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Path-integral treatment for localized electrons in the periodic Anderson model
ELSEVIER
Physica B 237-238 (1997) 231-233
Path-integral treatment for localized electrons in the periodic Anderson model V. I v a n o v a ' * ' 1, M .
Zhuravlev b, V. Yarunin c, R. Pucci d
alnstitute for Molecular Science, Myoda(ji-cho, Okazaki 444, Japan bN. S. Kurnakov Institute of General & Inorganic Chemistry of the Russian Academy of Sciences, Moscow, Russian Federation cJoint Institute for Nuclear Research, Dubna, Russian Federation dCatania University, Catania, Italy
Abstract
The partition function of the periodic Anderson model is expressed via the path integral over "slow" localized d-electron variables and those of "light" conduction s-electrons. The temperature-dependent d-electron structure is derived. The energy shift of the d-electron level demonstrates the Kondo-like temperature logarithm. At low temperature a Hubbard-like behaviour of d-electrons is found. Keywords: Anderson lattice; Band structure; Heavy fermions; Path integral
+ y~ [enf, + V(s;,d,, + b.c.)
1. Introduction
i,a
In mixed/intermediate valence compounds, heavy-fermion systems, chemisorbates, HTSCs and organics localized electrons at crystal sites and conduction band electrons are nearly energy degenerate. It gives the possibility for transitions between localized d-(f-, etc.) states and the band. Such compounds can be idealized as uncorrelated conduction electrons hybridizing with correlated ones localized at lattice, i.e. as the lattice version [1] of the single-impurity Anderson model: H=
+
~ tpsv ,a'p, a + U ~ neni,a a p, a
i
*Corresponding author. 1On leave from N. S. Kurnakov Institute of General & Inorganic Chemistry of the Russian Academy of Sciences, Moscow, Russian Federation.
- ~(nf. + n~o)].
(1)
The "d"-electrons with angular m o m e n t a 1 have N = 2 ( 2 / + 1)-fold orbital degeneracy denoted by m and the s - d mixing is m and (momentum p dependent given by V v, m ~ Yz,,,(P) (spherical harmonics). But the average ( V p *, m, Vp, m,) over Fermi surface is small unless m = m'. So, it is no contradicition to use a constant value (1Vp[ 2 ) = V2 [ 2 , 3]. In the Hamiltonian (1) this corresponds to the contact hybridization V in the vicinity of the chemical potential #. The H u b b a r d energy U and other intra-atomic correlations 1-4] are taken to be strong enough in order to make the d-electrons heavier than s-ones. Based on the assumption that the motion of narrow-band d-electrons is slower than those of
0921-4526/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 2 6 ( 9 7 ) 0 0 1 1 3 - 0
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K lvanov et al. / Physica B 2 3 7 - 2 3 8 (1997) 231 233
broad-band s-electrons, a functional integral analysis is employed to the model (1). 2. Energy spectrum
The partition function Q of the system (1) can be represented [5, 6] as Q = Tr[exp( - [~H)]
= fDtxexp[ fi (L~+L,l+L~d)dr] '
(fl=l)
6E of d-level energy E: 2 Va ate
fl(e
1
2
p)]
de
(5)
where nd is the d-electron number per site and ps the s-electron DOS. For a flat weighted DOS, p~(e)= 1/2w #(w 2 -e2). At low temperatures, fll~ - ~l > 1. Eq. (5) is
(2) 2w
via the functional measure D/~ for s- and d-electron trajectories with antiperiodicity at [0, fl]. The selectron action is Ss =
dz
V2nd
Kfpq -- ~ (f/,qqOpq + q)p*fpq) ,
rE-
P,q
where d
K
-
dr
f.~ = (s.~, \Sq/
M,
VZnd2wlog
+r
~- #
w---~'w - #
At high temperatures T > Iw - #l, # - T ~ w 2T, p + T ~ w Eq. (5) reduces to
(Ls + L~o)dr
= ~,
--
+
M
=
(.0) 0
tq -
lW
~w 3-w
2T__de e-12
~
__ __V2nd
log
2&.
