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Kondo lattice model a perspective of heavy electron phenomena
ELSEVIER
Physica B 230-232 (1997) 22-26
Kondo lattice model A perspective of heavy electron phenomena Kazuo Ueda Institute for Solid State Physics, University of Tokyo, Roppongi, Tokyo, 106 Japan
Abstract Recently, there has been great progress in the theoretical understanding of the Kondo lattice model which is a basic model for the heavy electron phenomena. The ground-state phase diagram of the Kondo lattice model has been completed in one dimension, where there are three different phases: the ferromagnetic metallic phase, the paramagnetic metallic one and the insulating spin liquid phase. The last one is a theoretical prototype for the Kondo insulators. The ferromagnetic metallic phase is an example of magnetic phases with reduced moment. The paramagnetic metallic phase is the region where we expect the typical heavy electron phenomena. In one dimension it belongs to the universality class of the Tomonaga-Luttinger liquids. From the period of the Friedel oscillations it is concluded that kv is determined by the density of conduction electrons and the f-spins. Keywords: Heavy electron system; Heavy fermion magnetism; Kondo insulators; Tomonaga-Luttinger liquid
I. Introduction Heavy fermion systems have been a source of interesting m a n y - b o d y phenomena. Heavy fermions are defined by the large effective mass of quasiparticles formed below a low-temperature energy scale, which is typically of the order of ten Kelvin. The effective mass ranges from several hundred to a thousand times bigger than the bare electron mass. The enormous mass is a consequence of renormalization of the quasiparticles due to the strong correlations. An intriguing feature of heavy fermions shared with the other strongly correlated systems is diversity of ground states. Its list includes the normal heavy fermion state, the magnetically ordered states with tiny moment, and the unusual superconducting states. Recently, new items like the K o n d o insulators and the low-carrierdensity K o n d o systems have been appended in the list.
One of the key questions in the heavy fermion physics is which control parameter determines the ground-state phase out of different possibilities. For this problem the first step of theoretical study is obviously to determine the ground-state phase diagram of the models for heavy fermions. The K o n d o lattice model is a standard model for the heavy fermions and the simplest one a m o n g the models which include the two degrees of freedom of the localized f-orbitals and the extended conduction electrons. It reads as
(ij)
s
i s,s'
where ~ = (a~, ay, az) are the Pauli matrices and St = ~ , , , ~as~,fzsf~s,1 t is the f-electron spin at site i. In this article we will review recent progress in theoretical understanding of the K o n d o lattice model. In particular, the ground-state phase diagram has been completed for the K o n d o lattice
0921-4526/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PII S092 1-4526(96)0053 5-2
K. Ueda /Physica B 230-232 (1997) 22 26
model in one dimension. It consists of three phases: a ferromagnetic metallic phase, a paramagnetic metallic phase, and an insulating spin liquid phase. They represent theoretical examples for a magnetically ordered phase, a normal heavy fermion state, and a Kondo insulator, respectively. Thus, detailed discussions are given for properties of each phase. Once the lattice structure is fixed the Kondo lattice model has only two parameters. One is the exchange coupling in units of the hopping matrix element (J/t) or the bandwidth and the other is the density of the conduction electrons no. Fig. 1 shows a schematic ground-state phase diagram of the Kondo lattice model in one dimension. The boundary was obtained by exact diagonalization study Eli.
