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Spin-triplet superconductivity in the Hubbard chains coupled with ferromagnetic exchange interaction
Physica C 460–462 (2007) 1059–1060 www.elsevier.com/locate/physc
Spin-triplet superconductivity in the Hubbard chains coupled with ferromagnetic exchange interaction S. Nishimoto a
a,*
, T. Shirakawa b, Y. Ohta
b,c
Max-Planck-Institut fu¨r Physik komplexer Systeme, No¨thnizer Straße 38, 01187 Dresden, Germany b Graduate School of Science and Technology, Chiba University, Chiba 263-8522, Japan c Department of Physics, Chiba University, Chiba 263-8522, Japan Available online 28 March 2007
Abstract Double-chain Hubbard model coupled with zigzag-type or rung-type ferromagnetic exchange interaction is studied by the densitymatrix renormalization group method. We establish the phase diagram based on the results of pair binding energy, spin gap, and ground-state total spin as well as local density. Ó 2007 Elsevier B.V. All rights reserved. PACS: 71.10.Fd; 71.30.+h; 74.20.Mn Keywords: Triplet superconductivity; Ferromagnetism; Hubbard model
Mechanism of the spin-triplet superconductivity has been one of the major issues in the field of strongly correlated electron systems. Recently, we have proposed a new mechanism of the spin-triplet superconductivity in two Hubbard chains coupled with zigzag hopping integrals [1]. In Ref. [1], it is believed that the ferromagnetic interaction between the two-chains is derived from the ring-exchange mechanism. We thus study two types of double-chain system as simple and effective models which take into account the ring-exchange ferromagnetic interaction; one is two Hubbard chains coupled with zigzag-type ferromagnetic exchange interaction (referred to as model I, see Fig. 1a) and the other is two Hubbard chains coupled with rungtype ferromagnetic exchange interaction (model II, see Fig. 1b). So far the model II has been studied theoretically [2–5]. The Hamiltonian of two Hubbard chains is defined as X y X H chain ¼ t ciþ1;kr ci;kr þ H :c: þ U ni;k" ni;k# : ð1Þ i;kr
*
i;k
Corresponding author. Tel.: +49 351 871 2412. E-mail address: [email protected] (S. Nishimoto).
0921-4534/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.03.216
The ferromagnetic exchange interaction is given by X ðSi;1 Si;2 þ Si;1 Siþ1;2 Þ H J ;zigzag ¼ J
ð2Þ
i
for model I, and by X H J ;rung ¼ J Si;1 Si;2
ð3Þ
i
for model II. Here cyi;kr creates an electron of spin r at site i on side k = 1 (left) or 2 (right), t is the hopping integrals along the 1D chains, U is the on-site Coulomb interaction, and J(P0) is the ferromagnetic exchange interaction. We set hereafter t = 1 as an energy unit. We apply the exact-diagonalization technique on small clusters and the density-matrix renormalization group (DMRG) method on larger clusters to calculate the pair binding energy, the spin gap, the local density, and the pair correlation functions as well as the anormalus Green’s functions. First, we consider the model I and estimate the boundary of phase separation which takes place in the large-J region. We can observe the phase-separated state visually from a plot of local density since it is a state with broken
1060
S. Nishimoto et al. / Physica C 460–462 (2007) 1059–1060
a
t
U J
py
b
dxy
U
J t Fig. 1. Schematic representation of the two-chain Hubbard model coupled with zigzag-type (a) and rung-type (b) ferromagnetic exchange interaction.