2w
(6) At low temperatures the tanh-term is negligible in Eq. (3):
I1 '
L~ = V 2 ~ d*dp/(l~- tp)(V2/2M < 1/fl < Iw - #1), p (PPq =
dq
Putting the action Ss into Eq. (2) we get the Qfunction over s-electron variables. Assuming that d-electrons are slower than s-electrons, d ~ 3, the amplitudes ~0 are z-independent and the effective interaction Lagrangian is
LSd= 2 4Op*qR(M)4opq,
and the d-system can be described by the Hubbardlike Hamiltonian: H~ ff =
_ t eff
2
+ di~djo + U~
(i,j), a
d d ni~ni.t
i
+ Z (Ear - #)n~.
(7)
i,¢r
with parameters
P,q
R(M) = V2/(2M)(I - 2/([3M) tanh([3M/2)).
(3)
The d-electron dispersion ~p depends on temperature and at Mfl < 1 it looks as follows: ~p=E
-2V2--(12 tanh fl(p_q- t . ) (4) # - t, \ fl(# - tp) ,. "
V 2, Eeff
tcff
[w -- p
E - 2ww log ~
,
3 [VEPlog w - - # = 4 L w~ ~
+
:
~] "
(8)
4. Conclusions 3. Temperature anomalies of d-electrons
According to Eqs. (3 and 4) the d-electron effective energy includes hopping tp and the energy shift
The Lagrangian and temperature-dependent electronic spectrum are derived. The energy shift of the d-electron level is governed by a Kondo-like
V. Ivanov et al. / Physica B 237-238 (1997) 231-233
logarithmic expression (Eq. (6)) and from the relation 0 = E - ~t + fiE the " K o n d o t e m p e r a t u r e " is given as follows: , w +/~exp{ TK=: 2
/~--E ~d~}'
(9)
AS fixed N and nd = 1 the shift f i E ( T ) was noticed as reported in Refs. [7, 8]. Expansion of d-electron concentration nd = nimp I Ond/OW[dw eff a r o u n d the impurity value, n~mp, leads to renormalization of the single impurity TK due to p a r a m e t e r V4/W3(Weff= zteff ~ VE/w). The latter is the new characteristic energy in the periodic Anderson lattice [9]. For/~ ~ E the energy shift of Eq. (6) provides the K o n d o effect as in Ref. [10]. If/~ is far from E the T~ in Eq. (9) is not the K o n d o t e m p e r a t u r e originating from s - d scattering. T h e present study provides new m o t i v a t i o n for the application of the periodic Anderson model to the n a r r o w energy b a n d materials. d
--
233
Acknowledgements The authors acknowledge the s u p p o r t of the Ministry of Education, Science, Sports and Culture of J a p a n (V. I.) and the Russian Ministry of Science and Technical Policy (Project No. 96149).
References [1] [2] [3] [4] 15] 16]
D. A. Smith, J. Phys. C 1 (1968) 1263• A. Bringer et al., Z. Phys. B 28 (1977) 213. O. Gunnarson et al., Phys. Rev. B 28 (1983) 4315. V. A. Ivanov, J. Phys.: Condens. Matter 6 (1994) 2065. V. Ivanov, et al., Mod. Phys. Lett. 7 (1993) 1733. V.N. Popov, V. S. Yarunin. Collective Effects in Quantum Statistics of Radiation and Matter (Kluwer, Amsterdam, 1988). 1-7-1A. F. Barabanov, et al., Fiz. Tverdogo Tela (Soy. Phys. Solid State) 21 (1979) 3214. 1-8] A. F. Barabanov, et al., Physica C 176 (1991) 511. [9] T. M. Hong et al., Physica B 186-188 (1993) 869. 1-10] Y. Nagaoka, Phys. Rev. A 138 (1963) 1112; J. Phys. Chem. Solids 27 (1966) 1139.
