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

TL;DR

This paper analyzes finite size effects on the excitation gap in the one-dimensional Hubbard model with attraction, identifying power law, exponential, and soliton-related corrections that dominate in different regimes.

Contribution

It provides a detailed characterization of finite size corrections to the excitation gap, including exponential and soliton contributions, in the weakly interacting regime of the 1D Hubbard model.

Findings

Exponential correction behaves as exp(-N_a Δ_infinity / 4t) in the weakly interacting limit.

Soliton contributions lead to additional non-exponential finite size effects.

Finite size effects can dominate the gap for small systems with thousands of particles.

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 type 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 ($|U|\ll t$) behaves as $\sim \exp (-N_a \Delta_{\infty}/4 t)$ 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 non-zero momenta, which provides an extra (non-exponential) 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 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.