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Theory of momentum-dependent magnetic response in heavy-fermion systems
Physica 148B (1987) 80-83 North-Holland, Amsterdam
T H E O R Y OF M O M E N T U M - D E P E N D E N T M A G N E T I C RESPONSE IN HEAVY-FERMION SYSTEMS Y. K U R A M O T O
and T. W A T A N A B E
Departmentof AppliedPhysics,Tohoku University,Sendai980,Japan Received 3 August 1987 A self-consistent dynamical theory is presented for the Anderson lattice. In a variational Lagrangian formulation a trial conduction-electron propagator is introduced. A variational principle leads to optimization of the propagator with the accuracy of U(I/zn) where z., is the number of interacting neighbors. The extended non-crossing approximation (XNCA) is reproduced as a special case of the present theory. An integral equation is obtained for the dynamical susceptibility which incorporates the RKKY interaction and the Kondo effect. The closed-form solution is given in a case where perturbation theory is applicable.
T h e dynamical m a g n e t i c response of heavyf e r m i o n systems reflects b o t h on-site and intersite correlations of f electrons. A t t e m p e r a t u r e T higher than the K o n d o t e m p e r a t u r e T K the dynamical susceptibility x ( q , o)) p r o b e d by neutron scattering and N M R shows characteristics very similar to those in dilute impurity systems. A t T < T K, h o w e v e r , significant intersite interactions have b e e n o b s e r v e d in some h e a v y - f e r m i o n systems such as C e C u 6 [1] and U P t 3 [2]. T h e interactions are frequently of a n t i f e r r o m a g n e t i c type. Theoretical study of x ( q , o2) is highly desired in o r d e r to u n d e r s t a n d the nature of intersite interactions. T h e enterprise is a hard o n e since o n e must treat the crossover f r o m the single-site b e h a v i o r to the b a n d - t y p e one taking a c c o u n t of the K o n d o effect. P e r t u r b a t i o n a l theories [3, 4] have not so far succeeded in achieving a self-consistency of the f r a m e w o r k . In a recent p a p e r [5] one of the present authors has presented a phenomenological analysis of x ( q , w) on the basis of the Mori formalism and the Fermi-liquid theory. A puzzling b e h a v i o r f o u n d in C e C u 6 [1] has b e e n explained in terms of nearly localized quasiparticles. It is of course desirable to have m o r e microscopic f o u n d a t i o n for the analysis. T h e p u r p o s e of the present p a p e r is to give a brief a c c o u n t of a first-principle and self-consistent t h e o r y for x ( q , w). We consider the A n d e r s o n - l a t t i c e m o d e l with 0378-4363/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) and Yamada Science Foundation
twofold spin degeneracy. Extension to a m o r e general m o d e l is straightforward. T h e (grand) partition function Z of the N-site system at t e m p e r a t u r e T = / 3 1 is expressed as a functional integral over G r a s s m a n n n u m b e r s [6] which corr e s p o n d to f e r m i o n fields. T h e conduction-electron part is easily integrated out. T h e remaining part is given by
Z/Zc= f DffDf exp{- f~dr + 2ehy ( )] } , ,
0
+ U 2 nit (')ni+ i
~hyb(T) = V2 Z
(~'),
(2)
dT' f ~ ( T )
l]cr
x
(1)
-
(3)
H e r e Z c is the partition function of c o n d u c t i o n + electrons without hybridization, fio- represents creation of an f electron with spin o- at site i, and g q ( ~ - - ~-') is the bare p r o p a g a t o r of c o n d u c t i o n electrons in the site representation. F o r simplicity we assume that the hybridization matrix elem e n t V does not d e p e n d on m o m e n t u m of con-
81
Y. Kuramoto and T. Watanabe / Magnetic response in heavy-fermion systems
duction electrons. The last term in (2) represents the Coulomb repulsion between f electrons at the same site. Our strategy to account for intersite interactions is in line with the mean-field theory; the effect of environment around a given f-electron site is represented by an effective field. In the present case the effective field is a site-diagonal trial propagator G ~ ( r - r ' ) of conduction electrons. The convenience of using the functional integral is that we can introduce a trial Lagrangian ~tr which, in contrast to the Hamiltonian formalism, can describe a retardation effect. Thus we make a new decomposition ~e + ~'hyb = ~t~ + ~i,t with /3
0
(4)
f[~(r)dc(Z - r')fi~(z') , 13
~i,,t ("r) = V 2 E,,..,,f dr' - a,jdc(
f~,~(r)[gij('r
- r')
(5)
- -
and consider a perturbation expansion in terms
(0) O(I/Z n)
0+0+©+ (b) O ( I / z 2) +
~
+
...
