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Physics Letters A

Arbitrariness and symmetry properties of the functional formulation of the Hubbard hamiltonian

Claudio Castellani, C. Di Castro

Published Feb 1, 1979• 19 citations
Publisher
DOI
10.1016/0375-9601(79)90320-7

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By making use of some symmetry properties of the relevant Hamiltonian, two fundamental relations between the ferromagnetic magnetization and a spin correlation function are derived for the d(=1,2,3)d(=1,2,3)d(=1,2,3)-dimensional Hubbard model at finite temperatures. These can be viewed as a kind of Ward-Takahashi identity. The properties of the magnetization as a function of the applied field are discussed. The results thus obtained hold true for both repulsive and attractive on-site Coulomb interactions, and for arbitrary electron fillings.
We present an algorithm for microscopic modeling of superconducting states of arbitrary pairing symmetry via an implementation of self-consistent perturbation theory, the fluctuation exchange approximation (FLEX), applied to Hubbard-like models. Like the original formulation of FLEX, all ring diagrams that are generated with the bare interaction vertex are included in the generating functional for the self-energy. Thus, this algorithm is suitable for FLEX-based Hubbard model calculations in both the attractive and repulsive cases.
Divergencies appearing in perturbation expansions of interacting many-body systems can often be removed by expanding around a suitably chosen renormalized (instead of the noninteracting) Hamiltonian. We describe such a renormalized perturbation expansion for interacting Fermi systems, which treats Fermi surface shifts and superconductivity with an arbitrary gap function via additive counterterms. The expansion is formulated explicitly for the Hubbard model to second order in the interaction. Numerical solutions of the self-consistency condition determining the Fermi surface and the gap function are calculated for the two-dimensional case. For the repulsive Hubbard model close to half-filling we find a superconducting state with d-wave symmetry, as expected. For Fermi levels close to the van Hove singularity a Pomeranchuk instability leads to Fermi surfaces with a broken square lattice symmetry, whose topology can be closed or open. For the attractive Hubbard model the second-order calculation yields an s-wave superconductivity with a weakly momentum dependent gap, whose size is reduced compared to the mean-field result.