2. Ferromagnetic metallic phase Concerning the low-carrier-density limit a rigorous theorem of the ferromagnetic ground state in the case of single electron has been proven independent of dimensionality [-2]. Existence of the finite ferromagnetic region in the parameter space of J and nc has been established in one dimension by the second theorem: the spin degeneracy of the one-dimensional Kondo lattice model at J/t = ~ is lifted in a perturbative sense towards a spin-polarized state for any electron density [-3]. Among heavy fermion materials there is a group of compounds with tiny magnetic moments. Typi-
(30
[
J/t
Insulating ,/4 Kondo spin liquid
Paramagnetic Metallic 0 0
Flc
1
Fig. 1. The ground-state phase diagram of the one-dimensional Kondo lattice model
23
cal examples are antiferromagnetically ordered UPt3 and URu2Si2. The ferromagnetic metallic phase is a theoretical example of magnetic orderings in one dimension, since the ferromagnetism is the only possible long-range order in one dimension. The strong coupling limit is the fixed point of this phase, where conduction electrons are combined into singlets with localized spins. Away from half-filling, there remain unpaired localized spins. The ferromagnetic state is realized so that these singlets or, equivalently, the unpaired localized spins can gain kinetic energy maximally, leading to the ferromagnetic alignment of the unpaired spins. Naturally, the magnetization is reduced by the formation of Kondo singlets. In particular, magnetization vanishes as approaching half-filling. Cerium monopnictites CeX (X: P, As, Sb, Bi) are typical examples of low-carrier-density Kondo systems [4]. The carriers consist of electrons and holes whose concentrations are only a few percent. Considering the smallness of number of carriers the 7-term of the electronic specific heat is relatively large, meaning a large mass enhancement for the carriers. In ordinary heavy fermions, anomalous behaviors are brought about through the process of screening localized moments by conduction electrons. An interesting question here is the origin of the heavy mass in the low-carrierdensity Kondo systems. Magnetic properties of CeX is also very intriguing and show series of transitions between different magnetic structures [-5 7]. Among the cerium monopnictites, CeSb is known to have the richest magnetic phase diagram ever observed. The key to understand the complex phase diagram is the existence of two types of carriers; electrons around the X point and holes around the P point. In this situation a new type of exchange interaction with f spins becomes possible in connection with transitions between electrons and holes, which may be called an off-diagonal exchange. This exchange favors an antiferromagnetic ordering since the momentum transfer is (0, 0, ~) typically. Thus, it causes frustration with the usual Kondo coupling which favors ferromagnetism in the case of low carrier density. This frustration is the origin of the complex magnetic phase diagram in CeX [8].
24
K. Ueda / Physica B 230-232 (1997) 22-26
3. Kondo spin liquid phase At half-filling the ground state is the Kondo spin liquid phase which is a theoretical prototype of the Kondo insulators in one dimension. The strong coupling limit is again a good starting point to understand the spin liquid phase. In this limit every localized spin forms a singlet with a conduction electron without overlapping with neighboring singlets. In this localized limit it is easy to see that the energy cost to create a triplet excitation is J. To create a charge excitation it is necessary to move an electron from one site to a neighboring site, forming an empty site and a doubly occupied site. On both the empty and doubly occupied sites the singlet formation energy 3 j is lost, leading to the charge gap of 3j. Therefore, the charge gap is 50% larger than the spin gap in the strong coupling limit. An important and interesting question arises in the weak coupling region. Kondo screening occurs in the energy scale of the Kondo temperature which is exponentially small in the weak coupling regime. On the other hand, magnetic ordering is expected due to the RKKY interaction whose energy scale is JZ/ev. At half-filling it is shown rigorously that the spin structure factor has the maximum at the antiferromagnetic wave vector [9]. Thus, in two or three dimension it is natural to expect a quantum phase transition between the antiferromagnetic state and the Kondo spin liquid phase. In one dimension it is not possible