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Low density electron pairs in the extended Hubbard model
ELSEVIER
Synthetic
Metals 90 (1997)
77-79
Low density electron pairs in the extended Hubbard model ’
Abstract We study the extended I-Iubbard model in the parameter regionwhereit exhibitssuperconducting instabilities. The modelis defined by onsiteandnearest-neighbor CoulombinteractionsU andV, andhoppingintegralr. Wecalculatepairbindingenergies for variousfiniteclusters by performing exact diagonalization of the Hamiltonian. We show results for one-dimensional chains and discuss results for 1 X 4.6 X 6, and 8 X 8 two-dimensional square lattices. Results of solving the BCS equation for the pair binding energy as functions of carrier density II, and Coulomb interactions U and V, are cotnpared with the exact diagonalization results. In the dilute concentration limit (~2+ 0)) the BCS equation is expected to be exact and, in the low electron density region, the results for the pair binding energy are found to be in fairly good agreement with the exact solutions. 0 1997 Elsevier Science S.A. Kqivords:
Many-body and quasiparticle
theories; Computer
simulations
1. Introduction The Hamiltonian of the extendedHubbard model is zF’= --I
C
(&c,,+h.c.)
l k= -2t Ccosk.6;
6
+ U C~z~~iz,~
(ij).u
I
+ V Cn,ii - p Cni i
(1)
where (ij) runs over nearest-neighborpairs, c,,(c;‘,) is the electron annihilation (creation) operator of spin c at site i, and 12~ = C,C$~, is the number density operator. The parametersof this Hamiltonian are as follows: t is the hopping integral for nearest-neighbors,U is theon-siteCoulombinteraction, V is the nearest-neighborCoulomb repulsion, and ,u is the chemicalpotential. Another important parameteris the band-filling IZ=N,/N, where N, is the total number of electronsand N is the total numberof lattice sites. In parameterspacewheresuperconductivity may exist, we apply the BCS approximation to Eq. (1) and obtain the reducedHamiltonian: ~~ * Corresponding author. Tel.: t 552 2609 6365: fax: f 552 2603 5204. ’ Paper presented at the ICSM ‘96 Conference, Snowbird, UT, USA, 28 July-2.4~. 1996. 0379-6779/97/$17.00 PIISO379-6779(97)03954-4
8 1997 Elsevier
Science S.A. AUrights
resrrvcd
where Ssumsover the coordinates. To solve Eq. (2), a mean-field approximation is usedby assumingthat the interactionbetweenpairsis of higherorder. Thus. one obtainsa gap equationat zero temperature: (3)
togetherwith the electron numberequation:
(5) Eqs. (3) and (5) are solvedself-consistentlyandthe form of the gap d, dependson the form of interaction Vkk,.Solutions with S-. P-, and D-wave symmetriesare all possible. The ground state pairing_symmetryis determinedby comparing energies,calculatedby
78
I3.Q. Lin et al. /Sguheric
The quantity of interest is the pair binding energy for two additional particles at density n = NJN: q,(n) =2E(N,+
1) -E(N,)
-E(N,+2)
(7)
The energy per particle that is required to break a pair, +,(n) / 2, is the minimum quasiparticle excitation energy that will be compared with the results of the BCS solution: do=min{Ek}
results
In this work, we concentrate on the region where superconducting correlation dominates. We calculate thepairbinding energy by diagonalizing the Hamiltonians in Eqs. ( 1) and (2) exactly on finite clusters [4], and by solving the BCS equation, Eq. (3). Periodic boundary conditions were used in all these calculations. In general, results obtained from the three equations are different, for xncs picks up particular terms in Eq. ( 1) , while Eq. (3) is the mean-field solution of ‘A+nCS. However, in the dilute limit, solutions of the BCS equation are expected to be exact since interactions between pairs are negligible. Table I shows the ground state energy of the extended Hubbard model and its BCS form as a function of particle numbers. We see that Eq. (2) agrees with Eq. ( 1) at small n and deviates from it as n increases. In fact, for two electrons on any finite clusters, it can be shown exactly that Zscs gives the same answer as Eq. ( 1)) and the results of the BCS gap equation tend to the same answer as n * 0. In Fig. 1, we show the pair binding energy as a function of band-filling for U= -2.0, I/= -0.5. Diamonds are exact diagonalization results on a one-dimensional l&site lattice Table 1 Ground energy of Eqs. ( 1) and (2) as a function U= -2.0, V= -0.5 on a 12-site latiice
2 4 6
90 (1997)
77-79
L/F\,.-..-0 .4..., e, 0.6 -._‘.
