Finite-size effects for the gap in the excitation spectrum of the one-dimensional Hubbard model

Abstract

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 ($|U|\ensuremath{\ll}t$) behaves as $~$$\mathrm{exp}(\ensuremath{-}{N}_{a}{\ensuremath{\Delta}}_{\ensuremath{\infty}}/4t)$ in the extreme limit of ${N}_{a}{\ensuremath{\Delta}}_{\ensuremath{\infty}}/t\ensuremath{\gg}1$, where $t$ is the hopping amplitude, $U$ is the on-site energy, and ${\ensuremath{\Delta}}_{\ensuremath{\infty}}$ 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 $\ensuremath{\delta}$. For moderate but still large values of ${N}_{a}{\ensuremath{\Delta}}_{\ensuremath{\infty}}/t$, these corrections significantly increase and may become comparable with the $1/{N}_{a}$ conformal correction. Moreover, in the case of weak interactions where ${\ensuremath{\Delta}}_{\ensuremath{\infty}}\ensuremath{\ll}t$, the exponential correction exceeds higher-order power law corrections in a wide range of parameters, namely for ${N}_{a}\ensuremath{\lesssim}(8t/{\ensuremath{\Delta}}_{\ensuremath{\infty}})\mathrm{ln}(4t/|U|)$, and so does $\ensuremath{\delta}$ 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.