Numerical Approach to Magnetic Mechanism of Superconductivity

  • Masatoshi Imada
Part of the NATO ASI Series book series (NSSB, volume 213)


Since the discovery of high-Tc superconductors, various models have been proposed and investigated torn different point of view to clarify the mechanism of high-Tc superconductivity. We start with a two-band d-p model with copper and oxygen orbitals in the CuO 2 plane of the high-Tc superconductors. The models investigated in the numerical analysis include the original two-band d-p model, the coupled spin-fermion model and the t-J model.

The d-p model given by
$$H = - {t_1}\sum\limits_{ < ij > \sigma } {\left( {d_{i\sigma }^\dag {p_{j\sigma }} + d_{i\sigma }^\dag {q_{j\sigma }} + h.c.} \right)} - {t_2}\sum\limits_{ < ij > \sigma } {\left( {p_{i\sigma }^\dag {q_{j\sigma }} + h.c.} \right)} + {U_d}\sum\limits_i {{n_{di \uparrow }}{n_{di \downarrow }} + } {U_p}\sum\limits_j {\left( {{n_{pj \uparrow }}{n_{pj \downarrow }} + {n_{qj \uparrow }}{n_{qj \downarrow }}} \right)} + V\sum\limits_{ < ij > } {{n_{di}}\left( {{n_{pj}} + {n_{qj}}} \right)} + {\varepsilon _d}\sum\limits_i {{n_{di}} + {\varepsilon _p}\sum\limits_j {\left( {{n_{pj}} + {n_{qj}}} \right),} }$$
has been investigated by the quantum simulation method 1), where fermion operators d, p and q represent holes in \(Cu - 3{d_{{x^2} - {y^2}}},{p_x}\,and\,{p_y}\) orbitals, respectively. The results show small but finite enhancement of the superconducting susceptibilities as compared to the noninteracting case.
The coupled spin-fermion model given by
$$H = - t\sum\limits_{ < i,j > \sigma } {\left( {c_{i\sigma }^\dag {c_{j\sigma }} + c_{j\sigma }^\dag {c_{i\sigma }}} \right) + {U_h}\sum\limits_i {{n_{i \uparrow }}{n_{i \downarrow }} - 2{J_K}\sum\limits_{ < i,l > } {{{\vec S}_l} \cdot {{\vec \sigma }_i} - 2{J_s}} \sum\limits_{ < l,m > } {{{\vec S}_l} \cdot {{\vec S}_m}} } } (Js < 0)\vec \sigma = \frac{1}{2}c_{i\sigma }^\dag {\left( {\vec \sigma } \right)_{\sigma {\sigma ^1}}}{c_{i{\sigma ^1}}}n = c_{i\sigma }^\dag {c_{i\sigma }}$$
is derived as the Kondo limit of the d-p model, where the charge fluctuation at the Copper sites is suppressed and the local moment is preserved. The ground state of this model has been examined 2) by the exact diagonalization of finite lattices. The system with one itinerant fermion shows the existence of the extended cloud around the fermion, when the transfer of the itinerant fermion and the Kondo coupling are large. In the cloud, the antiferromagnetic correlation in the substrate of the Heisenberg spins is reduced from the pure Heisenberg system. The fermion’s spin couples antiferromagnetically to the localized spins in the cloud. It shows the developement of totally singlet structure constructed from the fermion’s spin and the localized Heisenberg spins in the cloud. The interaction of two fermions has also been calculated. It seems to show attractive behavior irrespective of the detailed fermion lattice structure and the dimensionality. The origin of the attractive interaction is attributed to the fact that the total area of the reduced antiferromagnetic correlation is smaller if two clouds overlap. The developement of the singlet cloud and the attractive interaction seem to be the characteristic feature only in the case of the antiferromagnetic Kondo coupling. In fact, the distortion of the localized spins around the fermion is not remarkable in the case of ferromagnetic Kondo coupling.


  1. 1).
    M. Imada: J. Phys. Soc. Jpn. 56, 3793 (1987). M. Imada: J. Phys. Soc. Jpn. 57, 3128 (1988).CrossRefADSGoogle Scholar
  2. 2).
    M. Imada, N. Nagaosa and Y. Hatsugai: J. Phys. Soc. Jpn. 57, 2901 (1988). M. Imada, Y. Hatsugai and N. Nagaosa: Proceedings of the Adriático Conference “Towards the Theoretical Understanding of High — Tc Superconductors” ed. by S. Lundquist et al. (World Scientific, Singapore, 1988) p423. Y. Hatsugai, M. Imada and N. Nagaosa: J. Phys. Soc. Jpn. 58 No.4 (1989). M. Imada: Proceedings of the 2nd.NEC Symposium “Mechanism of High-Tc Superconductivity” (Springer Verlag, 1989). N. Nagaosa, Y. Hatsugai and M. Imada: J. Phys. Soc. Jpn. 58 No.3 (1989).CrossRefADSGoogle Scholar

Copyright information

© Plenum Press, New York 1989

Authors and Affiliations

  • Masatoshi Imada
    • 1
  1. 1.Department of Physics, College of Liberal ArtsUniversity of SaitamaUrawa 338Japan

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