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Weak-coupling theory of the periodic Anderson model an infinite dimensions
PHYSICAi ELSEVIER
Physica B 199&200 (1994) 206-208
Weak-coupling theory of the periodic Anderson model in infinite dimensions Dai S. Hirashima*, Tetsuya Mutou blstitute of Physics. University of Tst~kuba. Tsukuba. lbaraki 305. Japan
Abstract Second-order perturbation theory is applied to the periodic Anderson model in infinite dimensions. In the particle-hole symmetric case, the electron density of states (DOS) has a hybridization gap at the Fermi level at T = 0; the width ,t of the gap decreases as the interaction U increases. It is found that A -1 is roughly proportional to the mass enhancement. Furthermore, using this model, we discuss the Van Vleck susceptibility at T = 0. The Van Vleck term is found to be enhanced by the interaction in the weak coupling region.
1. lntroduc:ion Recently, a set of heavy-fermion materials that behave as small-gap semiconductors at low temperatures, i.e. " K o n d o insulators", has attracted much interest [1]. The simplest interpretation of the gap in these materials is that it is caused by the hybridization between f-levels and a conduction band. This situation can be described by the periodic Anderson model (PAM), In this paper, we report the results obtained for the non-degenerate PAM in infinite dimensions by the second-order perturbation theory (SOPT). In fact, we use the following two methods: (I) Infinitedimensional models can be mapped to an impurity model [2]. We study this impurity model by the SOPT around the Hartree-Fock solution [3]. (2) We apply the self-consistent SOPT (SCSOPT) to the infinite-dimensional PAM. The former method was applied to the infinite-dimensional models and shown to be a rather good approximation [4, 5]. The SCSOPT is a fully con*
Corresponding author.
serving approximation, and has been applied to the PAM to evaluate the one-particle DOS [6]. Although exact Q M C calculation of the infinite-dimensional PAM has already been carried out [51 approximate calculation is also useful to get an insight of the problem, (in particular, in the low-temperature and low-energy region) and to study the validity of the approximation.
2. Model We study the non-degenerate PAM: the hybridization between the f-electrons and the conduction electrons is denoted by V and the on-site Coulomb repulsion U works between the f-electrons. We assume that the free conduction electron DOS for V :: 0 is given by pO((,~) = (2/n).w/I _
((o/t)2
i.e. we consider the PAM on the Bethe lattice with infinite connectivity. In the following, we set t = 1. In this paper, we mainly deal with the particle hole symmetric case and
0921-4526/94/$07.00 (c) 1994 Elsevier Science B.V. All rights reserved SSDI 0921-4526193)E0259-J
D.S. Hirashima. T. Mutou / Physk'a B 199&200 (1994) 206-208
restrict ourselves to the paramagnetic phase: n,,,t = 2 and Plf = i.
3. D O S Infinite-dimensional models can be mapped to an impurity problem embedded in an effective "field", which has to be determined self-consistently. Here we evaluate the impurity self-energy by the SOPT around the H F solution [3-5]. The obtained f-electron D O S at T = 0 is shown in Fig. i. The D O S has a gap around the Fermi level, and has a satellite structure evolving as U increases. The mass enhancement m*/m and the inverse A - x of the gap at T = 0 are plotted in Fig. 2; m*/m = l d~'/do~l,o=o. It can be seen that A - t increases with U and
2 3 V
'
I
'
U=1,213
I
1
.
.
.
'
i ,,, , ,
.
Q.
2
1
0
2
co
3
Fig. 1. T h e f-electron density o f states for l" = 0.5 a n d U = 1, 2 a n d 3 at T = 0. T h e inset s h o w s the low-energy p a r t for U = 2 a n d T = 0, 0.1, and 0.2.
!
6
o
I
'
I
'
'
|
4, Spin susceptibility
The uniform spin susceptibility X" of the Kondo insulators remains finite and large at T = 0. Usually it is attributed to the Van Vleck susceptibility. Whether or not the Van Vleck term is also enhanced by U, however, is not a trivial problem [7"i. In the present model, if we set gc = Of, X" vanishes at T = 0 for n,o, = 2; X" "" e x p ( - A d T ) as T--* 0 where A~ ,,, A. If we assume ,q, # ,qr, however, ~~ remains finite at T = 0 (even for U = 0) even when the D O S is gapped at the Fermi level. This model can be realized when the degeneracy of the f-electron is reduced to 2 by strong spin--orbit interaction and crystalline field [8]. in this case X~ can be written as 7" = A~ ~/,qf = - Ag Z~/,q¢ when the Fermi level locates inside the gap, where X ~ is the contribution of the f-lconduction) electrons and Aq = ,qf - g~. We calculate y~ by the SCSOPT. In Fig. 3, we show ~ ( T = 0)/m* and X~( T = 0) A for ,Of/,q~ = 0.5 as function of U. It can be seen that Z~( T = 0) is enhanced by U and is roughly proportional to m* and J " The result thus shows that in this model :he Van Vlcck term is a!so enhanced by the interaction in the weakcoupling region. In Ref. [7] it was shown by the RPA that the enhancement of the non-quasiparticle part of X', (which is, strictly speakir~g, different from the Van Vleck term), is rather model-dependent. Study of more realistic models is also necessary.
