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Ground state of the kondo lattice
Physica B 1658~166 (1990) 397-398 North-Holland
GROUND STATE OF THE KONDO LATTICE Kazuo Ueda and Kouki Yamamoto Institute of Materials Science, University of Tsukuba, Tsukuba 305, Japan The one-dimensional Kondo lattice is diagonalized numerically. It is shown that the ground state is ferromagnetic at low densities of conduction electrons. On the other hand at half-filling the ground state is a singlet with stronq antiferromagnetic correlations which are the main mechanism to compensate the localized spins.
1. INTRODUCTION The key question about the normal Fermion heavy state is the why nonmagnetic state instead of some magnetically ordered state is stabilized in spite of the strong Coulomb repulsion. Mean field type theories, the lattice version of expansion and the l/N Gutzwiller type theory have succeeded in describing the normal heavy Fermion state with the agreement in the large degeneracy limit. However it is clear that a better theory than those mean field type theories is required to explain such magnetic properties as small antiferromagnetic moment in URu2Si2 (1) UPt3 and and metamagnetic-like (2) behavior in CeRu2Si2 (3). In the present paper we report on the results of exact numerical diagonalization of the onedimensional Kondo lattice. 2. MODEL The one-dimensional Kondo lattice is written in the usual form
(1) where L is the number of lattice sites. We impose the periodic boundary condition. As the exchange constant J= -0.2 is used in the following numerical results. We show the results of six and seven lattice sites. The exact numerical diagonalization was performed for all 0921-4526/90/$03.50
@ 1990 -
electron numbers N less than or equal to half filling, N
Spin of the ground state of the seven-sites Kondo lattice. N is the conduction electron number.
N
1234561
S
3
l/2
0
5/2
0
l/2
0
Spin of the ground state of the six-sites Kondo lattice.
Tabel II
N
1
2
3
4
5
6
S
5/2
0
l/2
2
l/2
0
The results for the six sites and the seven sites are very similar. In the following we discuss the case of L=7. To understand the ground state it is helpful to consider the energy levels of the conduction band at J=O Ek= -2 cos pk
Elsevier Science Publishers B.V. (North-Holland)
k=O,fl,
...
(2)
398
K. Ueda,K. Yamamoto
Table
III
On-site correlation between conduciton electorn spins
N
Tab.IV
Nearest N
<&f(i)&f(i+l)>
1
-0.054
2
3
4
-0.025 -0.088
neighbor 1
3
0.018 0.134 -0.069
Let us start with the case of N=l. The ground state is a incomplete ferromagnetic state. The conduction electron occupies the lowest level k=O. The gain of exchange energy is maximized aligning localized spin in one by direction and at the same time fixing the spin of the conduction electron in opposite direction. Actually spin correlation between a localized spin and conduction electrons on a single site is This -0.054. magnitude is slightly bigger than -l/4.7 =-0.036 which is a value expected when the spin alignment is static. The difference comes from quantum fluctuations due to the transverse components of the exchange term. For N=2, the two conduction electrons are accommodated in the lowest level k=O with antiparallel spins. Therefore the system is singlet as a whole. However the correlation between nearest neighbor localized spins are ferromagnetic, 0.134. This double means that the exchange mechanism which causes the ferromagnetic ground state when N=l is still effective. Now we turn to the opposite case of nearly half-filled conduction band (N=6, The ground states have the smallest 7). spin possible. In a single impurity Kondo problem it is well known that the localized spin is completely screened by conduction electron spins (Kondo cloud). As is pointed out by Nozieres (4), in a lattice Kondo cloud does not suffice to
5
between 4
spin and
6
-0.129 -0.089
correlation 2
the localized
I
-0.024
-0.100
the localized 5
0.026 -0.107
6
spins 7
-0.384
-0.396
compensate the localized spins since only a part (TK/W) of conduciton electrons take part in the Kondo cloud. The present results show that close to the halffilling the localized spins themselves are cancelling each other mainly via the antiferromagnetic nearest neighbor correlations. In the vicinity of quater filling, correlations and antiferromagneic the ferromagnetic correleations are competing. For N=3 and 5, singlet states are realized nearest neighbor but the antiferromagnetic correlations are not strong. The origin of the ferromagnetic ground state at N=4 may be traced back to a special level scheme of the conduction electrons. The highest occupied level is fold degenerate including spin four degrees of freedom. Therefore without much loss of kinetic evergy two conduction electrons in the highest level can have the same spin opposite to the localized spins aligned in one direction.
REFERENCES (1) C. Broholm et al. (1987) 1467. (2) G. Aeppli et al. (1988) 615 (3) P. Haen et al. J. (1987) 391. (4) P. Nozieres, Ann. (1985) 19.
