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Bogoliubov quasiparticle excitations in the negative-U Hubbard model: a small-cluster study
Physica C 235-240 (! 994) 2 ! 69-2170
PHYSlCA@
North-Holland
B O G O L I U B O V Q U A S I P A R T I C L E E X C I T A T I O N S IN T H E N E G A T I V E - U H U B B A R D MODEL: A SMALL-CLUSTER STUDY
Y. Ohta, A. Nakauchi, R. Eder, K. Tsutsui, and S. Maekawa
Department of Applied Physics, Nagoya University, Nagoya 464-01, Japan An exact-diagonalization technique on small clusters is used to study excitations and superconductivity in the two-dimensiona} negative-U Hubbard model. We calculate the Bogoliubov-quasiparticle spectrum, condensation amplitude, and coherence length as a function of the coupling strength (U/t), thereby working out how the picture of Bogoliubov quasiparticles in the BCS superconductors is affected by increasiag the attraction U[t. High-temperature superconductors are characteristic of the very short coherence length and may be in an intermediate regime between two limits, a BCS superconductor and a condensate of pre-formed bosons [1]. Although much theoretical effort has been devoted to examining this crossover regime, not very much is known, in particular, of the low-lying excitations. To consider this problem, we have studied the negative-U Hubbard model because of its controllable strength of attraction. This model, much discussed recently [2-8], may be a good reference system to more relevant models, such as the t - J model, for cuprate superconductivity. The Hamiltonian is written
n
.--
-t Z ( c + o c - + H + ) + v Z c itt c i t c i l c i la
i
with a negative value of U, where c~a is the electron-creation operator at site i and spin or, and summation is taken over all the nearestneighbor pairs on the two-dimensional square lattice. In this paper we examine the validity of the Bogoliubov-quasiparticle picture in the negative-U Hubbard model as a function of increasing attraction U/t. This is done by using a recently proposed [9] technique for examining low-lying exciations, i.e., an exact calculation of Bogoiiubov quasiparticle spectrum on small clusters. We thereby see whether low-lying states of the negative-U Hubbard cluster of a given attraction U/t can be described by the picture of Bogoliubov quasit>articles in the BCS pairing theory. For this purpose, we calculate the one-particle anomalous Green's function defined as G(k, z)
1 t,_,N ~ , o + 2.1 C u t z _ H + E 0 C _tk t l +.+,.N\ o]
where 1¢~) is the cluster ground state with N (even) electrons, E0 is the ground-state energy (averaged over N and N + 2 electron states), and Ctkc, is the Fourier transform of c++. We then define the spectral function F(k,w)=-(1/rc)ImG(k,w+io) with +7=+0 and its frequency integral
.Fk- (+,6'v++Ic+ktC+__ktl+'6+v>. This Green's function [9] should describe the excitation of a Bogoliubov quasiparticle in the cluster, i.e., F(k,w)=Fk6(W-Ek) with Fk:Ak/2Ek for the quasiparticle energy Ek and gap function A k , provide.] that the low-lying states can be described by the microcanonical version of the BCS pairing theory. We use a 4x 4 cluster with periodic boundary condition, where the ground states are at k=(0, 0) with point-group symmetry At for all fillings and parameter values U<0, consistent with the expected s-wave pairing. The calculated results for F ( k , w ) are shown in Figs. 1 (a) and (b). We find that, at U/t=-2, the spectrum fits fairly well with the BCS spectrum with s-wave gap function Ak=A0: pronounced low-energy peaks appear at Fermi momentum kF, smaller peaks appear at higher energies for other momenta, and observed gap is isotropic (see Fig. 1), and also the spectral weights are consistent with the BCS form of the condensation amplitude Fk with a maximum at kF (see Fig. 2). However, at U/t=-6, the peaks already extend to higher-energy regions and the dispersion becomes rather deformed, indicating that the Bogoliubov quasiparticle is becoming a less well-defined excitation although the lowest-energy peaks at kr are still sharp and well-defined. For much larger values of IUl/t the spectra are totally incoherent and the notion of Bogoliubov quasiparticles loses its significance. We also calculate the
0921-4534/94/S07.00 © 1994 - Elsevier Science B.V. All rights reserved. SSDI 0921-4534(94)01650-X
2170
Y.
