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Existence of the Fermi-liquid behaviour in the theory of intermediate valence
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Solid State Communications, Vol. 47, No. 5, pp. 3 6 7 - 3 6 9 , 1983. Printed in Great Britain.
EXISTENCE OF THE FERMI-LIQUID BEHAVIOUR IN THE THEORY OF INTERMEDIATE VALENCE S.G. Ovchinnikov and I.S. Sandalov Institute of Physics, Siberian Department o f the USSR Academy of Sciences, Krasnoyarsk, 660036, U.S.S.R.
(Received 24 Februa~ 1983; in revised form 9 March 1983 by G.S. Zhdanov) An exact equivalence o f the periodic Anderson model and generalized Hubbard model in atomic representation is shown. Energy spectrum of the quasiparticle is calculated by using diagram technique for the Hubbard operators. Conditions of Fermi-surface existence and Fermi-liquid behaviour are obtained in intermediate valence systems. INTERCONFIGURATIONAL FLUCTUATIONS (ICF) are known to be the main feature of intermediate valence (IV) compounds [1 ]. A scattering of band electron due to ICF is supposed in some papers [2] to be the reason of non-zero quasiparticle damping on the Fermi level. The finite lifetime on the Fermi level e = p is known also in one-impurity model of IV [I ]. in this paper we obtain the conditions of Fermi surface existence when the imaginary part of self-energy hn Z(e = p) = 0 at temperature T = O. The quasiparticle spectrum is calculated in the atomic limit of periodic Anderson model using diagram technique for tlubbard operators [3 l- We show exactly the equivalence o f this model and generalized I lubbard model, which is considered in gas approximation when tile electronic concentration ne is equal to n,~ = n -+ c, n = 0, 1,2 . . . . . c "~ 1. As in the llubbard model [41 we obtain tile results like in the theory of nonideal gas [51 but with calculated scattering amplitude instead of the phenomenological value in [5 ]. In the atomic representation the periodic Anderson model is described by tile tlamiltonian
When b0 = 0 the Hamiltonian (1) describes the number o f independent ions and may be diagonalized exactly [7]. All possible ionic states IP) (P = 1,2 . . . . . 12) are the mixtures o f f n, (fn+l, cfn), (cfn*t, c~fn), c2f "÷l configurations. We put forward a set of new tlubbard operators X f q = [ip)(iql and then write equation (1) as generalized Hubbard model in the atomic representation 12
.~c = E E ( + ' ; - , , . . ) x / ' " + E 1,,i E A , , ~ ' x p % , ' ¢ . p- 1 1
JCt = ~ {(E"-"u)z~ +~ [ E . " - ( " + ')ulzT°} (1)
to
pp:
qq.
(2)
The energy terms 1:~, and states Ip) were obtained in [7] for tile case e = ,u = E, --n,u = 0 but using of oneparticle fermion operators in [71 for describing transitions between Eo tends to the cinematic interactions in tile cinematic energy, tlubbard operators T technique allows to avoid these difficulties. Ttle only difference of equation (2) and tile usual Hubbard model Hamiltonian is the number of levels (12 instead of 4) and cinetic matrix/k (in the Hubbard model A~i = -- 1 [3 ]) but the operator structure is the same. Moreover, when co = En. ~ -- fi~ > 0 a group of 4 states lies below all other states [7] and is similar to the states in nondegenerate Hubbard model; hole 10), spin up It), spin down 15>and pair 12). If the origin of energy is e = 0 then these states are I (in paramagnetic state)
~c = Jcr + ~c~ +,~cot,
o = -+ 1/2.
lj
(I)
,~, = n,
I0> = Ic°f">,
(2)
ne = n + 1,
1;o = E. -
nu,
iio
1o) =aolc'tof~ +3olc°fff*b,
oO
E,, = Eo + ½ ( ~ -- vL) - - U , ia
Here E,, is a ground state energy of n-electron ion with spin s = 0;E,+i - - ( n + 1) electron term s = 1/2; z ~ = lieb(i3[ is Hubbard operator [6] on a site Rt; e is an energy of band electron (c-electron) in a crystal field, b u is the transfer integral and v is hybridization.