to have the antiferromagnetic long-range order due to strong quantum fluctuations. Thus, the question in one dimension is whether the excitations are gapful or gapless. If they are gapless the correlation functions decay only algebraically, leading to a quasi-long-range order. By using a finite-size scaling for the results obtained by the numerical exact diagonalization, it has been shown that in one dimension there is no critical J which separates the spin liquid phase from a state with a quasi-long-range order [10]. Similarly, there is no critical U for the periodic Anderson model [11]. For the Kondo spin liquid phase the magnitude of the spin gap defines the lowest energy scale of the system. Thus, it is important to determine the
functional form of the spin gap. Recent development of the density matrix renormalization group method introduced by White [12] has enabled us to study much larger systems compared to the exact diagonalization [13]. In the impurity Kondo problem all low-energy properties are scaled by the single energy scale, i.e. Kondo temperature TK = evexp(- 1~pJ) where p is the density of conduction electrons at the Fermi energy. A naive extension to the lattice problem is to include an enhancement of the exponent due to intersite correlations. Thus, the spin gap is expected to behave as
where e is the enhancement factor. The Gutzwiller approximation predicts an enhancement factor of two [14]. From the results by the DMRG method c~= 1.4(1) is obtained. Concerning the charge gap the DMRG calculation confirms that in the weak coupling limit it approaches to Ao = ½J
(3)
which is the result obtained by a simple mean field theory with antiferromagnetic long-range order. Of course, in one dimension there is no antiferromagnetic long-range order. The point is that the spin-spin correlation length is much longer than the charge--charge correlation length. Therefore, for the discussion of the charge gap we can treat the antiferromagnetic short-range order as if it is a long-range one. From these results it is concluded that the charge gap is larger than the spin gap and the ratio between the two diverges in the weak coupling limit of J. It means that for the periodic Anderson model the ratio diverges in the limit of strong Coulomb interaction under the symmetric condition. The charge gap may be best estimated from optical conductivity, while the spin gap by neutron inelastic scatterings. In Ce3Bi4Pt3 the ratio is estimated to be about 2 [15]. Guerrero and Yu [16] suggested that the not particularly large value should not be attributed to weak correlation but rather be considered as a consequence of the asymmetry.
K. Ueda / Physica B 230-232 (1997) 22 26
4. Paramagnetic metallic phase In the one-dimensional Kondo lattice model the paramagnetic metallic phase exists away from halffilling in the region where the exchange coupling is weak or intermediate (Fig. 1). Clearly related to the heavy fermion states, this paramagnetic phase is very interesting. However, compared with the other phases studied in the previous sections this phase is less understood. Concerning the difficulty we should note first that the weak coupling limit is not a trivial limit, because in this limit there remains macroscopic degeneracy 2L where L is the number of sites (f spins). Secondly, although the ground state at half-filling is well understood it is a highly non-trivial problem when it is doped with holes or electrons similar to the high-T~ problem. Finally, one cannot go to the strong coupling limit since there is a phase transition to the ferromagnetic metallic phase. In one dimension it has been established that most interacting metallic systems belong to the universality class of Tomonaga-Luttinger liquids [17]. The asymptotic forms of charge- and spincorrelation functions are (n(x) n(O)) = Kp/(rcx) 2 + Alcos(2kvx) x -1-K" + A2 COS(4kFX)X -41q', ( S ( x ) . S ( O ) ) = 1/(rex) 2 + Blcos(2kFX ) x - l - K p ,
(4) (5)
where kv = ½rtn, is the Fermi momentum and with n being the density of charge carriers, Kp the correlation exponent [18]. The anomalous power law decays of the correlation functions naturally reflect themselves in the Friedel oscillations: the asymptotic form of the charge density oscillations induced by an impurity potential is 6p(x) ~ C1 COS(2kFx) x {- 1 - K,,)/2
+ C2 COS(4kvx) x - 2K°
(6)
as a function of the distance x from the impurity [-19,20]. Analogously, the spin density oscillations induced by a local magnetic field behave as a(x) ~ DI cos(2kvx) x -K..