Fig. 1. Diagrams f o r / ~ , in the loop-wise expansion.
Aq = g # - 8#G c and the dots are cumulant averages at each site. We rearrange the perturbation series in/~int to ascending powers of 1/zn where z n is the number of interacting neighbors around a given site. For each loop made up of Aq we associate a factor 1/Zn [7]. This rearrangement corresponds to the loop-wise expansion in the field theory [6]. Thus in the leading order of 1 / z , we collect all sequence of diagrams in fig. l(a) which involve the site-diagonal Green function of f electrons as the two-point cumulant. There is no restriction on the site summation because one is dealing with cumulant averages. With Fourier transform to momentum and Matsubara frequency space, A/j changes into A(k, ien) where en = (2n + 1)~rT with n integer. We obtain g2~nt = ~ T ~ In[1 k n
VZF(ien)A(k, it,,)],
(8)
of
Since ~tr is diagonal with respect to site indices, the partition function without ~int is derived from that of the effective single-site system Zf. The thermodynamic potential/-2 of the whole system is then given as a functional of Gc by -fl/2{Gc} = In Z c + N l n Zf(Gc} - ~ i n t { S c )
,
(6)
d'r ~int(1")])cum - 1 0
1 ~] [1 N k
_
_
(t(ien) =
t~ -jS~(-]int{Gc} = ( e x p [ - f
where F(ien) is the site-diagonal part of the f-electron Green function. We postulate according to the variational principle that J2{Gc} be stationary against variation of G¢ around its optimum choice. This condition leads to the following equation:
,
(7)
where the bracket in (7) means the cumulant average with respect to the trial distribution specified by ~tr" SO far no approximations have been introduced. Fig. 1 shows Feynman diagrams for J~int" The lines with an arrow represent
t(ien)A(k,"
=1, (9)
V2F(ien)).
One can use one of a few theoretical methods to solve the effective single-site problem, such as the field theoretical method [8, 9], the quantum Monte Carlo simulation [10], and the resolvent method using the non-crossing approximation (NCA) [4, 7]. The present theory is flexible in that (9) is valid for any method to derive F(ien). If one uses the NCA, then (9) becomes identical
82
Y. Kuramoto and T. Watanabe / Magnetic response in heavy-fermion systems
to the relation (eq. (4.3) in [7]) between F(ien) and (~c(ien) of the extended N C A (XNCA). If time- and space-dependent magnetic field hi(r ) is included in the starting Lagrangian, the same procedure gives a generalization of ,0 to a non-equilibrium case. The magnetic response within the X N C A is derived by considering a change of resolvents against infinitesimal magnetic field. The stationary principle for ~ sets a condition for the functional derivative 8GffgF. Leaving the details for a separate paper, we quote here only the results in the case of infinite U in (2). The susceptibility x ( q , iVm) with even Matsubara frequency iv m is given by x ( q , iVm) = 2/*2 •
lfdz ~
e-t3~S(q, z + iVm, Z),
c
(10)
