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Perturbation theory of the one-dimensional Anderson lattice
455
Journal of Magnetism and Magnetic Materials 31-34 (1983) 455-456 PERTURBATION
THEORY
OF THE
ONE-DIMENSIONAL
ANDERSON LATTICE
H. K A G A a n d Y. S H I B U Y A D e p a r t m e n t o f Physics, Niigata University, Niigata 950-21, J a p a n
The ground-state densities of states and single-particle dispersions of the one-dimensional periodic Anderson Hamiltonian with Coulomb screening Us are studied by perturbation expansion in U and Us for the half-filled ( N , / N = 2) symmetric case. Despite considerable efforts the peculiar low-temperature behaviors, of much current interest, of valence fluctuation systems and dense Kondo systems have not been unveiled [1]. The model we address here is the one-dimensional version of the Anderson lattice which is recognized as describing the essence of these systems. A first full-fledged many-body study of this model has recently been made by Yamada and Yosida [2] who obtained the localized-electron density of states. We also investigate the effects of Coulomb screening Us of localized electrons by conduction electrons, which tends to partly screen Coulomb repulsion U between localized electrons and to modify the relative electron occupances of the two bands, thus being important for these systems [3]. We study the single-particle dispersions and the average densities of states of conduction as well as localized electrons in second-order self-energy renormalizations by both U and Us. We consider here only the symmetric Anderson lattice with two electrons per site. The symmetric Anderson lattice Hamiltonian augmented with Coulomb screening Us is
(7) (8)
o:o
aeeo oeX {[ 1 T OEo &k
(9)
The perturbed Green's functions are derived with second-order self-energy renormalizations ZIk~(~) by H 1 and Zk,,(~o 22" ) and Zko(~O 2c ) b y / / 2 : =
[iw - z k12" o(,~)
22'
- Zko(,~)
k
-
V2 ~o
] --I
it0 - ik -- ~ko(°~) -
,
- . V k o) :( °, o
I.
V2
H = Ho + H, + H2, -
+
no = E ~c~oc~° ko
..1¢. V
(l) ~
/
- - ~ 2., reqN iko
ikR
+
- ¼ N U - N Us, H,= UE(n,t
-
] -1 22"
(11)
~o(,O)
The average densities of states are calculated from these Green's functions G[o(~O) and G~
(3)
- ~ J) ( rt i, - ½ ) ,
o;(.,)
i
H2= U s E
12"
-
,,~- ~o(,~)-
'~,ocko + h.c.)
(1o)
(n~,o- ½)(n,o,- ½),
'~
aL(,o)
(4)
ion"
where the shifted symmetric conduction band gk
~ =-,~ + ~ = - (,~+~/~ + ~ ) = - ~ + ~ / ~ ,
(5)
and where the imaginary parts of the self-energies vanish, ~
2 ¢
and the localized levels
p L ( , ~ ) = a(;,ko) + ( v / , o ~ ) p~o(,~),
03)
Eo + U~ = - ( E o + U + U ~ ) , E o
~ o -= ,0 - Z ~12' o(,~)
(14)
= -½U-U~.
(6)
n~o and n~o are the number operators satisfying N ~ / N --~~.,io(ncJo)/N = 1, N ~ / N = Y~io(nio)/N = 1 for the symmetric case, where subscripts or superscripts £, c refer to as localized and conduction electrons hereafter. By first diagonalizing H 0 we obtain the two ( + ) hybridized bands E ~ and the unperturbed localized G2"to)~ ,x and conduction-electron Green's functions c(O) Gko (,o), 0304-8853/83/0000-0000/$03.00
2£ - ~k°(,o).
The perturbed hybridized bands are no longer well defined, but the effective single-particle dispersions o~± ( k ) can be determined from the peaks of the k-dependent densities of states, tre~o(~) and O~,o(~). The resultant dispersions ~ ± ( k ) would naturally possess finite life-time broadenings. Using a constant-density conduction band gk = 21kV~r - 1 ( - 1 ~
© 1983 N o r t h - H o l l a n d
H. Kaga, Y. Shibuya / One-dimensional periodic Anderson model
456
°I
,>--,_._° I
I- S-
I\!
•-Q5L ~ ~-2
0.5
r ; /f
~.
-,.ol/
'.::.....
I
, •¢ 1 2
%L3
0.:
.'~
, tO
0.5
0
i
"~i'-~
-0.5
•
>-
:7 ."~/2
-13t~ 0
(a)
I°
K ~
o
0
L3. . . .