Physica B 237-238 (1997) 231-233
Path-integral treatment for localized electrons in the periodic Anderson model V. I v a n o v a ' * ' 1, M .
Zhuravlev b, V. Yarunin c, R. Pucci d
alnstitute for Molecular Science, Myoda(ji-cho, Okazaki 444, Japan bN. S. Kurnakov Institute of General & Inorganic Chemistry of the Russian Academy of Sciences, Moscow, Russian Federation cJoint Institute for Nuclear Research, Dubna, Russian Federation dCatania University, Catania, Italy
Abstract
The partition function of the periodic Anderson model is expressed via the path integral over "slow" localized d-electron variables and those of "light" conduction s-electrons. The temperature-dependent d-electron structure is derived. The energy shift of the d-electron level demonstrates the Kondo-like temperature logarithm. At low temperature a Hubbard-like behaviour of d-electrons is found. Keywords: Anderson lattice; Band structure; Heavy fermions; Path integral
+ y~ [enf, + V(s;,d,, + b.c.)
1. Introduction
i,a
In mixed/intermediate valence compounds, heavy-fermion systems, chemisorbates, HTSCs and organics localized electrons at crystal sites and conduction band electrons are nearly energy degenerate. It gives the possibility for transitions between localized d-(f-, etc.) states and the band. Such compounds can be idealized as uncorrelated conduction electrons hybridizing with correlated ones localized at lattice, i.e. as the lattice version [1] of the single-impurity Anderson model: H=
+
~ tpsv ,a'p, a + U ~ neni,a a p, a
i
*Corresponding author. 1On leave from N. S. Kurnakov Institute of General & Inorganic Chemistry of the Russian Academy of Sciences, Moscow, Russian Federation.
- ~(nf. + n~o)].
(1)
The "d"-electrons with angular m o m e n t a 1 have N = 2 ( 2 / + 1)-fold orbital degeneracy denoted by m and the s - d mixing is m and (momentum p dependent given by V v, m ~ Yz,,,(P) (spherical harmonics). But the average ( V p *, m, Vp, m,) over Fermi surface is small unless m = m'. So, it is no contradicition to use a constant value (1Vp[ 2 ) = V2 [ 2 , 3]. In the Hamiltonian (1) this corresponds to the contact hybridization V in the vicinity of the chemical potential #. The H u b b a r d energy U and other intra-atomic correlations 1-4] are taken to be strong enough in order to make the d-electrons heavier than s-ones. Based on the assumption that the motion of narrow-band d-electrons is slower than those of
0921-4526/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 2 6 ( 9 7 ) 0 0 1 1 3 - 0
232
K lvanov et al. / Physica B 2 3 7 - 2 3 8 (1997) 231 233
broad-band s-electrons, a functional integral analysis is employed to the model (1). 2. Energy spectrum
The partition function Q of the system (1) can be represented [5, 6] as Q = Tr[exp( - [~H)]
= fDtxexp[ fi (L~+L,l+L~d)dr] '
(fl=l)
6E of d-level energy E: 2 Va ate
fl(e
1
2
p)]
de
(5)
where nd is the d-electron number per site and ps the s-electron DOS. For a flat weighted DOS, p~(e)= 1/2w #(w 2 -e2). At low temperatures, fll~ - ~l > 1. Eq. (5) is
(2) 2w
via the functional measure D/~ for s- and d-electron trajectories with antiperiodicity at [0, fl]. The selectron action is Ss =
dz
V2nd
Kfpq -- ~ (f/,qqOpq + q)p*fpq) ,
rE-
P,q
where d
K
-
dr
f.~ = (s.~, \Sq/
M,
VZnd2wlog
+r
~- #
w---~'w - #
At high temperatures T > Iw - #l, # - T ~ w 2T, p + T ~ w Eq. (5) reduces to
(Ls + L~o)dr
= ~,
--
+
M
=
(.0) 0
tq -
lW
~w 3-w
2T__de e-12
~
__ __V2nd
log
2&.
