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Journal of the Physical Society of Japan

Higher Order Perturbation Expansion for Pairing Interaction in Repulsive Hubbard Model

Takuji Nomura, Kōsaku Yamada

Published Sep 15, 2003• 2 citations
Publisher
DOI
10.1143/jpsj.72.2403

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We demonstrate that an expansion in powers of the strength of the interaction, of the free energy at fixed order parameter, can be used to generate and correct mean-field theories for interacting quantum many-body systems. The first two terms in the expansion generally yield ordinary Hartree-Fock mean-field theory and the next term gives an ``Onsager reaction field'' correction to Hartree-Fock theory. This method can be used to directly generate expansions for inverse susceptibilities. We illustrate the method for the one- and two-dimensional Hubbard model, for which we consider corrections to mean-field theories of antiferromagnetism for the repulsive-U half-filled case and superconductivity in the attractive-U case. These corrections give a quantitative account superior to that of the random-phase approximation (RPA) for the correlation energy at small and intermediate values of U. For susceptibilities, we recover from the first two terms in the expansion the usual RPA results, while higher-order terms give systematic corrections to the RPA susceptibilities. For the case of superconductivity in the repulsive-U Hubbard model, we show that the higher-order terms in the expansion must be considered to determine whether or not an instability exists. We find that there is no superconducting instability in the repulsive-U case, at least towards ordinary singlet or triplet pairing. We also find no evidence for a superconducting instability driven by a coexisting antiferromagnetic order.
We compute the perturbative expansion for the effective interaction WWW of the half-filled 2-dimensional Hubbard model. We derive extensions of standard random-phase-approximation resummations that include arbitrarily high order contributions in the ${W}_{\ensuremath{\uparrow}\ensuremath{\uparrow}}$ and ${W}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}$ basis. Using algorithmic tools we explore the static QQQ-dependent interaction as well as the same-time quantity both in momentum and real space. We emphasize the absence of screening in the Hubbard interaction where we find an enhanced repulsive local ${W}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}$ with a nonzero attractive ${W}_{\ensuremath{\uparrow}\ensuremath{\uparrow}}$. Finally, starting from only a locally repulsive bare interaction we find an emergent nonlocal nearest-neighbor attraction for low temperatures at sufficiently large values of U/tU/tU/t which may be key to understanding pairing processes in the model.
A linked-diagram perturbation expansion of the linear half-filled-band Hubbard model is developed and carried out through seventh order. The zero-order Hamiltonian is taken to be the intrasite Coulomb repulsion terms, and the perturbation is taken to be the intersite interaction. The degenerate perturbation theory applied here results in an effective Hamiltonian defined on a simple spin space, and is found to consist of terms involving nearest-neighbor, next-nearest-neighbor, and biquadratic spin operators.