translational symmetry under the open-end boundary conditions. In Fig. 2a we show the values of Jc at which the system separates into an electron-rich and a no-electron phase. Jc has the maximum value 11 around n = 0.5,
and there exists wide metallic phase in the small-J regime. We confirm that the spins are fully polarized in the phaseseparated region and the ground-state total spin is always zero in the metallic region. To study pairing state in the metallic phase, we calculate the pair binding energy DB and the spin gap Ds. Those for system size L are defined by DB ðLÞ ¼ EL ðN þ 2; 0Þ þ EL ðN ; 0Þ 2EL ðN þ 1; 1=2Þ;
ð4Þ
Ds ðLÞ ¼ EL ðN ; 1Þ EL ðN ; 0Þ;
ð5Þ
where EL(N, Sz) is the ground-state energy in the subspace with a given even number N of electrons and a given spin Sz. We extrapolate DB(L) and Ds(L) as a function of 1/L, and then DB and Ds are obtained in the thermodynamic limit (L ! 1). At high concentrations (n J 0.7), we find that DB < 0 and Ds > 0, i.e., spin-singlet pair binding occurs; whereas, at lower concentration (n [ 0.7), both DB and Ds are zero, i.e., the system is paramagnetic metal. Next, let us turn to the model II. This model can be regarded as a single-chain model with two degenerate orbitals if we identify the ferromagnetic exchange with the Hund’s coupling. Therefore, on the analogy of the Haldane gap, it would be expected that the spin gap opens at half filling. We confirm that there exists the spin gap at high concentrations and the spin-singlet dxy-like pair correlations are the most dominant there; whereas, at lower concentration, the spins are fully polarized and the spin-triplet py-like pair correlations are the most dominant. The boundary of these two regions is shown in Fig. 2b, for several U values. Acknowledgements This work was supported in part by Grants-in-Aid for Scientific Research from the Ministry of Education, Science, Sports, and Culture of Japan. A part of computations was carried out at the Research Center for Computational Science, Okazaki Research Facilities, and the Institute for Solid State Physics, University of Tokyo. References
Fig. 2. Schematic phase diagram of the model I with U = 10 (a) and of the model II (b).
[1] [2] [3] [4] [5]
Y. Ohta et al., Phys. Rev. B 72 (2005) 012503. S. Fujimoto, N. Kawakami, Phys. Rev. B 50 (1995) 6189. D.B. Shelton, A.M. Tsvelik, Phys. Rev. B 53 (1996) 252. B. Ammon, M. Imada, J. Phys. Soc. Jpn. 70 (2000) 547. H. Sakamoto et al., Phys. Rev. B 65 (2002) 224403.
Spin-triplet superconductivity in the Hubbard chains coupled with ferromagnetic exchange interaction S. Nishimoto a
a,*
, T. Shirakawa b, Y. Ohta
b,c
Max-Planck-Institut fu¨r Physik komplexer Systeme, No¨thnizer Straße 38, 01187 Dresden, Germany b Graduate School of Science and Technology, Chiba University, Chiba 263-8522, Japan c Department of Physics, Chiba University, Chiba 263-8522, Japan Available online 28 March 2007
Abstract Double-chain Hubbard model coupled with zigzag-type or rung-type ferromagnetic exchange interaction is studied by the densitymatrix renormalization group method. We establish the phase diagram based on the results of pair binding energy, spin gap, and ground-state total spin as well as local density. Ó 2007 Elsevier B.V. All rights reserved. PACS: 71.10.Fd; 71.30.+h; 74.20.Mn Keywords: Triplet superconductivity; Ferromagnetism; Hubbard model
Mechanism of the spin-triplet superconductivity has been one of the major issues in the field of strongly correlated electron systems. Recently, we have proposed a new mechanism of the spin-triplet superconductivity in two Hubbard chains coupled with zigzag hopping integrals [1]. In Ref. [1], it is believed that the ferromagnetic interaction between the two-chains is derived from the ring-exchange mechanism. We thus study two types of double-chain system as simple and effective models which take into account the ring-exchange ferromagnetic interaction; one is two Hubbard chains coupled with zigzag-type ferromagnetic exchange interaction (referred to as model I, see Fig. 1a) and the other is two Hubbard chains coupled with rungtype ferromagnetic exchange interaction (model II, see Fig. 1b). So far the model II has been studied theoretically [2–5]. The Hamiltonian of two Hubbard chains is defined as X y X H chain ¼ t ciþ1;kr ci;kr þ H :c: þ U ni;k" ni;k# : ð1Þ i;kr
*
i;k
Corresponding author. Tel.: +49 351 871 2412. E-mail address: [email protected] (S. Nishimoto).