<1
"R
0 v
®
m/m
•
is roughly proportional to m*/m. This result is consistent with the renormalized band theory. The inset of Fig, ! shows the temperature dependence of the low-energy part. As T increases, the DOS becomes finite inside the gap, and the gap turns into a pseudo-gap.
t
A(U=O)/A
m
_
i
0
-
207
I
1
I
'
o o
o n
~
9
o
g
V=0.5 8
1
X
V=0.4 o
,
0
I
i
I
2
,
: I
,
U
I
I
i
4
Fig. 2. T h e inverse A ~ o f the g a p and the m a s s e n h a n c e m e n t m * , m for V = 0.5 at T = 0.
O
v
to
×0
t
!
[
1
i
I
i
2
U
3
Fig. 3. / ' ( T = 0p.'m* (solid s y m b o l s l a n d Z't T = 0)A Wpen s y m bols) n o r m a l i z e d by the values al ( = 0 for qfq, = 0.5 a n d V = 0.5 {circles) and 0.4 (squares).
208
D.S. Hirashima, T. Mutou / Physica B 199&200 (1994) 206-208
References 1-I] G. Aeppli and Z. Fisk, Comments Cond. Mat. Phys. 16 (1992) 155. [2"1 V. Janis, Z. Phys. B 83 (1991) 227; A. Georges and G. Kotliar, Phys, Rev. B 45 (1992) 6479. [3] K. Yamada, Prog. Theor. Phys. 53 (1975) 970. [4] X.Y. Zhang, M.J. Rozenberg and G. Kotliar, Phys, Rev. Lett. 70 11993) 1666.
I'5] M. Jarrell, H. Akhlaghpour and Th. Pruschke, Phys. Rev. Lett. 70 (1993) 1670. [6"1 H. Schweitzer and G. Czycholl, Z, Phys. B 79 (1990) 377. [7] M. Nakano, Phys. Rev. B 44 (1991) 10300, and references therein. [8] K. Hanzawa, Y. Yosida and K. Yamada, Prog. Theor. Phys. 81 (1989) 960.
Physica B 199&200 (1994) 206-208
Weak-coupling theory of the periodic Anderson model in infinite dimensions Dai S. Hirashima*, Tetsuya Mutou blstitute of Physics. University of Tst~kuba. Tsukuba. lbaraki 305. Japan
Abstract Second-order perturbation theory is applied to the periodic Anderson model in infinite dimensions. In the particle-hole symmetric case, the electron density of states (DOS) has a hybridization gap at the Fermi level at T = 0; the width ,t of the gap decreases as the interaction U increases. It is found that A -1 is roughly proportional to the mass enhancement. Furthermore, using this model, we discuss the Van Vleck susceptibility at T = 0. The Van Vleck term is found to be enhanced by the interaction in the weak coupling region.
1. lntroduc:ion Recently, a set of heavy-fermion materials that behave as small-gap semiconductors at low temperatures, i.e. " K o n d o insulators", has attracted much interest [1]. The simplest interpretation of the gap in these materials is that it is caused by the hybridization between f-levels and a conduction band. This situation can be described by the periodic Anderson model (PAM), In this paper, we report the results obtained for the non-degenerate PAM in infinite dimensions by the second-order perturbation theory (SOPT). In fact, we use the following two methods: (I) Infinitedimensional models can be mapped to an impurity model [2]. We study this impurity model by the SOPT around the Hartree-Fock solution [3]. (2) We apply the self-consistent SOPT (SCSOPT) to the infinite-dimensional PAM. The former method was applied to the infinite-dimensional models and shown to be a rather good approximation [4, 5]. The SCSOPT is a fully con*
Corresponding author.
serving approximation, and has been applied to the PAM to evaluate the one-particle DOS [6]. Although exact Q M C calculation of the infinite-dimensional PAM has already been carried out [51 approximate calculation is also useful to get an insight of the problem, (in particular, in the low-temperature and low-energy region) and to study the validity of the approximation.
2. Model We study the non-degenerate PAM: the hybridization between the f-electrons and the conduction electrons is denoted by V and the on-site Coulomb repulsion U works between the f-electrons. We assume that the free conduction electron DOS for V :: 0 is given by pO((,~) = (2/n).w/I _
((o/t)2
i.e. we consider the PAM on the Bethe lattice with infinite connectivity. In the following, we set t = 1. In this paper, we mainly deal with the particle hole symmetric case and
0921-4526/94/$07.00 (c) 1994 Elsevier Science B.V. All rights reserved SSDI 0921-4526193)E0259-J
D.S. Hirashima. T. Mutou / Physk'a B 199&200 (1994) 206-208
restrict ourselves to the paramagnetic phase: n,,,t = 2 and Plf = i.