(3)
he=n+2,
1 2 ) = M c 2 / " ) + ~ ( [ c ¢ f i"
+ Ic//('b), E~ = Eo + ½(w - v.) - 2u. 367
,'1+1
>
(3)
FERMI-LIQUID BEHAVIOUR IN INTERMEDIATEVALENCE
368
Here v~ = co2 + 4v 2, u~ = w z + 8v 2 and coefficients in (3) are equal to
Vol. 47, No. 5
± {-~[--&oo--~:,-o + ( F ( O o ) A u -
F(--o2)Av_)b(k)l
2
+ F(Oo)F(--cr2)At. A-.I bZ(k)} 1'2.
ot~ =
I+
,
6~ = 1
In the atomic limit when halfbandwidth w = Ib(0)l '~ Ee spectrum (7) has a more simple form
1 ~o 2
E+(k) = Az,_ o + F(2, --o)A22(k) --ta + O(w2/E~),
A nonzero block 2 x 2 in the cinetic matrix/~ is given by (0, o)
(o0) (2.-o)
E - ( k ) = Aoo + F ( o O ) A l , ( k ) - U + O(w2/Eg2) • The exact Green function (4) may be expressed
(--o,2)
(A,t
Atz ) ,
Art
\A:t
A22
Az, = A~': = %(~_oa* +ILo~5*).
= laolL
A2z = I~,-oa*+/3_o~*lL
To use the diagram technique [3] we introduce vector a(p, q) which denotes transition Iq) --* [p), X,''q --" X~ and matrix Green function
D,~(k. COn) = •Z . tJoi l T e'"to. r-ik(.Ri-Ri)(T r .,~/~(r)~-a(O))dr. '-" in zero approximation on his
using the irreducible part I ! ~ [which cannot be divided into two parts by the interaction line A(k)]
Og~(k. ~ , )
= I t J ( k . , ~ , ) + n_~.~(/,-).
The irreducible part satisfies the Dyson equation
(4) I Ig~(k~,)
=
o-' (~.) Dg~
-- ~: (/,-. ~,)F-'(cO.
-GI-',
O~(w,,) = 6o~F(o0[--ico . +/:;,
o~ We restrict ourselves t o the case n e = n + c, c '~ I.
where
F(ot)
(8)
/ =
tit,
+
?lq,
tip
=
Then tile Fermi level is situated near tile bottom o f /'.'-(k)-band (/a ~ wc '~ w if we put ta = 0 at tile bottom of tile band)• In tile atomic limit w "~/:~, intraband scattering in the electron--electron channel ( 0 + ) -*" (0--) gives the main contribution to tile self-energy :~o.(k. OJn). The scattering amplitude
e-F'p/~I/Z e-FmlT. / rrl
There are two one-particle localized excitations: resonance (o, 0)
A°.o -- ta = I';o -- Fo = ~(~-
resonance (2, --o)
~',)-u,
i'(pl, 0 +; P2, 0--; P3, 0 +;/94, 0 --),
A~._o --,u = E2 --E_o =
depends only on P3 and P4 and is found as a sum of ladder diagrams with small summary momentum s' = Pt + P2 = P3 + P4 and in leading terms on bit and c is eqt, al to the two-particle amplitude of scattering in the
~(v,-v~)-u,
with a gap between them b~]° = ~ : . - , , - ~ o o = vl - ½(w +
v2)
vacuuln
P:
( 2V4/W3, V/CO~ O. {[(2--x/2)v, v / w '*'°°.
(5)
Hartree-Fock approximation on b// corresponds to the Hubbard I solution and to zero approximation of the self-consistent field [3 ]: DoC311F"~tv v,. wn) = O ~0" ( 6 % ) + A_a.t~(k ) (6) A~.tj(k) = Ao,.ab(k),
b(k) = ~ b(h) exp (ikh).