2w
(6) At low temperatures the tanh-term is negligible in Eq. (3):
I1 '
L~ = V 2 ~ d*dp/(l~- tp)(V2/2M < 1/fl < Iw - #1), p (PPq =
dq
Putting the action Ss into Eq. (2) we get the Qfunction over s-electron variables. Assuming that d-electrons are slower than s-electrons, d ~ 3, the amplitudes ~0 are z-independent and the effective interaction Lagrangian is
LSd= 2 4Op*qR(M)4opq,
and the d-system can be described by the Hubbardlike Hamiltonian: H~ ff =
_ t eff
2
+ di~djo + U~
(i,j), a
d d ni~ni.t
i
+ Z (Ear - #)n~.
(7)
i,¢r
with parameters
P,q
R(M) = V2/(2M)(I - 2/([3M) tanh([3M/2)).
(3)
The d-electron dispersion ~p depends on temperature and at Mfl < 1 it looks as follows: ~p=E
-2V2--(12 tanh fl(p_q- t . ) (4) # - t, \ fl(# - tp) ,. "
V 2, Eeff
tcff
[w -- p
E - 2ww log ~
,
3 [VEPlog w - - # = 4 L w~ ~
+
:
~] "
(8)
4. Conclusions 3. Temperature anomalies of d-electrons
According to Eqs. (3 and 4) the d-electron effective energy includes hopping tp and the energy shift
The Lagrangian and temperature-dependent electronic spectrum are derived. The energy shift of the d-electron level is governed by a Kondo-like
V. Ivanov et al. / Physica B 237-238 (1997) 231-233
logarithmic expression (Eq. (6)) and from the relation 0 = E - ~t + fiE the " K o n d o t e m p e r a t u r e " is given as follows: , w +/~exp{ TK=: 2
/~--E ~d~}'
(9)
AS fixed N and nd = 1 the shift f i E ( T ) was noticed as reported in Refs. [7, 8]. Expansion of d-electron concentration nd = nimp I Ond/OW[dw eff a r o u n d the impurity value, n~mp, leads to renormalization of the single impurity TK due to p a r a m e t e r V4/W3(Weff= zteff ~ VE/w). The latter is the new characteristic energy in the periodic Anderson lattice [9]. For/~ ~ E the energy shift of Eq. (6) provides the K o n d o effect as in Ref. [10]. If/~ is far from E the T~ in Eq. (9) is not the K o n d o t e m p e r a t u r e originating from s - d scattering. T h e present study provides new m o t i v a t i o n for the application of the periodic Anderson model to the n a r r o w energy b a n d materials. d
--
233
Acknowledgements The authors acknowledge the s u p p o r t of the Ministry of Education, Science, Sports and Culture of J a p a n (V. I.) and the Russian Ministry of Science and Technical Policy (Project No. 96149).
References [1] [2] [3] [4] 15] 16]
D. A. Smith, J. Phys. C 1 (1968) 1263• A. Bringer et al., Z. Phys. B 28 (1977) 213. O. Gunnarson et al., Phys. Rev. B 28 (1983) 4315. V. A. Ivanov, J. Phys.: Condens. Matter 6 (1994) 2065. V. Ivanov, et al., Mod. Phys. Lett. 7 (1993) 1733. V.N. Popov, V. S. Yarunin. Collective Effects in Quantum Statistics of Radiation and Matter (Kluwer, Amsterdam, 1988). 1-7-1A. F. Barabanov, et al., Fiz. Tverdogo Tela (Soy. Phys. Solid State) 21 (1979) 3214. 1-8] A. F. Barabanov, et al., Physica C 176 (1991) 511. [9] T. M. Hong et al., Physica B 186-188 (1993) 869. 1-10] Y. Nagaoka, Phys. Rev. A 138 (1963) 1112; J. Phys. Chem. Solids 27 (1966) 1139.