93, North Holland (1981). S. Doniach, Physica 91B, 231 (1977); H.J. Leder & B. Muhlschlegel, Z. Phys. B29, 341 (1978). C.M. Verma, Ref. [3], p. 277; S. Doniach, Phys. Rev. B35, 1814 (1987); Z. Tesanovic & O.T. Valls, Phys. Rev. B34, 1918 (1986). T.V. Ramakrishnan & K. Sur, Phys. Rev. B26, 1798 (1982); P. Coleman, Phys. Rev. B29, 3035 (1984); N. Read & D.M. Newns, Solid State Commun. 52, 993 (1984). K. Yamada & K. Yosida, J. Mag. Mag. Mat. 31-34, 461 (1983), in Ref [3] p. 183. B . H . Brandow, Phys. Rev. B33, 215 (1986); J. Mag. Mag. Mat. 63-64, 264 (1987). T.M. Rice & K. Ueda, Phys. Rev. Lett. 55, 995 (1985); Phys. Rev. B34, 6420 (1986). C.M. Verma, W. Weber & L.J. Randall, Phys. Rev. B33, 1015 (1986). A. Oguchi, Prog. Theor. Phys. 77, 278 (1987). T. Yanagisawa, Phys. Rev. B37, 2050 (1988); P. Fazekas, SolidState Commun. 60, 431 (1986). J.C. Fuggle, F.U. Hillebrecht, Z. Zolnierek, R. Lasser, C. Freiburg, O. Gunnarsson & K. Schonhammer, Phys. Rev. B27, 7330 (1983); O. Gunnarson & K. Schonhammer, Phys. Rev. B31, 4815 (1985).