(7)
25
It may be natural to expect that the paramagnetic metallic state of the one-dimensional Kondo lattice model also belongs to the class of Luttinger liquids. However, in this case the position of the Fermi points is already a nontrivial problem since the model contains two different types of electrons, the conduction electrons and the localized spins. Concerning the size of the Fermi volume two different viewpoints are possible. If the interaction between the conduction electrons and the localized spins is strictly zero, it is clear that the singularity of the momentum distribution function of the conduction electrons is determined only by the number of conduction electrons. Thus, if the singular points are not affected by the exchange interaction, kF = rtnc/2 would be expected. On the other hand, a different answer is obtained by identifying the Kondo lattice model as an effective model for the periodic Anderson model. In the periodic Anderson model the conduction electrons and the f-electrons are mixed with each other through the hybridization matrix elements. Therefore, it is natural to expect that kv is determined by the total density of both the conduction electrons and the f-electrons; kv = ½rt(l + no). Recently, we have calculated the Friedel oscillations of the one-dimensional Kondo lattice model by using the DMRG [21]. The charge density oscillations are induced naturally by open boundary conditions, and the spin density oscillations are introduced by applying local magnetic fields at the both ends. From the period of the Friedel oscillations we can determine the Fermi momentum. It is clearly seen that the Friedel oscillations are compatible with the large Fermi surface kv = ½rt(1 + nc) in spite of the fact that the charge degrees of freedom are completely suppressed for the f-electrons in the Kondo lattice model. These results are consistent with the previous work on the t-t' Kondo lattice model for which it was shown exactly that its strong coupling limit is described by a Luttinger liquid with the large Fermi surface [-22]. Therefore, it is natural to conclude that the paramagnetic metallic phase of the Kondo lattice model has a large Fermi surface in general.
26
K. Ueda /Physica B 230-232 (1997) 22 26
References [1] H. Tsunetsugu, M. Sigrist and K. Ueda, Phys. Rev. B 47 (1993) 8345. [2] M. Sigrist, H. Tsunetsugu and K. Ueda, Phys. Rev. Lett. 67 (1991) 2211. [3] M. Sigrist, K. Ueda and H. Tsunetsugu, Phys. Rev. B 46 (1992) 175. [4] T. Suzuki, Physica B 186-188 (1993) 347. [5] J. Rossat-Mignod et al., J. Magn. Magn. Mater. 31-34 (1983) 398. [6] J. Rossat-Mignod et al., J. Magn. Magn. Mater. 52 (1985) 111. [7] M. Kohgi et al., Phys. Rev. B 49 (1994) 7068. [8] N. Shibata, C. Ishii and K. Ueda, Phys. Rev. B 52 (1995) 10232. [9] G.-S. Tian, Phys. Rev. B 50 (1994) 6246. [10] H. Tsunetsugu, Y. Hatsugai, K. Ueda and M. Sigrist, Phys. Rev. B 46 (1992) 3175.
[11] T. Nishino and K. Ueda, Phys. Rev. B 47 (1993) 12451. [12] S.R. White, Phys. Rev. Lett. 69 (1992) 2863. [13] N. Shibata, T. Nishino, K. Ueda and C. Ishii, Phys. Rev. B 53 (1996) 8828. [14] T.M. Rice and K. Ueda, Phys. Rev. Lett. 55 (1985) 995; Phys. Rev. B 34 (1986) 6420. [15] B. Bucher, Z. Schlesinger, P.C. Canfield and Z. Fisk, Phys. Rev. Lett. 72 (1994) 522. 1-16] M. Guerrero and C.C. Yu, Phys. Rev. B 51 (1995) 10301. [17] F.D.M. Haldane, J. Phys. C 14 (1981) 2585. [18] HJ. Schulz, Phys. Rev. Lett. 64 (1990) 2831. [19] M. Fabrizio and A.O. Gogolin, Phys. Rev. B 51 (1995) 17827. [20] R. Egger and H. Grabert, Phys. Rev. Lett. 75 (1995) 3505. [21] N. Shibata, K. Ueda, T. Nishino and C. Ishii, Phys. Rev. B 54 (1996) 13495. [22] K. Ueda, T. Nishino and H. Tsunetsugu, Phys. Rev. B 50 (1994) 612.
