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Electronic properties of the periodic Anderson model
70
Journal of Magnetism and Magnetic Materials 76 & 77 (1988) 70 72 North-Holland. Amsterdam
ELECTRONIC PROPERTIES OF THE PERIODIC A N D E R S O N M O D E L V. Z L A T I C , A.A. A L I G I A and D. S C H U L Z
Institute of Physi~wof the Unit~ersiO' of Zagreh, P.O.B. 304. 4H)OI Zagreh. )'u~oslat,la Using the low order perturbation theory we study the interplay between the geometry of the electronic motion and the local Coulomb interaction for the periodic Anderson model. It is shown that in the vicinity of the Fermi level the f-electron self-energy varies rapidly as a function of both frequency and the wave vector. Functional form of the self-energy dcpends very sensitively on the shape of the unperturbed bands and the shape and the strength of the hybridisation. Large renormalization of the model properties resuh even for small values of the Coulomb correlation.
1. Introduction We are interested in the p r o p e r t i e s of t h e periodic A n d e r s o n model defined by the H a m i l t o n i a n
H , = Eeac[,,cao + E erf;ofao + U E £ +. f, r f f s J~ ~ ka
ko
+ ~'.(v~,'Lko +
i
h.c.)
(1)
ko
which describes a set of dispersionless, c o r r e l a t e d f-states and a set of u n c o r r e l a t e d ligand states with dispersion % and b a n d w i d t h W. The two sets of states are mixed by VA which, if the mixing is due to the off-site hopping, must be k - d e p e n d e n t . The main difficulties in o b t a i n i n g the model p r o p e r t i e s are associated with the presence of the C o u l o m b correlation U. However, the g e o m e t r y of the electronic motion, i.e. the structure of uncorrelated b a n d s and the shape of the h y b r i d i z a t i o n , is giving rise to large effects as well. T h e possibility to study the interplay of the geometrical effects and the correlation provides an interesting aspect of the P A M . F u r t h e r m o r e , it seems that the model exhibits certain interesting features, which m a k e it relevant for the description of heavy fermions a n d high T~ systems with metallic Cu O planes, even for relatively small values of U. F o r example, large mass e n h a n c e m e n t can be o b t a i n e d for f-electrons with small values of U, if we define the unperturbed b a n d s in an a p p r o p r i a t e way. Experimentally, the large differences in 7 values a m o n g various Ce (or U) intermetallics are p r o b a b l y due to the variations in the h y b r i d i z a t i o n matrix element. In this paper, the p r o p e r t i e s of the model are discussed by the p e r t u r b a t i o n theory in terms of U
[1 4]. Thus, we assume that the c o r r e l a t e d state is a normal F e r m i liquid which could be reached perturbatively. A l t h o u g h we calculate the model p r o p e r t i e s ill low o r d e r ill U only, we have little d o u b t that the features o b s e r v e d e x p e r i m e n t a l l y in "'correlated s y s t e m s " will emerge a l r e a d y for those U values which are within the reach of the pert u r b a t i o n theory. Note, that a similar a p p r o a c h allowed a qualitative d e s c r i p t i o n of the single imp u r i t y K o n d o and V F p r o b l e m [5 8], of the symmetric p e r i o d i c A n d e r s o n model [1 4] and of the H u b b a r d m o d e l [9]. We do not claim that the p e r t u r b a t i v e l y constructed c o r r e l a t e d g r o u n d state is the correct g r o u n d state of the P A M everywhere in the p a r a m e t e r space. It seems, however, that the nature of that exact g r o u n d state will remain unknown for quite some time to follow. Thus. the answers p r o v i d e d by the p e r t u r b a t i o n theory might help to u n d e r s t a n d the model p r o p e r t i e s and classify the e x p e r i m e n t a l data.
2. Calculations To generate the e x p a n s i o n we rewrite the H a m iltonian (1) by a d d i n g and s u b t r a c t i n g the H a r t r e e Fock term which gives
H x = H o+ I1'.
