- Email: [email protected]
One- and two-hole excitation spectra of the 1d Hubbard model
1029
Journal of Magnetism and Magnetic Materials 54-57 (1986) 1029-1030 ONE- AND TWO-HOLE
EXCITATION
SPECTRA
A . M . O L E S *, G . T R I ~ G L I A , D. S P A N J A A R D
OF THE
ld HUBBARD
MODEL
a n d R. J U L I E N
Laboratoire de Physique des Solides, Universitb de Paris-Sud, F-91405 Orsav, France
Modified periodic boundary conditions are used to obtain one- and two-hole excitation spectra by considering small chains described by the ld Hubbard model. A quasiparticle description and the nature of many-body effects are discussed. The results agree rather well with those of second order perturbation theory.
1. Finite cell calculations for the ld Hubbard model
a n d dispersion
The H u b b a r d model H
ok =
= --tY'.a+oajo + U ~ , n i t ni$, 0o
is quite realistic for quasi-one-dimensional solids [1]. Here we use it to study rigorously the m a n y - b o d y effects in one- a n d two-hole excitation spectra in one d i m e n s i o n b y p e r f o r m i n g calculations on finite systems. They are used to check the accuracy of second order p e r t u r b a t i o n theory (SOPT) [2,3]. The simulation of an infinite H u b b a r d chain with rigorous solutions of finite systems is possible by using modified periodic b o u n d a r y conditions [4]. They require that the wave function changes its p h a s e by an arbitrary value ¢ after m a k i n g a rotation along a ring of N sites. This allows to c o m p a r e the results o b t a i n e d with the chains of even a n d odd value N / 2 a n d extrapolate t h e m to a n infinite chain by finite cell scaling (FCS) [5]. As a test of our approach, we have p e r f o r m e d a FCS of the gap in the half-filled case ( n = 1) by using the chains up to N = 8 [6]. T h e gap has b e e n reproduced quite well for 0.5 < U / W < 2.0, where W = 4 t . 2. One-hole excitation spectrum in the half-filled case T h e one-hole excitation spectrum of a finite H u b b a r d chain is characterized by the spectral density function
Ak+(W)=~,[(et"~[akslCbo)126(o~-Eo+E~).
d~o ( o 3 + E k - - T U )
× (SLd, These quantities were found numerically for the chains N = 4, 6 [6]. As an example, the result for N = 6 a n d U / W = 0 . 5 , 2.0 is presented in fig. 1. If U = 0 , the energy E k has a physical m e a n i n g only for k ~< "n/2. W i t h increasing U / W a gradual transition is seen from the b a n d picture to the one resulting from an excitation of one hole in a n otherwise almost localized system. The presence of m i n i m u m k m in E k allows to define two regions in the k-space: for k < k m the quasiparticle states have mainly b a n d character, while the states with k > k m may be called scattering states. The latter states have first (at U / W = 0.5) rather small weight and m u c h larger o k2 t h a n the respective b a n d states. For larger U / W they become more weighted a n d n o difference in Ok2 between the two regions is observed. The transition from b a n d to localized states is also seen in the density of quasiparticle states p(~0). The hole spectrum at U = 0 is a half of the density of states of a one-dimensional infinite chain. It has a singularity at w = - W / 2 which disappears at finite U / W . A t the same time the weight of scattering states increases and they contribute to the low-energy tail of p(~0). The numerical result is presented in fig. 2. The structure
(2)
m
i
J
b
#'l
/
,,' 0.5
-
,
I
',, )~-, "
l
",,
~
- o.o5
(3) --'~"
o
* On leave of absence from the Institute of Physics, Jagellonian University, Cracow, Poland. 0304-8853/86/$03.50
i
~.s u/wo2.o
T h e system is assumed to b e paramagnetic, i.e. n t = n = ½. It is c o n v e n i e n t to describe the spectral density in terms of quasiparticles with energy
+ 7 ~s
Ak~(~
(1)
i
i
0.5 k E~]
i
0
1.O ,'-
Fig. 1. Quasiparticle quantities Ek and o~ at n =1 for N = 6 chain.
© E l s e v i e r S c i e n c e P u b l i s h e r s B.V.
1030
A.M. Ole~ et al. / Spectra of the ld Hubbard model .
I
T 2.8 _
n = 1.67
n=l.O
n = 1.67
~" 0.8
2.0
0.6
k2
0.4
o.
1.75 <~~0
0.2 -1 .o wlW
o
.
.
-3
~ l
•~ 1.25
0.4 -2.0
.