Phys. Rev. Lett. 58 Phys. Rev. Lett. Low Temp. Phys. Phys.(Paris)
10
60 67
GROUND STATE OF THE KONDO LATTICE Kazuo Ueda and Kouki Yamamoto Institute of Materials Science, University of Tsukuba, Tsukuba 305, Japan The one-dimensional Kondo lattice is diagonalized numerically. It is shown that the ground state is ferromagnetic at low densities of conduction electrons. On the other hand at half-filling the ground state is a singlet with stronq antiferromagnetic correlations which are the main mechanism to compensate the localized spins.
1. INTRODUCTION The key question about the normal Fermion heavy state is the why nonmagnetic state instead of some magnetically ordered state is stabilized in spite of the strong Coulomb repulsion. Mean field type theories, the lattice version of expansion and the l/N Gutzwiller type theory have succeeded in describing the normal heavy Fermion state with the agreement in the large degeneracy limit. However it is clear that a better theory than those mean field type theories is required to explain such magnetic properties as small antiferromagnetic moment in URu2Si2 (1) UPt3 and and metamagnetic-like (2) behavior in CeRu2Si2 (3). In the present paper we report on the results of exact numerical diagonalization of the onedimensional Kondo lattice. 2. MODEL The one-dimensional Kondo lattice is written in the usual form
(1) where L is the number of lattice sites. We impose the periodic boundary condition. As the exchange constant J= -0.2 is used in the following numerical results. We show the results of six and seven lattice sites. The exact numerical diagonalization was performed for all 0921-4526/90/$03.50
@ 1990 -
electron numbers N less than or equal to half filling, N
Spin of the ground state of the seven-sites Kondo lattice. N is the conduction electron number.
N
1234561
S
3
l/2
0
5/2
0
l/2
0
Spin of the ground state of the six-sites Kondo lattice.
Tabel II
N
1
2
3
4
5
6
S
5/2
0
l/2
2
l/2
0
The results for the six sites and the seven sites are very similar. In the following we discuss the case of L=7. To understand the ground state it is helpful to consider the energy levels of the conduction band at J=O Ek= -2 cos pk
Elsevier Science Publishers B.V. (North-Holland)
k=O,fl,
...
(2)
398
K. Ueda,K. Yamamoto
Table
III
On-site correlation between conduciton electorn spins
N
Tab.IV
Nearest N
<&f(i)&f(i+l)>
1
-0.054
2
3
4
-0.025 -0.088
neighbor 1
3
0.018 0.134 -0.069
Let us start with the case of N=l. The ground state is a incomplete ferromagnetic state. The conduction electron occupies the lowest level k=O. The gain of exchange energy is maximized aligning localized spin in one by direction and at the same time fixing the spin of the conduction electron in opposite direction. Actually spin correlation between a localized spin and conduction electrons on a single site is This -0.054. magnitude is slightly bigger than -l/4.7 =-0.036 which is a value expected when the spin alignment is static. The difference comes from quantum fluctuations due to the transverse components of the exchange term. For N=2, the two conduction electrons are accommodated in the lowest level k=O with antiparallel spins. Therefore the system is singlet as a whole. However the correlation between nearest neighbor localized spins are ferromagnetic, 0.134. This double means that the exchange mechanism which causes the ferromagnetic ground state when N=l is still effective. Now we turn to the opposite case of nearly half-filled conduction band (N=6, The ground states have the smallest 7). spin possible. In a single impurity Kondo problem it is well known that the localized spin is completely screened by conduction electron spins (Kondo cloud). As is pointed out by Nozieres (4), in a lattice Kondo cloud does not suffice to
5
between 4
spin and
6
-0.129 -0.089
correlation 2
the localized
I
-0.024
-0.100
the localized 5
0.026 -0.107
6
spins 7
-0.384
-0.396
compensate the localized spins since only a part (TK/W) of conduciton electrons take part in the Kondo cloud. The present results show that close to the halffilling the localized spins themselves are cancelling each other mainly via the antiferromagnetic nearest neighbor correlations. In the vicinity of quater filling, correlations and antiferromagneic the ferromagnetic correleations are competing. For N=3 and 5, singlet states are realized nearest neighbor but the antiferromagnetic correlations are not strong. The origin of the ferromagnetic ground state at N=4 may be traced back to a special level scheme of the conduction electrons. The highest occupied level is fold degenerate including spin four degrees of freedom. Therefore without much loss of kinetic evergy two conduction electrons in the highest level can have the same spin opposite to the localized spins aligned in one direction.
REFERENCES (1) C. Broholm et al. (1987) 1467. (2) G. Aeppli et al. (1988) 615 (3) P. Haen et al. J. (1987) 391. (4) P. Nozieres, Ann. (1985) 19.
Phys. Rev. Lett. 58 Phys. Rev. Lett. Low Temp. Phys. Phys.(Paris)
10
60 67