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Two-pole approach to the Hubbard bands
ELSEVIER
Physica B 206 & 207 (1995) 685-687
Two-pole approach to the Hubbard bands Andrzej M. Ole~ a, Henk Eskes b'* alnstitute of Physics, Jagellonian University, Reymonta 4, PL-30059 Krak6w, Poland bMax-Planck-lnstitut fiir FestkOrperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
Abstract
We test the accuracy of a two-delta function ansatz, as an approximate approach to the one-particle spectra in the Hubbard model, by comparing it with large-U perturbation theory and numerical calculations. It is found that the k-dependent correlation functions (the local spin order) are crucial to reproduce the numerical results.
moment contains as well the two-site correlation functions,
I. Introduction
The Hubbard model [1], H = - t ~'~ a~ai+~, ~ + U ~'~ n i t ni~ , i8o~
f.__(3) rtk,cr
= U3n,~ + U2[(2na + n~)E,2 _}_ Bk~] + 3UEkn, ,2
..~
(1)
O)
i
is generic for the physical properties of strongiy correlated systems• If the Coulomb interaction U is large, the excitation spectrum splits into two regions, the lower (LHB) and upper (UHB) Hubbard bands, separated by a gap. Recently, rather spectacular spectral weight transfers between these two bands have been observed experimentally [2] and found in numerical analysis [3,4]. Yet, the present understanding of these rapid changes of the spectral weights is still far from complete. The two Hubbard sub-bands are directly obtained using a so-called two-pole ansatz for the k-dependent spectral density [5], Ak,,(¢o) = ~] wi.,,,.3(w - e,.,¢).
(2)
i-1,2
The best approximation to the one-particle spectrum is obtained by conserving the first four moments, f"--~") nkcr , n = 0 , . . . ,3, of A,~(o)). The first three are determined by the dispersion, ~, = - t Z ~ exp[ik.(R~+sR~)], U, and the density n~ = (ni~), while the fourth
~k3
t
2
-- (ni~aisai+,.~n,.,,~) ] + ,k((ni~,ni+8,&) -- n a (a,~a,+,,oa,oa,+~.~) - (a~a~oa,+s,oa,+~,~)) .
-
(4) For a periodic system of N sites the expectation values are site independent. The quantity B,~ consists of a local term ~ t, which can be calculated selfconsistently [6], and a non-local term ~ , which has to be approximated or which is sometimes neglected• The ansatz (Eq. 2) can be viewed as an improved Hubbard I approximation [1], which is obtained by setting Bk, = 0. In one dimension (1D) for large U, however, analytic expressions can be found• Using the spincharge separated Ogata-Shiba wave function [7] we find B k ~ •
* Corresponding author. 0921-4526/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved S S D I 0921-4526(94)00556-7
sin(~rn)
~t
'11"
-
~k
[-~
_
( ~1 - - l n 2 ) ( n 2
sinZ---(~rn)-)]. xr 2 .J
(5)
686
A.M. O16s, H. Eskes / Physica B 206 & 207 (1995) 685-687
We have shown recently [8] that, for large U, sum rules for the individual bands can be obtained by performing a canonical transformation to new fermions ci~ which, unlike the a-fermions, do not change the number of double occupancies. Following Harris and Lange [9] the spectral moments can be defined. The weight of the LHB at momentum k is given by _{o} = :~( {a,~;o,a~;o , } ) where the fermion operators fflk; 0 conserve the number of double occupancies. The results are found to be identical to an expansion of Eqs. ( 2 ) - ( 4 ) to first order in t/U. In second order the two approaches start to deviate. The two-pole approach reproduces very accurately the numerical k-dependent weight in the LHB at half filling (Fig. la), equivalent in this case to the occupation number nk, and gives better results than the first-order perturbation theory [4,8]. For n = 0.9 the numerical spectra show a large Luttinger-like Fermi surface and the momentum dependent increase of the weight in the LHB (Fig. lb) is more sensitive to the hole dynamics at finite U. Only for very large U (U = 20) the transferred spectral weight is well described by the first order in t / U perturbation expression, and it distributes over all k values. On the contrary, with decreasing U the additional weight is
1,0
.