Physica B 230-232 (1997) 22-26
Kondo lattice model A perspective of heavy electron phenomena Kazuo Ueda Institute for Solid State Physics, University of Tokyo, Roppongi, Tokyo, 106 Japan
Abstract Recently, there has been great progress in the theoretical understanding of the Kondo lattice model which is a basic model for the heavy electron phenomena. The ground-state phase diagram of the Kondo lattice model has been completed in one dimension, where there are three different phases: the ferromagnetic metallic phase, the paramagnetic metallic one and the insulating spin liquid phase. The last one is a theoretical prototype for the Kondo insulators. The ferromagnetic metallic phase is an example of magnetic phases with reduced moment. The paramagnetic metallic phase is the region where we expect the typical heavy electron phenomena. In one dimension it belongs to the universality class of the Tomonaga-Luttinger liquids. From the period of the Friedel oscillations it is concluded that kv is determined by the density of conduction electrons and the f-spins. Keywords: Heavy electron system; Heavy fermion magnetism; Kondo insulators; Tomonaga-Luttinger liquid
I. Introduction Heavy fermion systems have been a source of interesting m a n y - b o d y phenomena. Heavy fermions are defined by the large effective mass of quasiparticles formed below a low-temperature energy scale, which is typically of the order of ten Kelvin. The effective mass ranges from several hundred to a thousand times bigger than the bare electron mass. The enormous mass is a consequence of renormalization of the quasiparticles due to the strong correlations. An intriguing feature of heavy fermions shared with the other strongly correlated systems is diversity of ground states. Its list includes the normal heavy fermion state, the magnetically ordered states with tiny moment, and the unusual superconducting states. Recently, new items like the K o n d o insulators and the low-carrierdensity K o n d o systems have been appended in the list.
One of the key questions in the heavy fermion physics is which control parameter determines the ground-state phase out of different possibilities. For this problem the first step of theoretical study is obviously to determine the ground-state phase diagram of the models for heavy fermions. The K o n d o lattice model is a standard model for the heavy fermions and the simplest one a m o n g the models which include the two degrees of freedom of the localized f-orbitals and the extended conduction electrons. It reads as
(ij)
s
i s,s'
where ~ = (a~, ay, az) are the Pauli matrices and St = ~ , , , ~as~,fzsf~s,1 t is the f-electron spin at site i. In this article we will review recent progress in theoretical understanding of the K o n d o lattice model. In particular, the ground-state phase diagram has been completed for the K o n d o lattice
0921-4526/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PII S092 1-4526(96)0053 5-2
K. Ueda /Physica B 230-232 (1997) 22 26
model in one dimension. It consists of three phases: a ferromagnetic metallic phase, a paramagnetic metallic phase, and an insulating spin liquid phase. They represent theoretical examples for a magnetically ordered phase, a normal heavy fermion state, and a Kondo insulator, respectively. Thus, detailed discussions are given for properties of each phase. Once the lattice structure is fixed the Kondo lattice model has only two parameters. One is the exchange coupling in units of the hopping matrix element (J/t) or the bandwidth and the other is the density of the conduction electrons no. Fig. 1 shows a schematic ground-state phase diagram of the Kondo lattice model in one dimension. The boundary was obtained by exact diagonalization study Eli.