(2)
where
,,,=uzt,,,,-<,,,,))(,,, i
and where < - . . > is tile g r a n d - e n s e m b l e average with respect to the H a r t r e e Fock H a m i l t o n i a n tt o. Since we are e x p a n d i n g above the n o n - m a g -
0 3 0 4 - 8 8 5 3 / 8 8 / $ 0 3 . 5 0 ,~C,Elsevier Science Publishers B.V.
71
V. Zlati( et al. / The periodic Anderson model
netic ground state, we take (n~o) = ( n f ) , which is site and spin independent. Note, that the discussion is not restricted to the electron-hole symmetry limit. Standard S-matrix expansion generates the usual diagrams and we see, by inspection, that the expressions obtained by H' assume exactly the same form as in the case of the single-impurity Anderson model. The only difference is that now the momentum conservation is supplementing the energy conservation at each interaction vertex. Thus, the expansion coefficients can be expressed in terms of determinants [10], built out of the Hartree-Fock Green's functions, which are integrated over the internal momentum and energy variables. The renormalized f-electrons Green's function is given by
tained from the H a r t r e e - F o c k Hamiltonian and are given by a~? = 1{1 _+ ( e r - % ) / [ ( e
r- %
where /~ is the chemical potential. Note, that the fully renormalized f-electron number ( ( n f)) appears in eq. (4) as a result of the exact summation of all the w-independent diagrams and that ~ k ( z ) is the sum of all the w-dependent terms. The finite temperature result for ~k, evaluated to second order, is given by - '
3"22)(ie°") =
(U)
2
a h c Oik+qOgpOlp+q
£
p,q a,h,c
×F.,,,(k.
i~o,,+ /1 -- E"
"
<
k+q -- E ; -[- E ; + q
p, q),
I]1/2} (6)
)2
-~ 1/2} +41veil
.
At T = 0 K, expression (5) reduces to the result of ref. [3] in which the PAM with the constant hybridization has been studied. The presence of the Fermi functions restricts the p and q summations and it is at this stage that the assumptions regarding the shape of the bands and the hybridization starts to influence the results. The density of states for the localized electrons is given by
qr
(4)
+ 411//2
and
pr(e ) = _ 2 Y'~Im G~(e + i'iT) z +/~ - %
2
(7)
k
and the chemical potential should be determined so as to obtain the required total number of particles when integrating 0r(e) and &(E). If the total number of electrons is given and the centre of the conduction band put at e 0` the relevant model parameters are the charge-transfer energy (% - er)/W, the average hybridisation V / W and the Coulomb correlation U / W . The general analysis of S;k and 0f(e) is of course impossible and the numerical evaluation of the expansion coefficients is very difficult. Nevertheless, some limiting cases could be investigated analytically and the 1-dimensional case can be discussed numerically without too much difficulties.
(5)
where
3. Discussion and conclusions
,<;,,,,.(,<, ,', qt = [ i( F-;+q- ,<) - /( F4+.- F4 )]
We notice first that most of the f-electron spectral weight is placed around er and er + U, which can be seen from the structure of the poles in ~ k ( z ) for ]z] > [V[. If we are close to the electron-hole symmetry limit, additional structure in the spectral density appears at the Fermi energy. This is the situation we encounter in heavy fermions. Here, the spin susceptibility is enhanced and so is the specific heat coefficient, because the self-energy is a very rapidly varying function of ~0
×[S(E;-,<)-:(F-;+.-,<)] and where S(x) is the Fermi function coming from the frequency integrations. The sum over p and q goes over the 1st Brillouin zone, while the a, b, c sums go over the bonding and antibonding (_+) Hartree-Fock solutions. The coefficients a~ and the excitation energies E ~ are simply ob-