.
.
.
i
.
.
.
.
°.2o
r+
~---~. o.6 E
-
n
1 + -~
(Ei~,+~
(6)
where the twiggled operators are expressed in terms of c-fermions. The same expression is obtained by expanding Qk~ to order t. It follows from Eq. (6) that the average dispersion of the LHB around half filling (n = 1) is determined by the nearest-neighbour spin correlation function. Thus, contrary to the frequently used simplifications [6], the dynamics is correctly described only if this term is included! Writing E LHB'ID = AEk + const, we find in 1D [8], A = 2 ~ -2 ~ [1 - n +
(Si'Si+,+l)~(n
2
sin/(~n))] -~ .j.
(7) The average ( . . . H refers to the spin degrees of freedom only, described by the Heisenberg model [7]• For the ground state ( S i ' S i + l ) n = l / 4 - 1 n 2 , independent of n. With increasing filling one finds a crossover at n = 0 . 8 6 , shown in Fig. 2, from the narrowed free-fermion dispersion to the inverted one at half filling, with the minimum of the (averaged) LHB at k = ~r. The latter dispersion may be seen as driven by quantum fluctuations in a Heisenberg antiferromagnet, as no dispersion would result in a N6el
~_.l_..oz._..._ U=5
0.0
,
,
(b)
I
2Ek /
0.4 0.2
~" ,E:
mainly distributed over k states in vicinity of the Luttinger Fermi level k F. The average energy in the LHB is ._.{l) ELHB fftko';O t 1 ~ , _ 2- n N (c,aci+~.~) ,,,~ = I.._(o) rftktr; 0 i~o"
0.2
,
o.1 ~
•
''
0
•
~
.
.
.
.
.
.
.
.
.
.
0.5 <
.
.¢-.
.
rc
1.0
.
\u=5 / /
•
.
:2,, 0.0
.
,
U=IO
-E~"
I
,
.
.
.
0.0
.
2~
k Fig. 1. Weight of the LHB ink; (0)0 at n = 1 (a) and k dependence of the weight transfer from UHB to LHB, ink; 0(°) (n = 0 • 9"~ i - - f_(o) r t k ; O (n = 1), (b), in 1D (t = 1). Symbols are numerical data for a 10-site Hubbard ring, and lines follow from the two-pole approach combined with the Ogata-Shiba wave function [7].
-0.5
0.0
................... 0.2 0.4 0.6 n
0.8
.0
Fig. 2. Dispersion coefficient A of the first moment of the LHB in 1D, E LHs = Aek + const, as a function of density n, for antiferromagnetic (full), N6ei (dashed-dotted), random (short dashed), and ferromagnetic (long dashed) spin order.
A.M. Ol~s, H. Eskes I Physica B 206 & 207 (1995) 685-687
state considered in the retraceable-path approach [10] (daslied-dotted curve, n -- 1). The reversed dispersion cannot be reproduced by a mean-field description of an antiferromagnet in a doubled unit-cell, or by slaveboson mean-field theories. In contrast, for a random or ferromagnetic spin order, the dispersion is not reversed (see Fig. 2). The crucial dependence of A on the local spin order implies, for instance, that one has to go down to temperatures k BT ~
n;O0 ... ,
-
.
.
,
.
1 f ~'~
.
.
We thank L.F. Feiner, P. Horsch and G . A . Sawatzky for valuable discussions. A . M . O . acknowledges the support by the Committee of Scientific Research (KBN) Project 2 P302 93 055 05.
.
