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Charge- and spin-susceptibilities in the one-dimensional hubbard model with long-range interaction
Physica C 235-240 (1994) 2303-2304 North-Holland
PHYSiCA
Charge- and Spin-Susceptibilities in the One-Dimensional Hubbard Model with Long-Range Interaction S. Yunoki, K. Tsutsui, Y. Ohta, and S. Maekawa
Department of Applied Physics, Nagoya University, Nagoya 464-01, Japan Effects of long-range Coulomb interaction in one-dimensional electron systems are studied. By using a quantum Monte Carlo method, we obtain that the zero-frequency wave-number dependent charge susceptibility at quarterfilling shows both 2kF and 4kF singularities, kF being the Fermi wave munber, in contrast with the Hubbard model with the on-site and nearest-neighbor interactions. The results are in accord with the X-ray exl)eriments in (NMP)~(Phen)t-~(TCNQ). The power-law exponent a in the spectral function p(w) ,-, [,z [" observed in the high-resolution photoemission experiments is calculated by using the exact diagonalization nwthod and found to increase with increasing the range of the interaction.
In the low-dimensional electron systems, the charge screening is less complete. Actually, several experiments in quasi-one-dimensional conductors suggest t h a t the long-range Coulomb interaction between electrons is of crucial importance. Epstein etal. [1] have shown in the X-ray diffuse scattering experiments in (NMP)~(Phen)a_~(TCNQ) that the crossover of the electronic instability from 2/,'r to 4kF occurs on the same T C N Q chains as :r decreases and both 2kF and 4kF instabilities are seen for 0.57 < x _< 2/3. This fact may be interpreted intuitively as the crossover from the Peierls instability to Wigner crystallization. In order to verify this scenario, it is necessary to show what model exhibits both instabilities at the same time. However, few study has been done so far [2]. The recent high-resolution photoenfission experiments in quasi-one-dimensional conductors, Ko.3MoO3, (TaSe4)2I, BaVS3 and (TMTSF)2X with X=C1Oa, PF6 and AsF0 [3-5] revealed that ~,,~ opt,t, . . . . . . . . . . . . . p(a,') shows ,Ue power-law behavior, I ~ ] a , and the exponent a is much larger than that in the Hubbard model [6-71, indicating the importance of the long-range interaction. The purposes of this paper are to analyze effects of the long-range Coulomb interaction on the zero-frequency wave-number-dependent charge- and spin-susc~ ptibilities, N(q) and \(q),
in one-dimensional electron systems in the quantum Monte Carlo method and to investigate the relation between the power-law exl)onent a and the interaction range in the exact diagonalization method. The Hamiltonian is written as
H = -, Z
+ l,.,-.)
i,o"
+U Z
tt,!tli: + X I" Z llllt~-+-j • i
j=l
i
where the notations are tile conventional ones. Let us first study N(q) and \(q) in the Monte Carlo method. By using the world line algorithm [8], simulations are carried out on I)eriodic rings with 60 sites at t/T=24, T being temperature, which corresponds to ~, 50K for 4t ,-~ 0.5eV. In Fig.l, the results of N(q) and \(q) at quarterfilling are plotted, respectively. In the figure, open circles show the results with the interactions up to the second-neighbors (U/t = 4, I'l/t = 2, I 2 / t = 1), whereas cross marks show those when the interaction between the rhir~l-neig, hbors (Va/t = 0.5) is further introduced into the al)ove mentioned system. As seen in the figure, both 2kF and 4/;,v singularities appear in N(q), although \(q) is ahnost independent of the interaction range. This is in strong contrast with the results by Hirsch et al. [2] in the extended Hubbard model with the nearest-neighbor interaction
0921-4534/94/S07.00 © 1994 - ElsevierScience B.V. All rights reserved. SSDI 0921-4534(94)01717-4
S. Yunoki et al./Ph.,,sica C 235-240 (1994) 2303-2304
2304
where one of the singularities, 2kF or 4kF, t:as been obtained, depending on the value of the interaction.
