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Spin-triplet superconductivity in two-dimensional Hubbard model on triangular lattice
Physica C 388–389 (2003) 57–58 www.elsevier.com/locate/physc
Spin-triplet superconductivity in two-dimensional Hubbard model on triangular lattice Yunori Nisikawa *, Kosaku Yamada Department of Physics, Kyoto University, Kyoto 606-8502, Japan
Abstract We discuss the possibility of spin-triplet (ST) superconductivity in two-dimensional Hubbard model on triangular lattice within the third order perturbation theory. We obtain ST superconducting state. It is found that the vertex correction terms which are not included in the interaction mediated by the spin-fluctuation is essential for realizing the ST pairing for our searched values of parameters. We argue that the present mechanism of ST superconductivity can be applied to heavy-fermion superconductors UNi2 Al3 , UPt3 . Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Spin-triplet superconductivity; Vertex correction; UNi2 Al3 ; UPt3
Spin-triplet (ST) superconductors (SC) have attracted much interest since the mechanism of these superconductivity has not been well known. In STSC Sr2 RuO4 , Nomura and Yamada have recognized that the momentum and frequency dependence of the effective interaction between electrons which is not included in the interaction mediated by the spin-fluctuation is essential for realizing the ST pairing and have explained the superconducting mechanism within the third order perturbation theory (TOPT) [1]. The perturbation approach is sensitive to the dispersion of the bare energy band, by its nature. Therefore it is important to evaluate superconducting transition temperature Tc on the basis of the detailed electronic structure in each system. Stimulated by the desire to explain the superconducting mechanism of UNi2 Al3 (STSC indicated recently [2]), UPt3 (STSC), we calculate here Tc of ST superconductivity in twodimensional Hubbard model on triangular lattice within TOPT. In a similar model, Kuroki and Arita have pro-
* Corresponding author. Tel.: +81-75-753-3771; fax: +81-75753-3891. E-mail address: [email protected] (Y. Nisikawa).
posed that spin fluctuation-mediated ST superconductivity can be realized in their model with disconnected Fermi surfaces, by using fluctuation-exchange approximation (FLEX) [3]. However, as mentioned later, our proposed mechanism of ST superconductivity is completely different from their proposed mechanism. WeP adopt the following Hubbard P P Hamiltonian, H ¼ k;r ððkÞ lÞaykr akr þ ðU =2NÞ r6¼r0 k1 k2 k3 k4 dk1 þk2 ;k3 þk4 ayk1 r ayk2 r0 ak3 r0 ak4 r , where l, U , and ðkÞ are the chemical potential, the Coulomb repulsion, and the dispersion of the bare energy band on two-dimensional triangular lattice, respectively. We obtain Tc by solving liashbergÕs equation (Fig. 1). We expand normal the E self-energy, effective interaction with respect to U up to third order, respectively (Fig. 2(a) and (b)). The diagrams enclosed by the dashed line in Fig. 2(b) are the vertex correction terms which are not direct contribution from spin-fluctuations. The other diagrams are included in RPA. We calculate here Tc of two models. At first, we consider a possible model of UNi2 Al3 . In the case of spinsinglet (SS) SC UPd2 Al3 , we have adopted following dispersion of the bare energy pffiffi band in the previous work [4], ðkÞ ¼ 4 cos 23kx cos 12ky 2tm cosðky Þ where tm 6¼ 1 reflecting the effect of AF order with a large
0921-4534/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4534(02)02623-0
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Y. Nisikawa, K. Yamada / Physica C 388–389 (2003) 57–58
liashbergÕs equation. The thick line represents Fig. 1. The E GreenÕs function with self-energy correction. The shaded rectangular represents the effective interaction.
Fig. 3. (a) The calculated Tc as U is varied for various values the electron number n per one spin site. (b) The calculated Tc as U is varied at n ¼ 0:55, t1 ¼ 0:5 (v ¼ : second) in these figures means that eigenvalue v calculated by second order perturbation theory is v ¼ at Tc calculated by TOPT.