configurations, respectively. It can easily be seen that w i t h o u t / / t h e susceptibility is reduced to the single-site expression Xl(iVm). It seems hard to reach (11) and (12), whose intricacy is the price to pay for achieving the consistency with thermodynamic properties, via a straightforward diagrammatic analysis. The results (10)-(13) take account of both spin and charge fluctuations. We shall show that the results reproduce the standard R K K Y interation in the case where charge fluctuations and the Kondo effect can be neglected. In this case the resolvent R 0 in (13) is approximated by 1/ef. Then we obtain the closed-form solution x ( q , iv,,,) = [xl(iVm) +
J2T ~ n
where ~B is the Bohr magneton and the contour C runs counterclockwise to enclose all singularities of the integrand. To write out the integral equation for S(q, z + iv m, z) we need to introduce the notations 1 P(q, z, z') : ~ ~ [A(p,
Z) -1 -- /(Z)] -1
×[A(p+q,z')
'-t(z')]
1 (11)
El(q, z, z') : [e(q, z, z') -1 + t(z)t(z')l 1 . (12) Then the integral equation reads S( q, z + iVm, Z) = RI(Z + ivm)R~(z)
× [ 1 - VaT ~. Ro(z + i v m - i e , ) H ( q , iu m,ien) tl
× zffl f 21vidz'e _ ~ , R o ( z , _ ie,)
1
II(q, ie. + iv m, ie~)
, (14)
with J = V2/gf. If one further neglects t in H, (14) reduces to the mean-field result with the R K K Y interaction. In the full result (13), the Kondo effect influences not only X1 but the intersite interaction. We note that (14) does not hold in the presence of the Kondo effect, since the approximation VZRo = J is then not justified even though charge fluctuations are negligible. Since 12 in (6) with the N C A for Zf is stationary against variation of either (7c or resolvents, the theory is "conserving" in the terminology of [11]. As a result the static and homogeneous limit of the dynamical susceptibility agrees with the second thermodynamic derivative of ~2 with respect to the magnetic field. In summary, we have given a fully self-consistent microscopic theory for the dynamical susceptibility. Since the theory can derive singleparticle spectrum within the same framework [12[, it seems especially useful for unified understanding of heavy-fermion systems. Numerical results for x ( q , oJ) will be presented in a future publication.
C
× S(q, z' + i~' m, Z')]
.
(13)
where R1 and R o are the resolvents for fl and fo
Acknowledgement This work is supported by a Grant-in-Aid for
Y. Kuramoto and T. Watanabe / Magnetic response in heavy-fermion systems
Scientific R e s e a r c h f r o m t h e M i n i s t r y o f E d u c a tion, Science and Culture of Japan.
References [1] G. Aeppli et al., Phys. Rev. Lett. 57 (1986) 122. L.P. Regnault et al., J. Magn. Magn. Mat. 63 & 64 (1987) 289. [2] G. Aeppli et al., Phys. Rev. Lett. 58 (1987) 808. [3] Y. Kuramoto, Z. Phys. B 40 (1981) 293. [4] N. Grewe and T. Pruschke, Z. Phys. B 60 (1985) 311. N. Grewe, Z. Phys. B 67 (1987) 323. [5] Y. Kuramoto, Solid State Commun. 63 (1987) 467.
83
[6] C. Itzykson and J.-B. Zuber, Quantum Field Theory, (McGraw-Hill, New York, 1980), p. 425. [7] Y. Kuramoto, in: Theory of Heavy-Fermions and Valence Fluctuations, T. Kasuya and T. Saso, eds. (Springer, Berlin, 1985), p. 152. [8] N. Read and D.M. Newns, J. Phys. C 16 (1983) 3273. P. Coleman, in ref. [7], p. 163. [9] T. Koyama and M. Tachiki, Phys. Rev. B 34 (1986) 3272. [10] J. Gubernatis, J.E. Hirsch and D.J. Scalapino, Phys. Rev. B 35 (1987) 8478. [11] G. Baym, Phys. Rev. 127 (1962) 1391. [12] C.-I. Kim, Y. Kuramoto and T. Kasuya, Solid State Commun. 62 (1987) 627. Y. Kuramoto, C.-I. Kim and T. Kasuya, Jpn. J. Appl. Phys. Suppl. 26-3 (1987) 459.