0,5 (a)
(b)
k=(O,O)
Ohta et al./Physica C 233-240 (1994) 2169-2170
is of the size of the nearest-neighbor to nextnearest-neighbor distance. The quasiparticles appropriate for intermediate to strong coupling regime, of which a spatially-extended compositeboson description works well, will be discussed elsewhere [10]. Summarizing, we have applied a recently proposed technique for examining the low-lying excitation spectrum, i.e., exact calculation of anomalous Green's function characteristic of superconductivity, to the two-dimensional negative-U Hubbard modeld, and have shown how the picture of Bogoliubov-quasiparticle excitations in the BCS superconductors loses its significance as the coupling strength increases. This work was supported by Priority-Areas Grants from the Ministry of Education, Science, and Culture of Japan. R. E. acknowledges financial support by the Japan Society for Promotion of Science. Computations were partly carried out in the Computer Center of Institute for Molecular Science, Okazaki National Research Institutes.
k=(O,O)
0 k=(~/2,=/2) C ,,L
-
k=(~,=)
k=(=,=)
]%
k=(=,=/2)
k=(=,~12)
0.5 c
0.5!
k=(~, O) ,L
=
0.5
x/2,0)
k=(=/2,0) ~JL
100
0
5
10
~lt
Fig. 1. Bogoliubov-quasiparticle spectrum F(k, w) at (a) U/t=-2 an'~,"',~) U/t=-6 for filling of N=10. The dotted curves. :tow the BCS spectral function obtained for Ao/t:=0.4. We use the value q/t=O.15.
0.61
,
,
.
2.0
0.5 0.4
)
o31
•t
T/
~o
loo
777" /
i
=
00~-
~
• 0
10
"(:,") - -
' 20
0.5 ---~0.0 30
IUII t
Fig. 2. Condensation amplitude Fk and coherence length ~ as a function of the attraction U/t calculated for the 4 x 4 cluster with filling of N=8. coherence length ~ defined as k
k
the result (see Fig. 2) shows a rapid but smooth crossover from a Cooper-paired state to a Bose condensed state of tightly bound pairs. At [U[/t~_3-4,
REFERENCES [1] See, e.g., a recent review article by R. MichaS, J. Ranninger, and S. Robaszkiewicz, Rev. Mod. Phys. 62, 113(1990). [2] R. T. Scalettar, E. Y. Loh, J. E. Gubernatis, A. Moreo, S. R. White, D. J. Scalapino, R. L. Sugar, and E. Dagotto, Phys. Rev. Lett. 62, 1407 (1989). [3] A. Moreo and D. J. Scalapino, Phys. Rev. Lett. 66, 946 (1991). [4] A. Moreo, D. J. Scalapino, and S. R. White, Phys. Rev. B 45, 7544 (1992). [5] M. Randeria, N. Trivedi, A. Moreo, and R. T. Scalettar, Phys. Rev. Lett. 69, 2001 (1992). [6] J. O. Solo, C. A. Balseiro, and H. E. Castillo, Phys. Rev. B 45, 9860 (1992). [7] H. E. Ca.stillo and C. A. Balseiro, Phys. Rev. B 45, 10549 (1992). [8] F. F. Assaad, W. Hanke, and D. J. Scalapino, Phys. Rev. Lett. 71, 1915 (1993) and Phys. Rev. B 49, 4327 (1994). [9] Y. Ohta, T. Shimozato, R. Eder, and S. Maekawa, Phys. Rev. Lett., at press. [10] Y. Ohta et al., in preparation.
PHYSlCA@
North-Holland
B O G O L I U B O V Q U A S I P A R T I C L E E X C I T A T I O N S IN T H E N E G A T I V E - U H U B B A R D MODEL: A SMALL-CLUSTER STUDY
Y. Ohta, A. Nakauchi, R. Eder, K. Tsutsui, and S. Maekawa
Department of Applied Physics, Nagoya University, Nagoya 464-01, Japan An exact-diagonalization technique on small clusters is used to study excitations and superconductivity in the two-dimensiona} negative-U Hubbard model. We calculate the Bogoliubov-quasiparticle spectrum, condensation amplitude, and coherence length as a function of the coupling strength (U/t), thereby working out how the picture of Bogoliubov quasiparticles in the BCS superconductors is affected by increasiag the attraction U[t. High-temperature superconductors are characteristic of the very short coherence length and may be in an intermediate regime between two limits, a BCS superconductor and a condensate of pre-formed bosons [1]. Although much theoretical effort has been devoted to examining this crossover regime, not very much is known, in particular, of the low-lying excitations. To consider this problem, we have studied the negative-U Hubbard model because of its controllable strength of attraction. This model, much discussed recently [2-8], may be a good reference system to more relevant models, such as the t - J model, for cuprate superconductivity. The Hamiltonian is written
n
.--
-t Z ( c + o c - + H + ) + v Z c itt c i t c i l c i l
i
with a negative value of U, where c~a is the electron-creation operator at site i and spin or, and summation
1 t,_,N ~ , o + 2.1 C u t z _ H + E 0 C _tk t l +.+,.N\ o]
where 1¢~) is the cluster ground state with N (even) electrons, E0 is the ground-state energy (averaged over N and N + 2 electron states), and Ctkc, is the Fourier transform of c++. We then define the spectral function F(k,w)=-(1/rc)ImG(k,w+io) with +7=+0 and its frequency integral
.Fk- (+,6'v++Ic+ktC+__ktl+'6+v>. This Green's function [9] should describe the excitation of a Bogoliubov quasiparticle in the cluster, i.e., F(k,w)=Fk6(W-Ek) with Fk:Ak/2Ek for the quasiparticle energy Ek and gap function A k , provide.] that the low-lying states can be described by the microcanonical version of the BCS pairing theory. We use a 4x 4 cluster with periodic boundary condition, where the ground states are at k=(0, 0) with point-group symmetry At for all fillings and parameter values U<0, consistent with the expected s-wave pairing. The calculated results for F ( k , w ) are shown in Figs. 1 (a) and (b). We find that, at U/t=-2, the spectrum fits fairly well with the BCS spectrum with s-wave gap function Ak=A0: pronounced low-energy peaks appear at Fermi momentum kF, smaller peaks appear at higher energies for other momenta, and observed gap is isotropic (see Fig. 1), and also the spectral weights are consistent with the BCS form of the condensation amplitude Fk with a maximum at kF (see Fig. 2). However, at U/t=-6, the peaks already extend to higher-energy regions and the dispersion becomes rather deformed, indicating that the Bogoliubov quasiparticle is becoming a less well-defined excitation although the lowest-energy peaks at kr are still sharp and well-defined. For much larger values of IUl/t the spectra are totally incoherent and the notion of Bogoliubov quasiparticles loses its significance. We also calculate the
0921-4534/94/S07.00 © 1994 - Elsevier Science B.V. All rights reserved. SSDI 0921-4534(94)01650-X
2170
Y.
