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Numerical renormalization group study of the one-dimensional Anderson lattice
ELSEVIER
Physica B 206 & 207 (1995) 151-153
Numerical renormalization group study of the one-dimensional Anderson lattice M. Guerrero*, Clare C. Yu Department of Physics, University of California, lrvine, CA 92717-4575, USA
Abstract
We have studied both the symmetric and asymmetric periodic Anderson model in one dimension using the density matrix formulation of the numerical renormalization group. We have calculated the charge gap, spin gap and quasiparticle gap as a function of the repulsive interaction U using open boundary conditions for lattices up to 24 sites at half filling. We find that the charge gap is larger than the spin gap for all U for both the symmetric and asymmetric cases.
In this work, we study the one-dimensional Anderson lattice at half filling using the density matrix formulation of the numerical renormalization group technique [1]. We consider both the symmetric case and the asymmetric case. We study the different excitations (spin, charge, quasiparticle) as a function of the parameters. We consider the standard Anderson Hamiltonian in one dimension: t-1 = --t Z (Ci~Ci+ * l t r + C~+,~Ci,~) + Ef Z ni,~
+u~
' ~ ' i~ + V Y ~ (Ci,~fi~ * + fio-Cio-) * ni~
(1)
where t is the hopping matrix element for electrons between neighboring sites, ef is the energy of the localized f-orbital, U is the on-site Coulomb repulsion of the f-electrons, and V is the on-site hybridization matrix element between f-elements and conduction electrons. For simplicity, we neglect orbital degeneracy. For U = 0, this Hamiltonian can be diagonalized exactly in k-space. We obtain two hybridized bands,
)t; = ½[(~,- 2t cos(ka)) -+ ~(Ef + 2t cos(ka)) z + 4VZ]. (2) When N = 2N, the lower band is full while the upper one is empty. Thus, the system is insulating. Here N~ is the number of electrons, N is the number of sites in the lattice and a is the lattice constant. The symmetric case corresponds to a value of ef = U/2. In addition to the usual SU(2) spin symmetry, this particular choice introduces a SU(2) charge pseudospin symmetry into the system [2]. The charge pseudospin operator has components [2] I z = (N'e,/2) - N , i ÷ = ~ ' ( - 1 ) ~(ci, , c,, , -f~,f~,),
and I - = (1+)*. Note that half filling corresponds to I z = 0. The ground state is a singlet both in spin and pseudospin space (S = 0, 1 = 0) [3]. The spin, charge [2] and quasiparticle gap are given by A =E(S=I,I=O)-E
o,
A c = E(S = 0 , I = 1 ) - E0, Aqp = E ( 2 N + 1) +
*Corresponding author.
(3)
i
E(ZN - 1) -
(4) 2Eo(ZU),
where E0 is the energy of the ground state. For U = 0, all the gaps coincide and are given by
0921-4526/95/$09.50 © 1995 Elsevier Science BN. All rights reserved S S D I : 0921-4526(94)00391-2
152
theh ~ i d i z a t i o n
M. Guerrero, C.C. Yu / Physica B 206 & 207 (1995) 151-153
gap: A, =
a c = Aqp
= h i ~ - - A.~ =
1.0
2Vt 2 + V 2 - 2t. As U--* 2, the f-electrons decouple from the conduction electrons, and all the gaps go to zero.
'
0.8
We now consider the asymmetric case in which ef can have any value. The pseudospin operator is no longer conserved. Only the total number of electrons (lz) remains a good quantum number. For all the parameters that we explored, we found that the ground state is still a spin singlet at half filling. We define the spin gap as
~
~~,,.'~, mqp \~::~
b
00
\
8 sites
16 sites
0.4
'~ A
"~ "- L~'--.. ED
"<
~-
(5)
To find the charge gap, we look for the lowest excited state In) with S = 0 and ( O l E q p q l n ) # O , where pq is the q-component of the Fourier transformed charge density operator and 10) is the ground state. We find that In) is the lowest S = 0 excited state. So the charge gap is now given by A~ = E, (S = 0) - E o .
8'sites 16 sites 24 sites
o--
0,2
A~ = e ( s = 1) - E,,.