2. Ferromagnetic metallic phase Concerning the low-carrier-density limit a rigorous theorem of the ferromagnetic ground state in the case of single electron has been proven independent of dimensionality [-2]. Existence of the finite ferromagnetic region in the parameter space of J and nc has been established in one dimension by the second theorem: the spin degeneracy of the one-dimensional Kondo lattice model at J/t = ~ is lifted in a perturbative sense towards a spin-polarized state for any electron density [-3]. Among heavy fermion materials there is a group of compounds with tiny magnetic moments. Typi-
(30
[
J/t
Insulating ,/4 Kondo spin liquid
Paramagnetic Metallic 0 0
Flc
1
Fig. 1. The ground-state phase diagram of the one-dimensional Kondo lattice model
23
cal examples are antiferromagnetically ordered UPt3 and URu2Si2. The ferromagnetic metallic phase is a theoretical example of magnetic orderings in one dimension, since the ferromagnetism is the only possible long-range order in one dimension. The strong coupling limit is the fixed point of this phase, where conduction electrons are combined into singlets with localized spins. Away from half-filling, there remain unpaired localized spins. The ferromagnetic state is realized so that these singlets or, equivalently, the unpaired localized spins can gain kinetic energy maximally, leading to the ferromagnetic alignment of the unpaired spins. Naturally, the magnetization is reduced by the formation of Kondo singlets. In particular, magnetization vanishes as approaching half-filling. Cerium monopnictites CeX (X: P, As, Sb, Bi) are typical examples of low-carrier-density Kondo systems [4]. The carriers consist of electrons and holes whose concentrations are only a few percent. Considering the smallness of number of carriers the 7-term of the electronic specific heat is relatively large, meaning a large mass enhancement for the carriers. In ordinary heavy fermions, anomalous behaviors are brought about through the process of screening localized moments by conduction electrons. An interesting question here is the origin of the heavy mass in the low-carrierdensity Kondo systems. Magnetic properties of CeX is also very intriguing and show series of transitions between different magnetic structures [-5 7]. Among the cerium monopnictites, CeSb is known to have the richest magnetic phase diagram ever observed. The key to understand the complex phase diagram is the existence of two types of carriers; electrons around the X point and holes around the P point. In this situation a new type of exchange interaction with f spins becomes possible in connection with transitions between electrons and holes, which may be called an off-diagonal exchange. This exchange favors an antiferromagnetic ordering since the momentum transfer is (0, 0, ~) typically. Thus, it causes frustration with the usual Kondo coupling which favors ferromagnetism in the case of low carrier density. This frustration is the origin of the complex magnetic phase diagram in CeX [8].
24
K. Ueda / Physica B 230-232 (1997) 22-26
3. Kondo spin liquid phase At half-filling the ground state is the Kondo spin liquid phase which is a theoretical prototype of the Kondo insulators in one dimension. The strong coupling limit is again a good starting point to understand the spin liquid phase. In this limit every localized spin forms a singlet with a conduction electron without overlapping with neighboring singlets. In this localized limit it is easy to see that the energy cost to create a triplet excitation is J. To create a charge excitation it is necessary to move an electron from one site to a neighboring site, forming an empty site and a doubly occupied site. On both the empty and doubly occupied sites the singlet formation energy 3 j is lost, leading to the charge gap of 3j. Therefore, the charge gap is 50% larger than the spin gap in the strong coupling limit. An important and interesting question arises in the weak coupling region. Kondo screening occurs in the energy scale of the Kondo temperature which is exponentially small in the weak coupling regime. On the other hand, magnetic ordering is expected due to the RKKY interaction whose energy scale is JZ/ev. At half-filling it is shown rigorously that the spin structure factor has the maximum at the antiferromagnetic wave vector [9]. Thus, in two or three dimension it is natural to expect a quantum phase transition between the antiferromagnetic state and the Kondo spin liquid phase. In one dimension it is not possible