m
O
R
a _~ 9/16, respectively. We have obtained that the value of c~ increases with increasing the range of the interaction. Then, c~ can be larger than one when the interaction is not screened and has the 1 / r dependence. In summary, we have studied effects of the longrange interaction in one-dimensional systems. We have found in the quantum Monte-Carlo method that the long-range Coulomb interaction causes both 2kF and 4kF singularities in the charge susceptibility. We also obtained in the exact diagonalization method that the power-law exponent a of the spectral function increases with increasing the range of the interaction and can be larger than one. The detailed results of susceptibilities [101 and the exponent [11] will be given in separate publications. We would like to thank Prof. A. Fujinmri for providing experimental data in ref.5 prior to publication. This work was supported by a Grant-in Aid for scientific Research on Priority Area from Ministry of Education, Science, and Culture of Japan. REFERENCES
0
r/2 q
Figure 1. (,,) Charge and (b) spin susceptibilities for U / t = 4, l / i / t - 2, V2/t = 1, Va/t = 0 (open circle) and U / t = 4, 1.5/t = 2, V2/t = 1, V a / t = 0.5 (cross mark) at quarter-filling.
Next, the value of ~ in p(~) is examined in the exact diagonalization method. We use the standard Lanczos method in the ring at quarter-filling with 8 and 12 sites. The periodic and antiperiodic boundary conditions are chosen for N / 2 = o d d and even, respectively, N being the ::,amber of electrons, in order to get a singlet ground state. The relations of the Tomonaga-Luttinger liquid are used to calculate a [9]. As is known, the Hubbard model and the extended Hubbard model with the nearest-neighbor interaction give a _< 1/8 and
1. A . J . Epstein et al., Phys. Rev. Lett. 47, 741 (1981). 2. J. Hirsch and D. J. Scalapino, Phys. Rev. Lett. 50, 1168 (1983); Phys. Rev. B 27, 7169 (1983); ibid. 29, 5554 (1984). 3. B. Dardel et al., Phys. Rev. Lett. 67, 3144 (1991); B. Dardel et al., Europhys. Lett. 24, 687 (1993). 4. Y. Hwu et al., Phys. Rev. B 46, 13624 (1992). 5. M. Nakamura et al., Phys. Rev. B, ill press. 6. N Kawakami and S. K. Yang, Phys. Lett. A 148, 359 (1990). 7. F. MiUa and X. Zotos, Europhys. Lett. 24, 133 (1993). 8. J. E. Hirsch et al.. Ptlys. Rev. B 26, 5033 (1982). 9. M. Ogata et al., Phys. Rev. Lett. 66, 2388 (1991). 10. S. Yunoki and S Maekawa, to be I)ublished. 11. S. Yunoki et al., to be published.
PHYSiCA
Charge- and Spin-Susceptibilities in the One-Dimensional Hubbard Model with Long-Range Interaction S. Yunoki, K. Tsutsui, Y. Ohta, and S. Maekawa
Department of Applied Physics, Nagoya University, Nagoya 464-01, Japan Effects of long-range Coulomb interaction in one-dimensional electron systems are studied. By using a quantum Monte Carlo method, we obtain that the zero-frequency wave-number dependent charge susceptibility at quarterfilling shows both 2kF and 4kF singularities, kF being the Fermi wave munber, in contrast with the Hubbard model with the on-site and nearest-neighbor interactions. The results are in accord with the X-ray exl)eriments in (NMP)~(Phen)t-~(TCNQ). The power-law exponent a in the spectral function p(w) ,-, [,z [" observed in the high-resolution photoemission experiments is calculated by using the exact diagonalization nwthod and found to increase with increasing the range of the interaction.
In the low-dimensional electron systems, the charge screening is less complete. Actually, several experiments in quasi-one-dimensional conductors suggest t h a t the long-range Coulomb interaction between electrons is of crucial importance. Epstein etal. [1] have shown in the X-ray diffuse scattering experiments in (NMP)~(Phen)a_~(TCNQ) that the crossover of the electronic instability from 2/,'r to 4kF occurs on the same T C N Q chains as :r decreases and both 2kF and 4kF instabilities are seen for 0.57 < x _< 2/3. This fact may be interpreted intuitively as the crossover from the Peierls instability to Wigner crystallization. In order to verify this scenario, it is necessary to show what model exhibits both instabilities at the same time. However, few study has been done so far [2]. The recent high-resolution photoenfission experiments in quasi-one-dimensional conductors, Ko.3MoO3, (TaSe4)2I, BaVS3 and (TMTSF)2X with X=C1Oa, PF6 and AsF0 [3-5] revealed that ~,,~ opt,t, . . . . . . . . . . . . . p(a,') shows ,Ue power-law behavior, I ~ ] a , and the exponent a is much larger than that in the Hubbard model [6-71, indicating the importance of the long-range interaction. The purposes of this paper are to analyze effects of the long-range Coulomb interaction on the zero-frequency wave-number-dependent charge- and spin-susc~ ptibilities, N(q) and \(q),
in one-dimensional electron systems in the quantum Monte Carlo method and to investigate the relation between the power-law exl)onent a and the interaction range in the exact diagonalization method. The Hamiltonian is written as
H = -, Z
+ l,.,-.)
