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Fermi liquid and non-Fermi liquid in M-channel: N fold degenerate Anderson lattice
Physica B 281&282 (2000) 408}409
Fermi liquid and non-Fermi liquid in M-channel: N fold degenerate Anderson lattice Atsushi Tsuruta!,*, Akito Kobayashi", Ken Deguchi#, Yoshiaki O1 no!,", Tamifusa Matsuura!, Yoshihiro Kuroda!," !Department of Physics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan "CREST, Japan Science and Technology Corporation (JST), Japan #Faculty of Education, Tokoha Gakuen University, Sena, Shizuoka 420-0911, Japan
Abstract We investigate electronic states in the M-channel N fold degenerate Anderson lattice with the help of 1/N-expansion. In the single-channel case, we showed that the imaginary part of the self-energy of the conduction electrons up to O(N~2) is exactly given by a form of the Fermi-liquid-type. In the multichannel case, we "nd a ¹-linear term of O(N~1) in the imaginary part of the self-energy of the conduction electrons, in contrast to a constant term found in the in"nite dimensions. Then we critically examine the results obtained in the in"nite dimensions. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Fermi liquid; Heavy fermion systems; Multichannel Anderson lattice; 1/N-Expansion
In the impurity Anderson model, it has been shown in the leading order of 1/N-expansion that the system behaves as a local Fermi liquid where the Friedel sum rule holds [1]. Matsuura et al. [2] explicitly examined the infrared divergence in the pseudo-particle spectra by including higher-order terms in the literal 1/N-expansion study where they collected all diagrams in each order of 1/N, and determined the threshold exponents. They also showed that contributions from all those divergent terms in the pseudo-particle spectra to the scattering rate of the conduction electrons (c-electrons) are cancelled out by the corresponding vertex corrections leaving only nondivergent terms, which satisfy the Friedel sum rule. Then Tsuruta et al. [3] showed that in the multichannel Anderson model, the situation is entirely di!erent from the above. The anomaly due to the pseudo-particle spectra appears in physical quantities, i.e., the system is a nonFermi liquid. In the Anderson lattice O1 no et al. [4] showed that the heavy electrons are well described in the leading order of
* Corresponding author. Fax: #81-52-789-2928. E-mail address: [email protected] (A. Tsuruta)
1/N-expansion where the Luttinger sum rule holds. However, in order to show unambiguously that the system behaves as a Fermi liquid, we need to investigate e!ects of the higher-order terms in the 1/N-expansion. On the other hand, Jarrell et al. [5] studied properties of the two-channel Kondo lattice model in the in"nite dimensions and found that the resistivity is "nite even at ¹"0. However, we believe that it happened because e!ects of the translational symmetry were not properly taken into account. We investigate such e!ects by including the intersite correlation e!ects. For these purposes, we study the M-channel N fold degenerate Anderson lattice model given by N M H" + + + e mq C`mq C mq #+ e f` f k k k f im im i,m m/1 q/1 kmq < # + + MC`mq b`6 f e~*kmq Ri k iq im JN i,m,q kmq L # f` b 6 C mq e*kmq Ri N. (1) im iq k We divide the phase space for k into NM/2 subspaces for k keeping the total degrees of freedom for spins, ormq bitals, and channels a constant as + "+ "2N . p,k m,q,kmq L
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 9 9 8 - 9
A. Tsuruta et al. / Physica B 281&282 (2000) 408}409
For guaranteeing physical equivalence between the present model, Eq. (1), and the original ;"R model, this Hamiltonian must be treated within the subspace where the local constraint, QK "+ b`b #+ f` f "1 holds. i q iq iq m im im In O(N0), the heavy electrons whose lifetime is in"nite and whose bandwidth is O(E ) (E is the binding energy 0 0 of the slave boson) are formed in both the cases of the single channel and the multichannel. The heavy electron band consists of c-electron component (c-component) and f-electron component. In the M-channel case, the band is NM fold degeneracy. In the single-channel case (M"1), we critically examine Fermi liquid properties. We collect all diagrams contributing to the self-energy of the c-component up to O(N~2) and show that the imaginary part of the self-energy of the c-component is exactly given by a form of the Fermi-liquid-type: Im R m (u#i0 )"!pa m (u2#p2¹2) k ` k where a m is a constant of O(D/NE2 ) (2D is the bandk 0 width of the c-electrons), and that higher-order corrections of O(N~1) to the real part of the self-energy only modi"es the fundamental parameters such as the characteristic energy E at u, ¹;E [6]. The Luttinger sum 0 0 rule is shown to hold up to O(N~1). In the multichannel case (M'1), we calculate all terms of the self-energy up to O(N~1) which includes the intersite correlation e!ects and show that the imaginary part of O(N~1) is given by Im R mq "Im &(FL) #Im R(NFL) , k kmq kmq
where a mq and b mq are constants of O(D/NE2 ) and k 0 k O(1/NE ), respectively. The Luttinger sum rule also 0 holds in the multichannel case. The localized spin is overscreened because of the degrees of the multichannel, and so the scattering rate of the c-component in the multichannel case is more singular than that in the single-channel case. In some intermetallic compounds with uranium, the ground state of a U-ion with (5f )2 is a non-Kramers doublet ! and the intermediate state with (5f )1 is 3 a Kramers doublet ! under a crystalline "eld with cubic 7 symmetry. We will explain ordered states of the quadrupolar moment and the tiny magnetization in URu Si et 2 2 al. with the use of the multichannel model. In conclusion, using the literal 1/N-expansion, we showed that the heavy electrons in the in"nite-; singlechannel Anderson lattice form the Fermi liquid and that the heavy electrons in the multichannel Anderson lattice form the non-Fermi liquid where the imaginary part of the self-energy of the c-component is ¹-linear.