~.' • • ........ • 0-- -- • 0 0 [] ........[]
(6)
The quasiparticle gap is defined as before. In using the density matrix renormalization group technique [1], we consider lattices up to 24 sites and we use open boundary conditions. We set t = 1 so that energies are measured in units of t. We fix V= 1 and vary U from 0 to 30. Although this value of V is somewhat larger than what occurs in real materials, it is convenient for numerical reasons and for comparison with previous calculations [2]. We typically keep 100 states per block, although in some cases greater accuracy is needed and we keep up to 200 states. For small U, the results are very accurate with truncation errors of the order of 10 -s, while for large U the accuracy is reduced and the truncation errors increase to 10 -4 . In Fig. 1 we plot the gaps versus U for different lattice sizes in the symmetric case. The spin gap decreases monotonically with increasing U. It compares well with the results of Nishino and Ueda in Ref. [2] who did exact diagonalization with periodic boundary conditions. On the other hand, the quasiparticle gap initially increases with U, goes through a maximum and then decreases. The spin gap shows small finite size effects while the quasiparticle gap has large finite size effects, especially for large U. These features can be interpreted as follows: the spin excitation has small dispersion because it is a spin flip that has mainly f-character. This makes it a local excitation which is not very sensitive to the length of the lattice or to the boundary conditions. On the other
0.0
0
'
5
10
15
20
U
Fig. 1. Gaps versus U for the symmetric case (t = 1, V= 1, ef -U/2). =
hand, the quasiparticle gap is obtained by adding a particle to the system (in the symmetric case Aqp = 2[E(2N + 1) - E(2N)]). For very small U, this particle goes predominantly into an f-orbital; so as U increases, the gap also increases due to the Coulomb repulsion. However, for large U, the extra particle goes mostly into the conduction band. Thus, the gap starts decreasing and, as U - ~ o% approaches the value of Aqp for a free electron band on a finite size lattice. We confirmed this finite size effect by calculating Aqp on finite size lattices for free electrons. In the symmetric case, the charge gap can be obtained by adding two particles to the half filled system. We find that it has the same features as the quasiparticle gap. The fact Ac is greater than aqp for any finite value of U means that the two added particles repel each other. So as N - - ~ they will be infinitely far apart, and A~ will equal Aqp. The charge gap differs from that found in Ref. [2] in the strong coupling regime because they use periodic boundary conditions while we use open boundary conditions. For the asymmetric case, we set ~f = 0, V = 1 and vary U. In Fig. 2 we show the gaps versus U for the asymmetric case. As U--->0% the gaps tend to a finite value [4]. In this case, U = oo suppresses the states with double f-occupancy but allows the system to fluctuate between states with no f-electrons in the local orbital and states with one f-electron. This hybridization results in the formation of a gap. For the spin gap, we are able to perform an extrapolation for N--->oo. We find that, for large U, the spin gap can be fitted by the form
M. Guerrero, C.C. Yu / Physica B 206 & 207 (1995) 151-153
1.0
"4
I
--
r
crossover to a state made out of two S = 1 excitations. This picture is supported by the fact that for small U, Ac - A q p but as U increases, Ac ~ 2As. In summary, we have calculated the spin gap, quasiparticle gap and charge gap for the one-dimensional A n d e r s o n lattice. In the symmetric case, we found As < Aqo < a¢ in agreement with Ref. [2]. In the asymmetric case, the spin gap is smaller than the charge gap and the quasiparticle gap. The spin gap approaches its U = ~ value as U - " with the t~ very close to 1.
I
A
0.B 1
Ill.._ "'-'O--~ ~ Ac "I~ ~
..... ......
-~L~
153
i
~- . . . .
0.6
0.4 8 sites 16 sites 0 - - - • 24 sites
II--I
0.2
Acknowledgements 0.0
0
5,
.
10,
1~5
20
U
Fig. 2. Gaps versus U for the asymmetric case (t = 1, V= 1, ~f =
0).
Const. a ~ ( u ) = a ~ ( u = oo) + u - - - 7 - ,
(7)
with a---~l as U -->°°. The behavior of the gap for short chains with e, < 0 suggests that the spin gap goes as (ef + U ) - '. In the asymmetric case, for U ~<2, z~c > Aqo, but for U > 2 , A c
We would like to thank Steve White for very helpful discussions. This work was supported in part by O N R grant N000014-91-J-1502 and an allocation of computer time from the University of California Irvine.
References
[1] S.R. White, Phys. Rev. B 48 (1993) 10345. [2] T. Nishino and K. Ueda, Phys. Rev. B 47 (1993) 12451. [3] K. Ueda, H. Tsunetsugu and M. Sigrist, Phys. Rev. Lett. 68 (1992) 1030. [4] R. JuUien and R.M. Martin, Phys Rev. B 26 (1982) 6173.