Fig. 2. (a) The Feynman diagrams of the normal self-energy up to third order. (b) The Feynman diagrams of the effective interaction up to third order. The solid and dashed lines correspond to the bare GreenÕs function and the interaction, respectively.
magnetic moment of 0.8lB . In the case of UNi2 Al3 , we assume here that tm ’ 1 reflecting SDW order with a tiny moment of 0.2lB , although the detailed electronic structure of UNi2 Al3 has not been investigated. So we try to investigate the possibility of ST superconductivity near the previous model of UPd2 Al3 . In the case of tm ¼ 1, we obtain the highest Tc of ST superconducting state (Fig. 3(a)). In the next, we adopt following disperpffiffi 3 sion of the bare energy band, ðkÞ ¼ 4 cos kx pffiffi 1 3 2 3 cos 2ky 2 cosðky Þ t1 4 cos 2 kx cos 2ky þ 2 pffiffiffi cos 3ky which reproduce quasi two-dimensional Fermi surface of UPt3 . In this case, we obtain the highest Tc of ST superconducting state (Fig. 3(b)). In all results mentioned above , the vertex correction terms is essential for realizing the ST pairing state, although only the second order term (which is only term included in RPA
within TOPT) does not give rise to the ST pairing state within the precision of our numerical calculations. From our calculation, it has been found that the vertex correction terms gives rise to the ST pairing state, while, in the case of SSSC [4–6], it has been found that the vertex correction terms reduce Tc within TOPT. The vertex correction is known to favor ST pairing in the case of two-dimensional electron gas where the electronhole symmetry is degraded [7,8]. This seems to be the case with the present result. In conclusion, we have discussed the possibility of ST superconductivity of two-dimensional Hubbard model on triangular lattice within TOPT and have obtained ST superconducting state. We have tried to apply the presented mechanism to heavy-fermion SC UNi2 Al3 , UPt3 . References [1] [2] [3] [4] [5] [6] [7] [8]
T. Nomura, K. Yamada, J. Phys. Soc. Jpn. 69 (2000) 3678. K. Ishida et al., Phys. Rev. Lett. 89 (2002) 037002. K. Kuroki, R. Arita, Phys. Rev. B 63 (2001) 174507. Y. Nisikawa, K. Yamada, J. Phys. Soc. Jpn. 71 (2002) 237. T. Jujo et al., J. Phys. Soc. Jpn. 68 (1999) 1331. Y. Nisikawa et al., J. Phys. Soc. Jpn. 71 (2002) 1140. A.V. Chubukov, Phys. Rev. B 48 (1993) 1097. J. Feldman et al., Helv. Phys. Acta 70 (1997) 154.
Spin-triplet superconductivity in two-dimensional Hubbard model on triangular lattice Yunori Nisikawa *, Kosaku Yamada Department of Physics, Kyoto University, Kyoto 606-8502, Japan
Abstract We discuss the possibility of spin-triplet (ST) superconductivity in two-dimensional Hubbard model on triangular lattice within the third order perturbation theory. We obtain ST superconducting state. It is found that the vertex correction terms which are not included in the interaction mediated by the spin-fluctuation is essential for realizing the ST pairing for our searched values of parameters. We argue that the present mechanism of ST superconductivity can be applied to heavy-fermion superconductors UNi2 Al3 , UPt3 . Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Spin-triplet superconductivity; Vertex correction; UNi2 Al3 ; UPt3
Spin-triplet (ST) superconductors (SC) have attracted much interest since the mechanism of these superconductivity has not been well known. In STSC Sr2 RuO4 , Nomura and Yamada have recognized that the momentum and frequency dependence of the effective interaction between electrons which is not included in the interaction mediated by the spin-fluctuation is essential for realizing the ST pairing and have explained the superconducting mechanism within the third order perturbation theory (TOPT) [1]. The perturbation approach is sensitive to the dispersion of the bare energy band, by its nature. Therefore it is important to evaluate superconducting transition temperature Tc on the basis of the detailed electronic structure in each system. Stimulated by the desire to explain the superconducting mechanism of UNi2 Al3 (STSC indicated recently [2]), UPt3 (STSC), we calculate here Tc of ST superconductivity in twodimensional Hubbard model on triangular lattice within TOPT. In a similar model, Kuroki and Arita have pro-
* Corresponding author. Tel.: +81-75-753-3771; fax: +81-75753-3891. E-mail address: [email protected] (Y. Nisikawa).