to have the antiferromagnetic long-range order due to strong quantum fluctuations. Thus, the question in one dimension is whether the excitations are gapful or gapless. If they are gapless the correlation functions decay only algebraically, leading to a quasi-long-range order. By using a finite-size scaling for the results obtained by the numerical exact diagonalization, it has been shown that in one dimension there is no critical J which separates the spin liquid phase from a state with a quasi-long-range order [10]. Similarly, there is no critical U for the periodic Anderson model [11]. For the Kondo spin liquid phase the magnitude of the spin gap defines the lowest energy scale of the system. Thus, it is important to determine the
functional form of the spin gap. Recent development of the density matrix renormalization group method introduced by White [12] has enabled us to study much larger systems compared to the exact diagonalization [13]. In the impurity Kondo problem all low-energy properties are scaled by the single energy scale, i.e. Kondo temperature TK = evexp(- 1~pJ) where p is the density of conduction electrons at the Fermi energy. A naive extension to the lattice problem is to include an enhancement of the exponent due to intersite correlations. Thus, the spin gap is expected to behave as
where e is the enhancement factor. The Gutzwiller approximation predicts an enhancement factor of two [14]. From the results by the DMRG method c~= 1.4(1) is obtained. Concerning the charge gap the DMRG calculation confirms that in the weak coupling limit it approaches to Ao = ½J
(3)
which is the result obtained by a simple mean field theory with antiferromagnetic long-range order. Of course, in one dimension there is no antiferromagnetic long-range order. The point is that the spin-spin correlation length is much longer than the charge--charge correlation length. Therefore, for the discussion of the charge gap we can treat the antiferromagnetic short-range order as if it is a long-range one. From these results it is concluded that the charge gap is larger than the spin gap and the ratio between the two diverges in the weak coupling limit of J. It means that for the periodic Anderson model the ratio diverges in the limit of strong Coulomb interaction under the symmetric condition. The charge gap may be best estimated from optical conductivity, while the spin gap by neutron inelastic scatterings. In Ce3Bi4Pt3 the ratio is estimated to be about 2 [15]. Guerrero and Yu [16] suggested that the not particularly large value should not be attributed to weak correlation but rather be considered as a consequence of the asymmetry.
K. Ueda / Physica B 230-232 (1997) 22 26
4. Paramagnetic metallic phase In the one-dimensional Kondo lattice model the paramagnetic metallic phase exists away from halffilling in the region where the exchange coupling is weak or intermediate (Fig. 1). Clearly related to the heavy fermion states, this paramagnetic phase is very interesting. However, compared with the other phases studied in the previous sections this phase is less understood. Concerning the difficulty we should note first that the weak coupling limit is not a trivial limit, because in this limit there remains macroscopic degeneracy 2L where L is the number of sites (f spins). Secondly, although the ground state at half-filling is well understood it is a highly non-trivial problem when it is doped with holes or electrons similar to the high-T~ problem. Finally, one cannot go to the strong coupling limit since there is a phase transition to the ferromagnetic metallic phase. In one dimension it has been established that most interacting metallic systems belong to the universality class of Tomonaga-Luttinger liquids [17]. The asymptotic forms of charge- and spincorrelation functions are (n(x) n(O)) = Kp/(rcx) 2 + Alcos(2kvx) x -1-K" + A2 COS(4kFX)X -41q', ( S ( x ) . S ( O ) ) = 1/(rex) 2 + Blcos(2kFX ) x - l - K p ,
(4) (5)
where kv = ½rtn, is the Fermi momentum and with n being the density of charge carriers, Kp the correlation exponent [18]. The anomalous power law decays of the correlation functions naturally reflect themselves in the Friedel oscillations: the asymptotic form of the charge density oscillations induced by an impurity potential is 6p(x) ~ C1 COS(2kFx) x {- 1 - K,,)/2
+ C2 COS(4kvx) x - 2K°
(6)
as a function of the distance x from the impurity [-19,20]. Analogously, the spin density oscillations induced by a local magnetic field behave as a(x) ~ DI cos(2kvx) x -K..