0921-4534/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.03.216
The ferromagnetic exchange interaction is given by X ðSi;1 Si;2 þ Si;1 Siþ1;2 Þ H J ;zigzag ¼ J
ð2Þ
i
for model I, and by X H J ;rung ¼ J Si;1 Si;2
ð3Þ
i
for model II. Here cyi;kr creates an electron of spin r at site i on side k = 1 (left) or 2 (right), t is the hopping integrals along the 1D chains, U is the on-site Coulomb interaction, and J(P0) is the ferromagnetic exchange interaction. We set hereafter t = 1 as an energy unit. We apply the exact-diagonalization technique on small clusters and the density-matrix renormalization group (DMRG) method on larger clusters to calculate the pair binding energy, the spin gap, the local density, and the pair correlation functions as well as the anormalus Green’s functions. First, we consider the model I and estimate the boundary of phase separation which takes place in the large-J region. We can observe the phase-separated state visually from a plot of local density since it is a state with broken
1060
S. Nishimoto et al. / Physica C 460–462 (2007) 1059–1060
a
t
U J
py
b
dxy
U
J t Fig. 1. Schematic representation of the two-chain Hubbard model coupled with zigzag-type (a) and rung-type (b) ferromagnetic exchange interaction.
translational symmetry under the open-end boundary conditions. In Fig. 2a we show the values of Jc at which the system separates into an electron-rich and a no-electron phase. Jc has the maximum value 11 around n = 0.5,
and there exists wide metallic phase in the small-J regime. We confirm that the spins are fully polarized in the phaseseparated region and the ground-state total spin is always zero in the metallic region. To study pairing state in the metallic phase, we calculate the pair binding energy DB and the spin gap Ds. Those for system size L are defined by DB ðLÞ ¼ EL ðN þ 2; 0Þ þ EL ðN ; 0Þ 2EL ðN þ 1; 1=2Þ;
ð4Þ
Ds ðLÞ ¼ EL ðN ; 1Þ EL ðN ; 0Þ;
ð5Þ
where EL(N, Sz) is the ground-state energy in the subspace with a given even number N of electrons and a given spin Sz. We extrapolate DB(L) and Ds(L) as a function of 1/L, and then DB and Ds are obtained in the thermodynamic limit (L ! 1). At high concentrations (n J 0.7), we find that DB < 0 and Ds > 0, i.e., spin-singlet pair binding occurs; whereas, at lower concentration (n [ 0.7), both DB and Ds are zero, i.e., the system is paramagnetic metal. Next, let us turn to the model II. This model can be regarded as a single-chain model with two degenerate orbitals if we identify the ferromagnetic exchange with the Hund’s coupling. Therefore, on the analogy of the Haldane gap, it would be expected that the spin gap opens at half filling. We confirm that there exists the spin gap at high concentrations and the spin-singlet dxy-like pair correlations are the most dominant there; whereas, at lower concentration, the spins are fully polarized and the spin-triplet py-like pair correlations are the most dominant. The boundary of these two regions is shown in Fig. 2b, for several U values. Acknowledgements This work was supported in part by Grants-in-Aid for Scientific Research from the Ministry of Education, Science, Sports, and Culture of Japan. A part of computations was carried out at the Research Center for Computational Science, Okazaki Research Facilities, and the Institute for Solid State Physics, University of Tokyo. References
Fig. 2. Schematic phase diagram of the model I with U = 10 (a) and of the model II (b).
[1] [2] [3] [4] [5]
Y. Ohta et al., Phys. Rev. B 72 (2005) 012503. S. Fujimoto, N. Kawakami, Phys. Rev. B 50 (1995) 6189. D.B. Shelton, A.M. Tsvelik, Phys. Rev. B 53 (1996) 252. B. Ammon, M. Imada, J. Phys. Soc. Jpn. 70 (2000) 547. H. Sakamoto et al., Phys. Rev. B 65 (2002) 224403.