Even-Odd and Super-Even Effects in the Attractive Hubbard Model

The canonical BCS wave function is tested for the attractive Hubbard model. Results are presented for one dimension, and are compared with the exact solutions by the Bethe ansatz and the results from the conventional grand canonical BCS approximation, for various chain lengths, electron densities, and coupling strengths. While the exact ground state energies are reproduced very well both by the canonical and grand canonical BCS approximations, the canonical method significantly improves the energy gaps for small systems and weak coupling. The parity'' effect due to the number of electrons being even or odd naturally emerges in our canonical results. Furthermore, we find a super-even’’ effect: the energy gap oscillates as a function of even electron number, depending on whether the number of electrons is $4m$ or $4m+2$ (m integer). Such oscillations as a function of electron number should be observable with tunneling measurements in ultrasmall metallic grains.