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Excitations and superconductivity in the two-dimensional t-J model
Physiea C 235-240 (! 994) 2277-2278 North-Holland
PHYSICA
EXCITATIONS AND SUPERCONDUCTIVITY IN THE TWO-DIMENSIONAL t - J MODEL Y. Ohta, T. Shimozato, R. Eder, and S. Maekawa Department of AppLied Physics, Nagoya University, Nagoya 464-01, Japan An exact-diagonalization technique on small clusters is used to study excitation spectra and superconductivity in the two-dimensional t - J model. We denmnstrate that the low-lying excitations in a wide parameter (J/t) and doping regions of the model are described well by the picture of dressed Bogoliubov quasiparticles within the BCS pairing theory. The pairing occurs predominantly in d~2_y2-wave channel and the energy gap has a size Ad"0.15J--0.27J between quarter and half fillings. One of the recent issues in the theory of hightemperature superconductivity is the question whether the two-dimensional (2D) t - J model contains a superconducting phase relevant to cuprate materials. Among many calculations made to address this question, Dagotto and Riera [1] have found indications of off-diagonal long-range order near the region of phase separation [2] by measuring equal-time pairing correlations directly in finite-size clusters; the existence of a dz2_u2-wave condensate has thereby been conjectured [1, 3]. We have recently proposed [4] a new technique for examining the low-lying excitations, i.e., the exact calculation of anomalous Green's functions (AGF) for the Bogoliubov-quasiparticle spectrum in finitesize clusters. The technique has so far been applied successfully to the 2D t - J model [4] and negativeU Hubbard model-[5]. In this paper we discuss the technique briefly and present some results for the 2D t - J model. The Hamiltonian is written
cr
where c~a is the projected electron-creation operator at site i and spin cr allowing no doubly occupied sites, Si is the electronic spin operator, and ni = nit+nil is the electron number operator. The summation is taken over nil the nearest-neighbor pairs on the square latt!':e. We adopt the working hypothesis th~tt the lowlying states of the cluster can be described by the microcanonical version of the BCS pairing theory. ttence, we assume that the cluster ground states with an even electron number N can be written as I¢0N)=P~IBCS), where IBCS> is the BCS wavefunction (formulated in terms of 'quasiparticles' so that the effects of strong correlations are already
included) and Pt¢ projects on the N-electron subspace. Similarly, we assume that the low-lying states with an odd electron-number N + I can be written as where 7kt~ cre-
l~+*)=P~+xT~=IBCS)
ates a Bogoliubov quasiparticle with m o m e n t u m k. Now, we define two AGF's as follows: 1
a ( k , z ) = (¢~+21Ctk! z - H + E0 G(k, k', z) = (~b0mlet_k, 1C-k t z - H + E~v n-kinkT 6kk, Z
where nk~--(~,~lc*k~l~5~'>, E N is the groundstate energy, and Cko is the Fourier transform of c~. We define the spectral function F(k,o~)= -(1/rc)ImG(k,w+irl) and its frequency integral Fk=(% +21C*ktC*_kt I ,g ) for one-particle AGF. The spectral function F(k, k', w) and its frequency integral Fkk, are also defined for two-particle AGF as above. Then, our working hypothesis predicts, via the Bogoliubov-Valatin transformation, that these AGF's describe the excitation of Bogoliubov quasiparticles in the cluster: i.e., F(k,w) = FkS(W- Ek) F ( k , k ' , w ) = FkF£,6(w- E k - Ek,) with Fk--ZkAk/2Ek, where Z k is the wave-function renormalization constant, Ek is tile renormalized quasiparticle energy, and Ak is the gap function. We have calculated AGF's for clusters of the size 4 x 4 and v / ~ × x / l ' 8 with periodic boundary condition. In Fig+ 1, we show the results for F(k, k', w). We find the following, all of which are
0921-4534/94/$07.00 © 1994 ElsevierScience B.V. All rights reserved. -
SSD!0921-4534(94)01704-2
lCkl
I! Ohtu et al. /Ph_vsica C 235-240
2278
0
4
80
4
80
4
8
w/t
Fig. 1. Bogoliubov-quasiparticle spectrum F k, k’, w) calculated for the 4x4 cluster with 8 ho I es at J/t=2.5. The momentum k dependence is shown with k’ fised at (0, -r/2). kF is located at (n/2,0) and its equivalent points. The dotted curves show the BCS s ectral function obtained for parameter values Ad f t=9.47, t,ff-/t=0.55, and zg=0.7.