• ._u--2°
n :0 9 I
i
t
U :5 i
i
!
i
n
i
i
.f, " ~~. . , ..- ~. ~
-2
n=l 0
.n.n. _. t n
"'~u=5 ~
order t/U, while second-order corrections - ~ ( t / U ) 2 are important at U = 5. Summarizing, we obtained a satisfactory description of the transferred spectral weight to the LHB and the momentum dependence of the one-particle spectrum by the ansatz (Eq. 2). The two poles have to be interpreted as describing the average energies and weights of the two bands. However, for intermediate U, the two-pole approach gives a much better description of the fast density-dependent changes around half filling in the k-dependent spectra than the perturbation theory. The k-dependence (energies ~ t) is given largely by the local spin order (determined on an energy scale J). We note that the incoherent width of the LHB is completely neglected in the two-pole approach. The incoherent processes are responsible for the observed deviations of the two g-function ansatz from the numerical results for n = 0.9. Although we have presented explicit results only in 1D, the qualitative conclusions hold for higher dimensions because the sum rules for the lower and upper Hubbard band are determined by local correlation functions. This also explains the nice agreement with the finite-size calculations.
Acknowledgement
0
-1
687
2r~
k Fig. 3. Average energy of the LHB E LHa for U = 5, 10 and 20 (t = 1), for n = 1, 0.9, and 0.6. Symbols and lines as Fig. 1.
References [1] J. Hubbard, Proc. R. Soc. London A 276 (1963) 238. [2] H. Romberg et al., Phys. Rev. (B) 42 (1990) 8768; C.T. Chen et al., Phys. Rev. Lett. 66 (1991) 104. [3] H. Eskes, M.B.J. Meinders and G.A. Sawatzky, Phys. Rev. Lett. 67 (1991) 1035; M.B.J. Meinders, H. Eskes and G.A. Sawatzky, Phys. Rev. (B) 48 (1993) 3916. [4] M.S. Hybertsen et al., Phys. Rev. (B) 45 (1992) 10032. [5] L. Roth, Phys. Rev. 184 (1969) 451. [6] G. Geipel and W. Nolting, Phys. Rev. B 38 (1988) 2608; W. Nolting and W. Borgiel, Phys. Rev. B 39 (1989) 6962. [7] M. Ogata and H. Shiba, Phys. Rev. B 41 (1990) 2326. [8] H. Eskes, A.M. Olrs, M.B.J. Meinders and W. Stephan, unpublished. [9] A.B. Harris and R.V. Lange, Phys. Rev. 157 (1967) 295. [10] W.F. Brinkman and T.M. Rice, Phys. Rev. B 2 (1970) 1324.
Physica B 206 & 207 (1995) 685-687
Two-pole approach to the Hubbard bands Andrzej M. Ole~ a, Henk Eskes b'* alnstitute of Physics, Jagellonian University, Reymonta 4, PL-30059 Krak6w, Poland bMax-Planck-lnstitut fiir FestkOrperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
Abstract
We test the accuracy of a two-delta function ansatz, as an approximate approach to the one-particle spectra in the Hubbard model, by comparing it with large-U perturbation theory and numerical calculations. It is found that the k-dependent correlation functions (the local spin order) are crucial to reproduce the numerical results.
moment contains as well the two-site correlation functions,
I. Introduction
The Hubbard model [1], H = - t ~'~ a~ai+~, ~ + U ~'~ n i t ni~ , i8o~
f.__(3) rtk,cr
= U3n,~ + U2[(2na + n~)E,2 _}_ Bk~] + 3UEkn, ,2
..~
(1)
O)
i
is generic for the physical properties of strongiy correlated systems• If the Coulomb interaction U is large, the excitation spectrum splits into two regions, the lower (LHB) and upper (UHB) Hubbard bands, separated by a gap. Recently, rather spectacular spectral weight transfers between these two bands have been observed experimentally [2] and found in numerical analysis [3,4]. Yet, the present understanding of these rapid changes of the spectral weights is still far from complete. The two Hubbard sub-bands are directly obtained using a so-called two-pole ansatz for the k-dependent spectral density [5], Ak,,(¢o) = ~] wi.,,,.3(w - e,.,¢).
(2)
i-1,2
The best approximation to the one-particle spectrum is obtained by conserving the first four moments, f"--~") nkcr , n = 0 , . . . ,3, of A,~(o)). The first three are determined by the dispersion, ~, = - t Z ~ exp[ik.(R~+sR~)], U, and the density n~ = (ni~), while the fourth
~k3
t
2
-- (ni~aisai+,.~n,.,,~) ] + ,k((ni~,ni+8,&) -- n a (a,~a,+,,oa,oa,+~.~) - (a~a~oa,+s,oa,+~,~)) .
