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/Na, where
Na 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/Na 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
Na. 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.
The Jordan-Wigner transformation connects spin operators in one-dimensional spin systems and fermionic operators. In this work, we elucidate the relationship between the finite-size corrections in the spin representation and the fermionic Casimir effect in the corresponding fermion representation. In particular, we focus on the ground-state energy of one-dimensional transverse-field Ising and XY models, and show that all finite-size corrections can be interpreted as lattice fermionic Casimir effects. We further find several types of Casimir phenomena, such as the conventional Casimir energy from massless fields, damping behavior from massive fields, vanishing behavior from flat or nonrelativistic bands, and oscillating behavior from the finite-density effect. Our findings establish a dictionary between finite-size corrections in spin chains and fermionic Casimir effects, and provide experimentally relevant platforms for the fermionic Casimir phenomena.