72
V. Zlati~ ~ et al. / The periodic Anderson model
in the vicinity of the Fermi level. The charge susceptibility is suppressed. We emphasize that the large density of f-states at e v and the large y enhancement are not necessarily the consequences of large U values, but rather result from the geometry of the unperturbed electronic motion. As we move away from the electron hole limit, e.g. by increasing the charge-transfer energy, the Kondo peak disappears and one of the broad peaks (el or Cr+ U peak) moves towards ~v-. At the same time the charge fluctuations become enhanced, which is what one would expect for valence fluctuators. Next, we notice that the selfenergy (5) is not only ~-dependent but is strongly k-dependent as well. For certain l-dimensional models [4] both aZ'k/~)~ and a,Y,k/ak diverge logarithmically at c v. In higher dimensions the divergency is removed but the strong k- and ~-dependence of the self-energy remains. The functional form of the self-energy depends sensitively on the shape of the unperturbed bands and the shape and strength of the hybridization. Thus, large renormalization of various correlation functions can result even from small values of U. Finally, we mention that the perturbative expressions for the spin susceptibility, charge susceptibility and the pair correlation function have been also derived. Detailed numerical analysis of
the model, based on these low-order perturbative expression, will be the subject of our subsequent work. Useful discussions with B. Horvatid and P. Entel are gratefully acknowledged. The financial support from the A. v. Humboldt Foundation is also gratefully acknowledged. References [1] K. Yamada and K. Yosida, in: Electron Correlation and Magnetism in Narrow-Band Systems, ed. T. Moriya, Springer Series in Solid-State Physics, vol. 29 (Springer, Berlin, 1981) p. 21//. [2] K. Yamada and K. Yosida, Prog. Theor. Phys. 76 (1986) 621. [3] K. Okada, K. Yamada and K. Yosida, Prog. Theor. Phys. 77 (1987) 1297. [4] V. Zlatid, S.K. Ghatak and K.H. Bennemann, Phys. Rcv. Left. 57 (1986) 1263. [5] K. Yamada, Prog. Theor. Phys. 53 (1975) 970. [6] K. Yosida and K. Yamada, Prog. Theor. Phys. 53 (1975) 1286. [7] B. Horvati,2 and V. Zlati& Phys. Stat. Sol. (b) 99 119801 251 and 111 (19821 65. [8] B. ttorvatid and V. Zlati,2, Phys. Rev. B 3(1 (1984)6717. [9] G. Traglia, F. Ducastelle and D. Spanjaard, J. de Phys. 41 (1980) 281. [10] V. Zlatid, P. lintel, B. Horvatid and D. Schulz. Proc. ttigh 7~: Workshop, Trieste, 1988 (World Scientific, Singapore, 1988) 1o appear.
Journal of Magnetism and Magnetic Materials 76 & 77 (1988) 70 72 North-Holland. Amsterdam
ELECTRONIC PROPERTIES OF THE PERIODIC A N D E R S O N M O D E L V. Z L A T I C , A.A. A L I G I A and D. S C H U L Z
Institute of Physi~wof the Unit~ersiO' of Zagreh, P.O.B. 304. 4H)OI Zagreh. )'u~oslat,la Using the low order perturbation theory we study the interplay between the geometry of the electronic motion and the local Coulomb interaction for the periodic Anderson model. It is shown that in the vicinity of the Fermi level the f-electron self-energy varies rapidly as a function of both frequency and the wave vector. Functional form of the self-energy dcpends very sensitively on the shape of the unperturbed bands and the shape and the strength of the hybridisation. Large renormalization of the model properties resuh even for small values of the Coulomb correlation.
1. Introduction We are interested in the p r o p e r t i e s of t h e periodic A n d e r s o n model defined by the H a m i l t o n i a n
H , = Eeac[,,cao + E erf;ofao + U E £ +. f, r f f s J~ ~ ka
ko
+ ~'.(v~,'Lko +
i
h.c.)