posed that spin fluctuation-mediated ST superconductivity can be realized in their model with disconnected Fermi surfaces, by using fluctuation-exchange approximation (FLEX) [3]. However, as mentioned later, our proposed mechanism of ST superconductivity is completely different from their proposed mechanism. WeP adopt the following Hubbard P P Hamiltonian, H ¼ k;r ððkÞ lÞaykr akr þ ðU =2NÞ r6¼r0 k1 k2 k3 k4 dk1 þk2 ;k3 þk4 ayk1 r ayk2 r0 ak3 r0 ak4 r , where l, U , and ðkÞ are the chemical potential, the Coulomb repulsion, and the dispersion of the bare energy band on two-dimensional triangular lattice, respectively. We obtain Tc by solving liashbergÕs equation (Fig. 1). We expand normal the E self-energy, effective interaction with respect to U up to third order, respectively (Fig. 2(a) and (b)). The diagrams enclosed by the dashed line in Fig. 2(b) are the vertex correction terms which are not direct contribution from spin-fluctuations. The other diagrams are included in RPA. We calculate here Tc of two models. At first, we consider a possible model of UNi2 Al3 . In the case of spinsinglet (SS) SC UPd2 Al3 , we have adopted following dispersion of the bare energy pffiffi band in the previous work [4], ðkÞ ¼ 4 cos 23kx cos 12ky 2tm cosðky Þ where tm 6¼ 1 reflecting the effect of AF order with a large
0921-4534/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4534(02)02623-0
58
Y. Nisikawa, K. Yamada / Physica C 388–389 (2003) 57–58
liashbergÕs equation. The thick line represents Fig. 1. The E GreenÕs function with self-energy correction. The shaded rectangular represents the effective interaction.
Fig. 3. (a) The calculated Tc as U is varied for various values the electron number n per one spin site. (b) The calculated Tc as U is varied at n ¼ 0:55, t1 ¼ 0:5 (v ¼ : second) in these figures means that eigenvalue v calculated by second order perturbation theory is v ¼ at Tc calculated by TOPT.
Fig. 2. (a) The Feynman diagrams of the normal self-energy up to third order. (b) The Feynman diagrams of the effective interaction up to third order. The solid and dashed lines correspond to the bare GreenÕs function and the interaction, respectively.
magnetic moment of 0.8lB . In the case of UNi2 Al3 , we assume here that tm ’ 1 reflecting SDW order with a tiny moment of 0.2lB , although the detailed electronic structure of UNi2 Al3 has not been investigated. So we try to investigate the possibility of ST superconductivity near the previous model of UPd2 Al3 . In the case of tm ¼ 1, we obtain the highest Tc of ST superconducting state (Fig. 3(a)). In the next, we adopt following disperpffiffi 3 sion of the bare energy band, ðkÞ ¼ 4 cos kx pffiffi 1 3 2 3 cos 2ky 2 cosðky Þ t1 4 cos 2 kx cos 2ky þ 2 pffiffiffi cos 3ky which reproduce quasi two-dimensional Fermi surface of UPt3 . In this case, we obtain the highest Tc of ST superconducting state (Fig. 3(b)). In all results mentioned above , the vertex correction terms is essential for realizing the ST pairing state, although only the second order term (which is only term included in RPA
within TOPT) does not give rise to the ST pairing state within the precision of our numerical calculations. From our calculation, it has been found that the vertex correction terms gives rise to the ST pairing state, while, in the case of SSSC [4–6], it has been found that the vertex correction terms reduce Tc within TOPT. The vertex correction is known to favor ST pairing in the case of two-dimensional electron gas where the electronhole symmetry is degraded [7,8]. This seems to be the case with the present result. In conclusion, we have discussed the possibility of ST superconductivity of two-dimensional Hubbard model on triangular lattice within TOPT and have obtained ST superconducting state. We have tried to apply the presented mechanism to heavy-fermion SC UNi2 Al3 , UPt3 . References [1] [2] [3] [4] [5] [6] [7] [8]
T. Nomura, K. Yamada, J. Phys. Soc. Jpn. 69 (2000) 3678. K. Ishida et al., Phys. Rev. Lett. 89 (2002) 037002. K. Kuroki, R. Arita, Phys. Rev. B 63 (2001) 174507. Y. Nisikawa, K. Yamada, J. Phys. Soc. Jpn. 71 (2002) 237. T. Jujo et al., J. Phys. Soc. Jpn. 68 (1999) 1331. Y. Nisikawa et al., J. Phys. Soc. Jpn. 71 (2002) 1140. A.V. Chubukov, Phys. Rev. B 48 (1993) 1097. J. Feldman et al., Helv. Phys. Acta 70 (1997) 154.