T H E O R Y OF M O M E N T U M - D E P E N D E N T M A G N E T I C RESPONSE IN HEAVY-FERMION SYSTEMS Y. K U R A M O T O
and T. W A T A N A B E
Departmentof AppliedPhysics,Tohoku University,Sendai980,Japan Received 3 August 1987 A self-consistent dynamical theory is presented for the Anderson lattice. In a variational Lagrangian formulation a trial conduction-electron propagator is introduced. A variational principle leads to optimization of the propagator with the accuracy of U(I/zn) where z., is the number of interacting neighbors. The extended non-crossing approximation (XNCA) is reproduced as a special case of the present theory. An integral equation is obtained for the dynamical susceptibility which incorporates the RKKY interaction and the Kondo effect. The closed-form solution is given in a case where perturbation theory is applicable.
T h e dynamical m a g n e t i c response of heavyf e r m i o n systems reflects b o t h on-site and intersite correlations of f electrons. A t t e m p e r a t u r e T higher than the K o n d o t e m p e r a t u r e T K the dynamical susceptibility x ( q , o)) p r o b e d by neutron scattering and N M R shows characteristics very similar to those in dilute impurity systems. A t T < T K, h o w e v e r , significant intersite interactions have b e e n o b s e r v e d in some h e a v y - f e r m i o n systems such as C e C u 6 [1] and U P t 3 [2]. T h e interactions are frequently of a n t i f e r r o m a g n e t i c type. Theoretical study of x ( q , o2) is highly desired in o r d e r to u n d e r s t a n d the nature of intersite interactions. T h e enterprise is a hard o n e since o n e must treat the crossover f r o m the single-site b e h a v i o r to the b a n d - t y p e one taking a c c o u n t of the K o n d o effect. P e r t u r b a t i o n a l theories [3, 4] have not so far succeeded in achieving a self-consistency of the f r a m e w o r k . In a recent p a p e r [5] one of the present authors has presented a phenomenological analysis of x ( q , w) on the basis of the Mori formalism and the Fermi-liquid theory. A puzzling b e h a v i o r f o u n d in C e C u 6 [1] has b e e n explained in terms of nearly localized quasiparticles. It is of course desirable to have m o r e microscopic f o u n d a t i o n for the analysis. T h e p u r p o s e of the present p a p e r is to give a brief a c c o u n t of a first-principle and self-consistent t h e o r y for x ( q , w). We consider the A n d e r s o n - l a t t i c e m o d e l with 0378-4363/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) and Yamada Science Foundation
twofold spin degeneracy. Extension to a m o r e general m o d e l is straightforward. T h e (grand) partition function Z of the N-site system at t e m p e r a t u r e T = / 3 1 is expressed as a functional integral over G r a s s m a n n n u m b e r s [6] which corr e s p o n d to f e r m i o n fields. T h e conduction-electron part is easily integrated out. T h e remaining part is given by
Z/Zc= f DffDf exp{- f~dr + 2ehy ( )] } , ,
0
+ U 2 nit (')ni+ i
~hyb(T) = V2 Z
(~'),
(2)
dT' f ~ ( T )
l]cr
x
(1)
-
(3)
H e r e Z c is the partition function of c o n d u c t i o n + electrons without hybridization, fio- represents creation of an f electron with spin o- at site i, and g q ( ~ - - ~-') is the bare p r o p a g a t o r of c o n d u c t i o n electrons in the site representation. F o r simplicity we assume that the hybridization matrix elem e n t V does not d e p e n d on m o m e n t u m of con-
81
Y. Kuramoto and T. Watanabe / Magnetic response in heavy-fermion systems
duction electrons. The last term in (2) represents the Coulomb repulsion between f electrons at the same site. Our strategy to account for intersite interactions is in line with the mean-field theory; the effect of environment around a given f-electron site is represented by an effective field. In the present case the effective field is a site-diagonal trial propagator G ~ ( r - r ' ) of conduction electrons. The convenience of using the functional integral is that we can introduce a trial Lagrangian ~tr which, in contrast to the Hamiltonian formalism, can describe a retardation effect. Thus we make a new decomposition ~e + ~'hyb = ~t~ + ~i,t with /3
0
(4)
f[~(r)dc(Z - r')fi~(z') , 13
~i,,t ("r) = V 2 E,,..,,f dr' - a,jdc(
f~,~(r)[gij('r
- r')
(5)
- -
and consider a perturbation expansion in terms
(0) O(I/Z n)
0+0+©+ (b) O ( I / z 2) +
~
+
...