(1)
ko
which describes a set of dispersionless, c o r r e l a t e d f-states and a set of u n c o r r e l a t e d ligand states with dispersion % and b a n d w i d t h W. The two sets of states are mixed by VA which, if the mixing is due to the off-site hopping, must be k - d e p e n d e n t . The main difficulties in o b t a i n i n g the model p r o p e r t i e s are associated with the presence of the C o u l o m b correlation U. However, the g e o m e t r y of the electronic motion, i.e. the structure of uncorrelated b a n d s and the shape of the h y b r i d i z a t i o n , is giving rise to large effects as well. T h e possibility to study the interplay of the geometrical effects and the correlation provides an interesting aspect of the P A M . F u r t h e r m o r e , it seems that the model exhibits certain interesting features, which m a k e it relevant for the description of heavy fermions a n d high T~ systems with metallic Cu O planes, even for relatively small values of U. F o r example, large mass e n h a n c e m e n t can be o b t a i n e d for f-electrons with small values of U, if we define the unperturbed b a n d s in an a p p r o p r i a t e way. Experimentally, the large differences in 7 values a m o n g various Ce (or U) intermetallics are p r o b a b l y due to the variations in the h y b r i d i z a t i o n matrix element. In this paper, the p r o p e r t i e s of the model are discussed by the p e r t u r b a t i o n theory in terms of U
[1 4]. Thus, we assume that the c o r r e l a t e d state is a normal F e r m i liquid which could be reached perturbatively. A l t h o u g h we calculate the model p r o p e r t i e s ill low o r d e r ill U only, we have little d o u b t that the features o b s e r v e d e x p e r i m e n t a l l y in "'correlated s y s t e m s " will emerge a l r e a d y for those U values which are within the reach of the pert u r b a t i o n theory. Note, that a similar a p p r o a c h allowed a qualitative d e s c r i p t i o n of the single imp u r i t y K o n d o and V F p r o b l e m [5 8], of the symmetric p e r i o d i c A n d e r s o n model [1 4] and of the H u b b a r d m o d e l [9]. We do not claim that the p e r t u r b a t i v e l y constructed c o r r e l a t e d g r o u n d state is the correct g r o u n d state of the P A M everywhere in the p a r a m e t e r space. It seems, however, that the nature of that exact g r o u n d state will remain unknown for quite some time to follow. Thus. the answers p r o v i d e d by the p e r t u r b a t i o n theory might help to u n d e r s t a n d the model p r o p e r t i e s and classify the e x p e r i m e n t a l data.
2. Calculations To generate the e x p a n s i o n we rewrite the H a m iltonian (1) by a d d i n g and s u b t r a c t i n g the H a r t r e e Fock term which gives
H x = H o+ I1'.
(2)
where
,,,=uzt,,,,-<,,,,))(,,, i
and where < - . . > is tile g r a n d - e n s e m b l e average with respect to the H a r t r e e Fock H a m i l t o n i a n tt o. Since we are e x p a n d i n g above the n o n - m a g -
0 3 0 4 - 8 8 5 3 / 8 8 / $ 0 3 . 5 0 ,~C,Elsevier Science Publishers B.V.
71
V. Zlati( et al. / The periodic Anderson model
netic ground state, we take (n~o) = ( n f ) , which is site and spin independent. Note, that the discussion is not restricted to the electron-hole symmetry limit. Standard S-matrix expansion generates the usual diagrams and we see, by inspection, that the expressions obtained by H' assume exactly the same form as in the case of the single-impurity Anderson model. The only difference is that now the momentum conservation is supplementing the energy conservation at each interaction vertex. Thus, the expansion coefficients can be expressed in terms of determinants [10], built out of the Hartree-Fock Green's functions, which are integrated over the internal momentum and energy variables. The renormalized f-electrons Green's function is given by
tained from the H a r t r e e - F o c k Hamiltonian and are given by a~? = 1{1 _+ ( e r - % ) / [ ( e
r- %
where /~ is the chemical potential. Note, that the fully renormalized f-electron number ( ( n f)) appears in eq. (4) as a result of the exact summation of all the w-independent diagrams and that ~ k ( z ) is the sum of all the w-dependent terms. The finite temperature result for ~k, evaluated to second order, is given by - '
3"22)(ie°") =
(U)
2
a h c Oik+qOgpOlp+q
£
p,q a,h,c
×F.,,,(k.
i~o,,+ /1 -- E"
"
<
k+q -- E ; -[- E ; + q
p, q),
I]1/2} (6)
)2
-~ 1/2} +41veil
.