Fo(p, s -- p)
P(p, s - p) = ~
2A, t(0)
1--
]
'
where to(p, p ' ) = A,t[b(p) + b(p')]. Due to c '~ 1 all the momenta are small p ~
=
NI
[
b(k)]-t
h
The Green-function (6) determines two narrow Hubbard bands of strongly correlated spinless fermions E±(k) = ½{/;~--/~o + [ F ( 0 o ) a t t + F(--oZ)A22]b(k)} (7)
Here/3 is a Watson integral. When b# 4= 0 only for the nearest neighbours/3 = 1.516 for the simple cubic lattice, 1.393 for b.c.c, and 1.345 for f.c.c. The imaginary part of Y. appears only in secondorder term on F:
(9)
Vol. 47, No. 5
FERMI-LIQUID BEHAVIOUR IN INTERMEDIATE VALENCE
1
r(p,,p:)F(k,p, + p : - - k )
369
f[E-(p,)If[E-(P2)]{I --f[E-(pt + P ~ - - k)]} f[E-(p,) +E-(p~)--E-(p, +p2--k)]
Z~)( k, tan) = ~-5 p,~p:E - ( p t ) +E-(p:)--E-(p~ + p:-- kl -- icon
(lO) where f ( E ) = [exp (E/T) + I ]-t. Quasiparticle spectrum renormalization is close to the results of the theory of non-ideal Fermi gas [5]. When T,~/a and e --t~ , ~ t then i m ~ (/~F,e) = ~lFIZNZ(lao)(e-la)z+OrT)2, /ao
(11)
where P is an angle averaged value of F, N(~o) is the energy density of states on the Fermi level ~to, Is0 = ts(T = 0). To summarize all the results of this paper we want to emphasize that it describes an IV-system with narrow (w < b~) bands, smaU concentration of heavy carriers (c < 1) and low value of Fermi energy Oi ~ wc ~ w). The Fermi surface is defined according to equation (! 1) but the Fermi liqt, id behavior takes place in narrow temperature region T < wc instead of T ' ~ w in a normal metal. The peculiar Fermi liquid properties of some IVcompot, nds are well-known, for example, in CeAIj and CcSn~ [ I l- Moreover, the Fermi surface existence is discovered rcccntly iq CcSn~ [8 ]. While the whole concen-
tration of electrons in CeSn3 is integer the gas situation in narrow band may appear as a result of intersection of the Fermi level with some other bands of light quasiparticles. This conclusion is in agreement with two kinds of Fermi surface branches (m~ = 4 - 9 m e and m~ ~ me) observed in CeSn3 [8]. REFERENCES
1. 2.
3. 4. 5. 6. 7. 8.
J.M. Robinson,Phys. Rep. 51,3 (1979); D.I. Khomskii, Uspechi Fiz. Nauk 129,434 (1979). A.M. Atoyan, A.F. Barabanov & L.A. Maksimov, ZhETF 74, 2220 (1978). R.O. Zaytzev, ZhETF 70, 1100 (1976). A.V. Vedyaev & V.A. lvanov, Theor. Mat. Fiz. 50, 415(1982). V.M. Galitsky,ZhETF34, 151 (1958). J.C. tlubbard, Proc. Roy. Soc. A285, 542 (1965). M.F. Foglio & L.M. Falicov, Phys. Rev. B20, 4554 (1979). W.R. Johanson, G.W. Grabtree, A.S. Edelstein & O.B. McMastcrs, Phys. Rev. Lett. 46,504 (1981).
Solid State Communications, Vol. 47, No. 5, pp. 3 6 7 - 3 6 9 , 1983. Printed in Great Britain.
EXISTENCE OF THE FERMI-LIQUID BEHAVIOUR IN THE THEORY OF INTERMEDIATE VALENCE S.G. Ovchinnikov and I.S. Sandalov Institute of Physics, Siberian Department o f the USSR Academy of Sciences, Krasnoyarsk, 660036, U.S.S.R.
(Received 24 Februa~ 1983; in revised form 9 March 1983 by G.S. Zhdanov) An exact equivalence o f the periodic Anderson model and generalized Hubbard model in atomic representation is shown. Energy spectrum of the quasiparticle is calculated by using diagram technique for the Hubbard operators. Conditions of Fermi-surface existence and Fermi-liquid behaviour are obtained in intermediate valence systems. INTERCONFIGURATIONAL FLUCTUATIONS (ICF) are known to be the main feature of intermediate valence (IV) compounds [1 ]. A scattering of band electron due to ICF is supposed in some papers [2] to be the reason of non-zero quasiparticle damping on the Fermi level. The finite lifetime on the Fermi level e = p is known also in one-impurity model of IV [I ]. in this paper we obtain the conditions of Fermi surface existence when the imaginary part of self-energy hn Z(e = p) = 0 at temperature T = O. The quasiparticle spectrum is calculated in the atomic limit of periodic Anderson model using diagram technique for tlubbard operators [3 l- We show exactly the equivalence o f this model and generalized I lubbard model, which is considered in gas approximation when tile electronic concentration ne is equal to n,~ = n -+ c, n = 0, 1,2 . . . . . c "~ 1. As in the llubbard model [41 we obtain tile results like in the theory of nonideal gas [51 but with calculated scattering amplitude instead of the phenomenological value in [5 ]. In the atomic representation the periodic Anderson model is described by tile tlamiltonian
When b0 = 0 the Hamiltonian (1) describes the number o f independent ions and may be diagonalized exactly [7]. All possible ionic states IP) (P = 1,2 . . . . . 12) are the mixtures o f f n, (fn+l, cfn), (cfn*t, c~fn), c2f "÷l configurations. We put forward a set of new tlubbard operators X f q = [ip)(iql and then write equation (1) as generalized Hubbard model in the atomic representation 12
.~c = E E ( + ' ; - , , . . ) x / ' " + E 1,,i E A , , ~ ' x p % , ' ¢ . p- 1 1
JCt = ~ {(E"-"u)z~ +~ [ E . " - ( " + ')ulzT°} (1)
to
pp:
qq.