-
(4) For a periodic system of N sites the expectation values are site independent. The quantity B,~ consists of a local term ~ t, which can be calculated selfconsistently [6], and a non-local term ~ , which has to be approximated or which is sometimes neglected• The ansatz (Eq. 2) can be viewed as an improved Hubbard I approximation [1], which is obtained by setting Bk, = 0. In one dimension (1D) for large U, however, analytic expressions can be found• Using the spincharge separated Ogata-Shiba wave function [7] we find B k ~ •
* Corresponding author. 0921-4526/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved S S D I 0921-4526(94)00556-7
sin(~rn)
~t
'11"
-
~k
[-~
_
( ~1 - - l n 2 ) ( n 2
sinZ---(~rn)-)]. xr 2 .J
(5)
686
A.M. O16s, H. Eskes / Physica B 206 & 207 (1995) 685-687
We have shown recently [8] that, for large U, sum rules for the individual bands can be obtained by performing a canonical transformation to new fermions ci~ which, unlike the a-fermions, do not change the number of double occupancies. Following Harris and Lange [9] the spectral moments can be defined. The weight of the LHB at momentum k is given by _{o} = :~( {a,~;o,a~;o , } ) where the fermion operators fflk; 0 conserve the number of double occupancies. The results are found to be identical to an expansion of Eqs. ( 2 ) - ( 4 ) to first order in t/U. In second order the two approaches start to deviate. The two-pole approach reproduces very accurately the numerical k-dependent weight in the LHB at half filling (Fig. la), equivalent in this case to the occupation number nk, and gives better results than the first-order perturbation theory [4,8]. For n = 0.9 the numerical spectra show a large Luttinger-like Fermi surface and the momentum dependent increase of the weight in the LHB (Fig. lb) is more sensitive to the hole dynamics at finite U. Only for very large U (U = 20) the transferred spectral weight is well described by the first order in t / U perturbation expression, and it distributes over all k values. On the contrary, with decreasing U the additional weight is
1,0
.
.
.
.
i
.
.
.
.
°.2o
r+
~---~. o.6 E
-
n
1 + -~
(Ei~,+~
(6)
where the twiggled operators are expressed in terms of c-fermions. The same expression is obtained by expanding Qk~ to order t. It follows from Eq. (6) that the average dispersion of the LHB around half filling (n = 1) is determined by the nearest-neighbour spin correlation function. Thus, contrary to the frequently used simplifications [6], the dynamics is correctly described only if this term is included! Writing E LHB'ID = AEk + const, we find in 1D [8], A = 2 ~ -2 ~ [1 - n +
(Si'Si+,+l)~(n
2
sin/(~n))] -~ .j.
(7) The average ( . . . H refers to the spin degrees of freedom only, described by the Heisenberg model [7]• For the ground state ( S i ' S i + l ) n = l / 4 - 1 n 2 , independent of n. With increasing filling one finds a crossover at n = 0 . 8 6 , shown in Fig. 2, from the narrowed free-fermion dispersion to the inverted one at half filling, with the minimum of the (averaged) LHB at k = ~r. The latter dispersion may be seen as driven by quantum fluctuations in a Heisenberg antiferromagnet, as no dispersion would result in a N6el
~_.l_..oz._..._ U=5
0.0
,
,
(b)
I
2Ek /
0.4 0.2
~" ,E:
mainly distributed over k states in vicinity of the Luttinger Fermi level k F. The average energy in the LHB is ._.{l) ELHB fftko';O t 1 ~ , _ 2- n N (c,aci+~.~) ,,,~ = I.._(o) rftktr; 0 i~o"
0.2
,
o.1 ~
•
''
0
•
~
.
.
.
.
.
.
.
.
.
.
0.5 <
.
.¢-.
.
rc
1.0
.
\u=5 / /
•
.
:2,, 0.0
.
,
U=IO
-E~"
I
,
.
.
.
0.0
.