--.,-1 0
-...
“--4----.~-6~~
OA
“-..*
. . ..- ---0
(8)
A positive value of eb implies that two particles tend to pair at the Fermi surface, so it is relevant to the question of existence of superconductivity. The extended Hubbard model exhibits rich phases corresponding to different symmetries. Extensive studies in one dimension have established that the charge-density-wave (CDW) phase, the spin-density-wave (SDW) phase, the singlet and triplet superconductingphases, and phase separation phases exist [ l-31. All of these phases are of great physical interest.
2. Numerical
Met&
.,I0.0 Fig. 1. Pair binding v= -0.5.
0.2
energy
0.4
n
as a function
0.6
0.6
of band-filling
1.0
for U= -2.0,
and the solid line is the solution of the BCS gap equation. It shows that the BCS solution overestimates pair binding but the overall trend is qualitatively the same as the exact results. We have also performed similar calculations on twodimensional 4 X 4, 6 X 6, and 8 X 8 lattices. On the 4 X 4 lattice, calculations could be done for any number of particles, while on the 6 X 6 and 8 X 8 lattices calculations were limited to six and four particles. respectively. Results in two dimensions are qualitatively the same as in one dimension, i.e., BCS solutions give agreeable answers as compared with exact diagonalization studies in the low density limit. Due to space limitation, we are unable to show any figures here. Compared to our previous studies ES], the agreements are not as good as reported there. For example, for fixed U the critical value of V (at which the pair binding energy is zero) obtained by the BCS gap equation is smaller (more negative) than that obtained by the exact diagonalization, One possibility is the finite size effect. Yet another possible reason is that in Ref. 151 superconductivity is driven by the occupation-dependent hopping term, At, and that term suppresses other symmetry breaking phases such as CDW and phase separations, while, in the extended Hubbard model, superconductivity is driven by Coulomb attractions and at the same time these attractions are also responsible for CDW or phase separations and thus correlations are much more important here than the case studied in Ref. [ 51.
3. Conclusions of band-filling
for
Eq. (1)
Eq. (2)
N,
Eq. (1)
Eq. (2)
-4.6563 - 9.2555 - 13.9039
-4.6564 -8.6752 - 12.1051
8 10 12
- 18.1816 - 22.7496 -26.5246
- 13.5500 - 16.4938 - 16.9524
In summary, we have studied the pair binding energy of the extended Hubbard model in one and two dimensions by using the exact diagonalization technique and BCS approach. It has been shown that, in the low density limit, i.e., for bandfilling II cO.2, BCS results for the pair binding energy are in fairly good agreement with the exact results.
H.Q. Lin et nl. /Sy~ii~eric
Acknowledgements This work was supported in part by the Direct Grant for Research from the Research Grants Council of the Hong Kong Government, and by the National Science Council of the Republic of China under Contract No. NSC85-2112-M032-004. Some of the computations were performed at the National Center for Supercomputing Applications, University of Illinois, at Urbana-Champaign, and the National Energy Research Supercomputer Center, Lawrence Livermore National Laboratory. We are grateful for their support.
Merols
90 (1997)
77-79
79
References [I] [2] [3j
[4] [5]
V.J. Emery, in J.T. Devreese et al. (eds.), Highly Conducting OneDimensional Solids, Plenum, New York, 1979, pp. 217-303. J. Solyom, Adv. Phys., 25 (1979) 201. H.Q. Lin, E.R. Gagliano, D.K. Campbell, E.H. Fradkin and J.E. Gubematis, in D. Baeriswyl et al. (eds.), The Hubbard Model: Proceedings of the 1993 NATO ARW on The Physics and Mathematical Physics of the Hubbard Model, Plenum, New York, 1995, to be published. H.Q. Lin and J.E. Gubematis, Comput. Phps.. 7 (1993) 100. H.Q. Lin and J.E. Hir& Phys. Rev. B, 52 (1995) 16 155.