Acknowledgements One of the authors (A.T.) is supported by a Research Fellowship of the Japan Society for the Promotion of Science for Young Scientists.
(2)
where R(FL) is the self-energy of the Fermi-liquid-type, kmq
References
(u#i0 )"!pa mq (u2#p2¹2) Im R(FL) kmq ` k
[1] [2] [3] [4]
(3)
and R(NFL) is the self-energy of the non-Fermi-liquidkmq type which does not make any contribution in the single-channel case, (u#i0 )"!pb mq ¹(1!M~2), Im R(NFL) ` k kmq
409
(4)
B. Jin, Y. Kuroda, J. Phys. Soc. Japan 57 (1988) 1687. T. Matsuura et al., J. Phys. Soc. Japan 66 (1997) 1245. A. Tsuruta et al., J. Phys. Soc. Japan 67 (1998) 2346. Y. OM no, T. Matsuura, Y. Kuroda, Physica C 159 (1989) 878. [5] M. Jarrell et al., Physica B 230}232 (1997) 557. [6] A. Tsuruta et al., J. Phys. Soc. Japan 68 (1999)
Fermi liquid and non-Fermi liquid in M-channel: N fold degenerate Anderson lattice Atsushi Tsuruta!,*, Akito Kobayashi", Ken Deguchi#, Yoshiaki O1 no!,", Tamifusa Matsuura!, Yoshihiro Kuroda!," !Department of Physics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan "CREST, Japan Science and Technology Corporation (JST), Japan #Faculty of Education, Tokoha Gakuen University, Sena, Shizuoka 420-0911, Japan
Abstract We investigate electronic states in the M-channel N fold degenerate Anderson lattice with the help of 1/N-expansion. In the single-channel case, we showed that the imaginary part of the self-energy of the conduction electrons up to O(N~2) is exactly given by a form of the Fermi-liquid-type. In the multichannel case, we "nd a ¹-linear term of O(N~1) in the imaginary part of the self-energy of the conduction electrons, in contrast to a constant term found in the in"nite dimensions. Then we critically examine the results obtained in the in"nite dimensions. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Fermi liquid; Heavy fermion systems; Multichannel Anderson lattice; 1/N-Expansion
In the impurity Anderson model, it has been shown in the leading order of 1/N-expansion that the system behaves as a local Fermi liquid where the Friedel sum rule holds [1]. Matsuura et al. [2] explicitly examined the infrared divergence in the pseudo-particle spectra by including higher-order terms in the literal 1/N-expansion study where they collected all diagrams in each order of 1/N, and determined the threshold exponents. They also showed that contributions from all those divergent terms in the pseudo-particle spectra to the scattering rate of the conduction electrons (c-electrons) are cancelled out by the corresponding vertex corrections leaving only nondivergent terms, which satisfy the Friedel sum rule. Then Tsuruta et al. [3] showed that in the multichannel Anderson model, the situation is entirely di!erent from the above. The anomaly due to the pseudo-particle spectra appears in physical quantities, i.e., the system is a nonFermi liquid. In the Anderson lattice O1 no et al. [4] showed that the heavy electrons are well described in the leading order of
* Corresponding author. Fax: #81-52-789-2928. E-mail address: [email protected] (A. Tsuruta)
1/N-expansion where the Luttinger sum rule holds. However, in order to show unambiguously that the system behaves as a Fermi liquid, we need to investigate e!ects of the higher-order terms in the 1/N-expansion. On the other hand, Jarrell et al. [5] studied properties of the two-channel Kondo lattice model in the in"nite dimensions and found that the resistivity is "nite even at ¹"0. However, we believe that it happened because e!ects of the translational symmetry were not properly taken into account. We investigate such e!ects by including the intersite correlation e!ects. For these purposes, we study the M-channel N fold degenerate Anderson lattice model given by N M H" + + + e mq C`mq C mq #+ e f` f k k k f im im i,m m/1 q/1 kmq < # + + MC`mq b`6 f e~*kmq Ri k iq im JN i,m,q kmq L # f` b 6 C mq e*kmq Ri N. (1) im iq k We divide the phase space for k into NM/2 subspaces for k keeping the total degrees of freedom for spins, ormq bitals, and channels a constant as + "+ "2N . p,k m,q,kmq L