Physica B 206 & 207 (1995) 151-153
Numerical renormalization group study of the one-dimensional Anderson lattice M. Guerrero*, Clare C. Yu Department of Physics, University of California, lrvine, CA 92717-4575, USA
Abstract
We have studied both the symmetric and asymmetric periodic Anderson model in one dimension using the density matrix formulation of the numerical renormalization group. We have calculated the charge gap, spin gap and quasiparticle gap as a function of the repulsive interaction U using open boundary conditions for lattices up to 24 sites at half filling. We find that the charge gap is larger than the spin gap for all U for both the symmetric and asymmetric cases.
In this work, we study the one-dimensional Anderson lattice at half filling using the density matrix formulation of the numerical renormalization group technique [1]. We consider both the symmetric case and the asymmetric case. We study the different excitations (spin, charge, quasiparticle) as a function of the parameters. We consider the standard Anderson Hamiltonian in one dimension: t-1 = --t Z (Ci~Ci+ * l t r + C~+,~Ci,~) + Ef Z ni,~
+u~
' ~ ' i~ + V Y ~ (Ci,~fi~ * + fio-Cio-) * ni~
(1)
where t is the hopping matrix element for electrons between neighboring sites, ef is the energy of the localized f-orbital, U is the on-site Coulomb repulsion of the f-electrons, and V is the on-site hybridization matrix element between f-elements and conduction electrons. For simplicity, we neglect orbital degeneracy. For U = 0, this Hamiltonian can be diagonalized exactly in k-space. We obtain two hybridized bands,
)t; = ½[(~,- 2t cos(ka)) -+ ~(Ef + 2t cos(ka)) z + 4VZ]. (2) When N = 2N, the lower band is full while the upper one is empty. Thus, the system is insulating. Here N~ is the number of electrons, N is the number of sites in the lattice and a is the lattice constant. The symmetric case corresponds to a value of ef = U/2. In addition to the usual SU(2) spin symmetry, this particular choice introduces a SU(2) charge pseudospin symmetry into the system [2]. The charge pseudospin operator has components [2] I z = (N'e,/2) - N , i ÷ = ~ ' ( - 1 ) ~(ci, , c,, , -f~,f~,),
and I - = (1+)*. Note that half filling corresponds to I z = 0. The ground state is a singlet both in spin and pseudospin space (S = 0, 1 = 0) [3]. The spin, charge [2] and quasiparticle gap are given by A =E(S=I,I=O)-E
o,
A c = E(S = 0 , I = 1 ) - E0, Aqp = E ( 2 N + 1) +
*Corresponding author.
(3)
i
E(ZN - 1) -
(4) 2Eo(ZU),
where E0 is the energy of the ground state. For U = 0, all the gaps coincide and are given by
0921-4526/95/$09.50 © 1995 Elsevier Science BN. All rights reserved S S D I : 0921-4526(94)00391-2
152
theh ~ i d i z a t i o n
M. Guerrero, C.C. Yu / Physica B 206 & 207 (1995) 151-153
gap: A, =
a c = Aqp
= h i ~ - - A.~ =
1.0
2Vt 2 + V 2 - 2t. As U--* 2, the f-electrons decouple from the conduction electrons, and all the gaps go to zero.
'
0.8
We now consider the asymmetric case in which ef can have any value. The pseudospin operator is no longer conserved. Only the total number of electrons (lz) remains a good quantum number. For all the parameters that we explored, we found that the ground state is still a spin singlet at half filling. We define the spin gap as
~
~~,,.'~, mqp \~::~
b
00
\
8 sites
16 sites
0.4
'~ A
"~ "- L~'--.. ED
"<
~-
(5)
To find the charge gap, we look for the lowest excited state In) with S = 0 and ( O l E q p q l n ) # O , where pq is the q-component of the Fourier transformed charge density operator and 10) is the ground state. We find that In) is the lowest S = 0 excited state. So the charge gap is now given by A~ = E, (S = 0) - E o .
8'sites 16 sites 24 sites
o--
0,2
A~ = e ( s = 1) - E,,.
