We study finite-size effects for the gap of the quasiparticle excitation spectrum in the weakly interacting regime one-dimensional Hubbard model with on-site attraction. Two types of corrections to the result of the thermodynamic limit are obtained. Aside from a power law (conformal) correction due to gapless excitations which behaves as $1/N_a$, where $N_a$ is the number of lattice sites, we obtain corrections related to the existence of gapped excitations. First of all, there is an exponential correction which in the weakly interacting regime ($\left|U\right|\ll t$) behaves as $~$$\exp (-N_a{\Delta }_{\infty }/4t)$ in the extreme limit of $N_a{\Delta }_{\infty }/t\gg 1$, where $t$ is the hopping amplitude, $U$ is the on-site energy, and ${\Delta }_{\infty }$ is the gap in the thermodynamic limit. Second, in a finite-size system a spin-flip producing unpaired fermions leads to the appearance of solitons with nonzero momenta, which provides an extra (nonexponential) contribution $\delta$. For moderate but still large values of $N_a{\Delta }_{\infty }/t$, these corrections significantly increase and may become comparable with the $1/N_a$ conformal correction. Moreover, in the case of weak interactions where ${\Delta }_{\infty }\ll t$, the exponential correction exceeds higher-order power law corrections in a wide range of parameters, namely for $N_a\lesssim (8t/{\Delta }_{\infty })\ln (4t/|U|)$, and so does $\delta$ even in a wider range of $N_a$. For a sufficiently small number of particles, which can be of the order of thousands in the weakly interacting regime, the gap is fully dominated by finite-size effects.

• Received 22 September 2008

DOI:https://doi.org/10.1103/PhysRevA.81.013611