consistent with expectations of the BCS pairing theory: A pronounced low-energy peak appears at kF and smaller peaks appear at higher energies for other momenta; the weights of the peaks are consistent with the BCS form of the condensation amplitude Fk with a ma?rimum at kF. The momentum dependence of F(k, k’, w), i.e., the change in sign under rotation by n/2 and almost vanishing weight along the X:“=ciy line, is a clear indication of dz2Ty 2-wave pairing. Although not shown here, the size of the energy gap scales well with J/t value, and with decreasing gap size, the peaks at, momenta other than kF lose their weight as espetted from the BCS theory. To make a quantitative comparison, we calculate the spectrum F(k, k’, w) given by the BCS predictions where we assume the quasiparticle energy &=t/Ei t- A: with one-electron energy and gap function &=- 2t,r(cos k, + cos ky)-p h-, - cos I;,). The effect ivc hopping nk = &&OS parameter t,r is used to take into account. the quasiparticle band narrowing 161, and he chemical potential 14 is chosen t,o guarantee vanishing
(1994) 2277-2275
fitting the positions of the low-energy peaks in F( k, k’, w). The renormalization factor :n (defined by assuming Zk to be k-independent) is estimated from weights of the low-energy peaks in F(k, k’, LJ). The parameters teff and Zn reflects the effects of strong correlation and imply the use of a dressed Bogoliubov-quasiparticle description. The fitted quasiparticle spectra are shown by dotted curves in Fig. 1. We find a fair agreement with the exact spectra, which demonstrates the validity of the BCS pairing theory for low-lying e?ccitations in the 2D t-J model. Detailed discussions on the parameter and doping dependences of the calculated spectra have been made in Ref. [4]. Summarizing, we have demonstrated that the low-energy escitations in a wide parameter and doping region of the 2D t-J model are described well by the picture of dressed Bogoliubov quasipartitles within the BCS pairing theory. The pairing occurs predominantly in the G?~Z -yz-wave channel and the gap energy scales with J and has a magnitude AdNo.I5.~--9.27.~ in a wide region between quarter and half fillings. This work provides the first demonstration of the validity of the BCS pairing theory fc>ra strongly-correlated electron model. 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 Iustitu tes. REFEREXCES [I] E. Dagotto and J. Riera, Phys. Rev. Lett. 70, G82 (1993) and Phys. Rev. B 46, 1.3084 (1992). [2] V. J. Emery, S. -4. Kivelson, and II. Q. Lin, Phys. Rev. Lett. 64, 475 (1990); W. 0. Putikka, XI. U. Luchini, and T. M. Rice, ibid. 68, 538 (1992). [3] E. Dagotto, J. Riera, Y. C. Che \, -4. Moreo, A. Kazarenko, F. Alcaraz, and F. Ortolani, Phys. Rev. B 49, 3548 (1994). [4] Y. Ohta, T. Shimozato, R. Eder, and S. Maekawa, Phyvs. Rev. Lett., at press. [5] 1’. Ohta. A. Nakauchi, R. Eder, K. Tsutsui, and S. 1Iaekakva. this volume. [G] Y. Ohta, Ii. Tsutsui, W. Iioshibae, T. Shirnfvato --__--.--.-,
2nd C ._...a “.
14022 (1992).
>.IF~~~!;xI.~+ Phys.
PLev.
B
46
7
PHYSICA
EXCITATIONS AND SUPERCONDUCTIVITY IN THE TWO-DIMENSIONAL t - J MODEL Y. Ohta, T. Shimozato, R. Eder, and S. Maekawa Department of AppLied Physics, Nagoya University, Nagoya 464-01, Japan An exact-diagonalization technique on small clusters is used to study excitation spectra and superconductivity in the two-dimensional t - J model. We denmnstrate that the low-lying excitations in a wide parameter (J/t) and doping regions of the model are described well by the picture of dressed Bogoliubov quasiparticles within the BCS pairing theory. The pairing occurs predominantly in d~2_y2-wave channel and the energy gap has a size Ad"0.15J--0.27J between quarter and half fillings. One of the recent issues in the theory of hightemperature superconductivity is the question whether the two-dimensional (2D) t - J model contains a superconducting phase relevant to cuprate materials. Among many calculations made to address this question, Dagotto and Riera [1] have found indications of off-diagonal long-range order near the region of phase separation [2] by measuring equal-time pairing correlations directly in finite-size clusters; the existence of a dz2_u2-wave condensate has thereby been conjectured [1, 3]. We have recently proposed [4] a new technique for examining the low-lying excitations, i.e., the exact calculation of anomalous Green's functions (AGF) for the Bogoliubov-quasiparticle spectrum in finitesize clusters. The technique has so far been applied successfully to the 2D t - J model [4] and negativeU Hubbard model-[5]. In this paper we discuss the technique briefly and present some results for the 2D t - J model. The Hamiltonian is written
where c~a is the projected electron-creation operator at site i and spin cr allowing no doubly occupied sites, Si is the electronic spin operator, and ni = nit+nil is the electron number operator. The summation
