Trap effects and the continuum limit of the Hubbard model in the presence of a harmonic potential

We discuss how to perform the continuum limit of the d-dimensional Hubbard model in the presence of a harmonic confining potential at zero temperature and fixed particle number. While for d≥3 the system can be mapped into a noninteracting two-component Fermi gas in a harmonic trap, for d=1 and 2 the two-body interaction is described by a Dirac δ. We show that the properties of this continuum limit can be put in one-to-one correspondence with those obtained by applying the trap-size scaling formalism to the confined Hubbard model in the so called dilute regime (fixed number of particles and weak confinement). The correspondence for d<3 has been tested, in the case of two particles with opposite spin, by comparing numerical exact-diagonalization results of the lattice system with those obtained in the continuum limit.