Fig. 1. Diagrams f o r / ~ , in the loop-wise expansion.
Aq = g # - 8#G c and the dots are cumulant averages at each site. We rearrange the perturbation series in/~int to ascending powers of 1/zn where z n is the number of interacting neighbors around a given site. For each loop made up of Aq we associate a factor 1/Zn [7]. This rearrangement corresponds to the loop-wise expansion in the field theory [6]. Thus in the leading order of 1 / z , we collect all sequence of diagrams in fig. l(a) which involve the site-diagonal Green function of f electrons as the two-point cumulant. There is no restriction on the site summation because one is dealing with cumulant averages. With Fourier transform to momentum and Matsubara frequency space, A/j changes into A(k, ien) where en = (2n + 1)~rT with n integer. We obtain g2~nt = ~ T ~ In[1 k n
VZF(ien)A(k, it,,)],
(8)
of
Since ~tr is diagonal with respect to site indices, the partition function without ~int is derived from that of the effective single-site system Zf. The thermodynamic potential/-2 of the whole system is then given as a functional of Gc by -fl/2{Gc} = In Z c + N l n Zf(Gc} - ~ i n t { S c )
,
(6)
d'r ~int(1")])cum - 1 0
1 ~] [1 N k
_
_
(t(ien) =
t~ -jS~(-]int{Gc} = ( e x p [ - f
where F(ien) is the site-diagonal part of the f-electron Green function. We postulate according to the variational principle that J2{Gc} be stationary against variation of G¢ around its optimum choice. This condition leads to the following equation:
,
(7)
where the bracket in (7) means the cumulant average with respect to the trial distribution specified by ~tr" SO far no approximations have been introduced. Fig. 1 shows Feynman diagrams for J~int" The lines with an arrow represent
t(ien)A(k,"
=1, (9)
V2F(ien)).
One can use one of a few theoretical methods to solve the effective single-site problem, such as the field theoretical method [8, 9], the quantum Monte Carlo simulation [10], and the resolvent method using the non-crossing approximation (NCA) [4, 7]. The present theory is flexible in that (9) is valid for any method to derive F(ien). If one uses the NCA, then (9) becomes identical
82
Y. Kuramoto and T. Watanabe / Magnetic response in heavy-fermion systems
to the relation (eq. (4.3) in [7]) between F(ien) and (~c(ien) of the extended N C A (XNCA). If time- and space-dependent magnetic field hi(r ) is included in the starting Lagrangian, the same procedure gives a generalization of ,0 to a non-equilibrium case. The magnetic response within the X N C A is derived by considering a change of resolvents against infinitesimal magnetic field. The stationary principle for ~ sets a condition for the functional derivative 8GffgF. Leaving the details for a separate paper, we quote here only the results in the case of infinite U in (2). The susceptibility x ( q , iVm) with even Matsubara frequency iv m is given by x ( q , iVm) = 2/*2 •
lfdz ~
e-t3~S(q, z + iVm, Z),
c
(10)