0.5 (b)
L1,CI L2,C~
f~'c({°)
1.0 N d
Fig. 1. (a) The effective single-particle dispersions to-(k) ( < 0) and (b) the average densities of states of localized (solid) and conduction (dotted) electrons below the gap (¢(k)~- 0). Horizontal short lines denote the measure of life-time broadening. p~(to), which are shown in fig. ia and b for the symmetric half b a n d ( t o - ( k ) < 0) for the two parameter sets ( U = 1.0, Us--O, V = 0.2) and ( U = 1.0, Us = 1.0, V = 0.2). Generally, a three-peak spectrum with some additional structures is found to exist in both ~ ( t o ) and p~(o~), the former being in accord with ref. [2]. While the main peak (L3) at to = _+ U / 2 on p~(to) corresponds to
the virtual local-moment states caused by electron correlation H~, this momentarily localized state gives rise to a dip (D) on p~(to) just at the same energy (to -- + U/2) due to a sort of many-body antiresonance. The main peak (C3) on p~(to) is the resonance of the L3-peak of pz(to). However, these peaks are not good single-particle states because of small life times (fig. I a). The two sharp peaks (L1, L2) of p~,(to) and (C1, C2) of #~(to), instead of the unperturbed single peak of t~,(°)(~), result due to the fact that the unperturbed indirect gap ( ~ 2 V 2) between k = 7r and k---0 becomes a more direct and smaller perturbed one near k ~ ~r/2. These states close to the Fermi energy OF(= 0) are well-defined quasi-particle states. There are also good quasi-particle states in the energy regions (1~1> U/2) outside the virtual-moment states, however the single-particle states in the other regions are not well defined due to the spin fluctuations. Coulomb screening Us does not alter the above general features of the symmetric case, but has the following important effects on 0~(to), P~(~) and 0~± (k). This interaction causes both spectra ~ ( t o ) and p~,(~) broader to the range - 1 - Us ~< ~ < 1 + US, the L3-peak closer to _ U/2, and the gap narrower and closer to k = ~r/2. A more significant US effect is to form the near-gap quasi-particle states in the wider k-space and energy region, as if their dispersion would tend to be separated from the rest to open up a new gap in between. References [1] Valence Fluctuations in Solids, eds. L.M. Falicov et al. (North-Holland, Amsterdam, 1981). [2] K. Yamada and K. Yosida, in: Electron Correlation and Magnetism in Narrow-Band Systems, ed. T. Moriya (Springer-Vedag, Berlin, 1981) p. 210. [3] H. Kaga, Progr. Theoret. Phys. 65 (1981) 1485. H. Kaga and I. Sato, ibid. 65 (1981) 105. H. Kaga, ibid. 67 (1982) 1659.
Journal of Magnetism and Magnetic Materials 31-34 (1983) 455-456 PERTURBATION
THEORY
OF THE
ONE-DIMENSIONAL
ANDERSON LATTICE
H. K A G A a n d Y. S H I B U Y A D e p a r t m e n t o f Physics, Niigata University, Niigata 950-21, J a p a n
The ground-state densities of states and single-particle dispersions of the one-dimensional periodic Anderson Hamiltonian with Coulomb screening Us are studied by perturbation expansion in U and Us for the half-filled ( N , / N = 2) symmetric case. Despite considerable efforts the peculiar low-temperature behaviors, of much current interest, of valence fluctuation systems and dense Kondo systems have not been unveiled [1]. The model we address here is the one-dimensional version of the Anderson lattice which is recognized as describing the essence of these systems. A first full-fledged many-body study of this model has recently been made by Yamada and Yosida [2] who obtained the localized-electron density of states. We also investigate the effects of Coulomb screening Us of localized electrons by conduction electrons, which tends to partly screen Coulomb repulsion U between localized electrons and to modify the relative electron occupances of the two bands, thus being important for these systems [3]. We study the single-particle dispersions and the average densities of states of conduction as well as localized electrons in second-order self-energy renormalizations by both U and Us. We consider here only the symmetric Anderson lattice with two electrons per site. The symmetric Anderson lattice Hamiltonian augmented with Coulomb screening Us is
(7) (8)
o:o
aeeo oeX {[ 1 T OEo &k
(9)
The perturbed Green's functions are derived with second-order self-energy renormalizations ZIk~(~) by H 1 and Zk,,(~o 22" ) and Zko(~O 2c ) b y / / 2 : =
[iw - z k12" o(,~)
22'
- Zko(,~)
k
-
V2 ~o
] --I
it0 - ik -- ~ko(°~) -
,
- . V k o) :( °, o
I.
V2
H = Ho + H, + H2, -
+
no = E ~c~oc~° ko
..1¢. V
(l) ~
/
- - ~ 2., reqN iko
ikR
+
- ¼ N U - N Us, H,= UE(n,t
-
] -1 22"
(11)
~o(,O)
The average densities of states are calculated from these Green's functions G[o(~O) and G~
(3)
- ~ J) ( rt i, - ½ ) ,
o;(.,)
i
H2= U s E
12"
-
,,~- ~o(,~)-
'~,ocko + h.c.)