3. D O S Infinite-dimensional models can be mapped to an impurity problem embedded in an effective "field", which has to be determined self-consistently. Here we evaluate the impurity self-energy by the SOPT around the H F solution [3-5]. The obtained f-electron D O S at T = 0 is shown in Fig. i. The D O S has a gap around the Fermi level, and has a satellite structure evolving as U increases. The mass enhancement m*/m and the inverse A - x of the gap at T = 0 are plotted in Fig. 2; m*/m = l d~'/do~l,o=o. It can be seen that A - t increases with U and
2 3 V
'
I
'
U=1,213
I
1
.
.
.
'
i ,,, , ,
.
Q.
2
1
0
2
co
3
Fig. 1. T h e f-electron density o f states for l" = 0.5 a n d U = 1, 2 a n d 3 at T = 0. T h e inset s h o w s the low-energy p a r t for U = 2 a n d T = 0, 0.1, and 0.2.
!
6
o
I
'
I
'
'
|
4, Spin susceptibility
The uniform spin susceptibility X" of the Kondo insulators remains finite and large at T = 0. Usually it is attributed to the Van Vleck susceptibility. Whether or not the Van Vleck term is also enhanced by U, however, is not a trivial problem [7"i. In the present model, if we set gc = Of, X" vanishes at T = 0 for n,o, = 2; X" "" e x p ( - A d T ) as T--* 0 where A~ ,,, A. If we assume ,q, # ,qr, however, ~~ remains finite at T = 0 (even for U = 0) even when the D O S is gapped at the Fermi level. This model can be realized when the degeneracy of the f-electron is reduced to 2 by strong spin--orbit interaction and crystalline field [8]. in this case X~ can be written as 7" = A~ ~/,qf = - Ag Z~/,q¢ when the Fermi level locates inside the gap, where X ~ is the contribution of the f-lconduction) electrons and Aq = ,qf - g~. We calculate y~ by the SCSOPT. In Fig. 3, we show ~ ( T = 0)/m* and X~( T = 0) A for ,Of/,q~ = 0.5 as function of U. It can be seen that Z~( T = 0) is enhanced by U and is roughly proportional to m* and J " The result thus shows that in this model :he Van Vlcck term is a!so enhanced by the interaction in the weakcoupling region. In Ref. [7] it was shown by the RPA that the enhancement of the non-quasiparticle part of X', (which is, strictly speakir~g, different from the Van Vleck term), is rather model-dependent. Study of more realistic models is also necessary.
<1
"R
0 v
®
m/m
•
is roughly proportional to m*/m. This result is consistent with the renormalized band theory. The inset of Fig, ! shows the temperature dependence of the low-energy part. As T increases, the DOS becomes finite inside the gap, and the gap turns into a pseudo-gap.
t
A(U=O)/A
m
_
i
0
-
207
I
1
I
'
o o
o n
~
9
o
g
V=0.5 8
1
X
V=0.4 o
,
0
I
i
I
2
,
: I
,
U
I
I
i
4
Fig. 2. T h e inverse A ~ o f the g a p and the m a s s e n h a n c e m e n t m * , m for V = 0.5 at T = 0.
O
v
to
×0
t
!
[
1
i
I
i
2
U
3
Fig. 3. / ' ( T = 0p.'m* (solid s y m b o l s l a n d Z't T = 0)A Wpen s y m bols) n o r m a l i z e d by the values al ( = 0 for qfq, = 0.5 a n d V = 0.5 {circles) and 0.4 (squares).
208
D.S. Hirashima, T. Mutou / Physica B 199&200 (1994) 206-208
References 1-I] G. Aeppli and Z. Fisk, Comments Cond. Mat. Phys. 16 (1992) 155. [2"1 V. Janis, Z. Phys. B 83 (1991) 227; A. Georges and G. Kotliar, Phys, Rev. B 45 (1992) 6479. [3] K. Yamada, Prog. Theor. Phys. 53 (1975) 970. [4] X.Y. Zhang, M.J. Rozenberg and G. Kotliar, Phys, Rev. Lett. 70 11993) 1666.
I'5] M. Jarrell, H. Akhlaghpour and Th. Pruschke, Phys. Rev. Lett. 70 (1993) 1670. [6"1 H. Schweitzer and G. Czycholl, Z, Phys. B 79 (1990) 377. [7] M. Nakano, Phys. Rev. B 44 (1991) 10300, and references therein. [8] K. Hanzawa, Y. Yosida and K. Yamada, Prog. Theor. Phys. 81 (1989) 960.