Universal scaling of three-dimensional bosonic gases in a trapping potential

We investigate the critical properties of cold bosonic gases in three dimensions, confined by an external quadratic potential coupled to the particle density, and realistically described by the Bose-Hubbard (BH) model. The trapping potential is often included in experiments with cold atoms and modifies the critical finite-size scaling of the homogeneous system in a non trivial way. The trap-size scaling (TSS) theory accounts for this effect through the exponent $\theta$. We perform extensive simulations of the BH model at the critical temperature, in the presence of harmonic traps. We find that the TSS predictions are universal once we account for the effective way in which the trap locally modifies the chemical potential $\mu$ of the system. The trap exponent for the BH model at $\mu=0$ is the one corresponding to an effective quartic potential. At positive $\mu$, evidence suggests that TSS breaks down sufficiently far from the centre of the trap, as the system encounters an effective phase boundary.