1.0
-2
-1 0 t u l W ~
1.00 i
Fig. 2. One-hole spectrum at U / W = 1 obtained with N 6 chain (solid line) and by using SOPT for an infinite chain (dashed line).
0
=
Fig. 3. One-hole (solid line) and two-hole (dashed line) excitation spectrum obtained with N = 6 chain at U / W = 1.
0.5
i
I
1.0 1.5 W/U
i
2.0
Fig. 4. Energy difference between the centers of gravity of the band and satellite part of the one-hole paramagnetic (A~), ferromagnetic (A F) and two-hole (A 2) spectrum for N = 6. × 8(o~- E0 + Em + 2U).
with two sharp m a x i m a on the right is due to the b a n d states. They occupy a narrower region that the hole s p e c t r u m at U = 0. T h e tail of scattering states is in principle well reproduced by SOPT results for an infinite chain up to U / W = 1. Obviously, SOPT fails to reproduce the gap. The essential features of quasiparticle spectra do not d e p e n d on the size of considered chain. FCS p e r f o r m e d on the weights of the b a n d (scattering) part of the s p e c t r u m gave the weight of an infinite chain which were increased (reduced) from those of chain N = 6 by a b o u t 20% [6].
3. Satellite structures in the one- and two-hole excitation spectra
The shape of the one-hole excitation spectra at n > 1 changes even more drastically with increasing U / W since new satellite structures a p p e a r to the left from the b a n d part for sufficiently large U / W . Their weight increases with increasing U / W up to the atomic limit value 2 - n. The convergence is, however, quite slow. This indicates that the dynamics of electrons having the opposite spin to the one of the excited hole plays a n i m p o r t a n t role. In fig. 3 the spectrum received with the chain N = 6 at n = 1.67 is presented. The satellite is well separated from the m a i n line a n d has still a rather small weight w~ = 0.09. If we make a n excitation in a majority s u b b a n d of a ferromagnetic system ( n ~ = 1, n t = n - 1), the electron d y n a n u c s is partially suppressed a n d gives ws = 0.16 for the same p a r a m e t e r s as in fig. 3. A t the same time the exchange splitting A is found to b e substantially reduced from the H a r t r e e - F o c k value AHF - - ( 1 - - n t ) U to ,A/AHV = 0.43, 0.31 for U / W = 0.5, 1.0, respectively. The b o u n d two-hole states m a y b e directly investigated by calculating the density of two-hole excitations D(o~) = ~ l i q ' m l a o r a o , I4)05 I 2 m
2.5
(5)
The exact result at n = 2 gives a quasiatomic narrow line at finite U [7]. This feature is rather well reproduced by finite chain calculations. At n < 2 no rigorous result is known. We have received the spectra which exhibit substantially b r o a d e r structures than in the filled case. D(0~) is large at ~0 which is close to the satellite in one-hole spectrum, indicating clearly the physical origin of the m a x i m u m due to two-hole b o u n d states (see fig. 3). T h e one- a n d two-hole rigorous excitation spectra of short chains are f o u n d to agree rather well with those o b t a i n e d by using SOPT [2,3] for U / W < 1. This proves that SOPT may b e successfully used for qualitative i n t e r p r e t a t i o n of experimental data for transition metals [2]. In particular, satellite structures have been observed in photoemission experiments on Ni. The distance between the satellite a n d b a n d part of b o t h one- and two-hole excitation spectrum may be used to estimate the value of U. This distance is always larger for the one-hole spectrum, as seen in fig. 4. Therefore, the estimation of U from Auger spectrum should be more accurate t h a n the one from photoemission. In summary, we have d e m o n s t r a t e d that the manyb o d y effects present in one-hole excitation spectra are quite different for n = 1 a n d n > 1. W e believe that our results may be used to interpret the photoemission a n d Auger spectra of certain quasi-one-dimensional solids in the near future. It is a pleasure to t h a n k Prof. J. Friedel for his interest in this work a n d for valuabl~ discussions. [1] [2] [3] [4] [5] [6] [7]
J. Hubbard, Phys. Rev. B 17 (1978) 494. G. Tr6glia et al., J. de Phys. 41 (1980) 281. G. Tr6glia et al., J. Phys. C 14 (1981) 4347. R. Jullien and R.M. Martin, Phys. Rev. B 26 (1982) 6173. L. Sneddon, J. Phys. C 11 (1978) 2823. A.M. Oleg et al., Phys. Rev. B 32 (1985) 2167. G.A. Sawatzky, Phys. Rev. Lett. 39 (1977) 504.