(2)
The energy terms 1:~, and states Ip) were obtained in [7] for tile case e = ,u = E, --n,u = 0 but using of oneparticle fermion operators in [71 for describing transitions between Eo tends to the cinematic interactions in tile cinematic energy, tlubbard operators T technique allows to avoid these difficulties. Ttle only difference of equation (2) and tile usual Hubbard model Hamiltonian is the number of levels (12 instead of 4) and cinetic matrix/k (in the Hubbard model A~i = -- 1 [3 ]) but the operator structure is the same. Moreover, when co = En. ~ -- fi~ > 0 a group of 4 states lies below all other states [7] and is similar to the states in nondegenerate Hubbard model; hole 10), spin up It), spin down 15>and pair 12). If the origin of energy is e = 0 then these states are I (in paramagnetic state)
~c = Jcr + ~c~ +,~cot,
o = -+ 1/2.
lj
(I)
,~, = n,
I0> = Ic°f">,
(2)
ne = n + 1,
1;o = E. -
nu,
iio
1o) =aolc'tof~ +3olc°fff*b,
oO
E,, = Eo + ½ ( ~ -- vL) - - U , ia
Here E,, is a ground state energy of n-electron ion with spin s = 0;E,+i - - ( n + 1) electron term s = 1/2; z ~ = lieb(i3[ is Hubbard operator [6] on a site Rt; e is an energy of band electron (c-electron) in a crystal field, b u is the transfer integral and v is hybridization.
(3)
he=n+2,
1 2 ) = M c 2 / " ) + ~ ( [ c ¢ f i"
+ Ic//('b), E~ = Eo + ½(w - v.) - 2u. 367
,'1+1
>
(3)
FERMI-LIQUID BEHAVIOUR IN INTERMEDIATEVALENCE
368
Here v~ = co2 + 4v 2, u~ = w z + 8v 2 and coefficients in (3) are equal to
Vol. 47, No. 5
± {-~[--&oo--~:,-o + ( F ( O o ) A u -
F(--o2)Av_)b(k)l
2
+ F(Oo)F(--cr2)At. A-.I bZ(k)} 1'2.
ot~ =
I+
,
6~ = 1
In the atomic limit when halfbandwidth w = Ib(0)l '~ Ee spectrum (7) has a more simple form
1 ~o 2
E+(k) = Az,_ o + F(2, --o)A22(k) --ta + O(w2/E~),
A nonzero block 2 x 2 in the cinetic matrix/~ is given by (0, o)
(o0) (2.-o)
E - ( k ) = Aoo + F ( o O ) A l , ( k ) - U + O(w2/Eg2) • The exact Green function (4) may be expressed
(--o,2)
(A,t
Atz ) ,
Art
\A:t
A22
Az, = A~': = %(~_oa* +ILo~5*).
= laolL
A2z = I~,-oa*+/3_o~*lL
To use the diagram technique [3] we introduce vector a(p, q) which denotes transition Iq) --* [p), X,''q --" X~ and matrix Green function
D,~(k. COn) = •Z . tJoi l T e'"to. r-ik(.Ri-Ri)(T r .,~/~(r)~-a(O))dr. '-" in zero approximation on his
using the irreducible part I ! ~ [which cannot be divided into two parts by the interaction line A(k)]
Og~(k. ~ , )
= I t J ( k . , ~ , ) + n_~.~(/,-).
The irreducible part satisfies the Dyson equation
(4) I Ig~(k~,)
=
o-' (~.) Dg~
-- ~: (/,-. ~,)F-'(cO.