0,5 (a)
(b)
k=(O,O)
Ohta et al./Physica C 233-240 (1994) 2169-2170
is of the size of the nearest-neighbor to nextnearest-neighbor distance. The quasiparticles appropriate for intermediate to strong coupling regime, of which a spatially-extended compositeboson description works well, will be discussed elsewhere [10]. Summarizing, we have applied a recently proposed technique for examining the low-lying excitation spectrum, i.e., exact calculation of anomalous Green's function characteristic of superconductivity, to the two-dimensional negative-U Hubbard modeld, and have shown how the picture of Bogoliubov-quasiparticle excitations in the BCS superconductors loses its significance as the coupling strength increases. This work was supported by Priority-Areas Grants from the Ministry of Education, Science, and Culture of Japan. R. E. acknowledges financial support by the Japan Society for Promotion of Science. Computations were partly carried out in the Computer Center of Institute for Molecular Science, Okazaki National Research Institutes.
k=(O,O)
0 k=(~/2,=/2) C ,,L
-
k=(~,=)
k=(=,=)
]%
k=(=,=/2)
k=(=,~12)
0.5 c
0.5!
k=(~, O) ,L
=
0.5
x/2,0)
k=(=/2,0) ~JL
100
0
5
10
~lt
Fig. 1. Bogoliubov-quasiparticle spectrum F(k, w) at (a) U/t=-2 an'~,"',~) U/t=-6 for filling of N=10. The dotted curves. :tow the BCS spectral function obtained for Ao/t:=0.4. We use the value q/t=O.15.
0.61
,
,
.
2.0
0.5 0.4
)
o31
•t
T/
~o
loo
777" /
i
=
00~-
~
• 0
10
"(:,") - -
' 20
0.5 ---~0.0 30
IUII t
Fig. 2. Condensation amplitude Fk and coherence length ~ as a function of the attraction U/t calculated for the 4 x 4 cluster with filling of N=8. coherence length ~ defined as k
k
the result (see Fig. 2) shows a rapid but smooth crossover from a Cooper-paired state to a Bose condensed state of tightly bound pairs. At [U[/t~_3-4,
REFERENCES [1] See, e.g., a recent review article by R. MichaS, J. Ranninger, and S. Robaszkiewicz, Rev. Mod. Phys. 62, 113(1990). [2] R. T. Scalettar, E. Y. Loh, J. E. Gubernatis, A. Moreo, S. R. White, D. J. Scalapino, R. L. Sugar, and E. Dagotto, Phys. Rev. Lett. 62, 1407 (1989). [3] A. Moreo and D. J. Scalapino, Phys. Rev. Lett. 66, 946 (1991). [4] A. Moreo, D. J. Scalapino, and S. R. White, Phys. Rev. B 45, 7544 (1992). [5] M. Randeria, N. Trivedi, A. Moreo, and R. T. Scalettar, Phys. Rev. Lett. 69, 2001 (1992). [6] J. O. Solo, C. A. Balseiro, and H. E. Castillo, Phys. Rev. B 45, 9860 (1992). [7] H. E. Ca.stillo and C. A. Balseiro, Phys. Rev. B 45, 10549 (1992). [8] F. F. Assaad, W. Hanke, and D. J. Scalapino, Phys. Rev. Lett. 71, 1915 (1993) and Phys. Rev. B 49, 4327 (1994). [9] Y. Ohta, T. Shimozato, R. Eder, and S. Maekawa, Phys. Rev. Lett., at press. [10] Y. Ohta et al., in preparation.