included) and Pt¢ projects on the N-electron subspace. Similarly, we assume that the low-lying states with an odd electron-number N + I can be written as where 7kt~ cre-
l~+*)=P~+xT~=IBCS)
ates a Bogoliubov quasiparticle with m o m e n t u m k. Now, we define two AGF's as follows: 1
a ( k , z ) = (¢~+21Ctk! z - H + E0 G(k, k', z) = (~b0mlet_k, 1C-k t z - H + E~v n-kinkT 6kk, Z
where nk~--(~,~lc*k~l~5~'>, E N is the groundstate energy, and Cko is the Fourier transform of c~. We define the spectral function F(k,o~)= -(1/rc)ImG(k,w+irl) and its frequency integral Fk=(% +21C*ktC*_kt I ,g ) for one-particle AGF. The spectral function F(k, k', w) and its frequency integral Fkk, are also defined for two-particle AGF as above. Then, our working hypothesis predicts, via the Bogoliubov-Valatin transformation, that these AGF's describe the excitation of Bogoliubov quasiparticles in the cluster: i.e., F(k,w) = FkS(W- Ek) F ( k , k ' , w ) = FkF£,6(w- E k - Ek,) with Fk--ZkAk/2Ek, where Z k is the wave-function renormalization constant, Ek is tile renormalized quasiparticle energy, and Ak is the gap function. We have calculated AGF's for clusters of the size 4 x 4 and v / ~ × x / l ' 8 with periodic boundary condition. In Fig+ 1, we show the results for F(k, k', w). We find the following, all of which are
0921-4534/94/$07.00 © 1994 ElsevierScience B.V. All rights reserved. -
SSD!0921-4534(94)01704-2
lCkl
I! Ohtu et al. /Ph_vsica C 235-240
2278
0
4
80
4
80
4
8
w/t
Fig. 1. Bogoliubov-quasiparticle spectrum F k, k’, w) calculated for the 4x4 cluster with 8 ho I es at J/t=2.5. The momentum k dependence is shown with k’ fised at (0, -r/2). kF is located at (n/2,0) and its equivalent points. The dotted curves show the BCS s ectral function obtained for parameter values Ad f t=9.47, t,ff-/t=0.55, and zg=0.7.
consistent with expectations of the BCS pairing theory: A pronounced low-energy peak appears at kF and smaller peaks appear at higher energies for other momenta; the weights of the peaks are consistent with the BCS form of the condensation amplitude Fk with a ma?rimum at kF. The momentum dependence of F(k, k’, w), i.e., the change in sign under rotation by n/2 and almost vanishing weight along the X:“=ciy line, is a clear indication of dz2Ty 2-wave pairing. Although not shown here, the size of the energy gap scales well with J/t value, and with decreasing gap size, the peaks at, momenta other than kF lose their weight as espetted from the BCS theory. To make a quantitative comparison, we calculate the spectrum F(k, k’, w) given by the BCS predictions where we assume the quasiparticle energy &=t/Ei t- A: with one-electron energy and gap function &=- 2t,r(cos k, + cos ky)-p h-, - cos I;,). The effect ivc hopping nk = &&OS parameter t,r is used to take into account. the quasiparticle band narrowing 161, and he chemical potential 14 is chosen t,o guarantee vanishing
(1994) 2277-2275
fitting the positions of the low-energy peaks in F( k, k’, w). The renormalization factor :n (defined by assuming Zk to be k-independent) is estimated from weights of the low-energy peaks in F(k, k’, LJ). The parameters teff and Zn reflects the effects of strong correlation and imply the use of a dressed Bogoliubov-quasiparticle description. The fitted quasiparticle spectra are shown by dotted curves in Fig. 1. We find a fair agreement with the exact spectra, which demonstrates the validity of the BCS pairing theory for low-lying e?ccitations in the 2D t-J model. Detailed discussions on the parameter and doping dependences of the calculated spectra have been made in Ref. [4]. Summarizing, we have demonstrated that the low-energy escitations in a wide parameter and doping region of the 2D t-J model are described well by the picture of dressed Bogoliubov quasipartitles within the BCS pairing theory. The pairing occurs predominantly in the G?~Z -yz-wave channel and the gap energy scales with J and has a magnitude AdNo.I5.~--9.27.~ in a wide region between quarter and half fillings. This work provides the first demonstration of the validity of the BCS pairing theory fc>ra strongly-correlated electron model. 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 Iustitu tes. REFEREXCES [I] E. Dagotto and J. Riera, Phys. Rev. Lett. 70, G82 (1993) and Phys. Rev. B 46, 1.3084 (1992). [2] V. J. Emery, S. -4. Kivelson, and II. Q. Lin, Phys. Rev. Lett. 64, 475 (1990); W. 0. Putikka, XI. U. Luchini, and T. M. Rice, ibid. 68, 538 (1992). [3] E. Dagotto, J. Riera, Y. C. Che \, -4. Moreo, A. Kazarenko, F. Alcaraz, and F. Ortolani, Phys. Rev. B 49, 3548 (1994). [4] Y. Ohta, T. Shimozato, R. Eder, and S. Maekawa, Phyvs. Rev. Lett., at press. [5] 1’. Ohta. A. Nakauchi, R. Eder, K. Tsutsui, and S. 1Iaekakva. this volume. [G] Y. Ohta, Ii. Tsutsui, W. Iioshibae, T. Shirnfvato --__--.--.-,
2nd C ._...a “.
14022 (1992).
>.IF~~~!;xI.~+ Phys.
PLev.
B
46
7