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

We give a prescription to perform the continuum limit of the $d$-dimensional Hubbard model in the presence of a harmonic trap at zero temperature. We perform the continuum limit at fixed number of particles. In $d\geq3$ the lattice system of spin-1/2 particles is mapped into a non-interacting two-component Fermi gas in a harmonic trap. In $d=1$ and $d=2$ the particles with opposite spin interact via a Dirac delta interaction. We show that the properties of this continuum limit can be put in correspondence with those derived applying the Trap-Size scaling (TSS) formalism to the confined Hubbard model in the so called Dilute Regime (fixed number of particles and weak confinement). The correspondence in $d=1$ and $d=2$ has been tested comparing the numerical results obtained for lattice system with those of the continuum limit in the case of two-particle and in absence of spin-polarization ($N=2$,$N_{\uparrow}=N_{\downarrow}=1$).