At T = 0 K, expression (5) reduces to the result of ref. [3] in which the PAM with the constant hybridization has been studied. The presence of the Fermi functions restricts the p and q summations and it is at this stage that the assumptions regarding the shape of the bands and the hybridization starts to influence the results. The density of states for the localized electrons is given by
qr
(4)
+ 411//2
and
pr(e ) = _ 2 Y'~Im G~(e + i'iT) z +/~ - %
2
(7)
k
and the chemical potential should be determined so as to obtain the required total number of particles when integrating 0r(e) and &(E). If the total number of electrons is given and the centre of the conduction band put at e 0` the relevant model parameters are the charge-transfer energy (% - er)/W, the average hybridisation V / W and the Coulomb correlation U / W . The general analysis of S;k and 0f(e) is of course impossible and the numerical evaluation of the expansion coefficients is very difficult. Nevertheless, some limiting cases could be investigated analytically and the 1-dimensional case can be discussed numerically without too much difficulties.
(5)
where
3. Discussion and conclusions
,<;,,,,.(,<, ,', qt = [ i( F-;+q- ,<) - /( F4+.- F4 )]
We notice first that most of the f-electron spectral weight is placed around er and er + U, which can be seen from the structure of the poles in ~ k ( z ) for ]z] > [V[. If we are close to the electron-hole symmetry limit, additional structure in the spectral density appears at the Fermi energy. This is the situation we encounter in heavy fermions. Here, the spin susceptibility is enhanced and so is the specific heat coefficient, because the self-energy is a very rapidly varying function of ~0
×[S(E;-,<)-:(F-;+.-,<)] and where S(x) is the Fermi function coming from the frequency integrations. The sum over p and q goes over the 1st Brillouin zone, while the a, b, c sums go over the bonding and antibonding (_+) Hartree-Fock solutions. The coefficients a~ and the excitation energies E ~ are simply ob-
72
V. Zlati~ ~ et al. / The periodic Anderson model
in the vicinity of the Fermi level. The charge susceptibility is suppressed. We emphasize that the large density of f-states at e v and the large y enhancement are not necessarily the consequences of large U values, but rather result from the geometry of the unperturbed electronic motion. As we move away from the electron hole limit, e.g. by increasing the charge-transfer energy, the Kondo peak disappears and one of the broad peaks (el or Cr+ U peak) moves towards ~v-. At the same time the charge fluctuations become enhanced, which is what one would expect for valence fluctuators. Next, we notice that the selfenergy (5) is not only ~-dependent but is strongly k-dependent as well. For certain l-dimensional models [4] both aZ'k/~)~ and a,Y,k/ak diverge logarithmically at c v. In higher dimensions the divergency is removed but the strong k- and ~-dependence of the self-energy remains. The functional form of the self-energy depends sensitively on the shape of the unperturbed bands and the shape and strength of the hybridization. Thus, large renormalization of various correlation functions can result even from small values of U. Finally, we mention that the perturbative expressions for the spin susceptibility, charge susceptibility and the pair correlation function have been also derived. Detailed numerical analysis of
the model, based on these low-order perturbative expression, will be the subject of our subsequent work. Useful discussions with B. Horvatid and P. Entel are gratefully acknowledged. The financial support from the A. v. Humboldt Foundation is also gratefully acknowledged. References [1] K. Yamada and K. Yosida, in: Electron Correlation and Magnetism in Narrow-Band Systems, ed. T. Moriya, Springer Series in Solid-State Physics, vol. 29 (Springer, Berlin, 1981) p. 21//. [2] K. Yamada and K. Yosida, Prog. Theor. Phys. 76 (1986) 621. [3] K. Okada, K. Yamada and K. Yosida, Prog. Theor. Phys. 77 (1987) 1297. [4] V. Zlatid, S.K. Ghatak and K.H. Bennemann, Phys. Rcv. Left. 57 (1986) 1263. [5] K. Yamada, Prog. Theor. Phys. 53 (1975) 970. [6] K. Yosida and K. Yamada, Prog. Theor. Phys. 53 (1975) 1286. [7] B. Horvati,2 and V. Zlati& Phys. Stat. Sol. (b) 99 119801 251 and 111 (19821 65. [8] B. ttorvatid and V. Zlati,2, Phys. Rev. B 3(1 (1984)6717. [9] G. Traglia, F. Ducastelle and D. Spanjaard, J. de Phys. 41 (1980) 281. [10] V. Zlatid, P. lintel, B. Horvatid and D. Schulz. Proc. ttigh 7~: Workshop, Trieste, 1988 (World Scientific, Singapore, 1988) 1o appear.