i,o"
+U Z
tt,!tli: + X I" Z llllt~-+-j • i
j=l
i
where the notations are tile conventional ones. Let us first study N(q) and \(q) in the Monte Carlo method. By using the world line algorithm [8], simulations are carried out on I)eriodic rings with 60 sites at t/T=24, T being temperature, which corresponds to ~, 50K for 4t ,-~ 0.5eV. In Fig.l, the results of N(q) and \(q) at quarterfilling are plotted, respectively. In the figure, open circles show the results with the interactions up to the second-neighbors (U/t = 4, I'l/t = 2, I 2 / t = 1), whereas cross marks show those when the interaction between the rhir~l-neig, hbors (Va/t = 0.5) is further introduced into the al)ove mentioned system. As seen in the figure, both 2kF and 4/;,v singularities appear in N(q), although \(q) is ahnost independent of the interaction range. This is in strong contrast with the results by Hirsch et al. [2] in the extended Hubbard model with the nearest-neighbor interaction
0921-4534/94/S07.00 © 1994 - ElsevierScience B.V. All rights reserved. SSDI 0921-4534(94)01717-4
S. Yunoki et al./Ph.,,sica C 235-240 (1994) 2303-2304
2304
where one of the singularities, 2kF or 4kF, t:as been obtained, depending on the value of the interaction.
m
O
R
a _~ 9/16, respectively. We have obtained that the value of c~ increases with increasing the range of the interaction. Then, c~ can be larger than one when the interaction is not screened and has the 1 / r dependence. In summary, we have studied effects of the longrange interaction in one-dimensional systems. We have found in the quantum Monte-Carlo method that the long-range Coulomb interaction causes both 2kF and 4kF singularities in the charge susceptibility. We also obtained in the exact diagonalization method that the power-law exponent a of the spectral function increases with increasing the range of the interaction and can be larger than one. The detailed results of susceptibilities [101 and the exponent [11] will be given in separate publications. We would like to thank Prof. A. Fujinmri for providing experimental data in ref.5 prior to publication. This work was supported by a Grant-in Aid for scientific Research on Priority Area from Ministry of Education, Science, and Culture of Japan. REFERENCES
0
r/2 q
Figure 1. (,,) Charge and (b) spin susceptibilities for U / t = 4, l / i / t - 2, V2/t = 1, Va/t = 0 (open circle) and U / t = 4, 1.5/t = 2, V2/t = 1, V a / t = 0.5 (cross mark) at quarter-filling.
Next, the value of ~ in p(~) is examined in the exact diagonalization method. We use the standard Lanczos method in the ring at quarter-filling with 8 and 12 sites. The periodic and antiperiodic boundary conditions are chosen for N / 2 = o d d and even, respectively, N being the ::,amber of electrons, in order to get a singlet ground state. The relations of the Tomonaga-Luttinger liquid are used to calculate a [9]. As is known, the Hubbard model and the extended Hubbard model with the nearest-neighbor interaction give a _< 1/8 and
1. A . J . Epstein et al., Phys. Rev. Lett. 47, 741 (1981). 2. J. Hirsch and D. J. Scalapino, Phys. Rev. Lett. 50, 1168 (1983); Phys. Rev. B 27, 7169 (1983); ibid. 29, 5554 (1984). 3. B. Dardel et al., Phys. Rev. Lett. 67, 3144 (1991); B. Dardel et al., Europhys. Lett. 24, 687 (1993). 4. Y. Hwu et al., Phys. Rev. B 46, 13624 (1992). 5. M. Nakamura et al., Phys. Rev. B, ill press. 6. N Kawakami and S. K. Yang, Phys. Lett. A 148, 359 (1990). 7. F. MiUa and X. Zotos, Europhys. Lett. 24, 133 (1993). 8. J. E. Hirsch et al.. Ptlys. Rev. B 26, 5033 (1982). 9. M. Ogata et al., Phys. Rev. Lett. 66, 2388 (1991). 10. S. Yunoki and S Maekawa, to be I)ublished. 11. S. Yunoki et al., to be published.