Synthetic
Metals 90 (1997)
77-79
Low density electron pairs in the extended Hubbard model ’
Abstract We study the extended I-Iubbard model in the parameter regionwhereit exhibitssuperconducting instabilities. The modelis defined by onsiteandnearest-neighbor CoulombinteractionsU andV, andhoppingintegralr. Wecalculatepairbindingenergies for variousfiniteclusters by performing exact diagonalization of the Hamiltonian. We show results for one-dimensional chains and discuss results for 1 X 4.6 X 6, and 8 X 8 two-dimensional square lattices. Results of solving the BCS equation for the pair binding energy as functions of carrier density II, and Coulomb interactions U and V, are cotnpared with the exact diagonalization results. In the dilute concentration limit (~2+ 0)) the BCS equation is expected to be exact and, in the low electron density region, the results for the pair binding energy are found to be in fairly good agreement with the exact solutions. 0 1997 Elsevier Science S.A. Kqivords:
Many-body and quasiparticle
theories; Computer
simulations
1. Introduction The Hamiltonian of the extendedHubbard model is zF’= --I
C
(&c,,+h.c.)
l k= -2t Ccosk.6;
6
+ U C~z~~iz,~
(ij).u
I
+ V Cn,ii - p Cni i
(1)
where (ij) runs over nearest-neighborpairs, c,,(c;‘,) is the electron annihilation (creation) operator of spin c at site i, and 12~ = C,C$~, is the number density operator. The parametersof this Hamiltonian are as follows: t is the hopping integral for nearest-neighbors,U is theon-siteCoulombinteraction, V is the nearest-neighborCoulomb repulsion, and ,u is the chemicalpotential. Another important parameteris the band-filling IZ=N,/N, where N, is the total number of electronsand N is the total numberof lattice sites. In parameterspacewheresuperconductivity may exist, we apply the BCS approximation to Eq. (1) and obtain the reducedHamiltonian: ~~ * Corresponding author. Tel.: t 552 2609 6365: fax: f 552 2603 5204. ’ Paper presented at the ICSM ‘96 Conference, Snowbird, UT, USA, 28 July-2.4~. 1996. 0379-6779/97/$17.00 PIISO379-6779(97)03954-4
8 1997 Elsevier
Science S.A. AUrights
resrrvcd
where Ssumsover the coordinates. To solve Eq. (2), a mean-field approximation is usedby assumingthat the interactionbetweenpairsis of higherorder. Thus. one obtainsa gap equationat zero temperature: (3)
togetherwith the electron numberequation:
(5) Eqs. (3) and (5) are solvedself-consistentlyandthe form of the gap d, dependson the form of interaction Vkk,.Solutions with S-. P-, and D-wave symmetriesare all possible. The ground state pairing_symmetryis determinedby comparing energies,calculatedby
78
I3.Q. Lin et al. /Sguheric
The quantity of interest is the pair binding energy for two additional particles at density n = NJN: q,(n) =2E(N,+
1) -E(N,)
-E(N,+2)
(7)
The energy per particle that is required to break a pair, +,(n) / 2, is the minimum quasiparticle excitation energy that will be compared with the results of the BCS solution: do=min{Ek}
results
In this work, we concentrate on the region where superconducting correlation dominates. We calculate thepairbinding energy by diagonalizing the Hamiltonians in Eqs. ( 1) and (2) exactly on finite clusters [4], and by solving the BCS equation, Eq. (3). Periodic boundary conditions were used in all these calculations. In general, results obtained from the three equations are different, for xncs picks up particular terms in Eq. ( 1) , while Eq. (3) is the mean-field solution of ‘A+nCS. However, in the dilute limit, solutions of the BCS equation are expected to be exact since interactions between pairs are negligible. Table I shows the ground state energy of the extended Hubbard model and its BCS form as a function of particle numbers. We see that Eq. (2) agrees with Eq. ( 1) at small n and deviates from it as n increases. In fact, for two electrons on any finite clusters, it can be shown exactly that Zscs gives the same answer as Eq. ( 1)) and the results of the BCS gap equation tend to the same answer as n * 0. In Fig. 1, we show the pair binding energy as a function of band-filling for U= -2.0, I/= -0.5. Diamonds are exact diagonalization results on a one-dimensional l&site lattice Table 1 Ground energy of Eqs. ( 1) and (2) as a function U= -2.0, V= -0.5 on a 12-site latiice
2 4 6
90 (1997)
77-79
L/F\,.-..-0 .4..., e, 0.6 -._‘.