(7)
25
It may be natural to expect that the paramagnetic metallic state of the one-dimensional Kondo lattice model also belongs to the class of Luttinger liquids. However, in this case the position of the Fermi points is already a nontrivial problem since the model contains two different types of electrons, the conduction electrons and the localized spins. Concerning the size of the Fermi volume two different viewpoints are possible. If the interaction between the conduction electrons and the localized spins is strictly zero, it is clear that the singularity of the momentum distribution function of the conduction electrons is determined only by the number of conduction electrons. Thus, if the singular points are not affected by the exchange interaction, kF = rtnc/2 would be expected. On the other hand, a different answer is obtained by identifying the Kondo lattice model as an effective model for the periodic Anderson model. In the periodic Anderson model the conduction electrons and the f-electrons are mixed with each other through the hybridization matrix elements. Therefore, it is natural to expect that kv is determined by the total density of both the conduction electrons and the f-electrons; kv = ½rt(l + no). Recently, we have calculated the Friedel oscillations of the one-dimensional Kondo lattice model by using the DMRG [21]. The charge density oscillations are induced naturally by open boundary conditions, and the spin density oscillations are introduced by applying local magnetic fields at the both ends. From the period of the Friedel oscillations we can determine the Fermi momentum. It is clearly seen that the Friedel oscillations are compatible with the large Fermi surface kv = ½rt(1 + nc) in spite of the fact that the charge degrees of freedom are completely suppressed for the f-electrons in the Kondo lattice model. These results are consistent with the previous work on the t-t' Kondo lattice model for which it was shown exactly that its strong coupling limit is described by a Luttinger liquid with the large Fermi surface [-22]. Therefore, it is natural to conclude that the paramagnetic metallic phase of the Kondo lattice model has a large Fermi surface in general.
26
K. Ueda /Physica B 230-232 (1997) 22 26
References [1] H. Tsunetsugu, M. Sigrist and K. Ueda, Phys. Rev. B 47 (1993) 8345. [2] M. Sigrist, H. Tsunetsugu and K. Ueda, Phys. Rev. Lett. 67 (1991) 2211. [3] M. Sigrist, K. Ueda and H. Tsunetsugu, Phys. Rev. B 46 (1992) 175. [4] T. Suzuki, Physica B 186-188 (1993) 347. [5] J. Rossat-Mignod et al., J. Magn. Magn. Mater. 31-34 (1983) 398. [6] J. Rossat-Mignod et al., J. Magn. Magn. Mater. 52 (1985) 111. [7] M. Kohgi et al., Phys. Rev. B 49 (1994) 7068. [8] N. Shibata, C. Ishii and K. Ueda, Phys. Rev. B 52 (1995) 10232. [9] G.-S. Tian, Phys. Rev. B 50 (1994) 6246. [10] H. Tsunetsugu, Y. Hatsugai, K. Ueda and M. Sigrist, Phys. Rev. B 46 (1992) 3175.
[11] T. Nishino and K. Ueda, Phys. Rev. B 47 (1993) 12451. [12] S.R. White, Phys. Rev. Lett. 69 (1992) 2863. [13] N. Shibata, T. Nishino, K. Ueda and C. Ishii, Phys. Rev. B 53 (1996) 8828. [14] T.M. Rice and K. Ueda, Phys. Rev. Lett. 55 (1985) 995; Phys. Rev. B 34 (1986) 6420. [15] B. Bucher, Z. Schlesinger, P.C. Canfield and Z. Fisk, Phys. Rev. Lett. 72 (1994) 522. 1-16] M. Guerrero and C.C. Yu, Phys. Rev. B 51 (1995) 10301. [17] F.D.M. Haldane, J. Phys. C 14 (1981) 2585. [18] HJ. Schulz, Phys. Rev. Lett. 64 (1990) 2831. [19] M. Fabrizio and A.O. Gogolin, Phys. Rev. B 51 (1995) 17827. [20] R. Egger and H. Grabert, Phys. Rev. Lett. 75 (1995) 3505. [21] N. Shibata, K. Ueda, T. Nishino and C. Ishii, Phys. Rev. B 54 (1996) 13495. [22] K. Ueda, T. Nishino and H. Tsunetsugu, Phys. Rev. B 50 (1994) 612.