We investigate the ground-state properties of trapped fermion systems described by the Hubbard model with an external confining potential. We discuss the universal behaviors of systems in different regimes: from few particles, i.e., in dilute regimes, to the trap thermodynamic limit. The asymptotic trap-size (TS) dependence in the dilute regime (increasing the trap size $\ell$ keeping the particle number $N$ fixed) is described by a universal TS scaling controlled by the dilute fixed point associated with the metal-to-vacuum quantum transition. This scaling behavior is numerically checked by DMRG simulations of the one-dimensional (1D) Hubbard model. In particular, the particle density and its correlations show crossovers among different regimes: for strongly repulsive interactions they approach those of a spinless Fermi gas, for weak interactions those of a free Fermi gas, and for strongly attractive interactions they match those of a gas of hard-core bosonic molecules. The large-$N$ limit keeping the ratio $N/\ell$ fixed corresponds to a 1D trap thermodynamic limit. We address issues related to the accuracy of the local density approximation (LDA). We show that the particle density approaches its LDA in the large-$\ell$ limit. When the trapped system is in the metallic phase, corrections at finite $\ell$ are $O\left({\ell }^{-1}\right)$ and oscillating around the center of the trap. They become significantly larger at the boundary of the fermion cloud, where they get suppressed as $O\left({\ell }^{-1/3}\right)$ only. This anomalous behavior arises from the nontrivial scaling at the metal-to-vacuum transition occurring at the boundaries of the fermion cloud.

• Received 18 January 2014

DOI:https://doi.org/10.1103/PhysRevA.89.023635