2~
k Fig. 1. Weight of the LHB ink; (0)0 at n = 1 (a) and k dependence of the weight transfer from UHB to LHB, ink; 0(°) (n = 0 • 9"~ i - - f_(o) r t k ; O (n = 1), (b), in 1D (t = 1). Symbols are numerical data for a 10-site Hubbard ring, and lines follow from the two-pole approach combined with the Ogata-Shiba wave function [7].
-0.5
0.0
................... 0.2 0.4 0.6 n
0.8
.0
Fig. 2. Dispersion coefficient A of the first moment of the LHB in 1D, E LHs = Aek + const, as a function of density n, for antiferromagnetic (full), N6ei (dashed-dotted), random (short dashed), and ferromagnetic (long dashed) spin order.
A.M. Ol~s, H. Eskes I Physica B 206 & 207 (1995) 685-687
state considered in the retraceable-path approach [10] (daslied-dotted curve, n -- 1). The reversed dispersion cannot be reproduced by a mean-field description of an antiferromagnet in a doubled unit-cell, or by slaveboson mean-field theories. In contrast, for a random or ferromagnetic spin order, the dispersion is not reversed (see Fig. 2). The crucial dependence of A on the local spin order implies, for instance, that one has to go down to temperatures k BT ~
n;O0 ... ,
-
.
.
,
.
1 f ~'~
.
.
We thank L.F. Feiner, P. Horsch and G . A . Sawatzky for valuable discussions. A . M . O . acknowledges the support by the Committee of Scientific Research (KBN) Project 2 P302 93 055 05.
.
• ._u--2°
n :0 9 I
i
t
U :5 i
i
!
i
n
i
i
.f, " ~~. . , ..- ~. ~
-2
n=l 0
.n.n. _. t n
"'~u=5 ~
order t/U, while second-order corrections - ~ ( t / U ) 2 are important at U = 5. Summarizing, we obtained a satisfactory description of the transferred spectral weight to the LHB and the momentum dependence of the one-particle spectrum by the ansatz (Eq. 2). The two poles have to be interpreted as describing the average energies and weights of the two bands. However, for intermediate U, the two-pole approach gives a much better description of the fast density-dependent changes around half filling in the k-dependent spectra than the perturbation theory. The k-dependence (energies ~ t) is given largely by the local spin order (determined on an energy scale J). We note that the incoherent width of the LHB is completely neglected in the two-pole approach. The incoherent processes are responsible for the observed deviations of the two g-function ansatz from the numerical results for n = 0.9. Although we have presented explicit results only in 1D, the qualitative conclusions hold for higher dimensions because the sum rules for the lower and upper Hubbard band are determined by local correlation functions. This also explains the nice agreement with the finite-size calculations.
Acknowledgement
0
-1
687
2r~
k Fig. 3. Average energy of the LHB E LHa for U = 5, 10 and 20 (t = 1), for n = 1, 0.9, and 0.6. Symbols and lines as Fig. 1.
References [1] J. Hubbard, Proc. R. Soc. London A 276 (1963) 238. [2] H. Romberg et al., Phys. Rev. (B) 42 (1990) 8768; C.T. Chen et al., Phys. Rev. Lett. 66 (1991) 104. [3] H. Eskes, M.B.J. Meinders and G.A. Sawatzky, Phys. Rev. Lett. 67 (1991) 1035; M.B.J. Meinders, H. Eskes and G.A. Sawatzky, Phys. Rev. (B) 48 (1993) 3916. [4] M.S. Hybertsen et al., Phys. Rev. (B) 45 (1992) 10032. [5] L. Roth, Phys. Rev. 184 (1969) 451. [6] G. Geipel and W. Nolting, Phys. Rev. B 38 (1988) 2608; W. Nolting and W. Borgiel, Phys. Rev. B 39 (1989) 6962. [7] M. Ogata and H. Shiba, Phys. Rev. B 41 (1990) 2326. [8] H. Eskes, A.M. Olrs, M.B.J. Meinders and W. Stephan, unpublished. [9] A.B. Harris and R.V. Lange, Phys. Rev. 157 (1967) 295. [10] W.F. Brinkman and T.M. Rice, Phys. Rev. B 2 (1970) 1324.