BEC Transition Temperature of a Dilute Homogeneous Imperfect Bose Gas

The leading-order effect of interactions on a homogeneous Bose gas is theoretically predicted to shift the critical temperature by an amount $\Delta T_c\simeq {\mathrm{ca}}_{\mathrm{sc}}{n}^{1/3}{T}_0$ from the ideal gas result $T_0$, where $a_{\mathrm{sc}}$ is the scattering length, $n$ is the density, and $c$ is a pure number. There have been several different theoretical estimates for $c$. We claim to settle the issue by measuring the numerical coefficient in a lattice simulation of O(2) ${\phi }^4$ field theory in three dimensions—an effective theory which, as observed previously in the literature, can be systematically matched to the dilute Bose gas problem to reproduce nonuniversal quantities such as the critical temperature. We find $c=1.32\mathrm{}\pm 0.02$.