--.,-1 0
-...
“--4----.~-6~~
OA
“-..*
. . ..- ---0
(8)
A positive value of eb implies that two particles tend to pair at the Fermi surface, so it is relevant to the question of existence of superconductivity. The extended Hubbard model exhibits rich phases corresponding to different symmetries. Extensive studies in one dimension have established that the charge-density-wave (CDW) phase, the spin-density-wave (SDW) phase, the singlet and triplet superconductingphases, and phase separation phases exist [ l-31. All of these phases are of great physical interest.
2. Numerical
Met&
.,I0.0 Fig. 1. Pair binding v= -0.5.
0.2
energy
0.4
n
as a function
0.6
0.6
of band-filling
1.0
for U= -2.0,
and the solid line is the solution of the BCS gap equation. It shows that the BCS solution overestimates pair binding but the overall trend is qualitatively the same as the exact results. We have also performed similar calculations on twodimensional 4 X 4, 6 X 6, and 8 X 8 lattices. On the 4 X 4 lattice, calculations could be done for any number of particles, while on the 6 X 6 and 8 X 8 lattices calculations were limited to six and four particles. respectively. Results in two dimensions are qualitatively the same as in one dimension, i.e., BCS solutions give agreeable answers as compared with exact diagonalization studies in the low density limit. Due to space limitation, we are unable to show any figures here. Compared to our previous studies ES], the agreements are not as good as reported there. For example, for fixed U the critical value of V (at which the pair binding energy is zero) obtained by the BCS gap equation is smaller (more negative) than that obtained by the exact diagonalization, One possibility is the finite size effect. Yet another possible reason is that in Ref. 151 superconductivity is driven by the occupation-dependent hopping term, At, and that term suppresses other symmetry breaking phases such as CDW and phase separations, while, in the extended Hubbard model, superconductivity is driven by Coulomb attractions and at the same time these attractions are also responsible for CDW or phase separations and thus correlations are much more important here than the case studied in Ref. [ 51.
3. Conclusions of band-filling
for
Eq. (1)
Eq. (2)
N,
Eq. (1)
Eq. (2)
-4.6563 - 9.2555 - 13.9039
-4.6564 -8.6752 - 12.1051
8 10 12
- 18.1816 - 22.7496 -26.5246
- 13.5500 - 16.4938 - 16.9524
In summary, we have studied the pair binding energy of the extended Hubbard model in one and two dimensions by using the exact diagonalization technique and BCS approach. It has been shown that, in the low density limit, i.e., for bandfilling II cO.2, BCS results for the pair binding energy are in fairly good agreement with the exact results.
H.Q. Lin et nl. /Sy~ii~eric
Acknowledgements This work was supported in part by the Direct Grant for Research from the Research Grants Council of the Hong Kong Government, and by the National Science Council of the Republic of China under Contract No. NSC85-2112-M032-004. Some of the computations were performed at the National Center for Supercomputing Applications, University of Illinois, at Urbana-Champaign, and the National Energy Research Supercomputer Center, Lawrence Livermore National Laboratory. We are grateful for their support.
Merols
90 (1997)
77-79
79
References [I] [2] [3j
[4] [5]
V.J. Emery, in J.T. Devreese et al. (eds.), Highly Conducting OneDimensional Solids, Plenum, New York, 1979, pp. 217-303. J. Solyom, Adv. Phys., 25 (1979) 201. H.Q. Lin, E.R. Gagliano, D.K. Campbell, E.H. Fradkin and J.E. Gubematis, in D. Baeriswyl et al. (eds.), The Hubbard Model: Proceedings of the 1993 NATO ARW on The Physics and Mathematical Physics of the Hubbard Model, Plenum, New York, 1995, to be published. H.Q. Lin and J.E. Gubematis, Comput. Phps.. 7 (1993) 100. H.Q. Lin and J.E. Hir& Phys. Rev. B, 52 (1995) 16 155.