(1o)
(n~,o- ½)(n,o,- ½),
'~
aL(,o)
(4)
ion"
where the shifted symmetric conduction band gk
~ =-,~ + ~ = - (,~+~/~ + ~ ) = - ~ + ~ / ~ ,
(5)
and where the imaginary parts of the self-energies vanish, ~
2 ¢
and the localized levels
p L ( , ~ ) = a(;,ko) + ( v / , o ~ ) p~o(,~),
03)
Eo + U~ = - ( E o + U + U ~ ) , E o
~ o -= ,0 - Z ~12' o(,~)
(14)
= -½U-U~.
(6)
n~o and n~o are the number operators satisfying N ~ / N --~~.,io(ncJo)/N = 1, N ~ / N = Y~io(nio)/N = 1 for the symmetric case, where subscripts or superscripts £, c refer to as localized and conduction electrons hereafter. By first diagonalizing H 0 we obtain the two ( + ) hybridized bands E ~ and the unperturbed localized G2"to)~ ,x and conduction-electron Green's functions c(O) Gko (,o), 0304-8853/83/0000-0000/$03.00
2£ - ~k°(,o).
The perturbed hybridized bands are no longer well defined, but the effective single-particle dispersions o~± ( k ) can be determined from the peaks of the k-dependent densities of states, tre~o(~) and O~,o(~). The resultant dispersions ~ ± ( k ) would naturally possess finite life-time broadenings. Using a constant-density conduction band gk = 21kV~r - 1 ( - 1 ~
© 1983 N o r t h - H o l l a n d
H. Kaga, Y. Shibuya / One-dimensional periodic Anderson model
456
°I
,>--,_._° I
I- S-
I\!
•-Q5L ~ ~-2
0.5
r ; /f
~.
-,.ol/
'.::.....
I
, •¢ 1 2
%L3
0.:
.'~
, tO
0.5
0
i
"~i'-~
-0.5
•
>-
:7 ."~/2
-13t~ 0
(a)
I°
K ~
o
0
L3. . . .
0.5 (b)
L1,CI L2,C~
f~'c({°)
1.0 N d
Fig. 1. (a) The effective single-particle dispersions to-(k) ( < 0) and (b) the average densities of states of localized (solid) and conduction (dotted) electrons below the gap (¢(k)~- 0). Horizontal short lines denote the measure of life-time broadening. p~(to), which are shown in fig. ia and b for the symmetric half b a n d ( t o - ( k ) < 0) for the two parameter sets ( U = 1.0, Us--O, V = 0.2) and ( U = 1.0, Us = 1.0, V = 0.2). Generally, a three-peak spectrum with some additional structures is found to exist in both ~ ( t o ) and p~(o~), the former being in accord with ref. [2]. While the main peak (L3) at to = _+ U / 2 on p~(to) corresponds to
the virtual local-moment states caused by electron correlation H~, this momentarily localized state gives rise to a dip (D) on p~(to) just at the same energy (to -- + U/2) due to a sort of many-body antiresonance. The main peak (C3) on p~(to) is the resonance of the L3-peak of pz(to). However, these peaks are not good single-particle states because of small life times (fig. I a). The two sharp peaks (L1, L2) of p~,(to) and (C1, C2) of #~(to), instead of the unperturbed single peak of t~,(°)(~), result due to the fact that the unperturbed indirect gap ( ~ 2 V 2) between k = 7r and k---0 becomes a more direct and smaller perturbed one near k ~ ~r/2. These states close to the Fermi energy OF(= 0) are well-defined quasi-particle states. There are also good quasi-particle states in the energy regions (1~1> U/2) outside the virtual-moment states, however the single-particle states in the other regions are not well defined due to the spin fluctuations. Coulomb screening Us does not alter the above general features of the symmetric case, but has the following important effects on 0~(to), P~(~) and 0~± (k). This interaction causes both spectra ~ ( t o ) and p~,(~) broader to the range - 1 - Us ~< ~ < 1 + US, the L3-peak closer to _ U/2, and the gap narrower and closer to k = ~r/2. A more significant US effect is to form the near-gap quasi-particle states in the wider k-space and energy region, as if their dispersion would tend to be separated from the rest to open up a new gap in between. References [1] Valence Fluctuations in Solids, eds. L.M. Falicov et al. (North-Holland, Amsterdam, 1981). [2] K. Yamada and K. Yosida, in: Electron Correlation and Magnetism in Narrow-Band Systems, ed. T. Moriya (Springer-Vedag, Berlin, 1981) p. 210. [3] H. Kaga, Progr. Theoret. Phys. 65 (1981) 1485. H. Kaga and I. Sato, ibid. 65 (1981) 105. H. Kaga, ibid. 67 (1982) 1659.