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 9 9 8 - 9
A. Tsuruta et al. / Physica B 281&282 (2000) 408}409
For guaranteeing physical equivalence between the present model, Eq. (1), and the original ;"R model, this Hamiltonian must be treated within the subspace where the local constraint, QK "+ b`b #+ f` f "1 holds. i q iq iq m im im In O(N0), the heavy electrons whose lifetime is in"nite and whose bandwidth is O(E ) (E is the binding energy 0 0 of the slave boson) are formed in both the cases of the single channel and the multichannel. The heavy electron band consists of c-electron component (c-component) and f-electron component. In the M-channel case, the band is NM fold degeneracy. In the single-channel case (M"1), we critically examine Fermi liquid properties. We collect all diagrams contributing to the self-energy of the c-component up to O(N~2) and show that the imaginary part of the self-energy of the c-component is exactly given by a form of the Fermi-liquid-type: Im R m (u#i0 )"!pa m (u2#p2¹2) k ` k where a m is a constant of O(D/NE2 ) (2D is the bandk 0 width of the c-electrons), and that higher-order corrections of O(N~1) to the real part of the self-energy only modi"es the fundamental parameters such as the characteristic energy E at u, ¹;E [6]. The Luttinger sum 0 0 rule is shown to hold up to O(N~1). In the multichannel case (M'1), we calculate all terms of the self-energy up to O(N~1) which includes the intersite correlation e!ects and show that the imaginary part of O(N~1) is given by Im R mq "Im &(FL) #Im R(NFL) , k kmq kmq
where a mq and b mq are constants of O(D/NE2 ) and k 0 k O(1/NE ), respectively. The Luttinger sum rule also 0 holds in the multichannel case. The localized spin is overscreened because of the degrees of the multichannel, and so the scattering rate of the c-component in the multichannel case is more singular than that in the single-channel case. In some intermetallic compounds with uranium, the ground state of a U-ion with (5f )2 is a non-Kramers doublet ! and the intermediate state with (5f )1 is 3 a Kramers doublet ! under a crystalline "eld with cubic 7 symmetry. We will explain ordered states of the quadrupolar moment and the tiny magnetization in URu Si et 2 2 al. with the use of the multichannel model. In conclusion, using the literal 1/N-expansion, we showed that the heavy electrons in the in"nite-; singlechannel Anderson lattice form the Fermi liquid and that the heavy electrons in the multichannel Anderson lattice form the non-Fermi liquid where the imaginary part of the self-energy of the c-component is ¹-linear.
Acknowledgements One of the authors (A.T.) is supported by a Research Fellowship of the Japan Society for the Promotion of Science for Young Scientists.
(2)
where R(FL) is the self-energy of the Fermi-liquid-type, kmq
References
(u#i0 )"!pa mq (u2#p2¹2) Im R(FL) kmq ` k
[1] [2] [3] [4]
(3)
and R(NFL) is the self-energy of the non-Fermi-liquidkmq type which does not make any contribution in the single-channel case, (u#i0 )"!pb mq ¹(1!M~2), Im R(NFL) ` k kmq
409
(4)
B. Jin, Y. Kuroda, J. Phys. Soc. Japan 57 (1988) 1687. T. Matsuura et al., J. Phys. Soc. Japan 66 (1997) 1245. A. Tsuruta et al., J. Phys. Soc. Japan 67 (1998) 2346. Y. OM no, T. Matsuura, Y. Kuroda, Physica C 159 (1989) 878. [5] M. Jarrell et al., Physica B 230}232 (1997) 557. [6] A. Tsuruta et al., J. Phys. Soc. Japan 68 (1999)