Journal of Magnetism and Magnetic Materials 54-57 (1986) 1029-1030 ONE- AND TWO-HOLE
EXCITATION
SPECTRA
A . M . O L E S *, G . T R I ~ G L I A , D. S P A N J A A R D
OF THE
ld HUBBARD
MODEL
a n d R. J U L I E N
Laboratoire de Physique des Solides, Universitb de Paris-Sud, F-91405 Orsav, France
Modified periodic boundary conditions are used to obtain one- and two-hole excitation spectra by considering small chains described by the ld Hubbard model. A quasiparticle description and the nature of many-body effects are discussed. The results agree rather well with those of second order perturbation theory.
1. Finite cell calculations for the ld Hubbard model
a n d dispersion
The H u b b a r d model H
ok =
= --tY'.a+oajo + U ~ , n i t ni$, 0o
is quite realistic for quasi-one-dimensional solids [1]. Here we use it to study rigorously the m a n y - b o d y effects in one- a n d two-hole excitation spectra in one d i m e n s i o n b y p e r f o r m i n g calculations on finite systems. They are used to check the accuracy of second order p e r t u r b a t i o n theory (SOPT) [2,3]. The simulation of an infinite H u b b a r d chain with rigorous solutions of finite systems is possible by using modified periodic b o u n d a r y conditions [4]. They require that the wave function changes its p h a s e by an arbitrary value ¢ after m a k i n g a rotation along a ring of N sites. This allows to c o m p a r e the results o b t a i n e d with the chains of even a n d odd value N / 2 a n d extrapolate t h e m to a n infinite chain by finite cell scaling (FCS) [5]. As a test of our approach, we have p e r f o r m e d a FCS of the gap in the half-filled case ( n = 1) by using the chains up to N = 8 [6]. T h e gap has b e e n reproduced quite well for 0.5 < U / W < 2.0, where W = 4 t . 2. One-hole excitation spectrum in the half-filled case T h e one-hole excitation spectrum of a finite H u b b a r d chain is characterized by the spectral density function
Ak+(W)=~,[(et"~[akslCbo)126(o~-Eo+E~).
d~o ( o 3 + E k - - T U )
× (SLd, These quantities were found numerically for the chains N = 4, 6 [6]. As an example, the result for N = 6 a n d U / W = 0 . 5 , 2.0 is presented in fig. 1. If U = 0 , the energy E k has a physical m e a n i n g only for k ~< "n/2. W i t h increasing U / W a gradual transition is seen from the b a n d picture to the one resulting from an excitation of one hole in a n otherwise almost localized system. The presence of m i n i m u m k m in E k allows to define two regions in the k-space: for k < k m the quasiparticle states have mainly b a n d character, while the states with k > k m may be called scattering states. The latter states have first (at U / W = 0.5) rather small weight and m u c h larger o k2 t h a n the respective b a n d states. For larger U / W they become more weighted a n d n o difference in Ok2 between the two regions is observed. The transition from b a n d to localized states is also seen in the density of quasiparticle states p(~0). The hole spectrum at U = 0 is a half of the density of states of a one-dimensional infinite chain. It has a singularity at w = - W / 2 which disappears at finite U / W . A t the same time the weight of scattering states increases and they contribute to the low-energy tail of p(~0). The numerical result is presented in fig. 2. The structure
(2)
m
i
J
b
#'l
/
,,' 0.5
-
,
I
',, )~-, "
l
",,
~
- o.o5
(3) --'~"
o
* On leave of absence from the Institute of Physics, Jagellonian University, Cracow, Poland. 0304-8853/86/$03.50
i
~.s u/wo2.o
T h e system is assumed to b e paramagnetic, i.e. n t = n = ½. It is c o n v e n i e n t to describe the spectral density in terms of quasiparticles with energy
+ 7 ~s
Ak~(~
(1)
i
i
0.5 k E~]
i
0
1.O ,'-
Fig. 1. Quasiparticle quantities Ek and o~ at n =1 for N = 6 chain.
© E l s e v i e r S c i e n c e P u b l i s h e r s B.V.
1030
A.M. Ole~ et al. / Spectra of the ld Hubbard model .
I
T 2.8 _
n = 1.67
n=l.O
n = 1.67
~" 0.8
2.0
0.6
k2
0.4
o.
1.75 <~~0
0.2 -1 .o wlW
o
.
.
-3
~ l
•~ 1.25
0.4 -2.0
.