-GI-',
O~(w,,) = 6o~F(o0[--ico . +/:;,
o~ We restrict ourselves t o the case n e = n + c, c '~ I.
where
F(ot)
(8)
/ =
tit,
+
?lq,
tip
=
Then tile Fermi level is situated near tile bottom o f /'.'-(k)-band (/a ~ wc '~ w if we put ta = 0 at tile bottom of tile band)• In tile atomic limit w "~/:~, intraband scattering in the electron--electron channel ( 0 + ) -*" (0--) gives the main contribution to tile self-energy :~o.(k. OJn). The scattering amplitude
e-F'p/~I/Z e-FmlT. / rrl
There are two one-particle localized excitations: resonance (o, 0)
A°.o -- ta = I';o -- Fo = ~(~-
resonance (2, --o)
~',)-u,
i'(pl, 0 +; P2, 0--; P3, 0 +;/94, 0 --),
A~._o --,u = E2 --E_o =
depends only on P3 and P4 and is found as a sum of ladder diagrams with small summary momentum s' = Pt + P2 = P3 + P4 and in leading terms on bit and c is eqt, al to the two-particle amplitude of scattering in the
~(v,-v~)-u,
with a gap between them b~]° = ~ : . - , , - ~ o o = vl - ½(w +
v2)
vacuuln
P:
( 2V4/W3, V/CO~ O. {[(2--x/2)v, v / w '*'°°.
(5)
Hartree-Fock approximation on b// corresponds to the Hubbard I solution and to zero approximation of the self-consistent field [3 ]: DoC311F"~tv v,. wn) = O ~0" ( 6 % ) + A_a.t~(k ) (6) A~.tj(k) = Ao,.ab(k),
b(k) = ~ b(h) exp (ikh).
Fo(p, s -- p)
P(p, s - p) = ~
2A, t(0)
1--
]
'
where to(p, p ' ) = A,t[b(p) + b(p')]. Due to c '~ 1 all the momenta are small p ~
=
NI
[
b(k)]-t
h
The Green-function (6) determines two narrow Hubbard bands of strongly correlated spinless fermions E±(k) = ½{/;~--/~o + [ F ( 0 o ) a t t + F(--oZ)A22]b(k)} (7)
Here/3 is a Watson integral. When b# 4= 0 only for the nearest neighbours/3 = 1.516 for the simple cubic lattice, 1.393 for b.c.c, and 1.345 for f.c.c. The imaginary part of Y. appears only in secondorder term on F:
(9)
Vol. 47, No. 5
FERMI-LIQUID BEHAVIOUR IN INTERMEDIATE VALENCE
1
r(p,,p:)F(k,p, + p : - - k )
369
f[E-(p,)If[E-(P2)]{I --f[E-(pt + P ~ - - k)]} f[E-(p,) +E-(p~)--E-(p, +p2--k)]
Z~)( k, tan) = ~-5 p,~p:E - ( p t ) +E-(p:)--E-(p~ + p:-- kl -- icon
(lO) where f ( E ) = [exp (E/T) + I ]-t. Quasiparticle spectrum renormalization is close to the results of the theory of non-ideal Fermi gas [5]. When T,~/a and e --t~ , ~ t then i m ~ (/~F,e) = ~lFIZNZ(lao)(e-la)z+OrT)2, /ao
(11)
where P is an angle averaged value of F, N(~o) is the energy density of states on the Fermi level ~to, Is0 = ts(T = 0). To summarize all the results of this paper we want to emphasize that it describes an IV-system with narrow (w < b~) bands, smaU concentration of heavy carriers (c < 1) and low value of Fermi energy Oi ~ wc ~ w). The Fermi surface is defined according to equation (! 1) but the Fermi liqt, id behavior takes place in narrow temperature region T < wc instead of T ' ~ w in a normal metal. The peculiar Fermi liquid properties of some IVcompot, nds are well-known, for example, in CeAIj and CcSn~ [ I l- Moreover, the Fermi surface existence is discovered rcccntly iq CcSn~ [8 ]. While the whole concen-
tration of electrons in CeSn3 is integer the gas situation in narrow band may appear as a result of intersection of the Fermi level with some other bands of light quasiparticles. This conclusion is in agreement with two kinds of Fermi surface branches (m~ = 4 - 9 m e and m~ ~ me) observed in CeSn3 [8]. REFERENCES
1. 2.
3. 4. 5. 6. 7. 8.
J.M. Robinson,Phys. Rep. 51,3 (1979); D.I. Khomskii, Uspechi Fiz. Nauk 129,434 (1979). A.M. Atoyan, A.F. Barabanov & L.A. Maksimov, ZhETF 74, 2220 (1978). R.O. Zaytzev, ZhETF 70, 1100 (1976). A.V. Vedyaev & V.A. lvanov, Theor. Mat. Fiz. 50, 415(1982). V.M. Galitsky,ZhETF34, 151 (1958). J.C. tlubbard, Proc. Roy. Soc. A285, 542 (1965). M.F. Foglio & L.M. Falicov, Phys. Rev. B20, 4554 (1979). W.R. Johanson, G.W. Grabtree, A.S. Edelstein & O.B. McMastcrs, Phys. Rev. Lett. 46,504 (1981).