~.' • • ........ • 0-- -- • 0 0 [] ........[]
(6)
The quasiparticle gap is defined as before. In using the density matrix renormalization group technique [1], we consider lattices up to 24 sites and we use open boundary conditions. We set t = 1 so that energies are measured in units of t. We fix V= 1 and vary U from 0 to 30. Although this value of V is somewhat larger than what occurs in real materials, it is convenient for numerical reasons and for comparison with previous calculations [2]. We typically keep 100 states per block, although in some cases greater accuracy is needed and we keep up to 200 states. For small U, the results are very accurate with truncation errors of the order of 10 -s, while for large U the accuracy is reduced and the truncation errors increase to 10 -4 . In Fig. 1 we plot the gaps versus U for different lattice sizes in the symmetric case. The spin gap decreases monotonically with increasing U. It compares well with the results of Nishino and Ueda in Ref. [2] who did exact diagonalization with periodic boundary conditions. On the other hand, the quasiparticle gap initially increases with U, goes through a maximum and then decreases. The spin gap shows small finite size effects while the quasiparticle gap has large finite size effects, especially for large U. These features can be interpreted as follows: the spin excitation has small dispersion because it is a spin flip that has mainly f-character. This makes it a local excitation which is not very sensitive to the length of the lattice or to the boundary conditions. On the other
0.0
0
'
5
10
15
20
U
Fig. 1. Gaps versus U for the symmetric case (t = 1, V= 1, ef -U/2). =
hand, the quasiparticle gap is obtained by adding a particle to the system (in the symmetric case Aqp = 2[E(2N + 1) - E(2N)]). For very small U, this particle goes predominantly into an f-orbital; so as U increases, the gap also increases due to the Coulomb repulsion. However, for large U, the extra particle goes mostly into the conduction band. Thus, the gap starts decreasing and, as U - ~ o% approaches the value of Aqp for a free electron band on a finite size lattice. We confirmed this finite size effect by calculating Aqp on finite size lattices for free electrons. In the symmetric case, the charge gap can be obtained by adding two particles to the half filled system. We find that it has the same features as the quasiparticle gap. The fact Ac is greater than aqp for any finite value of U means that the two added particles repel each other. So as N - - ~ they will be infinitely far apart, and A~ will equal Aqp. The charge gap differs from that found in Ref. [2] in the strong coupling regime because they use periodic boundary conditions while we use open boundary conditions. For the asymmetric case, we set ~f = 0, V = 1 and vary U. In Fig. 2 we show the gaps versus U for the asymmetric case. As U--->0% the gaps tend to a finite value [4]. In this case, U = oo suppresses the states with double f-occupancy but allows the system to fluctuate between states with no f-electrons in the local orbital and states with one f-electron. This hybridization results in the formation of a gap. For the spin gap, we are able to perform an extrapolation for N--->oo. We find that, for large U, the spin gap can be fitted by the form
M. Guerrero, C.C. Yu / Physica B 206 & 207 (1995) 151-153
1.0
"4
I
--
r
crossover to a state made out of two S = 1 excitations. This picture is supported by the fact that for small U, Ac - A q p but as U increases, Ac ~ 2As. In summary, we have calculated the spin gap, quasiparticle gap and charge gap for the one-dimensional A n d e r s o n lattice. In the symmetric case, we found As < Aqo < a¢ in agreement with Ref. [2]. In the asymmetric case, the spin gap is smaller than the charge gap and the quasiparticle gap. The spin gap approaches its U = ~ value as U - " with the t~ very close to 1.
I
A
0.B 1
Ill.._ "'-'O--~ ~ Ac "I~ ~
..... ......
-~L~
153
i
~- . . . .
0.6
0.4 8 sites 16 sites 0 - - - • 24 sites
II--I
0.2
Acknowledgements 0.0
0
5,
.
10,
1~5
20
U
Fig. 2. Gaps versus U for the asymmetric case (t = 1, V= 1, ~f =
0).
Const. a ~ ( u ) = a ~ ( u = oo) + u - - - 7 - ,
(7)
with a---~l as U -->°°. The behavior of the gap for short chains with e, < 0 suggests that the spin gap goes as (ef + U ) - '. In the asymmetric case, for U ~<2, z~c > Aqo, but for U > 2 , A c
We would like to thank Steve White for very helpful discussions. This work was supported in part by O N R grant N000014-91-J-1502 and an allocation of computer time from the University of California Irvine.
References
[1] S.R. White, Phys. Rev. B 48 (1993) 10345. [2] T. Nishino and K. Ueda, Phys. Rev. B 47 (1993) 12451. [3] K. Ueda, H. Tsunetsugu and M. Sigrist, Phys. Rev. Lett. 68 (1992) 1030. [4] R. JuUien and R.M. Martin, Phys Rev. B 26 (1982) 6173.