1.0
-2
-1 0 t u l W ~
1.00 i
Fig. 2. One-hole spectrum at U / W = 1 obtained with N 6 chain (solid line) and by using SOPT for an infinite chain (dashed line).
0
=
Fig. 3. One-hole (solid line) and two-hole (dashed line) excitation spectrum obtained with N = 6 chain at U / W = 1.
0.5
i
I
1.0 1.5 W/U
i
2.0
Fig. 4. Energy difference between the centers of gravity of the band and satellite part of the one-hole paramagnetic (A~), ferromagnetic (A F) and two-hole (A 2) spectrum for N = 6. × 8(o~- E0 + Em + 2U).
with two sharp m a x i m a on the right is due to the b a n d states. They occupy a narrower region that the hole s p e c t r u m at U = 0. T h e tail of scattering states is in principle well reproduced by SOPT results for an infinite chain up to U / W = 1. Obviously, SOPT fails to reproduce the gap. The essential features of quasiparticle spectra do not d e p e n d on the size of considered chain. FCS p e r f o r m e d on the weights of the b a n d (scattering) part of the s p e c t r u m gave the weight of an infinite chain which were increased (reduced) from those of chain N = 6 by a b o u t 20% [6].
3. Satellite structures in the one- and two-hole excitation spectra
The shape of the one-hole excitation spectra at n > 1 changes even more drastically with increasing U / W since new satellite structures a p p e a r to the left from the b a n d part for sufficiently large U / W . Their weight increases with increasing U / W up to the atomic limit value 2 - n. The convergence is, however, quite slow. This indicates that the dynamics of electrons having the opposite spin to the one of the excited hole plays a n i m p o r t a n t role. In fig. 3 the spectrum received with the chain N = 6 at n = 1.67 is presented. The satellite is well separated from the m a i n line a n d has still a rather small weight w~ = 0.09. If we make a n excitation in a majority s u b b a n d of a ferromagnetic system ( n ~ = 1, n t = n - 1), the electron d y n a n u c s is partially suppressed a n d gives ws = 0.16 for the same p a r a m e t e r s as in fig. 3. A t the same time the exchange splitting A is found to b e substantially reduced from the H a r t r e e - F o c k value AHF - - ( 1 - - n t ) U to ,A/AHV = 0.43, 0.31 for U / W = 0.5, 1.0, respectively. The b o u n d two-hole states m a y b e directly investigated by calculating the density of two-hole excitations D(o~) = ~ l i q ' m l a o r a o , I4)05 I 2 m
2.5
(5)
The exact result at n = 2 gives a quasiatomic narrow line at finite U [7]. This feature is rather well reproduced by finite chain calculations. At n < 2 no rigorous result is known. We have received the spectra which exhibit substantially b r o a d e r structures than in the filled case. D(0~) is large at ~0 which is close to the satellite in one-hole spectrum, indicating clearly the physical origin of the m a x i m u m due to two-hole b o u n d states (see fig. 3). T h e one- a n d two-hole rigorous excitation spectra of short chains are f o u n d to agree rather well with those o b t a i n e d by using SOPT [2,3] for U / W < 1. This proves that SOPT may b e successfully used for qualitative i n t e r p r e t a t i o n of experimental data for transition metals [2]. In particular, satellite structures have been observed in photoemission experiments on Ni. The distance between the satellite a n d b a n d part of b o t h one- and two-hole excitation spectrum may be used to estimate the value of U. This distance is always larger for the one-hole spectrum, as seen in fig. 4. Therefore, the estimation of U from Auger spectrum should be more accurate t h a n the one from photoemission. In summary, we have d e m o n s t r a t e d that the manyb o d y effects present in one-hole excitation spectra are quite different for n = 1 a n d n > 1. W e believe that our results may be used to interpret the photoemission a n d Auger spectra of certain quasi-one-dimensional solids in the near future. It is a pleasure to t h a n k Prof. J. Friedel for his interest in this work a n d for valuabl~ discussions. [1] [2] [3] [4] [5] [6] [7]
J. Hubbard, Phys. Rev. B 17 (1978) 494. G. Tr6glia et al., J. de Phys. 41 (1980) 281. G. Tr6glia et al., J. Phys. C 14 (1981) 4347. R. Jullien and R.M. Martin, Phys. Rev. B 26 (1982) 6173. L. Sneddon, J. Phys. C 11 (1978) 2823. A.M. Oleg et al., Phys. Rev. B 32 (1985) 2167. G.A. Sawatzky, Phys. Rev. Lett. 39 (1977) 504.