configurations, respectively. It can easily be seen that w i t h o u t / / t h e susceptibility is reduced to the single-site expression Xl(iVm). It seems hard to reach (11) and (12), whose intricacy is the price to pay for achieving the consistency with thermodynamic properties, via a straightforward diagrammatic analysis. The results (10)-(13) take account of both spin and charge fluctuations. We shall show that the results reproduce the standard R K K Y interation in the case where charge fluctuations and the Kondo effect can be neglected. In this case the resolvent R 0 in (13) is approximated by 1/ef. Then we obtain the closed-form solution x ( q , iv,,,) = [xl(iVm) +
J2T ~ n
where ~B is the Bohr magneton and the contour C runs counterclockwise to enclose all singularities of the integrand. To write out the integral equation for S(q, z + iv m, z) we need to introduce the notations 1 P(q, z, z') : ~ ~ [A(p,
Z) -1 -- /(Z)] -1
×[A(p+q,z')
'-t(z')]
1 (11)
El(q, z, z') : [e(q, z, z') -1 + t(z)t(z')l 1 . (12) Then the integral equation reads S( q, z + iVm, Z) = RI(Z + ivm)R~(z)
× [ 1 - VaT ~. Ro(z + i v m - i e , ) H ( q , iu m,ien) tl
× zffl f 21vidz'e _ ~ , R o ( z , _ ie,)
1
II(q, ie. + iv m, ie~)
, (14)
with J = V2/gf. If one further neglects t in H, (14) reduces to the mean-field result with the R K K Y interaction. In the full result (13), the Kondo effect influences not only X1 but the intersite interaction. We note that (14) does not hold in the presence of the Kondo effect, since the approximation VZRo = J is then not justified even though charge fluctuations are negligible. Since 12 in (6) with the N C A for Zf is stationary against variation of either (7c or resolvents, the theory is "conserving" in the terminology of [11]. As a result the static and homogeneous limit of the dynamical susceptibility agrees with the second thermodynamic derivative of ~2 with respect to the magnetic field. In summary, we have given a fully self-consistent microscopic theory for the dynamical susceptibility. Since the theory can derive singleparticle spectrum within the same framework [12[, it seems especially useful for unified understanding of heavy-fermion systems. Numerical results for x ( q , oJ) will be presented in a future publication.
C
× S(q, z' + i~' m, Z')]
.
(13)
where R1 and R o are the resolvents for fl and fo
Acknowledgement This work is supported by a Grant-in-Aid for
Y. Kuramoto and T. Watanabe / Magnetic response in heavy-fermion systems
Scientific R e s e a r c h f r o m t h e M i n i s t r y o f E d u c a tion, Science and Culture of Japan.
References [1] G. Aeppli et al., Phys. Rev. Lett. 57 (1986) 122. L.P. Regnault et al., J. Magn. Magn. Mat. 63 & 64 (1987) 289. [2] G. Aeppli et al., Phys. Rev. Lett. 58 (1987) 808. [3] Y. Kuramoto, Z. Phys. B 40 (1981) 293. [4] N. Grewe and T. Pruschke, Z. Phys. B 60 (1985) 311. N. Grewe, Z. Phys. B 67 (1987) 323. [5] Y. Kuramoto, Solid State Commun. 63 (1987) 467.
83
[6] C. Itzykson and J.-B. Zuber, Quantum Field Theory, (McGraw-Hill, New York, 1980), p. 425. [7] Y. Kuramoto, in: Theory of Heavy-Fermions and Valence Fluctuations, T. Kasuya and T. Saso, eds. (Springer, Berlin, 1985), p. 152. [8] N. Read and D.M. Newns, J. Phys. C 16 (1983) 3273. P. Coleman, in ref. [7], p. 163. [9] T. Koyama and M. Tachiki, Phys. Rev. B 34 (1986) 3272. [10] J. Gubernatis, J.E. Hirsch and D.J. Scalapino, Phys. Rev. B 35 (1987) 8478. [11] G. Baym, Phys. Rev. 127 (1962) 1391. [12] C.-I. Kim, Y. Kuramoto and T. Kasuya, Solid State Commun. 62 (1987) 627. Y. Kuramoto, C.-I. Kim and T. Kasuya, Jpn. J. Appl. Phys. Suppl. 26-3 (1987) 459.